rhetoric is the counterpart of dialectic. both alike are concerned
with such things as come, more or less, within the general ken of
all men and belong to no definite science. accordingly all men make
use, more or less, of both; for to a certain extent all men attempt
to discuss statements and to maintain them, to defend themselves and
to attack others. ordinary people do this either at random or through
practice and from acquired habit. both ways being possible, the subject
can plainly be handled systematically, for it is possible to inquire
the reason why some speakers succeed through practice and others spontaneously;
and every one will at once agree that such an inquiry is the function
now, the framers of the current treatises on rhetoric have constructed
but a small portion of that art. the modes of persuasion are the only
true constituents of the art: everything else is merely accessory.
these writers, however, say nothing about enthymemes, which are the
substance of rhetorical persuasion, but deal mainly with non-essentials.
the arousing of prejudice, pity, anger, and similar emotions has nothing
to do with the essential facts, but is merely a personal appeal to
the man who is judging the case. consequently if the rules for trials
which are now laid down some states-especially in well-governed states-were
applied everywhere, such people would have nothing to say. all men,
no doubt, think that the laws should prescribe such rules, but some,
as in the court of areopagus, give practical effect to their thoughts
and forbid talk about non-essentials. this is sound law and custom.
it is not right to pervert the judge by moving him to anger or envy
or pity-one might as well warp a carpenter's rule before using it.
again, a litigant has clearly nothing to do but to show that the alleged
fact is so or is not so, that it has or has not happened. as to whether
a thing is important or unimportant, just or unjust, the judge must
surely refuse to take his instructions from the litigants: he must
decide for himself all such points as the law-giver has not already
now, it is of great moment that well-drawn laws should themselves
define all the points they possibly can and leave as few as may be
to the decision of the judges; and this for several reasons. first,
to find one man, or a few men, who are sensible persons and capable
of legislating and administering justice is easier than to find a
large number. next, laws are made after long consideration, whereas
decisions in the courts are given at short notice, which makes it
hard for those who try the case to satisfy the claims of justice and
expediency. the weightiest reason of all is that the decision of the
lawgiver is not particular but prospective and general, whereas members
of the assembly and the jury find it their duty to decide on definite
cases brought before them. they will often have allowed themselves
to be so much influenced by feelings of friendship or hatred or self-interest
that they lose any clear vision of the truth and have their judgement
obscured by considerations of personal pleasure or pain. in general,
then, the judge should, we say, be allowed to decide as few things
as possible. but questions as to whether something has happened or
has not happened, will be or will not be, is or is not, must of necessity
be left to the judge, since the lawgiver cannot foresee them. if this
is so, it is evident that any one who lays down rules about other
matters, such as what must be the contents of the 'introduction' or
the 'narration' or any of the other divisions of a speech, is theorizing
about non-essentials as if they belonged to the art. the only question
with which these writers here deal is how to put the judge into a
given frame of mind. about the orator's proper modes of persuasion
they have nothing to tell us; nothing, that is, about how to gain
hence it comes that, although the same systematic principles apply
to political as to forensic oratory, and although the former is a
nobler business, and fitter for a citizen, than that which concerns
the relations of private individuals, these authors say nothing about
political oratory, but try, one and all, to write treatises on the
way to plead in court. the reason for this is that in political oratory
there is less inducement to talk about nonessentials. political oratory
is less given to unscrupulous practices than forensic, because it
treats of wider issues. in a political debate the man who is forming
a judgement is making a decision about his own vital interests. there
is no need, therefore, to prove anything except that the facts are
what the supporter of a measure maintains they are. in forensic oratory
this is not enough; to conciliate the listener is what pays here.
it is other people's affairs that are to be decided, so that the judges,
intent on their own satisfaction and listening with partiality, surrender
themselves to the disputants instead of judging between them. hence
in many places, as we have said already, irrelevant speaking is forbidden
in the law-courts: in the public assembly those who have to form a
judgement are themselves well able to guard against that.
it is clear, then, that rhetorical study, in its strict sense, is
concerned with the modes of persuasion. persuasion is clearly a sort
of demonstration, since we are most fully persuaded when we consider
a thing to have been demonstrated. the orator's demonstration is an
enthymeme, and this is, in general, the most effective of the modes
of persuasion. the enthymeme is a sort of syllogism, and the consideration
of syllogisms of all kinds, without distinction, is the business of
dialectic, either of dialectic as a whole or of one of its branches.
it follows plainly, therefore, that he who is best able to see how
and from what elements a syllogism is produced will also be best skilled
in the enthymeme, when he has further learnt what its subject-matter
is and in what respects it differs from the syllogism of strict logic.
the true and the approximately true are apprehended by the same faculty;
it may also be noted that men have a sufficient natural instinct for
what is true, and usually do arrive at the truth. hence the man who
makes a good guess at truth is likely to make a good guess at probabilities.
it has now been shown that the ordinary writers on rhetoric treat
of non-essentials; it has also been shown why they have inclined more
rhetoric is useful (1) because things that are true and things that
are just have a natural tendency to prevail over their opposites,
so that if the decisions of judges are not what they ought to be,
the defeat must be due to the speakers themselves, and they must be
blamed accordingly. moreover, (2) before some audiences not even the
possession of the exactest knowledge will make it easy for what we
say to produce conviction. for argument based on knowledge implies
instruction, and there are people whom one cannot instruct. here,
then, we must use, as our modes of persuasion and argument, notions
possessed by everybody, as we observed in the topics when dealing
with the way to handle a popular audience. further, (3) we must be
able to employ persuasion, just as strict reasoning can be employed,
on opposite sides of a question, not in order that we may in practice
employ it in both ways (for we must not make people believe what is
wrong), but in order that we may see clearly what the facts are, and
that, if another man argues unfairly, we on our part may be able to
confute him. no other of the arts draws opposite conclusions: dialectic
and rhetoric alone do this. both these arts draw opposite conclusions
impartially. nevertheless, the underlying facts do not lend themselves
equally well to the contrary views. no; things that are true and things
that are better are, by their nature, practically always easier to
prove and easier to believe in. again, (4) it is absurd to hold that
a man ought to be ashamed of being unable to defend himself with his
limbs, but not of being unable to defend himself with speech and reason,
when the use of rational speech is more distinctive of a human being
than the use of his limbs. and if it be objected that one who uses
such power of speech unjustly might do great harm, that is a charge
which may be made in common against all good things except virtue,
and above all against the things that are most useful, as strength,
health, wealth, generalship. a man can confer the greatest of benefits
by a right use of these, and inflict the greatest of injuries by using
it is clear, then, that rhetoric is not bound up with a single definite
class of subjects, but is as universal as dialectic; it is clear,
also, that it is useful. it is clear, further, that its function is
not simply to succeed in persuading, but rather to discover the means
of coming as near such success as the circumstances of each particular
case allow. in this it resembles all other arts. for example, it is
not the function of medicine simply to make a man quite healthy, but
to put him as far as may be on the road to health; it is possible
to give excellent treatment even to those who can never enjoy sound
health. furthermore, it is plain that it is the function of one and
the same art to discern the real and the apparent means of persuasion,
just as it is the function of dialectic to discern the real and the
apparent syllogism. what makes a man a 'sophist' is not his faculty,
but his moral purpose. in rhetoric, however, the term 'rhetorician'
may describe either the speaker's knowledge of the art, or his moral
purpose. in dialectic it is different: a man is a 'sophist' because
he has a certain kind of moral purpose, a 'dialectician' in respect,
not of his moral purpose, but of his faculty.
let us now try to give some account of the systematic principles of
rhetoric itself-of the right method and means of succeeding in the
object we set before us. we must make as it were a fresh start, and
before going further define what rhetoric is.
rhetoric may be defined as the faculty of observing in any given case
the available means of persuasion. this is not a function of any other
art. every other art can instruct or persuade about its own particular
subject-matter; for instance, medicine about what is healthy and unhealthy,
geometry about the properties of magnitudes, arithmetic about numbers,
and the same is true of the other arts and sciences. but rhetoric
we look upon as the power of observing the means of persuasion on
almost any subject presented to us; and that is why we say that, in
its technical character, it is not concerned with any special or definite
of the modes of persuasion some belong strictly to the art of rhetoric
and some do not. by the latter i mean such things as are not supplied
by the speaker but are there at the outset-witnesses, evidence given
under torture, written contracts, and so on. by the former i mean
such as we can ourselves construct by means of the principles of rhetoric.
the one kind has merely to be used, the other has to be invented.
of the modes of persuasion furnished by the spoken word there are
three kinds. the first kind depends on the personal character of the
speaker; the second on putting the audience into a certain frame of
mind; the third on the proof, or apparent proof, provided by the words
of the speech itself. persuasion is achieved by the speaker's personal
character when the speech is so spoken as to make us think him credible.
we believe good men more fully and more readily than others: this
is true generally whatever the question is, and absolutely true where
exact certainty is impossible and opinions are divided. this kind
of persuasion, like the others, should be achieved by what the speaker
says, not by what people think of his character before he begins to
speak. it is not true, as some writers assume in their treatises on
rhetoric, that the personal goodness revealed by the speaker contributes
nothing to his power of persuasion; on the contrary, his character
may almost be called the most effective means of persuasion he possesses.
secondly, persuasion may come through the hearers, when the speech
stirs their emotions. our judgements when we are pleased and friendly
are not the same as when we are pained and hostile. it is towards
producing these effects, as we maintain, that present-day writers
on rhetoric direct the whole of their efforts. this subject shall
be treated in detail when we come to speak of the emotions. thirdly,
persuasion is effected through the speech itself when we have proved
a truth or an apparent truth by means of the persuasive arguments
there are, then, these three means of effecting persuasion. the man
who is to be in command of them must, it is clear, be able (1) to
reason logically, (2) to understand human character and goodness in
their various forms, and (3) to understand the emotions-that is, to
name them and describe them, to know their causes and the way in which
they are excited. it thus appears that rhetoric is an offshoot of
dialectic and also of ethical studies. ethical studies may fairly
be called political; and for this reason rhetoric masquerades as political
science, and the professors of it as political experts-sometimes from
want of education, sometimes from ostentation, sometimes owing to
other human failings. as a matter of fact, it is a branch of dialectic
and similar to it, as we said at the outset. neither rhetoric nor
dialectic is the scientific study of any one separate subject: both
are faculties for providing arguments. this is perhaps a sufficient
account of their scope and of how they are related to each other.
with regard to the persuasion achieved by proof or apparent proof:
just as in dialectic there is induction on the one hand and syllogism
or apparent syllogism on the other, so it is in rhetoric. the example
is an induction, the enthymeme is a syllogism, and the apparent enthymeme
is an apparent syllogism. i call the enthymeme a rhetorical syllogism,
and the example a rhetorical induction. every one who effects persuasion
through proof does in fact use either enthymemes or examples: there
is no other way. and since every one who proves anything at all is
bound to use either syllogisms or inductions (and this is clear to
us from the analytics), it must follow that enthymemes are syllogisms
and examples are inductions. the difference between example and enthymeme
is made plain by the passages in the topics where induction and syllogism
have already been discussed. when we base the proof of a proposition
on a number of similar cases, this is induction in dialectic, example
in rhetoric; when it is shown that, certain propositions being true,
a further and quite distinct proposition must also be true in consequence,
whether invariably or usually, this is called syllogism in dialectic,
enthymeme in rhetoric. it is plain also that each of these types of
oratory has its advantages. types of oratory, i say: for what has
been said in the methodics applies equally well here; in some oratorical
styles examples prevail, in others enthymemes; and in like manner,
some orators are better at the former and some at the latter. speeches
that rely on examples are as persuasive as the other kind, but those
which rely on enthymemes excite the louder applause. the sources of
examples and enthymemes, and their proper uses, we will discuss later.
our next step is to define the processes themselves more clearly.
a statement is persuasive and credible either because it is directly
self-evident or because it appears to be proved from other statements
that are so. in either case it is persuasive because there is somebody
whom it persuades. but none of the arts theorize about individual
cases. medicine, for instance, does not theorize about what will help
to cure socrates or callias, but only about what will help to cure
any or all of a given class of patients: this alone is business: individual
cases are so infinitely various that no systematic knowledge of them
is possible. in the same way the theory of rhetoric is concerned not
with what seems probable to a given individual like socrates or hippias,
but with what seems probable to men of a given type; and this is true
of dialectic also. dialectic does not construct its syllogisms out
of any haphazard materials, such as the fancies of crazy people, but
out of materials that call for discussion; and rhetoric, too, draws
upon the regular subjects of debate. the duty of rhetoric is to deal
with such matters as we deliberate upon without arts or systems to
guide us, in the hearing of persons who cannot take in at a glance
a complicated argument, or follow a long chain of reasoning. the subjects
of our deliberation are such as seem to present us with alternative
possibilities: about things that could not have been, and cannot now
or in the future be, other than they are, nobody who takes them to
be of this nature wastes his time in deliberation.
it is possible to form syllogisms and draw conclusions from the results
of previous syllogisms; or, on the other hand, from premisses which
have not been thus proved, and at the same time are so little accepted
that they call for proof. reasonings of the former kind will necessarily
be hard to follow owing to their length, for we assume an audience
of untrained thinkers; those of the latter kind will fail to win assent,
because they are based on premisses that are not generally admitted
the enthymeme and the example must, then, deal with what is in the
main contingent, the example being an induction, and the enthymeme
a syllogism, about such matters. the enthymeme must consist of few
propositions, fewer often than those which make up the normal syllogism.
for if any of these propositions is a familiar fact, there is no need
even to mention it; the hearer adds it himself. thus, to show that
dorieus has been victor in a contest for which the prize is a crown,
it is enough to say 'for he has been victor in the olympic games',
without adding 'and in the olympic games the prize is a crown', a
there are few facts of the 'necessary' type that can form the basis
of rhetorical syllogisms. most of the things about which we make decisions,
and into which therefore we inquire, present us with alternative possibilities.
for it is about our actions that we deliberate and inquire, and all
our actions have a contingent character; hardly any of them are determined
by necessity. again, conclusions that state what is merely usual or
possible must be drawn from premisses that do the same, just as 'necessary'
conclusions must be drawn from 'necessary' premisses; this too is
clear to us from the analytics. it is evident, therefore, that the
propositions forming the basis of enthymemes, though some of them
may be 'necessary', will most of them be only usually true. now the
materials of enthymemes are probabilities and signs, which we can
see must correspond respectively with the propositions that are generally
and those that are necessarily true. a probability is a thing that
usually happens; not, however, as some definitions would suggest,
anything whatever that usually happens, but only if it belongs to
the class of the 'contingent' or 'variable'. it bears the same relation
to that in respect of which it is probable as the universal bears
to the particular. of signs, one kind bears the same relation to the
statement it supports as the particular bears to the universal, the
other the same as the universal bears to the particular. the infallible
kind is a 'complete proof' (tekmerhiou); the fallible kind has no
specific name. by infallible signs i mean those on which syllogisms
proper may be based: and this shows us why this kind of sign is called
'complete proof': when people think that what they have said cannot
be refuted, they then think that they are bringing forward a 'complete
proof', meaning that the matter has now been demonstrated and completed
(peperhasmeuou); for the word 'perhas' has the same meaning (of 'end'
or 'boundary') as the word 'tekmarh' in the ancient tongue. now the
one kind of sign (that which bears to the proposition it supports
the relation of particular to universal) may be illustrated thus.
suppose it were said, 'the fact that socrates was wise and just is
a sign that the wise are just'. here we certainly have a sign; but
even though the proposition be true, the argument is refutable, since
it does not form a syllogism. suppose, on the other hand, it were
said, 'the fact that he has a fever is a sign that he is ill', or,
'the fact that she is giving milk is a sign that she has lately borne
a child'. here we have the infallible kind of sign, the only kind
that constitutes a complete proof, since it is the only kind that,
if the particular statement is true, is irrefutable. the other kind
of sign, that which bears to the proposition it supports the relation
of universal to particular, might be illustrated by saying, 'the fact
that he breathes fast is a sign that he has a fever'. this argument
also is refutable, even if the statement about the fast breathing
be true, since a man may breathe hard without having a fever.
it has, then, been stated above what is the nature of a probability,
of a sign, and of a complete proof, and what are the differences between
them. in the analytics a more explicit description has been given
of these points; it is there shown why some of these reasonings can
the 'example' has already been described as one kind of induction;
and the special nature of the subject-matter that distinguishes it
from the other kinds has also been stated above. its relation to the
proposition it supports is not that of part to whole, nor whole to
part, nor whole to whole, but of part to part, or like to like. when
two statements are of the same order, but one is more familiar than
the other, the former is an 'example'. the argument may, for instance,
be that dionysius, in asking as he does for a bodyguard, is scheming
to make himself a despot. for in the past peisistratus kept asking
for a bodyguard in order to carry out such a scheme, and did make
himself a despot as soon as he got it; and so did theagenes at megara;
and in the same way all other instances known to the speaker are made
into examples, in order to show what is not yet known, that dionysius
has the same purpose in making the same request: all these being instances
of the one general principle, that a man who asks for a bodyguard
is scheming to make himself a despot. we have now described the sources
of those means of persuasion which are popularly supposed to be demonstrative.
there is an important distinction between two sorts of enthymemes
that has been wholly overlooked by almost everybody-one that also
subsists between the syllogisms treated of in dialectic. one sort
of enthymeme really belongs to rhetoric, as one sort of syllogism
really belongs to dialectic; but the other sort really belongs to
other arts and faculties, whether to those we already exercise or
to those we have not yet acquired. missing this distinction, people
fail to notice that the more correctly they handle their particular
subject the further they are getting away from pure rhetoric or dialectic.
this statement will be clearer if expressed more fully. i mean that
the proper subjects of dialectical and rhetorical syllogisms are the
things with which we say the regular or universal lines of argument
are concerned, that is to say those lines of argument that apply equally
to questions of right conduct, natural science, politics, and many
other things that have nothing to do with one another. take, for instance,
the line of argument concerned with 'the more or less'. on this line
of argument it is equally easy to base a syllogism or enthymeme about
any of what nevertheless are essentially disconnected subjects-right
conduct, natural science, or anything else whatever. but there are
also those special lines of argument which are based on such propositions
as apply only to particular groups or classes of things. thus there
are propositions about natural science on which it is impossible to
base any enthymeme or syllogism about ethics, and other propositions
about ethics on which nothing can be based about natural science.
the same principle applies throughout. the general lines of argument
have no special subject-matter, and therefore will not increase our
understanding of any particular class of things. on the other hand,
the better the selection one makes of propositions suitable for special
lines of argument, the nearer one comes, unconsciously, to setting
up a science that is distinct from dialectic and rhetoric. one may
succeed in stating the required principles, but one's science will
be no longer dialectic or rhetoric, but the science to which the principles
thus discovered belong. most enthymemes are in fact based upon these
particular or special lines of argument; comparatively few on the
common or general kind. as in the therefore, so in this work, we must
distinguish, in dealing with enthymemes, the special and the general
lines of argument on which they are to be founded. by special lines
of argument i mean the propositions peculiar to each several class
of things, by general those common to all classes alike. we may begin
with the special lines of argument. but, first of all, let us classify
rhetoric into its varieties. having distinguished these we may deal
with them one by one, and try to discover the elements of which each
is composed, and the propositions each must employ.
rhetoric falls into three divisions, determined by the three classes
of listeners to speeches. for of the three elements in speech-making--speaker,
subject, and person addressed--it is the last one, the hearer, that
determines the speech's end and object. the hearer must be either
a judge, with a decision to make about things past or future, or an
observer. a member of the assembly decides about future events, a
juryman about past events: while those who merely decide on the orator's
skill are observers. from this it follows that there are three divisions
of oratory-(1) political, (2) forensic, and (3) the ceremonial oratory
political speaking urges us either to do or not to do something: one
of these two courses is always taken by private counsellors, as well
as by men who address public assemblies. forensic speaking either
attacks or defends somebody: one or other of these two things must
always be done by the parties in a case. the ceremonial oratory of
display either praises or censures somebody. these three kinds of
rhetoric refer to three different kinds of time. the political orator
is concerned with the future: it is about things to be done hereafter
that he advises, for or against. the party in a case at law is concerned
with the past; one man accuses the other, and the other defends himself,
with reference to things already done. the ceremonial orator is, properly
speaking, concerned with the present, since all men praise or blame
in view of the state of things existing at the time, though they often
find it useful also to recall the past and to make guesses at the
rhetoric has three distinct ends in view, one for each of its three
kinds. the political orator aims at establishing the expediency or
the harmfulness of a proposed course of action; if he urges its acceptance,
he does so on the ground that it will do good; if he urges its rejection,
he does so on the ground that it will do harm; and all other points,
such as whether the proposal is just or unjust, honourable or dishonourable,
he brings in as subsidiary and relative to this main consideration.
parties in a law-case aim at establishing the justice or injustice
of some action, and they too bring in all other points as subsidiary
and relative to this one. those who praise or attack a man aim at
proving him worthy of honour or the reverse, and they too treat all
other considerations with reference to this one.
that the three kinds of rhetoric do aim respectively at the three
ends we have mentioned is shown by the fact that speakers will sometimes
not try to establish anything else. thus, the litigant will sometimes
not deny that a thing has happened or that he has done harm. but that
he is guilty of injustice he will never admit; otherwise there would
be no need of a trial. so too, political orators often make any concession
short of admitting that they are recommending their hearers to take
an inexpedient course or not to take an expedient one. the question
whether it is not unjust for a city to enslave its innocent neighbours
often does not trouble them at all. in like manner those who praise
or censure a man do not consider whether his acts have been expedient
or not, but often make it a ground of actual praise that he has neglected
his own interest to do what was honourable. thus, they praise achilles
because he championed his fallen friend patroclus, though he knew
that this meant death, and that otherwise he need not die: yet while
to die thus was the nobler thing for him to do, the expedient thing
it is evident from what has been said that it is these three subjects,
more than any others, about which the orator must be able to have
propositions at his command. now the propositions of rhetoric are
complete proofs, probabilities, and signs. every kind of syllogism
is composed of propositions, and the enthymeme is a particular kind
of syllogism composed of the aforesaid propositions.
since only possible actions, and not impossible ones, can ever have
been done in the past or the present, and since things which have
not occurred, or will not occur, also cannot have been done or be
going to be done, it is necessary for the political, the forensic,
and the ceremonial speaker alike to be able to have at their command
propositions about the possible and the impossible, and about whether
a thing has or has not occurred, will or will not occur. further,
all men, in giving praise or blame, in urging us to accept or reject
proposals for action, in accusing others or defending themselves,
attempt not only to prove the points mentioned but also to show that
the good or the harm, the honour or disgrace, the justice or injustice,
is great or small, either absolutely or relatively; and therefore
it is plain that we must also have at our command propositions about
greatness or smallness and the greater or the lesser-propositions
both universal and particular. thus, we must be able to say which
is the greater or lesser good, the greater or lesser act of justice
such, then, are the subjects regarding which we are inevitably bound
to master the propositions relevant to them. we must now discuss each
particular class of these subjects in turn, namely those dealt with
in political, in ceremonial, and lastly in legal, oratory.
first, then, we must ascertain what are the kinds of things, good
or bad, about which the political orator offers counsel. for he does
not deal with all things, but only with such as may or may not take
place. concerning things which exist or will exist inevitably, or
which cannot possibly exist or take place, no counsel can be given.
nor, again, can counsel be given about the whole class of things which
may or may not take place; for this class includes some good things
that occur naturally, and some that occur by accident; and about these
it is useless to offer counsel. clearly counsel can only be given
on matters about which people deliberate; matters, namely, that ultimately
depend on ourselves, and which we have it in our power to set going.
for we turn a thing over in our mind until we have reached the point
now to enumerate and classify accurately the usual subjects of public
business, and further to frame, as far as possible, true definitions
of them is a task which we must not attempt on the present occasion.
for it does not belong to the art of rhetoric, but to a more instructive
art and a more real branch of knowledge; and as it is, rhetoric has
been given a far wider subject-matter than strictly belongs to it.
the truth is, as indeed we have said already, that rhetoric is a combination
of the science of logic and of the ethical branch of politics; and
it is partly like dialectic, partly like sophistical reasoning. but
the more we try to make either dialectic rhetoric not, what they really
are, practical faculties, but sciences, the more we shall inadvertently
be destroying their true nature; for we shall be re-fashioning them
and shall be passing into the region of sciences dealing with definite
subjects rather than simply with words and forms of reasoning. even
here, however, we will mention those points which it is of practical
importance to distinguish, their fuller treatment falling naturally
the main matters on which all men deliberate and on which political
speakers make speeches are some five in number: ways and means, war
and peace, national defence, imports and exports, and legislation.
as to ways and means, then, the intending speaker will need to know
the number and extent of the country's sources of revenue, so that,
if any is being overlooked, it may be added, and, if any is defective,
it may be increased. further, he should know all the expenditure of
the country, in order that, if any part of it is superfluous, it may
be abolished, or, if any is too large, it may be reduced. for men
become richer not only by increasing their existing wealth but also
by reducing their expenditure. a comprehensive view of these questions
cannot be gained solely from experience in home affairs; in order
to advise on such matters a man must be keenly interested in the methods
as to peace and war, he must know the extent of the military strength
of his country, both actual and potential, and also the mature of
that actual and potential strength; and further, what wars his country
has waged, and how it has waged them. he must know these facts not
only about his own country, but also about neighbouring countries;
and also about countries with which war is likely, in order that peace
may be maintained with those stronger than his own, and that his own
may have power to make war or not against those that are weaker. he
should know, too, whether the military power of another country is
like or unlike that of his own; for this is a matter that may affect
their relative strength. with the same end in view he must, besides,
have studied the wars of other countries as well as those of his own,
and the way they ended; similar causes are likely to have similar
with regard to national defence: he ought to know all about the methods
of defence in actual use, such as the strength and character of the
defensive force and the positions of the forts-this last means that
he must be well acquainted with the lie of the country-in order that
a garrison may be increased if it is too small or removed if it is
not wanted, and that the strategic points may be guarded with special
with regard to the food supply: he must know what outlay will meet
the needs of his country; what kinds of food are produced at home
and what imported; and what articles must be exported or imported.
this last he must know in order that agreements and commercial treaties
may be made with the countries concerned. there are, indeed, two sorts
of state to which he must see that his countrymen give no cause for
offence, states stronger than his own, and states with which it is
but while he must, for security's sake, be able to take all this into
account, he must before all things understand the subject of legislation;
for it is on a country's laws that its whole welfare depends. he must,
therefore, know how many different forms of constitution there are;
under what conditions each of these will prosper and by what internal
developments or external attacks each of them tends to be destroyed.
when i speak of destruction through internal developments i refer
to the fact that all constitutions, except the best one of all, are
destroyed both by not being pushed far enough and by being pushed
too far. thus, democracy loses its vigour, and finally passes into
oligarchy, not only when it is not pushed far enough, but also when
it is pushed a great deal too far; just as the aquiline and the snub
nose not only turn into normal noses by not being aquiline or snub
enough, but also by being too violently aquiline or snub arrive at
a condition in which they no longer look like noses at all. it is
useful, in framing laws, not only to study the past history of one's
own country, in order to understand which constitution is desirable
for it now, but also to have a knowledge of the constitutions of other
nations, and so to learn for what kinds of nation the various kinds
of constitution are suited. from this we can see that books of travel
are useful aids to legislation, since from these we may learn the
laws and customs of different races. the political speaker will also
find the researches of historians useful. but all this is the business
of political science and not of rhetoric.
these, then, are the most important kinds of information which the
political speaker must possess. let us now go back and state the premisses
from which he will have to argue in favour of adopting or rejecting
measures regarding these and other matters.
it may be said that every individual man and all men in common aim
at a certain end which determines what they choose and what they avoid.
this end, to sum it up briefly, is happiness and its constituents.
let us, then, by way of illustration only, ascertain what is in general
the nature of happiness, and what are the elements of its constituent
parts. for all advice to do things or not to do them is concerned
with happiness and with the things that make for or against it; whatever
creates or increases happiness or some part of happiness, we ought
to do; whatever destroys or hampers happiness, or gives rise to its
we may define happiness as prosperity combined with virtue; or as
independence of life; or as the secure enjoyment of the maximum of
pleasure; or as a good condition of property and body, together with
the power of guarding one's property and body and making use of them.
that happiness is one or more of these things, pretty well everybody
from this definition of happiness it follows that its constituent
parts are:-good birth, plenty of friends, good friends, wealth, good
children, plenty of children, a happy old age, also such bodily excellences
as health, beauty, strength, large stature, athletic powers, together
with fame, honour, good luck, and virtue. a man cannot fail to be
completely independent if he possesses these internal and these external
goods; for besides these there are no others to have. (goods of the
soul and of the body are internal. good birth, friends, money, and
honour are external.) further, we think that he should possess resources
and luck, in order to make his life really secure. as we have already
ascertained what happiness in general is, so now let us try to ascertain
now good birth in a race or a state means that its members are indigenous
or ancient: that its earliest leaders were distinguished men, and
that from them have sprung many who were distinguished for qualities
the good birth of an individual, which may come either from the male
or the female side, implies that both parents are free citizens, and
that, as in the case of the state, the founders of the line have been
notable for virtue or wealth or something else which is highly prized,
and that many distinguished persons belong to the family, men and
the phrases 'possession of good children' and 'of many children' bear
a quite clear meaning. applied to a community, they mean that its
young men are numerous and of good a quality: good in regard to bodily
excellences, such as stature, beauty, strength, athletic powers; and
also in regard to the excellences of the soul, which in a young man
are temperance and courage. applied to an individual, they mean that
his own children are numerous and have the good qualities we have
described. both male and female are here included; the excellences
of the latter are, in body, beauty and stature; in soul, self-command
and an industry that is not sordid. communities as well as individuals
should lack none of these perfections, in their women as well as in
their men. where, as among the lacedaemonians, the state of women
is bad, almost half of human life is spoilt.
the constituents of wealth are: plenty of coined money and territory;
the ownership of numerous, large, and beautiful estates; also the
ownership of numerous and beautiful implements, live stock, and slaves.
all these kinds of property are our own, are secure, gentlemanly,
and useful. the useful kinds are those that are productive, the gentlemanly
kinds are those that provide enjoyment. by 'productive' i mean those
from which we get our income; by 'enjoyable', those from which we
get nothing worth mentioning except the use of them. the criterion
of 'security' is the ownership of property in such places and under
such conditions that the use of it is in our power; and it is 'our
own' if it is in our own power to dispose of it or keep it. by 'disposing
of it' i mean giving it away or selling it. wealth as a whole consists
in using things rather than in owning them; it is really the activity-that
is, the use-of property that constitutes wealth.
fame means being respected by everybody, or having some quality that
is desired by all men, or by most, or by the good, or by the wise.
honour is the token of a man's being famous for doing good. it is
chiefly and most properly paid to those who have already done good;
but also to the man who can do good in future. doing good refers either
to the preservation of life and the means of life, or to wealth, or
to some other of the good things which it is hard to get either always
or at that particular place or time-for many gain honour for things
which seem small, but the place and the occasion account for it. the
constituents of honour are: sacrifices; commemoration, in verse or
prose; privileges; grants of land; front seats at civic celebrations;
state burial; statues; public maintenance; among foreigners, obeisances
and giving place; and such presents as are among various bodies of
men regarded as marks of honour. for a present is not only the bestowal
of a piece of property, but also a token of honour; which explains
why honour-loving as well as money-loving persons desire it. the present
brings to both what they want; it is a piece of property, which is
what the lovers of money desire; and it brings honour, which is what
the excellence of the body is health; that is, a condition which allows
us, while keeping free from disease, to have the use of our bodies;
for many people are 'healthy' as we are told herodicus was; and these
no one can congratulate on their 'health', for they have to abstain
from everything or nearly everything that men do.-beauty varies with
the time of life. in a young man beauty is the possession of a body
fit to endure the exertion of running and of contests of strength;
which means that he is pleasant to look at; and therefore all-round
athletes are the most beautiful, being naturally adapted both for
contests of strength and for speed also. for a man in his prime, beauty
is fitness for the exertion of warfare, together with a pleasant but
at the same time formidable appearance. for an old man, it is to be
strong enough for such exertion as is necessary, and to be free from
all those deformities of old age which cause pain to others. strength
is the power of moving some one else at will; to do this, you must
either pull, push, lift, pin, or grip him; thus you must be strong
in all of those ways or at least in some. excellence in size is to
surpass ordinary people in height, thickness, and breadth by just
as much as will not make one's movements slower in consequence. athletic
excellence of the body consists in size, strength, and swiftness;
swiftness implying strength. he who can fling forward his legs in
a certain way, and move them fast and far, is good at running; he
who can grip and hold down is good at wrestling; he who can drive
an adversary from his ground with the right blow is a good boxer:
he who can do both the last is a good pancratiast, while he who can
happiness in old age is the coming of old age slowly and painlessly;
for a man has not this happiness if he grows old either quickly, or
tardily but painfully. it arises both from the excellences of the
body and from good luck. if a man is not free from disease, or if
he is strong, he will not be free from suffering; nor can he continue
to live a long and painless life unless he has good luck. there is,
indeed, a capacity for long life that is quite independent of health
or strength; for many people live long who lack the excellences of
the body; but for our present purpose there is no use in going into
the terms 'possession of many friends' and 'possession of good friends'
need no explanation; for we define a 'friend' as one who will always
try, for your sake, to do what he takes to be good for you. the man
towards whom many feel thus has many friends; if these are worthy
'good luck' means the acquisition or possession of all or most, or
the most important, of those good things which are due to luck. some
of the things that are due to luck may also be due to artificial contrivance;
but many are independent of art, as for example those which are due
to nature-though, to be sure, things due to luck may actually be contrary
to nature. thus health may be due to artificial contrivance, but beauty
and stature are due to nature. all such good things as excite envy
are, as a class, the outcome of good luck. luck is also the cause
of good things that happen contrary to reasonable expectation: as
when, for instance, all your brothers are ugly, but you are handsome
yourself; or when you find a treasure that everybody else has overlooked;
or when a missile hits the next man and misses you; or when you are
the only man not to go to a place you have gone to regularly, while
the others go there for the first time and are killed. all such things
as to virtue, it is most closely connected with the subject of eulogy,
and therefore we will wait to define it until we come to discuss that
it is now plain what our aims, future or actual, should be in urging,
and what in depreciating, a proposal; the latter being the opposite
of the former. now the political or deliberative orator's aim is utility:
deliberation seeks to determine not ends but the means to ends, i.e.
what it is most useful to do. further, utility is a good thing. we
ought therefore to assure ourselves of the main facts about goodness
we may define a good thing as that which ought to be chosen for its
own sake; or as that for the sake of which we choose something else;
or as that which is sought after by all things, or by all things that
have sensation or reason, or which will be sought after by any things
that acquire reason; or as that which must be prescribed for a given
individual by reason generally, or is prescribed for him by his individual
reason, this being his individual good; or as that whose presence
brings anything into a satisfactory and self-sufficing condition;
or as self-sufficiency; or as what produces, maintains, or entails
characteristics of this kind, while preventing and destroying their
opposites. one thing may entail another in either of two ways-(1)
simultaneously, (2) subsequently. thus learning entails knowledge
subsequently, health entails life simultaneously. things are productive
of other things in three senses: first as being healthy produces health;
secondly, as food produces health; and thirdly, as exercise does-i.e.
it does so usually. all this being settled, we now see that both the
acquisition of good things and the removal of bad things must be good;
the latter entails freedom from the evil things simultaneously, while
the former entails possession of the good things subsequently. the
acquisition of a greater in place of a lesser good, or of a lesser
in place of a greater evil, is also good, for in proportion as the
greater exceeds the lesser there is acquisition of good or removal
of evil. the virtues, too, must be something good; for it is by possessing
these that we are in a good condition, and they tend to produce good
works and good actions. they must be severally named and described
elsewhere. pleasure, again, must be a good thing, since it is the
nature of all animals to aim at it. consequently both pleasant and
beautiful things must be good things, since the former are productive
of pleasure, while of the beautiful things some are pleasant and some
the following is a more detailed list of things that must be good.
happiness, as being desirable in itself and sufficient by itself,
and as being that for whose sake we choose many other things. also
justice, courage, temperance, magnanimity, magnificence, and all such
qualities, as being excellences of the soul. further, health, beauty,
and the like, as being bodily excellences and productive of many other
good things: for instance, health is productive both of pleasure and
of life, and therefore is thought the greatest of goods, since these
two things which it causes, pleasure and life, are two of the things
most highly prized by ordinary people. wealth, again: for it is the
excellence of possession, and also productive of many other good things.
friends and friendship: for a friend is desirable in himself and also
productive of many other good things. so, too, honour and reputation,
as being pleasant, and productive of many other good things, and usually
accompanied by the presence of the good things that cause them to
be bestowed. the faculty of speech and action; since all such qualities
are productive of what is good. further-good parts, strong memory,
receptiveness, quickness of intuition, and the like, for all such
faculties are productive of what is good. similarly, all the sciences
and arts. and life: since, even if no other good were the result of
life, it is desirable in itself. and justice, as the cause of good
the above are pretty well all the things admittedly good. in dealing
with things whose goodness is disputed, we may argue in the following
ways:-that is good of which the contrary is bad. that is good the
contrary of which is to the advantage of our enemies; for example,
if it is to the particular advantage of our enemies that we should
be cowards, clearly courage is of particular value to our countrymen.
and generally, the contrary of that which our enemies desire, or of
that at which they rejoice, is evidently valuable. hence the passage
this principle usually holds good, but not always, since it may well
be that our interest is sometimes the same as that of our enemies.
hence it is said that 'evils draw men together'; that is, when the
further: that which is not in excess is good, and that which is greater
than it should be is bad. that also is good on which much labour or
money has been spent; the mere fact of this makes it seem good, and
such a good is assumed to be an end-an end reached through a long
chain of means; and any end is a good. hence the lines beginning:
"and for priam (and troy-town's folk) should
"to have tarried so long and return empty-handed
and there is also the proverb about 'breaking the pitcher at the
that which most people seek after, and which is obviously an object
of contention, is also a good; for, as has been shown, that is good
which is sought after by everybody, and 'most people' is taken to
be equivalent to 'everybody'. that which is praised is good, since
no one praises what is not good. so, again, that which is praised
by our enemies or by the worthless for when even those who have
a grievance think a thing good, it is at once felt that every one
must agree with them; our enemies can admit the fact only because
it is evident, just as those must be worthless whom their friends
censure and their enemies do not. (for this reason the corinthians
conceived themselves to be insulted by simonides when he wrote:
"against the corinthians hath ilium no complaint.) "
again, that is good which has been distinguished by the favour of
a discerning or virtuous man or woman, as odysseus was distinguished
by athena, helen by theseus, paris by the goddesses, and achilles
by homer. and, generally speaking, all things are good which men deliberately
choose to do; this will include the things already mentioned, and
also whatever may be bad for their enemies or good for their friends,
and at the same time practicable. things are 'practicable' in two
senses: (1) it is possible to do them, (2) it is easy to do them.
things are done 'easily' when they are done either without pain or
quickly: the 'difficulty' of an act lies either in its painfulness
or in the long time it takes. again, a thing is good if it is as men
wish; and they wish to have either no evil at an or at least a balance
of good over evil. this last will happen where the penalty is either
imperceptible or slight. good, too, are things that are a man's very
own, possessed by no one else, exceptional; for this increases the
credit of having them. so are things which befit the possessors, such
as whatever is appropriate to their birth or capacity, and whatever
they feel they ought to have but lack-such things may indeed be trifling,
but none the less men deliberately make them the goal of their action.
and things easily effected; for these are practicable (in the sense
of being easy); such things are those in which every one, or most
people, or one's equals, or one's inferiors have succeeded. good also
are the things by which we shall gratify our friends or annoy our
enemies; and the things chosen by those whom we admire: and the things
for which we are fitted by nature or experience, since we think we
shall succeed more easily in these: and those in which no worthless
man can succeed, for such things bring greater praise: and those which
we do in fact desire, for what we desire is taken to be not only pleasant
but also better. further, a man of a given disposition makes chiefly
for the corresponding things: lovers of victory make for victory,
lovers of honour for honour, money-loving men for money, and so with
the rest. these, then, are the sources from which we must derive our
means of persuasion about good and utility.
since, however, it often happens that people agree that two things
are both useful but do not agree about which is the more so, the next
step will be to treat of relative goodness and relative utility.
a thing which surpasses another may be regarded as being that other
thing plus something more, and that other thing which is surpassed
as being what is contained in the first thing. now to call a thing
'greater' or 'more' always implies a comparison of it with one that
is 'smaller' or 'less', while 'great' and 'small', 'much' and 'little',
are terms used in comparison with normal magnitude. the 'great' is
that which surpasses the normal, the 'small' is that which is surpassed
by the normal; and so with 'many' and 'few'.
now we are applying the term 'good' to what is desirable for its own
sake and not for the sake of something else; to that at which all
things aim; to what they would choose if they could acquire understanding
and practical wisdom; and to that which tends to produce or preserve
such goods, or is always accompanied by them. moreover, that for the
sake of which things are done is the end (an end being that for the
sake of which all else is done), and for each individual that thing
is a good which fulfils these conditions in regard to himself. it
follows, then, that a greater number of goods is a greater good than
one or than a smaller number, if that one or that smaller number is
included in the count; for then the larger number surpasses the smaller,
and the smaller quantity is surpassed as being contained in the larger.
again, if the largest member of one class surpasses the largest member
of another, then the one class surpasses the other; and if one class
surpasses another, then the largest member of the one surpasses the
largest member of the other. thus, if the tallest man is taller than
the tallest woman, then men in general are taller than women. conversely,
if men in general are taller than women, then the tallest man is taller
than the tallest woman. for the superiority of class over class is
proportionate to the superiority possessed by their largest specimens.
again, where one good is always accompanied by another, but does not
always accompany it, it is greater than the other, for the use of
the second thing is implied in the use of the first. a thing may be
accompanied by another in three ways, either simultaneously, subsequently,
or potentially. life accompanies health simultaneously (but not health
life), knowledge accompanies the act of learning subsequently, cheating
accompanies sacrilege potentially, since a man who has committed sacrilege
is always capable of cheating. again, when two things each surpass
a third, that which does so by the greater amount is the greater of
the two; for it must surpass the greater as well as the less of the
other two. a thing productive of a greater good than another is productive
of is itself a greater good than that other. for this conception of
'productive of a greater' has been implied in our argument. likewise,
that which is produced by a greater good is itself a greater good;
thus, if what is wholesome is more desirable and a greater good than
what gives pleasure, health too must be a greater good than pleasure.
again, a thing which is desirable in itself is a greater good than
a thing which is not desirable in itself, as for example bodily strength
than what is wholesome, since the latter is not pursued for its own
sake, whereas the former is; and this was our definition of the good.
again, if one of two things is an end, and the other is not, the former
is the greater good, as being chosen for its own sake and not for
the sake of something else; as, for example, exercise is chosen for
the sake of physical well-being. and of two things that which stands
less in need of the other, or of other things, is the greater good,
since it is more self-sufficing. (that which stands 'less' in need
of others is that which needs either fewer or easier things.) so when
one thing does not exist or cannot come into existence without a second,
while the second can exist without the first, the second is the better.
that which does not need something else is more self-sufficing than
that which does, and presents itself as a greater good for that reason.
again, that which is a beginning of other things is a greater good
than that which is not, and that which is a cause is a greater good
than that which is not; the reason being the same in each case, namely
that without a cause and a beginning nothing can exist or come into
existence. again, where there are two sets of consequences arising
from two different beginnings or causes, the consequences of the more
important beginning or cause are themselves the more important; and
conversely, that beginning or cause is itself the more important which
has the more important consequences. now it is plain, from all that
has been said, that one thing may be shown to be more important than
another from two opposite points of view: it may appear the more important
(1) because it is a beginning and the other thing is not, and also
(2) because it is not a beginning and the other thing is-on the ground
that the end is more important and is not a beginning. so leodamas,
when accusing callistratus, said that the man who prompted the deed
was more guilty than the doer, since it would not have been done if
he had not planned it. on the other hand, when accusing chabrias he
said that the doer was worse than the prompter, since there would
have been no deed without some one to do it; men, said he, plot a
further, what is rare is a greater good than what is plentiful. thus,
gold is a better thing than iron, though less useful: it is harder
to get, and therefore better worth getting. reversely, it may be argued
that the plentiful is a better thing than the rare, because we can
make more use of it. for what is often useful surpasses what is seldom
more generally: the hard thing is better than the easy, because it
is rarer: and reversely, the easy thing is better than the hard, for
it is as we wish it to be. that is the greater good whose contrary
is the greater evil, and whose loss affects us more. positive goodness
and badness are more important than the mere absence of goodness and
badness: for positive goodness and badness are ends, which the mere
absence of them cannot be. further, in proportion as the functions
of things are noble or base, the things themselves are good or bad:
conversely, in proportion as the things themselves are good or bad,
their functions also are good or bad; for the nature of results corresponds
with that of their causes and beginnings, and conversely the nature
of causes and beginnings corresponds with that of their results. moreover,
those things are greater goods, superiority in which is more desirable
or more honourable. thus, keenness of sight is more desirable than
keenness of smell, sight generally being more desirable than smell
generally; and similarly, unusually great love of friends being more
honourable than unusually great love of money, ordinary love of friends
is more honourable than ordinary love of money. conversely, if one
of two normal things is better or nobler than the other, an unusual
degree of that thing is better or nobler than an unusual degree of
the other. again, one thing is more honourable or better than another
if it is more honourable or better to desire it; the importance of
the object of a given instinct corresponds to the importance of the
instinct itself; and for the same reason, if one thing is more honourable
or better than another, it is more honourable and better to desire
it. again, if one science is more honourable and valuable than another,
the activity with which it deals is also more honourable and valuable;
as is the science, so is the reality that is its object, each science
being authoritative in its own sphere. so, also, the more valuable
and honourable the object of a science, the more valuable and honourable
the science itself is-in consequence. again, that which would be judged,
or which has been judged, a good thing, or a better thing than something
else, by all or most people of understanding, or by the majority of
men, or by the ablest, must be so; either without qualification, or
in so far as they use their understanding to form their judgement.
this is indeed a general principle, applicable to all other judgements
also; not only the goodness of things, but their essence, magnitude,
and general nature are in fact just what knowledge and understanding
will declare them to be. here the principle is applied to judgements
of goodness, since one definition of 'good' was 'what beings that
acquire understanding will choose in any given case': from which it
clearly follows that that thing is hetter which understanding declares
to be so. that, again, is a better thing which attaches to better
men, either absolutely, or in virtue of their being better; as courage
is better than strength. and that is a greater good which would be
chosen by a better man, either absolutely, or in virtue of his being
better: for instance, to suffer wrong rather than to do wrong, for
that would be the choice of the juster man. again, the pleasanter
of two things is the better, since all things pursue pleasure, and
things instinctively desire pleasurable sensation for its own sake;
and these are two of the characteristics by which the 'good' and the
'end' have been defined. one pleasure is greater than another if it
is more unmixed with pain, or more lasting. again, the nobler thing
is better than the less noble, since the noble is either what is pleasant
or what is desirable in itself. and those things also are greater
goods which men desire more earnestly to bring about for themselves
or for their friends, whereas those things which they least desire
to bring about are greater evils. and those things which are more
lasting are better than those which are more fleeting, and the more
secure than the less; the enjoyment of the lasting has the advantage
of being longer, and that of the secure has the advantage of suiting
our wishes, being there for us whenever we like. further, in accordance
with the rule of co-ordinate terms and inflexions of the same stem,
what is true of one such related word is true of all. thus if the
action qualified by the term 'brave' is more noble and desirable than
the action qualified by the term 'temperate', then 'bravery' is more
desirable than 'temperance' and 'being brave' than 'being temperate'.
that, again, which is chosen by all is a greater good than that which
is not, and that chosen by the majority than that chosen by the minority.
for that which all desire is good, as we have said;' and so, the more
a thing is desired, the better it is. further, that is the better
thing which is considered so by competitors or enemies, or, again,
by authorized judges or those whom they select to represent them.
in the first two cases the decision is virtually that of every one,
in the last two that of authorities and experts. and sometimes it
may be argued that what all share is the better thing, since it is
a dishonour not to share in it; at other times, that what none or
few share is better, since it is rarer. the more praiseworthy things
are, the nobler and therefore the better they are. so with the things
that earn greater honours than others-honour is, as it were, a measure
of value; and the things whose absence involves comparatively heavy
penalties; and the things that are better than others admitted or
believed to be good. moreover, things look better merely by being
divided into their parts, since they then seem to surpass a greater
number of things than before. hence homer says that meleager was roused
"all horrors that light on a folk whose city
"when they slaughter the men, when the burg is
"when strangers are haling young children to thraldom,
the same effect is produced by piling up facts in a climax after the
manner of epicharmus. the reason is partly the same as in the case
of division (for combination too makes the impression of great superiority),
and partly that the original thing appears to be the cause and origin
of important results. and since a thing is better when it is harder
or rarer than other things, its superiority may be due to seasons,
ages, places, times, or one's natural powers. when a man accomplishes
something beyond his natural power, or beyond his years, or beyond
the measure of people like him, or in a special way, or at a special
place or time, his deed will have a high degree of nobleness, goodness,
and justice, or of their opposites. hence the epigram on the victor
"in time past, hearing a yoke on my shoulders,
"i carried my loads of fish from, argos to tegea town. "
so iphicrates used to extol himself by describing the low estate from
which he had risen. again, what is natural is better than what is
acquired, since it is harder to come by. hence the words of homer:
and the best part of a good thing is particularly good; as when pericles
in his funeral oration said that the country's loss of its young men
in battle was 'as if the spring were taken out of the year'. so with
those things which are of service when the need is pressing; for example,
in old age and times of sickness. and of two things that which leads
more directly to the end in view is the better. so too is that which
is better for people generally as well as for a particular individual.
again, what can be got is better than what cannot, for it is good
in a given case and the other thing is not. and what is at the end
of life is better than what is not, since those things are ends in
a greater degree which are nearer the end. what aims at reality is
better than what aims at appearance. we may define what aims at appearance
as what a man will not choose if nobody is to know of his having it.
this would seem to show that to receive benefits is more desirable
than to confer them, since a man will choose the former even if nobody
is to know of it, but it is not the general view that he will choose
the latter if nobody knows of it. what a man wants to be is better
than what a man wants to seem, for in aiming at that he is aiming
more at reality. hence men say that justice is of small value, since
it is more desirable to seem just than to be just, whereas with health
it is not so. that is better than other things which is more useful
than they are for a number of different purposes; for example, that
which promotes life, good life, pleasure, and noble conduct. for this
reason wealth and health are commonly thought to be of the highest
value, as possessing all these advantages. again, that is better than
other things which is accompanied both with less pain and with actual
pleasure; for here there is more than one advantage; and so here we
have the good of feeling pleasure and also the good of not feeling
pain. and of two good things that is the better whose addition to
a third thing makes a better whole than the addition of the other
to the same thing will make. again, those things which we are seen
to possess are better than those which we are not seen to possess,
since the former have the air of reality. hence wealth may be regarded
as a greater good if its existence is known to others. that which
is dearly prized is better than what is not-the sort of thing that
some people have only one of, though others have more like it. accordingly,
blinding a one-eyed man inflicts worse injury than half-blinding a
man with two eyes; for the one-eyed man has been robbed of what he
the grounds on which we must base our arguments, when we are speaking
for or against a proposal, have now been set forth more or less completely.
the most important and effective qualification for success in persuading
audiences and speaking well on public affairs is to understand all
the forms of government and to discriminate their respective customs,
institutions, and interests. for all men are persuaded by considerations
of their interest, and their interest lies in the maintenance of the
established order. further, it rests with the supreme authority to
give authoritative decisions, and this varies with each form of government;
there are as many different supreme authorities as there are different
forms of government. the forms of government are four-democracy, oligarchy,
aristocracy, monarchy. the supreme right to judge and decide always
rests, therefore, with either a part or the whole of one or other
a democracy is a form of government under which the citizens distribute
the offices of state among themselves by lot, whereas under oligarchy
there is a property qualification, under aristocracy one of education.
by education i mean that education which is laid down by the law;
for it is those who have been loyal to the national institutions that
hold office under an aristocracy. these are bound to be looked upon
as 'the best men', and it is from this fact that this form of government
has derived its name ('the rule of the best'). monarchy, as the word
implies, is the constitution a in which one man has authority over
all. there are two forms of monarchy: kingship, which is limited by
prescribed conditions, and 'tyranny', which is not limited by anything.
we must also notice the ends which the various forms of government
pursue, since people choose in practice such actions as will lead
to the realization of their ends. the end of democracy is freedom;
of oligarchy, wealth; of aristocracy, the maintenance of education
and national institutions; of tyranny, the protection of the tyrant.
it is clear, then, that we must distinguish those particular customs,
institutions, and interests which tend to realize the ideal of each
constitution, since men choose their means with reference to their
ends. but rhetorical persuasion is effected not only by demonstrative
but by ethical argument; it helps a speaker to convince us, if we
believe that he has certain qualities himself, namely, goodness, or
goodwill towards us, or both together. similarly, we should know the
moral qualities characteristic of each form of government, for the
special moral character of each is bound to provide us with our most
effective means of persuasion in dealing with it. we shall learn the
qualities of governments in the same way as we learn the qualities
of individuals, since they are revealed in their deliberate acts of
choice; and these are determined by the end that inspires them.
we have now considered the objects, immediate or distant, at which
we are to aim when urging any proposal, and the grounds on which we
are to base our arguments in favour of its utility. we have also briefly
considered the means and methods by which we shall gain a good knowledge
of the moral qualities and institutions peculiar to the various forms
of government-only, however, to the extent demanded by the present
occasion; a detailed account of the subject has been given in the
we have now to consider virtue and vice, the noble and the base, since
these are the objects of praise and blame. in doing so, we shall at
the same time be finding out how to make our hearers take the required
view of our own characters-our second method of persuasion. the ways
in which to make them trust the goodness of other people are also
the ways in which to make them trust our own. praise, again, may be
serious or frivolous; nor is it always of a human or divine being
but often of inanimate things, or of the humblest of the lower animals.
here too we must know on what grounds to argue, and must, therefore,
now discuss the subject, though by way of illustration only.
the noble is that which is both desirable for its own sake and also
worthy of praise; or that which is both good and also pleasant because
good. if this is a true definition of the noble, it follows that virtue
must be noble, since it is both a good thing and also praiseworthy.
virtue is, according to the usual view, a faculty of providing and
preserving good things; or a faculty of conferring many great benefits,
and benefits of all kinds on all occasions. the forms of virtue are
justice, courage, temperance, magnificence, magnanimity, liberality,
gentleness, prudence, wisdom. if virtue is a faculty of beneficence,
the highest kinds of it must be those which are most useful to others,
and for this reason men honour most the just and the courageous, since
courage is useful to others in war, justice both in war and in peace.
next comes liberality; liberal people let their money go instead of
fighting for it, whereas other people care more for money than for
anything else. justice is the virtue through which everybody enjoys
his own possessions in accordance with the law; its opposite is injustice,
through which men enjoy the possessions of others in defiance of the
law. courage is the virtue that disposes men to do noble deeds in
situations of danger, in accordance with the law and in obedience
to its commands; cowardice is the opposite. temperance is the virtue
that disposes us to obey the law where physical pleasures are concerned;
incontinence is the opposite. liberality disposes us to spend money
for others' good; illiberality is the opposite. magnanimity is the
virtue that disposes us to do good to others on a large scale; its
opposite is meanness of spirit . magnificence is a virtue productive
of greatness in matters involving the spending of money. the opposites
of these two are smallness of spirit and meanness respectively. prudence
is that virtue of the understanding which enables men to come to wise
decisions about the relation to happiness of the goods and evils that
the above is a sufficient account, for our present purpose, of virtue
and vice in general, and of their various forms. as to further aspects
of the subject, it is not difficult to discern the facts; it is evident
that things productive of virtue are noble, as tending towards virtue;
and also the effects of virtue, that is, the signs of its presence
and the acts to which it leads. and since the signs of virtue, and
such acts as it is the mark of a virtuous man to do or have done to
him, are noble, it follows that all deeds or signs of courage, and
everything done courageously, must be noble things; and so with what
is just and actions done justly. (not, however, actions justly done
to us; here justice is unlike the other virtues; 'justly' does not
always mean 'nobly'; when a man is punished, it is more shameful that
this should be justly than unjustly done to him). the same is true
of the other virtues. again, those actions are noble for which the
reward is simply honour, or honour more than money. so are those in
which a man aims at something desirable for some one else's sake;
actions good absolutely, such as those a man does for his country
without thinking of himself; actions good in their own nature; actions
that are not good simply for the individual, since individual interests
are selfish. noble also are those actions whose advantage may be enjoyed
after death, as opposed to those whose advantage is enjoyed during
one's lifetime: for the latter are more likely to be for one's own
sake only. also, all actions done for the sake of others, since less
than other actions are done for one's own sake; and all successes
which benefit others and not oneself; and services done to one's benefactors,
for this is just; and good deeds generally, since they are not directed
to one's own profit. and the opposites of those things of which men
feel ashamed, for men are ashamed of saying, doing, or intending to
do shameful things. so when alcacus said
"if for things good and noble thou wert yearning,
"if to speak baseness were thy tongue not burning,
"no load of shame would on thine eyelids weigh;
"what thou with honour wishest thou wouldst say. "
those things, also, are noble for which men strive anxiously, without
feeling fear; for they feel thus about the good things which lead
to fair fame. again, one quality or action is nobler than another
if it is that of a naturally finer being: thus a man's will be nobler
than a woman's. and those qualities are noble which give more pleasure
to other people than to their possessors; hence the nobleness of justice
and just actions. it is noble to avenge oneself on one's enemies and
not to come to terms with them; for requital is just, and the just
is noble; and not to surrender is a sign of courage. victory, too,
and honour belong to the class of noble things, since they are desirable
even when they yield no fruits, and they prove our superiority in
good qualities. things that deserve to be remembered are noble, and
the more they deserve this, the nobler they are. so are the things
that continue even after death; those which are always attended by
honour; those which are exceptional; and those which are possessed
by one person alone-these last are more readily remembered than others.
so again are possessions that bring no profit, since they are more
fitting than others for a gentleman. so are the distinctive qualities
of a particular people, and the symbols of what it specially admires,
like long hair in sparta, where this is a mark of a free man, as it
is not easy to perform any menial task when one's hair is long. again,
it is noble not to practise any sordid craft, since it is the mark
of a free man not to live at another's beck and call. we are also
to assume when we wish either to praise a man or blame him that qualities
closely allied to those which he actually has are identical with them;
for instance, that the cautious man is cold-blooded and treacherous,
and that the stupid man is an honest fellow or the thick-skinned man
a good-tempered one. we can always idealize any given man by drawing
on the virtues akin to his actual qualities; thus we may say that
the passionate and excitable man is 'outspoken'; or that the arrogant
man is 'superb' or 'impressive'. those who run to extremes will be
said to possess the corresponding good qualities; rashness will be
called courage, and extravagance generosity. that will be what most
people think; and at the same time this method enables an advocate
to draw a misleading inference from the motive, arguing that if a
man runs into danger needlessly, much more will he do so in a noble
cause; and if a man is open-handed to any one and every one, he will
be so to his friends also, since it is the extreme form of goodness
we must also take into account the nature of our particular audience
when making a speech of praise; for, as socrates used to say, 'it
is not difficult to praise the athenians to an athenian audience.'
if the audience esteems a given quality, we must say that our hero
has that quality, no matter whether we are addressing scythians or
spartans or philosophers. everything, in fact, that is esteemed we
are to represent as noble. after all, people regard the two things
all actions are noble that are appropriate to the man who does them:
if, for instance, they are worthy of his ancestors or of his own past
career. for it makes for happiness, and is a noble thing, that he
should add to the honour he already has. even inappropriate actions
are noble if they are better and nobler than the appropriate ones
would be; for instance, if one who was just an average person when
all went well becomes a hero in adversity, or if he becomes better
and easier to get on with the higher he rises. compare the saying
of lphicrates, 'think what i was and what i am'; and the epigram on
"in time past, bearing a yoke on my shoulders,
"a woman whose father, whose husband, whose
since we praise a man for what he has actually done, and fine actions
are distinguished from others by being intentionally good, we must
try to prove that our hero's noble acts are intentional. this is all
the easier if we can make out that he has often acted so before, and
therefore we must assert coincidences and accidents to have been intended.
produce a number of good actions, all of the same kind, and people
will think that they must have been intended, and that they prove
the good qualities of the man who did them.
praise is the expression in words of the eminence of a man's good
qualities, and therefore we must display his actions as the product
of such qualities. encomium refers to what he has actually done; the
mention of accessories, such as good birth and education, merely helps
to make our story credible-good fathers are likely to have good sons,
and good training is likely to produce good character. hence it is
only when a man has already done something that we bestow encomiums
upon him. yet the actual deeds are evidence of the doer's character:
even if a man has not actually done a given good thing, we shall bestow
praise on him, if we are sure that he is the sort of man who would
do it. to call any one blest is, it may be added, the same thing as
to call him happy; but these are not the same thing as to bestow praise
and encomium upon him; the two latter are a part of 'calling happy',
just as goodness is a part of happiness.
to praise a man is in one respect akin to urging a course of action.
the suggestions which would be made in the latter case become encomiums
when differently expressed. when we know what action or character
is required, then, in order to express these facts as suggestions
for action, we have to change and reverse our form of words. thus
the statement 'a man should be proud not of what he owes to fortune
but of what he owes to himself', if put like this, amounts to a suggestion;
to make it into praise we must put it thus, 'since he is proud not
of what he owes to fortune but of what he owes to himself.' consequently,
whenever you want to praise any one, think what you would urge people
to do; and when you want to urge the doing of anything, think what
you would praise a man for having done. since suggestion may or may
not forbid an action, the praise into which we convert it must have
one or other of two opposite forms of expression accordingly.
there are, also, many useful ways of heightening the effect of praise.
we must, for instance, point out that a man is the only one, or the
first, or almost the only one who has done something, or that he has
done it better than any one else; all these distinctions are honourable.
and we must, further, make much of the particular season and occasion
of an action, arguing that we could hardly have looked for it just
then. if a man has often achieved the same success, we must mention
this; that is a strong point; he himself, and not luck, will then
be given the credit. so, too, if it is on his account that observances
have been devised and instituted to encourage or honour such achievements
as his own: thus we may praise hippolochus because the first encomium
ever made was for him, or harmodius and aristogeiton because their
statues were the first to be put up in the market-place. and we may
censure bad men for the opposite reason.
again, if you cannot find enough to say of a man himself, you may
pit him against others, which is what isocrates used to do owing to
his want of familiarity with forensic pleading. the comparison should
be with famous men; that will strengthen your case; it is a noble
thing to surpass men who are themselves great. it is only natural
that methods of 'heightening the effect' should be attached particularly
to speeches of praise; they aim at proving superiority over others,
and any such superiority is a form of nobleness. hence if you cannot
compare your hero with famous men, you should at least compare him
with other people generally, since any superiority is held to reveal
excellence. and, in general, of the lines of argument which are common
to all speeches, this 'heightening of effect' is most suitable for
declamations, where we take our hero's actions as admitted facts,
and our business is simply to invest these with dignity and nobility.
'examples' are most suitable to deliberative speeches; for we judge
of future events by divination from past events. enthymemes are most
suitable to forensic speeches; it is our doubts about past events
that most admit of arguments showing why a thing must have happened
the above are the general lines on which all, or nearly all, speeches
of praise or blame are constructed. we have seen the sort of thing
we must bear in mind in making such speeches, and the materials out
of which encomiums and censures are made. no special treatment of
censure and vituperation is needed. knowing the above facts, we know
their contraries; and it is out of these that speeches of censure
we have next to treat of accusation and defence, and to enumerate
and describe the ingredients of the syllogisms used therein. there
are three things we must ascertain first, the nature and number of
the incentives to wrong-doing; second, the state of mind of wrongdoers;
third, the kind of persons who are wronged, and their condition. we
will deal with these questions in order. but before that let us define
we may describe 'wrong-doing' as injury voluntarily inflicted contrary
to law. 'law' is either special or general. by special law i mean
that written law which regulates the life of a particular community;
by general law, all those unwritten principles which are supposed
to be acknowledged everywhere. we do things 'voluntarily' when we
do them consciously and without constraint. (not all voluntary acts
are deliberate, but all deliberate acts are conscious-no one is ignorant
of what he deliberately intends.) the causes of our deliberately intending
harmful and wicked acts contrary to law are (1) vice, (2) lack of
self-control. for the wrongs a man does to others will correspond
to the bad quality or qualities that he himself possesses. thus it
is the mean man who will wrong others about money, the profligate
in matters of physical pleasure, the effeminate in matters of comfort,
and the coward where danger is concerned-his terror makes him abandon
those who are involved in the same danger. the ambitious man does
wrong for sake of honour, the quick-tempered from anger, the lover
of victory for the sake of victory, the embittered man for the sake
of revenge, the stupid man because he has misguided notions of right
and wrong, the shameless man because he does not mind what people
think of him; and so with the rest-any wrong that any one does to
others corresponds to his particular faults of character.
however, this subject has already been cleared up in part in our discussion
of the virtues and will be further explained later when we treat of
the emotions. we have now to consider the motives and states of mind
of wrongdoers, and to whom they do wrong.
let us first decide what sort of things people are trying to get or
avoid when they set about doing wrong to others. for it is plain that
the prosecutor must consider, out of all the aims that can ever induce
us to do wrong to our neighbours, how many, and which, affect his
adversary; while the defendant must consider how many, and which,
do not affect him. now every action of every person either is or is
not due to that person himself. of those not due to himself some are
due to chance, the others to necessity; of these latter, again, some
are due to compulsion, the others to nature. consequently all actions
that are not due to a man himself are due either to chance or to nature
or to compulsion. all actions that are due to a man himself and caused
by himself are due either to habit or to rational or irrational craving.
rational craving is a craving for good, i.e. a wish-nobody wishes
for anything unless he thinks it good. irrational craving is twofold,
thus every action must be due to one or other of seven causes: chance,
nature, compulsion, habit, reasoning, anger, or appetite. it is superfluous
further to distinguish actions according to the doers' ages, moral
states, or the like; it is of course true that, for instance, young
men do have hot tempers and strong appetites; still, it is not through
youth that they act accordingly, but through anger or appetite. nor,
again, is action due to wealth or poverty; it is of course true that
poor men, being short of money, do have an appetite for it, and that
rich men, being able to command needless pleasures, do have an appetite
for such pleasures: but here, again, their actions will be due not
to wealth or poverty but to appetite. similarly, with just men, and
unjust men, and all others who are said to act in accordance with
their moral qualities, their actions will really be due to one of
the causes mentioned-either reasoning or emotion: due, indeed, sometimes
to good dispositions and good emotions, and sometimes to bad; but
that good qualities should be followed by good emotions, and bad by
bad, is merely an accessory fact-it is no doubt true that the temperate
man, for instance, because he is temperate, is always and at once
attended by healthy opinions and appetites in regard to pleasant things,
and the intemperate man by unhealthy ones. so we must ignore such
distinctions. still we must consider what kinds of actions and of
people usually go together; for while there are no definite kinds
of action associated with the fact that a man is fair or dark, tall
or short, it does make a difference if he is young or old, just or
unjust. and, generally speaking, all those accessory qualities that
cause distinctions of human character are important: e.g. the sense
of wealth or poverty, of being lucky or unlucky. this shall be dealt
with later-let us now deal first with the rest of the subject before
the things that happen by chance are all those whose cause cannot
be determined, that have no purpose, and that happen neither always
nor usually nor in any fixed way. the definition of chance shows just
what they are. those things happen by nature which have a fixed and
internal cause; they take place uniformly, either always or usually.
there is no need to discuss in exact detail the things that happen
contrary to nature, nor to ask whether they happen in some sense naturally
or from some other cause; it would seem that chance is at least partly
the cause of such events. those things happen through compulsion which
take place contrary to the desire or reason of the doer, yet through
his own agency. acts are done from habit which men do because they
have often done them before. actions are due to reasoning when, in
view of any of the goods already mentioned, they appear useful either
as ends or as means to an end, and are performed for that reason:
'for that reason,' since even licentious persons perform a certain
number of useful actions, but because they are pleasant and not because
they are useful. to passion and anger are due all acts of revenge.
revenge and punishment are different things. punishment is inflicted
for the sake of the person punished; revenge for that of the punisher,
to satisfy his feelings. (what anger is will be made clear when we
come to discuss the emotions.) appetite is the cause of all actions
that appear pleasant. habit, whether acquired by mere familiarity
or by effort, belongs to the class of pleasant things, for there are
many actions not naturally pleasant which men perform with pleasure,
once they have become used to them. to sum up then, all actions due
to ourselves either are or seem to be either good or pleasant. moreover,
as all actions due to ourselves are done voluntarily and actions not
due to ourselves are done involuntarily, it follows that all voluntary
actions must either be or seem to be either good or pleasant; for
i reckon among goods escape from evils or apparent evils and the exchange
of a greater evil for a less (since these things are in a sense positively
desirable), and likewise i count among pleasures escape from painful
or apparently painful things and the exchange of a greater pain for
a less. we must ascertain, then, the number and nature of the things
that are useful and pleasant. the useful has been previously examined
in connexion with political oratory; let us now proceed to examine
the pleasant. our various definitions must be regarded as adequate,
even if they are not exact, provided they are clear.
we may lay it down that pleasure is a movement, a movement by which
the soul as a whole is consciously brought into its normal state of
being; and that pain is the opposite. if this is what pleasure is,
it is clear that the pleasant is what tends to produce this condition,
while that which tends to destroy it, or to cause the soul to be brought
into the opposite state, is painful. it must therefore be pleasant
as a rule to move towards a natural state of being, particularly when
a natural process has achieved the complete recovery of that natural
state. habits also are pleasant; for as soon as a thing has become
habitual, it is virtually natural; habit is a thing not unlike nature;
what happens often is akin to what happens always, natural events
happening always, habitual events often. again, that is pleasant which
is not forced on us; for force is unnatural, and that is why what
is compulsory, painful, and it has been rightly said
"all that is done on compulsion is bitterness unto the soul.
so all acts of concentration, strong effort, and strain are necessarily
painful; they all involve compulsion and force, unless we are accustomed
to them, in which case it is custom that makes them pleasant. the
opposites to these are pleasant; and hence ease, freedom from toil,
relaxation, amusement, rest, and sleep belong to the class of pleasant
things; for these are all free from any element of compulsion. everything,
too, is pleasant for which we have the desire within us, since desire
is the craving for pleasure. of the desires some are irrational, some
associated with reason. by irrational i mean those which do not arise
from any opinion held by the mind. of this kind are those known as
'natural'; for instance, those originating in the body, such as the
desire for nourishment, namely hunger and thirst, and a separate kind
of desire answering to each kind of nourishment; and the desires connected
with taste and sex and sensations of touch in general; and those of
smell, hearing, and vision. rational desires are those which we are
induced to have; there are many things we desire to see or get because
we have been told of them and induced to believe them good. further,
pleasure is the consciousness through the senses of a certain kind
of emotion; but imagination is a feeble sort of sensation, and there
will always be in the mind of a man who remembers or expects something
an image or picture of what he remembers or expects. if this is so,
it is clear that memory and expectation also, being accompanied by
sensation, may be accompanied by pleasure. it follows that anything
pleasant is either present and perceived, past and remembered, or
future and expected, since we perceive present pleasures, remember
past ones, and expect future ones. now the things that are pleasant
to remember are not only those that, when actually perceived as present,
were pleasant, but also some things that were not, provided that their
results have subsequently proved noble and good. hence the words
"sweet 'tis when rescued to remember pain, "
"even his griefs are a joy long after to one that remembers
the reason of this is that it is pleasant even to be merely free
from evil. the things it is pleasant to expect are those that when
present are felt to afford us either great delight or great but not
painful benefit. and in general, all the things that delight us when
they are present also do so, as a rule, when we merely remember or
expect them. hence even being angry is pleasant-homer said of wrath
"sweeter it is by far than the honeycomb dripping with sweetness-
for no one grows angry with a person on whom there is no prospect
of taking vengeance, and we feel comparatively little anger, or none
at all, with those who are much our superiors in power. some pleasant
feeling is associated with most of our appetites we are enjoying either
the memory of a past pleasure or the expectation of a future one,
just as persons down with fever, during their attacks of thirst, enjoy
remembering the drinks they have had and looking forward to having
more. so also a lover enjoys talking or writing about his loved one,
or doing any little thing connected with him; all these things recall
him to memory and make him actually present to the eye of imagination.
indeed, it is always the first sign of love, that besides enjoying
some one's presence, we remember him when he is gone, and feel pain
as well as pleasure, because he is there no longer. similarly there
is an element of pleasure even in mourning and lamentation for the
departed. there is grief, indeed, at his loss, but pleasure in remembering
him and as it were seeing him before us in his deeds and in his life.
we can well believe the poet when he says
"he spake, and in each man's heart he awakened
revenge, too, is pleasant; it is pleasant to get anything that it
is painful to fail to get, and angry people suffer extreme pain when
they fail to get their revenge; but they enjoy the prospect of getting
it. victory also is pleasant, and not merely to 'bad losers', but
to every one; the winner sees himself in the light of a champion,
and everybody has a more or less keen appetite for being that. the
pleasantness of victory implies of course that combative sports and
intellectual contests are pleasant (since in these it often happens
that some one wins) and also games like knuckle-bones, ball, dice,
and draughts. and similarly with the serious sports; some of these
become pleasant when one is accustomed to them; while others are pleasant
from the first, like hunting with hounds, or indeed any kind of hunting.
for where there is competition, there is victory. that is why forensic
pleading and debating contests are pleasant to those who are accustomed
to them and have the capacity for them. honour and good repute are
among the most pleasant things of all; they make a man see himself
in the character of a fine fellow, especially when he is credited
with it by people whom he thinks good judges. his neighbours are better
judges than people at a distance; his associates and fellow-countrymen
better than strangers; his contemporaries better than posterity; sensible
persons better than foolish ones; a large number of people better
than a small number: those of the former class, in each case, are
the more likely to be good judges of him. honour and credit bestowed
by those whom you think much inferior to yourself-e.g. children or
animals-you do not value: not for its own sake, anyhow: if you do
value it, it is for some other reason. friends belong to the class
of pleasant things; it is pleasant to love-if you love wine, you certainly
find it delightful: and it is pleasant to be loved, for this too makes
a man see himself as the possessor of goodness, a thing that every
being that has a feeling for it desires to possess: to be loved means
to be valued for one's own personal qualities. to be admired is also
pleasant, simply because of the honour implied. flattery and flatterers
are pleasant: the flatterer is a man who, you believe, admires and
likes to do the same thing often is pleasant, since, as we saw, anything
habitual is pleasant. and to change is also pleasant: change means
an approach to nature, whereas invariable repetition of anything causes
the excessive prolongation of a settled condition: therefore, says
that is why what comes to us only at long intervals is pleasant,
whether it be a person or a thing; for it is a change from what we
had before, and, besides, what comes only at long intervals has the
value of rarity. learning things and wondering at things are also
pleasant as a rule; wondering implies the desire of learning, so that
the object of wonder is an object of desire; while in learning one
is brought into one's natural condition. conferring and receiving
benefits belong to the class of pleasant things; to receive a benefit
is to get what one desires; to confer a benefit implies both posses
sion and superiority, both of which are things we try to attain. it
is because beneficent acts are pleasant that people find it pleasant
to put their neighbours straight again and to supply what they lack.
again, since learning and wondering are pleasant, it follows that
such things as acts of imitation must be pleasant-for instance, painting,
sculpture, poetry and every product of skilful imitation; this latter,
even if the object imitated is not itself pleasant; for it is not
the object itself which here gives delight; the spectator draws inferences
('that is a so-and-so') and thus learns something fresh. dramatic
turns of fortune and hairbreadth escapes from perils are pleasant,
because we feel all such things are wonderful.
and since what is natural is pleasant, and things akin to each other
seem natural to each other, therefore all kindred and similar things
are usually pleasant to each other; for instance, one man, horse,
or young person is pleasant to another man, horse, or young person.
hence the proverbs 'mate delights mate', 'like to like', 'beast knows
beast', 'jackdaw to jackdaw', and the rest of them. but since everything
like and akin to oneself is pleasant, and since every man is himself
more like and akin to himself than any one else is, it follows that
all of us must be more or less fond of ourselves. for all this resemblance
and kinship is present particularly in the relation of an individual
to himself. and because we are all fond of ourselves, it follows that
what is our own is pleasant to all of us, as for instance our own
deeds and words. that is why we are usually fond of our flatterers,
our lovers, and honour; also of our children, for our children are
our own work. it is also pleasant to complete what is defective, for
the whole thing thereupon becomes our own work. and since power over
others is very pleasant, it is pleasant to be thought wise, for practical
wisdom secures us power over others. (scientific wisdom is also pleasant,
because it is the knowledge of many wonderful things.) again, since
most of us are ambitious, it must be pleasant to disparage our neighbours
as well as to have power over them. it is pleasant for a man to spend
his time over what he feels he can do best; just as the poet says,
"to that each day allots most time, wherein
"he is indeed the best part of himself. "
similarly, since amusement and every kind of relaxation and laughter
too belong to the class of pleasant things, it follows that ludicrous
things are pleasant, whether men, words, or deeds. we have discussed
the ludicrous separately in the treatise on the art of poetry.
so much for the subject of pleasant things: by considering their opposites
we can easily see what things are unpleasant.
the above are the motives that make men do wrong to others; we are
next to consider the states of mind in which they do it, and the persons
they must themselves suppose that the thing can be done, and done
by them: either that they can do it without being found out, or that
if they are found out they can escape being punished, or that if they
are punished the disadvantage will be less than the gain for themselves
or those they care for. the general subject of apparent possibility
and impossibility will be handled later on, since it is relevant not
only to forensic but to all kinds of speaking. but it may here be
said that people think that they can themselves most easily do wrong
to others without being punished for it if they possess eloquence,
or practical ability, or much legal experience, or a large body of
friends, or a great deal of money. their confidence is greatest if
they personally possess the advantages mentioned: but even without
them they are satisfied if they have friends or supporters or partners
who do possess them: they can thus both commit their crimes and escape
being found out and punished for committing them. they are also safe,
they think, if they are on good terms with their victims or with the
judges who try them. their victims will in that case not be on their
guard against being wronged, and will make some arrangement with them
instead of prosecuting; while their judges will favour them because
they like them, either letting them off altogether or imposing light
sentences. they are not likely to be found out if their appearance
contradicts the charges that might be brought against them: for instance,
a weakling is unlikely to be charged with violent assault, or a poor
and ugly man with adultery. public and open injuries are the easiest
to do, because nobody could at all suppose them possible, and therefore
no precautions are taken. the same is true of crimes so great and
terrible that no man living could be suspected of them: here too no
precautions are taken. for all men guard against ordinary offences,
just as they guard against ordinary diseases; but no one takes precautions
against a disease that nobody has ever had. you feel safe, too, if
you have either no enemies or a great many; if you have none, you
expect not to be watched and therefore not to be detected; if you
have a great many, you will be watched, and therefore people will
think you can never risk an attempt on them, and you can defend your
innocence by pointing out that you could never have taken such a risk.
you may also trust to hide your crime by the way you do it or the
place you do it in, or by some convenient means of disposal.
you may feel that even if you are found out you can stave off a trial,
or have it postponed, or corrupt your judges: or that even if you
are sentenced you can avoid paying damages, or can at least postpone
doing so for a long time: or that you are so badly off that you will
have nothing to lose. you may feel that the gain to be got by wrong-doing
is great or certain or immediate, and that the penalty is small or
uncertain or distant. it may be that the advantage to be gained is
greater than any possible retribution: as in the case of despotic
power, according to the popular view. you may consider your crimes
as bringing you solid profit, while their punishment is nothing more
than being called bad names. or the opposite argument may appeal to
you: your crimes may bring you some credit (thus you may, incidentally,
be avenging your father or mother, like zeno), whereas the punishment
may amount to a fine, or banishment, or something of that sort. people
may be led on to wrong others by either of these motives or feelings;
but no man by both-they will affect people of quite opposite characters.
you may be encouraged by having often escaped detection or punishment
already; or by having often tried and failed; for in crime, as in
war, there are men who will always refuse to give up the struggle.
you may get your pleasure on the spot and the pain later, or the gain
on the spot and the loss later. that is what appeals to weak-willed
persons--and weakness of will may be shown with regard to all the
objects of desire. it may on the contrary appeal to you as it does
appeal to self-controlled and sensible people--that the pain and loss
are immediate, while the pleasure and profit come later and last longer.
you may feel able to make it appear that your crime was due to chance,
or to necessity, or to natural causes, or to habit: in fact, to put
it generally, as if you had failed to do right rather than actually
done wrong. you may be able to trust other people to judge you equitably.
you may be stimulated by being in want: which may mean that you want
necessaries, as poor people do, or that you want luxuries, as rich
people do. you may be encouraged by having a particularly good reputation,
because that will save you from being suspected: or by having a particularly
bad one, because nothing you are likely to do will make it worse.
the above, then, are the various states of mind in which a man sets
about doing wrong to others. the kind of people to whom he does wrong,
and the ways in which he does it, must be considered next. the people
to whom he does it are those who have what he wants himself, whether
this means necessities or luxuries and materials for enjoyment. his
victims may be far off or near at hand. if they are near, he gets
his profit quickly; if they are far off, vengeance is slow, as those
think who plunder the carthaginians. they may be those who are trustful
instead of being cautious and watchful, since all such people are
easy to elude. or those who are too easy-going to have enough energy
to prosecute an offender. or sensitive people, who are not apt to
show fight over questions of money. or those who have been wronged
already by many people, and yet have not prosecuted; such men must
surely be the proverbial 'mysian prey'. or those who have either never
or often been wronged before; in neither case will they take precautions;
if they have never been wronged they think they never will, and if
they have often been wronged they feel that surely it cannot happen
again. or those whose character has been attacked in the past, or
is exposed to attack in the future: they will be too much frightened
of the judges to make up their minds to prosecute, nor can they win
their case if they do: this is true of those who are hated or unpopular.
another likely class of victim is those who their injurer can pretend
have, themselves or through their ancestors or friends, treated badly,
or intended to treat badly, the man himself, or his ancestors, or
those he cares for; as the proverb says, 'wickedness needs but a pretext'.
a man may wrong his enemies, because that is pleasant: he may equally
wrong his friends, because that is easy. then there are those who
have no friends, and those who lack eloquence and practical capacity;
these will either not attempt to prosecute, or they will come to terms,
or failing that they will lose their case. there are those whom it
does not pay to waste time in waiting for trial or damages, such as
foreigners and small farmers; they will settle for a trifle, and always
be ready to leave off. also those who have themselves wronged others,
either often, or in the same way as they are now being wronged themselves-for
it is felt that next to no wrong is done to people when it is the
same wrong as they have often themselves done to others: if, for instance,
you assault a man who has been accustomed to behave with violence
to others. so too with those who have done wrong to others, or have
meant to, or mean to, or are likely to do so; there is something fine
and pleasant in wronging such persons, it seems as though almost no
wrong were done. also those by doing wrong to whom we shall be gratifying
our friends, or those we admire or love, or our masters, or in general
the people by reference to whom we mould our lives. also those whom
we may wrong and yet be sure of equitable treatment. also those against
whom we have had any grievance, or any previous differences with them,
as callippus had when he behaved as he did to dion: here too it seems
as if almost no wrong were being done. also those who are on the point
of being wronged by others if we fail to wrong them ourselves, since
here we feel we have no time left for thinking the matter over. so
aenesidemus is said to have sent the 'cottabus' prize to gelon, who
had just reduced a town to slavery, because gelon had got there first
and forestalled his own attempt. also those by wronging whom we shall
be able to do many righteous acts; for we feel that we can then easily
cure the harm done. thus jason the thessalian said that it is a duty
to do some unjust acts in order to be able to do many just ones.
among the kinds of wrong done to others are those that are done universally,
or at least commonly: one expects to be forgiven for doing these.
also those that can easily be kept dark, as where things that can
rapidly be consumed like eatables are concerned, or things that can
easily be changed in shape, colour, or combination, or things that
can easily be stowed away almost anywhere-portable objects that you
can stow away in small corners, or things so like others of which
you have plenty already that nobody can tell the difference. there
are also wrongs of a kind that shame prevents the victim speaking
about, such as outrages done to the women in his household or to himself
or to his sons. also those for which you would be thought very litigious
to prosecute any one-trifling wrongs, or wrongs for which people are
the above is a fairly complete account of the circumstances under
which men do wrong to others, of the sort of wrongs they do, of the
sort of persons to whom they do them, and of their reasons for doing
it will now be well to make a complete classification of just and
unjust actions. we may begin by observing that they have been defined
relatively to two kinds of law, and also relatively to two classes
of persons. by the two kinds of law i mean particular law and universal
law. particular law is that which each community lays down and applies
to its own members: this is partly written and partly unwritten. universal
law is the law of nature. for there really is, as every one to some
extent divines, a natural justice and injustice that is binding on
all men, even on those who have no association or covenant with each
other. it is this that sophocles' antigone clearly means when she
says that the burial of polyneices was a just act in spite of the
prohibition: she means that it was just by nature.
"but lives eternal: none can date its birth. "
and so empedocles, when he bids us kill no living creature, says
that doing this is not just for some people while unjust for others,
"nay, but, an all-embracing law, through the realms of the sky
"unbroken it stretcheth, and over the earth's immensity.
and as alcidamas says in his messeniac oration....
the actions that we ought to do or not to do have also been divided
into two classes as affecting either the whole community or some one
of its members. from this point of view we can perform just or unjust
acts in either of two ways-towards one definite person, or towards
the community. the man who is guilty of adultery or assault is doing
wrong to some definite person; the man who avoids service in the army
thus the whole class of unjust actions may be divided into two classes,
those affecting the community, and those affecting one or more other
persons. we will next, before going further, remind ourselves of what
'being wronged' means. since it has already been settled that 'doing
a wrong' must be intentional, 'being wronged' must consist in having
an injury done to you by some one who intends to do it. in order to
be wronged, a man must (1) suffer actual harm, (2) suffer it against
his will. the various possible forms of harm are clearly explained
by our previous, separate discussion of goods and evils. we have also
seen that a voluntary action is one where the doer knows what he is
doing. we now see that every accusation must be of an action affecting
either the community or some individual. the doer of the action must
either understand and intend the action, or not understand and intend
it. in the former case, he must be acting either from deliberate choice
or from passion. (anger will be discussed when we speak of the passions
the motives for crime and the state of mind of the criminal have already
been discussed.) now it often happens that a man will admit an act,
but will not admit the prosecutor's label for the act nor the facts
which that label implies. he will admit that he took a thing but not
that he 'stole' it; that he struck some one first, but not that he
committed 'outrage'; that he had intercourse with a woman, but not
that he committed 'adultery'; that he is guilty of theft, but not
that he is guilty of 'sacrilege', the object stolen not being consecrated;
that he has encroached, but not that he has 'encroached on state lands';
that he has been in communication with the enemy, but not that he
has been guilty of 'treason'. here therefore we must be able to distinguish
what is theft, outrage, or adultery, from what is not, if we are to
be able to make the justice of our case clear, no matter whether our
aim is to establish a man's guilt or to establish his innocence. wherever
such charges are brought against a man, the question is whether he
is or is not guilty of a criminal offence. it is deliberate purpose
that constitutes wickedness and criminal guilt, and such names as
'outrage' or 'theft' imply deliberate purpose as well as the mere
action. a blow does not always amount to 'outrage', but only if it
is struck with some such purpose as to insult the man struck or gratify
the striker himself. nor does taking a thing without the owner's knowledge
always amount to 'theft', but only if it is taken with the intention
of keeping it and injuring the owner. and as with these charges, so
we saw that there are two kinds of right and wrong conduct towards
others, one provided for by written ordinances, the other by unwritten.
we have now discussed the kind about which the laws have something
to say. the other kind has itself two varieties. first, there is the
conduct that springs from exceptional goodness or badness, and is
visited accordingly with censure and loss of honour, or with praise
and increase of honour and decorations: for instance, gratitude to,
or requital of, our benefactors, readiness to help our friends, and
the like. the second kind makes up for the defects of a community's
written code of law. this is what we call equity; people regard it
as just; it is, in fact, the sort of justice which goes beyond the
written law. its existence partly is and partly is not intended by
legislators; not intended, where they have noticed no defect in the
law; intended, where find themselves unable to define things exactly,
and are obliged to legislate as if that held good always which in
fact only holds good usually; or where it is not easy to be complete
owing to the endless possible cases presented, such as the kinds and
sizes of weapons that may be used to inflict wounds-a lifetime would
be too short to make out a complete list of these. if, then, a precise
statement is impossible and yet legislation is necessary, the law
must be expressed in wide terms; and so, if a man has no more than
a finger-ring on his hand when he lifts it to strike or actually strikes
another man, he is guilty of a criminal act according to the unwritten
words of the law; but he is innocent really, and it is equity that
declares him to be so. from this definition of equity it is plain
what sort of actions, and what sort of persons, are equitable or the
reverse. equity must be applied to forgivable actions; and it must
make us distinguish between criminal acts on the one hand, and errors
of judgement, or misfortunes, on the other. (a 'misfortune' is an
act, not due to moral badness, that has unexpected results: an 'error
of judgement' is an act, also not due to moral badness, that has results
that might have been expected: a 'criminal act' has results that might
have been expected, but is due to moral badness, for that is the source
of all actions inspired by our appetites.) equity bids us be merciful
to the weakness of human nature; to think less about the laws than
about the man who framed them, and less about what he said than about
what he meant; not to consider the actions of the accused so much
as his intentions, nor this or that detail so much as the whole story;
to ask not what a man is now but what he has always or usually been.
it bids us remember benefits rather than injuries, and benefits received
rather than benefits conferred; to be patient when we are wronged;
to settle a dispute by negotiation and not by force; to prefer arbitration
to motion-for an arbitrator goes by the equity of a case, a judge
by the strict law, and arbitration was invented with the express purpose
the above may be taken as a sufficient account of the nature of equity.
the worse of two acts of wrong done to others is that which is prompted
by the worse disposition. hence the most trifling acts may be the
worst ones; as when callistratus charged melanopus with having cheated
the temple-builders of three consecrated half-obols. the converse
is true of just acts. this is because the greater is here potentially
contained in the less: there is no crime that a man who has stolen
three consecrated half-obols would shrink from committing. sometimes,
however, the worse act is reckoned not in this way but by the greater
harm that it does. or it may be because no punishment for it is severe
enough to be adequate; or the harm done may be incurable-a difficult
and even hopeless crime to defend; or the sufferer may not be able
to get his injurer legally punished, a fact that makes the harm incurable,
since legal punishment and chastisement are the proper cure. or again,
the man who has suffered wrong may have inflicted some fearful punishment
on himself; then the doer of the wrong ought in justice to receive
a still more fearful punishment. thus sophocles, when pleading for
retribution to euctemon, who had cut his own throat because of the
outrage done to him, said he would not fix a penalty less than the
victim had fixed for himself. again, a man's crime is worse if he
has been the first man, or the only man, or almost the only man, to
commit it: or if it is by no means the first time he has gone seriously
wrong in the same way: or if his crime has led to the thinking-out
and invention of measures to prevent and punish similar crimes-thus
in argos a penalty is inflicted on a man on whose account a law is
passed, and also on those on whose account the prison was built: or
if a crime is specially brutal, or specially deliberate: or if the
report of it awakes more terror than pity. there are also such rhetorically
effective ways of putting it as the following: that the accused has
disregarded and broken not one but many solemn obligations like oaths,
promises, pledges, or rights of intermarriage between states-here
the crime is worse because it consists of many crimes; and that the
crime was committed in the very place where criminals are punished,
as for example perjurers do-it is argued that a man who will commit
a crime in a law-court would commit it anywhere. further, the worse
deed is that which involves the doer in special shame; that whereby
a man wrongs his benefactors-for he does more than one wrong, by not
merely doing them harm but failing to do them good; that which breaks
the unwritten laws of justice-the better sort of man will be just
without being forced to be so, and the written laws depend on force
while the unwritten ones do not. it may however be argued otherwise,
that the crime is worse which breaks the written laws: for the man
who commits crimes for which terrible penalties are provided will
not hesitate over crimes for which no penalty is provided at all.-so
much, then, for the comparative badness of criminal actions.
there are also the so-called 'non-technical' means of persuasion;
and we must now take a cursory view of these, since they are specially
characteristic of forensic oratory. they are five in number: laws,
first, then, let us take laws and see how they are to be used in persuasion
and dissuasion, in accusation and defence. if the written law tells
against our case, clearly we must appeal to the universal law, and
insist on its greater equity and justice. we must argue that the juror's
oath 'i will give my verdict according to honest opinion' means that
one will not simply follow the letter of the written law. we must
urge that the principles of equity are permanent and changeless, and
that the universal law does not change either, for it is the law of
nature, whereas written laws often do change. this is the bearing
the lines in sophocles' antigone, where antigone pleads that in burying
her brother she had broken creon's law, but not the unwritten law:
"but live eternal: (none can date their birth.)
"(and brave god's vengeance) for defying these. "
we shall argue that justice indeed is true and profitable, but that
sham justice is not, and that consequently the written law is not,
because it does not fulfil the true purpose of law. or that justice
is like silver, and must be assayed by the judges, if the genuine
is to be distinguished from the counterfeit. or that the better a
man is, the more he will follow and abide by the unwritten law in
preference to the written. or perhaps that the law in question contradicts
some other highly-esteemed law, or even contradicts itself. thus it
may be that one law will enact that all contracts must be held binding,
while another forbids us ever to make illegal contracts. or if a law
is ambiguous, we shall turn it about and consider which construction
best fits the interests of justice or utility, and then follow that
way of looking at it. or if, though the law still exists, the situation
to meet which it was passed exists no longer, we must do our best
to prove this and to combat the law thereby. if however the written
law supports our case, we must urge that the oath 'to give my verdict
according to my honest opinion' not meant to make the judges give
a verdict that is contrary to the law, but to save them from the guilt
of perjury if they misunderstand what the law really means. or that
no one chooses what is absolutely good, but every one what is good
for himself. or that not to use the laws is as ahas to have no laws
at all. or that, as in the other arts, it does not pay to try to be
cleverer than the doctor: for less harm comes from the doctor's mistakes
than from the growing habit of disobeying authority. or that trying
to be cleverer than the laws is just what is forbidden by those codes
of law that are accounted best.-so far as the laws are concerned,
the above discussion is probably sufficient.
as to witnesses, they are of two kinds, the ancient and the recent;
and these latter, again, either do or do not share in the risks of
the trial. by 'ancient' witnesses i mean the poets and all other notable
persons whose judgements are known to all. thus the athenians appealed
to homer as a witness about salamis; and the men of tenedos not long
ago appealed to periander of corinth in their dispute with the people
of sigeum; and cleophon supported his accusation of critias by quoting
the elegiac verse of solon, maintaining that discipline had long been
slack in the family of critias, or solon would never have written,
"pray thee, bid the red-haired critias do what
these witnesses are concerned with past events. as to future events
we shall also appeal to soothsayers: thus themistocles quoted the
oracle about 'the wooden wall' as a reason for engaging the enemy's
fleet. further, proverbs are, as has been said, one form of evidence.
thus if you are urging somebody not to make a friend of an old man,
or if you are urging that he who has made away with fathers should
"fool, who slayeth the father and leaveth his sons to avenge him.
'recent' witnesses are well-known people who have expressed their
opinions about some disputed matter: such opinions will be useful
support for subsequent disputants on the same oints: thus eubulus
used in the law-courts against the reply plato had made to archibius,
'it has become the regular custom in this country to admit that one
is a scoundrel'. there are also those witnesses who share the risk
of punishment if their evidence is pronounced false. these are valid
witnesses to the fact that an action was or was not done, that something
is or is not the case; they are not valid witnesses to the quality
of an action, to its being just or unjust, useful or harmful. on such
questions of quality the opinion of detached persons is highly trustworthy.
most trustworthy of all are the 'ancient' witnesses, since they cannot
in dealing with the evidence of witnesses, the following are useful
arguments. if you have no witnesses on your side, you will argue that
the judges must decide from what is probable; that this is meant by
'giving a verdict in accordance with one's honest opinion'; that probabilities
cannot be bribed to mislead the court; and that probabilities are
never convicted of perjury. if you have witnesses, and the other man
has not, you will argue that probabilities cannot be put on their
trial, and that we could do without the evidence of witnesses altogether
if we need do no more than balance the pleas advanced on either side.
the evidence of witnesses may refer either to ourselves or to our
opponent; and either to questions of fact or to questions of personal
character: so, clearly, we need never be at a loss for useful evidence.
for if we have no evidence of fact supporting our own case or telling
against that of our opponent, at least we can always find evidence
to prove our own worth or our opponent's worthlessness. other arguments
about a witness-that he is a friend or an enemy or neutral, or has
a good, bad, or indifferent reputation, and any other such distinctions-we
must construct upon the same general lines as we use for the regular
concerning contracts argument can be so far employed as to increase
or diminish their importance and their credibility; we shall try to
increase both if they tell in our favour, and to diminish both if
they tell in favour of our opponent. now for confirming or upsetting
the credibility of contracts the procedure is just the same as for
dealing with witnesses, for the credit to be attached to contracts
depends upon the character of those who have signed them or have the
custody of them. the contract being once admitted genuine, we must
insist on its importance, if it supports our case. we may argue that
a contract is a law, though of a special and limited kind; and that,
while contracts do not of course make the law binding, the law does
make any lawful contract binding, and that the law itself as a whole
is a of contract, so that any one who disregards or repudiates any
contract is repudiating the law itself. further, most business relations-those,
namely, that are voluntary-are regulated by contracts, and if these
lose their binding force, human intercourse ceases to exist. we need
not go very deep to discover the other appropriate arguments of this
kind. if, however, the contract tells against us and for our opponents,
in the first place those arguments are suitable which we can use to
fight a law that tells against us. we do not regard ourselves as bound
to observe a bad law which it was a mistake ever to pass: and it is
ridiculous to suppose that we are bound to observe a bad and mistaken
contract. again, we may argue that the duty of the judge as umpire
is to decide what is just, and therefore he must ask where justice
lies, and not what this or that document means. and that it is impossible
to pervert justice by fraud or by force, since it is founded on nature,
but a party to a contract may be the victim of either fraud or force.
moreover, we must see if the contract contravenes either universal
law or any written law of our own or another country; and also if
it contradicts any other previous or subsequent contract; arguing
that the subsequent is the binding contract, or else that the previous
one was right and the subsequent one fraudulent-whichever way suits
us. further, we must consider the question of utility, noting whether
the contract is against the interest of the judges or not; and so
on-these arguments are as obvious as the others.
examination by torture is one form of evidence, to which great weight
is often attached because it is in a sense compulsory. here again
it is not hard to point out the available grounds for magnifying its
value, if it happens to tell in our favour, and arguing that it is
the only form of evidence that is infallible; or, on the other hand,
for refuting it if it tells against us and for our opponent, when
we may say what is true of torture of every kind alike, that people
under its compulsion tell lies quite as often as they tell the truth,
sometimes persistently refusing to tell the truth, sometimes recklessly
making a false charge in order to be let off sooner. we ought to be
able to quote cases, familiar to the judges, in which this sort of
thing has actually happened. we must say that evidence under torture
is not trustworthy, the fact being that many men whether thick-witted,
tough-skinned, or stout of heart endure their ordeal nobly, while
cowards and timid men are full of boldness till they see the ordeal
of these others: so that no trust can be placed in evidence under
in regard to oaths, a fourfold division can be made. a man may either
both offer and accept an oath, or neither, or one without the other-that
is, he may offer an oath but not accept one, or accept an oath but
not offer one. there is also the situation that arises when an oath
has already been sworn either by himself or by his opponent.
if you refuse to offer an oath, you may argue that men do not hesitate
to perjure themselves; and that if your opponent does swear, you lose
your money, whereas, if he does not, you think the judges will decide
against him; and that the risk of an unfavourable verdict is prefer,
able, since you trust the judges and do not trust him.
if you refuse to accept an oath, you may argue that an oath is always
paid for; that you would of course have taken it if you had been a
rascal, since if you are a rascal you had better make something by
it, and you would in that case have to swear in order to succeed.
thus your refusal, you argue, must be due to high principle, not to
fear of perjury: and you may aptly quote the saying of xenophanes,
"'tis not fair that he who fears not god
it is as if a strong man were to challenge a weakling to strike,
if you agree to accept an oath, you may argue that you trust yourself
but not your opponent; and that (to invert the remark of xenophanes)
the fair thing is for the impious man to offer the oath and for the
pious man to accept it; and that it would be monstrous if you yourself
were unwilling to accept an oath in a case where you demand that the
judges should do so before giving their verdict. if you wish to offer
an oath, you may argue that piety disposes you to commit the issue
to the gods; and that your opponent ought not to want other judges
than himself, since you leave the decision with him; and that it is
outrageous for your opponents to refuse to swear about this question,
when they insist that others should do so.
now that we see how we are to argue in each case separately, we see
also how we are to argue when they occur in pairs, namely, when you
are willing to accept the oath but not to offer it; to offer it but
not to accept it; both to accept and to offer it; or to do neither.
these are of course combinations of the cases already mentioned, and
so your arguments also must be combinations of the arguments already
if you have already sworn an oath that contradicts your present one,
you must argue that it is not perjury, since perjury is a crime, and
a crime must be a voluntary action, whereas actions due to the force
or fraud of others are involuntary. you must further reason from this
that perjury depends on the intention and not on the spoken words.
but if it is your opponent who has already sworn an oath that contradicts
his present one, you must say that if he does not abide by his oaths
he is the enemy of society, and that this is the reason why men take
an oath before administering the laws. 'my opponents insist that you,
the judges, must abide by the oath you have sworn, and yet they are
not abiding by their own oaths.' and there are other arguments which
may be used to magnify the importance of the oath. so much, then,
for the 'non-technical' modes of persuasion.
we have now considered the materials to be used in supporting or
opposing a political measure, in pronouncing eulogies or censures,
and for prosecution and defence in the law courts. we have considered
the received opinions on which we may best base our arguments so as
to convince our hearers-those opinions with which our enthymemes deal,
and out of which they are built, in each of the three kinds of oratory,
according to what may be called the special needs of each.
but since rhetoric exists to affect the giving of decisions-the hearers
decide between one political speaker and another, and a legal verdict
is a decision-the orator must not only try to make the argument of
his speech demonstrative and worthy of belief; he must also make his
own character look right and put his hearers, who are to decide, into
the right frame of mind. particularly in political oratory, but also
in lawsuits, it adds much to an orator's influence that his own character
should look right and that he should be thought to entertain the right
feelings towards his hearers; and also that his hearers themselves
should be in just the right frame of mind. that the orator's own character
should look right is particularly important in political speaking:
that the audience should be in the right frame of mind, in lawsuits.
when people are feeling friendly and placable, they think one sort
of thing; when they are feeling angry or hostile, they think either
something totally different or the same thing with a different intensity:
when they feel friendly to the man who comes before them for judgement,
they regard him as having done little wrong, if any; when they feel
hostile, they take the opposite view. again, if they are eager for,
and have good hopes of, a thing that will be pleasant if it happens,
they think that it certainly will happen and be good for them: whereas
if they are indifferent or annoyed, they do not think so.
there are three things which inspire confidence in the orator's own
character-the three, namely, that induce us to believe a thing apart
from any proof of it: good sense, good moral character, and goodwill.
false statements and bad advice are due to one or more of the following
three causes. men either form a false opinion through want of good
sense; or they form a true opinion, but because of their moral badness
do not say what they really think; or finally, they are both sensible
and upright, but not well disposed to their hearers, and may fail
in consequence to recommend what they know to be the best course.
these are the only possible cases. it follows that any one who is
thought to have all three of these good qualities will inspire trust
in his audience. the way to make ourselves thought to be sensible
and morally good must be gathered from the analysis of goodness already
given: the way to establish your own goodness is the same as the way
to establish that of others. good will and friendliness of disposition
will form part of our discussion of the emotions, to which we must
the emotions are all those feelings that so change men as to affect
their judgements, and that are also attended by pain or pleasure.
such are anger, pity, fear and the like, with their opposites. we
must arrange what we have to say about each of them under three heads.
take, for instance, the emotion of anger: here we must discover (1)
what the state of mind of angry people is, (2) who the people are
with whom they usually get angry, and (3) on what grounds they get
angry with them. it is not enough to know one or even two of these
points; unless we know all three, we shall be unable to arouse anger
in any one. the same is true of the other emotions. so just as earlier
in this work we drew up a list of useful propositions for the orator,
let us now proceed in the same way to analyse the subject before us.
anger may be defined as an impulse, accompanied by pain, to a conspicuous
revenge for a conspicuous slight directed without justification towards
what concerns oneself or towards what concerns one's friends. if this
is a proper definition of anger, it must always be felt towards some
particular individual, e.g. cleon, and not 'man' in general. it must
be felt because the other has done or intended to do something to
him or one of his friends. it must always be attended by a certain
pleasure-that which arises from the expectation of revenge. for since
nobody aims at what he thinks he cannot attain, the angry man is aiming
at what he can attain, and the belief that you will attain your aim
is pleasant. hence it has been well said about wrath,
"sweeter it is by far than the honeycomb
"and spreads through the hearts of men. "
it is also attended by a certain pleasure because the thoughts dwell
upon the act of vengeance, and the images then called up cause pleasure,
now slighting is the actively entertained opinion of something as
obviously of no importance. we think bad things, as well as good ones,
have serious importance; and we think the same of anything that tends
to produce such things, while those which have little or no such tendency
we consider unimportant. there are three kinds of slighting-contempt,
spite, and insolence. (1) contempt is one kind of slighting: you feel
contempt for what you consider unimportant, and it is just such things
that you slight. (2) spite is another kind; it is a thwarting another
man's wishes, not to get something yourself but to prevent his getting
it. the slight arises just from the fact that you do not aim at something
for yourself: clearly you do not think that he can do you harm, for
then you would be afraid of him instead of slighting him, nor yet
that he can do you any good worth mentioning, for then you would be
anxious to make friends with him. (3) insolence is also a form of
slighting, since it consists in doing and saying things that cause
shame to the victim, not in order that anything may happen to yourself,
or because anything has happened to yourself, but simply for the pleasure
involved. (retaliation is not 'insolence', but vengeance.) the cause
of the pleasure thus enjoyed by the insolent man is that he thinks
himself greatly superior to others when ill-treating them. that is
why youths and rich men are insolent; they think themselves superior
when they show insolence. one sort of insolence is to rob people of
the honour due to them; you certainly slight them thus; for it is
the unimportant, for good or evil, that has no honour paid to it.
meaning that this is why he is angry. a man expects to be specially
respected by his inferiors in birth, in capacity, in goodness, and
generally in anything in which he is much their superior: as where
money is concerned a wealthy man looks for respect from a poor man;
where speaking is concerned, the man with a turn for oratory looks
for respect from one who cannot speak; the ruler demands the respect
of the ruled, and the man who thinks he ought to be a ruler demands
the respect of the man whom he thinks he ought to be ruling. hence
"great is the wrath of kings, whose father is zeus almighty,
"yea, but his rancour abideth long afterward also, "
their great resentment being due to their great superiority. then
again a man looks for respect from those who he thinks owe him good
treatment, and these are the people whom he has treated or is treating
well, or means or has meant to treat well, either himself, or through
his friends, or through others at his request.
it will be plain by now, from what has been said, (1) in what frame
of mind, (2) with what persons, and (3) on what grounds people grow
angry. (1) the frame of mind is that of one in which any pain is being
felt. in that condition, a man is always aiming at something. whether,
then, another man opposes him either directly in any way, as by preventing
him from drinking when he is thirsty, or indirectly, the act appears
to him just the same; whether some one works against him, or fails
to work with him, or otherwise vexes him while he is in this mood,
he is equally angry in all these cases. hence people who are afflicted
by sickness or poverty or love or thirst or any other unsatisfied
desires are prone to anger and easily roused: especially against those
who slight their present distress. thus a sick man is angered by disregard
of his illness, a poor man by disregard of his poverty, a man aging
war by disregard of the war he is waging, a lover by disregard of
his love, and so throughout, any other sort of slight being enough
if special slights are wanting. each man is predisposed, by the emotion
now controlling him, to his own particular anger. further, we are
angered if we happen to be expecting a contrary result: for a quite
unexpected evil is specially painful, just as the quite unexpected
fulfilment of our wishes is specially pleasant. hence it is plain
what seasons, times, conditions, and periods of life tend to stir
men easily to anger, and where and when this will happen; and it is
plain that the more we are under these conditions the more easily
these, then, are the frames of mind in which men are easily stirred
to anger. the persons with whom we get angry are those who laugh,
mock, or jeer at us, for such conduct is insolent. also those who
inflict injuries upon us that are marks of insolence. these injuries
must be such as are neither retaliatory nor profitable to the doers:
for only then will they be felt to be due to insolence. also those
who speak ill of us, and show contempt for us, in connexion with the
things we ourselves most care about: thus those who are eager to win
fame as philosophers get angry with those who show contempt for their
philosophy; those who pride themselves upon their appearance get angry
with those who show contempt for their appearance and so on in other
cases. we feel particularly angry on this account if we suspect that
we are in fact, or that people think we are, lacking completely or
to any effective extent in the qualities in question. for when we
are convinced that we excel in the qualities for which we are jeered
at, we can ignore the jeering. again, we are angrier with our friends
than with other people, since we feel that our friends ought to treat
us well and not badly. we are angry with those who have usually treated
us with honour or regard, if a change comes and they behave to us
otherwise: for we think that they feel contempt for us, or they would
still be behaving as they did before. and with those who do not return
our kindnesses or fail to return them adequately, and with those who
oppose us though they are our inferiors: for all such persons seem
to feel contempt for us; those who oppose us seem to think us inferior
to themselves, and those who do not return our kindnesses seem to
think that those kindnesses were conferred by inferiors. and we feel
particularly angry with men of no account at all, if they slight us.
for, by our hypothesis, the anger caused by the slight is felt towards
people who are not justified in slighting us, and our inferiors are
not thus justified. again, we feel angry with friends if they do not
speak well of us or treat us well; and still more, if they do the
contrary; or if they do not perceive our needs, which is why plexippus
is angry with meleager in antiphon's play; for this want of perception
shows that they are slighting us-we do not fail to perceive the needs
of those for whom we care. again we are angry with those who rejoice
at our misfortunes or simply keep cheerful in the midst of our misfortunes,
since this shows that they either hate us or are slighting us. also
with those who are indifferent to the pain they give us: this is why
we get angry with bringers of bad news. and with those who listen
to stories about us or keep on looking at our weaknesses; this seems
like either slighting us or hating us; for those who love us share
in all our distresses and it must distress any one to keep on looking
at his own weaknesses. further, with those who slight us before five
classes of people: namely, (1) our rivals, (2) those whom we admire,
(3) those whom we wish to admire us, (4) those for whom we feel reverence,
(5) those who feel reverence for us: if any one slights us before
such persons, we feel particularly angry. again, we feel angry with
those who slight us in connexion with what we are as honourable men
bound to champion-our parents, children, wives, or subjects. and with
those who do not return a favour, since such a slight is unjustifiable.
also with those who reply with humorous levity when we are speaking
seriously, for such behaviour indicates contempt. and with those who
treat us less well than they treat everybody else; it is another mark
of contempt that they should think we do not deserve what every one
else deserves. forgetfulness, too, causes anger, as when our own names
are forgotten, trifling as this may be; since forgetfulness is felt
to be another sign that we are being slighted; it is due to negligence,
the persons with whom we feel anger, the frame of mind in which we
feel it, and the reasons why we feel it, have now all been set forth.
clearly the orator will have to speak so as to bring his hearers into
a frame of mind that will dispose them to anger, and to represent
his adversaries as open to such charges and possessed of such qualities
since growing calm is the opposite of growing angry, and calmness
the opposite of anger, we must ascertain in what frames of mind men
are calm, towards whom they feel calm, and by what means they are
made so. growing calm may be defined as a settling down or quieting
of anger. now we get angry with those who slight us; and since slighting
is a voluntary act, it is plain that we feel calm towards those who
do nothing of the kind, or who do or seem to do it involuntarily.
also towards those who intended to do the opposite of what they did
do. also towards those who treat themselves as they have treated us:
since no one can be supposed to slight himself. also towards those
who admit their fault and are sorry: since we accept their grief at
what they have done as satisfaction, and cease to be angry. the punishment
of servants shows this: those who contradict us and deny their offence
we punish all the more, but we cease to be incensed against those
who agree that they deserved their punishment. the reason is that
it is shameless to deny what is obvious, and those who are shameless
towards us slight us and show contempt for us: anyhow, we do not feel
shame before those of whom we are thoroughly contemptuous. also we
feel calm towards those who humble themselves before us and do not
gainsay us; we feel that they thus admit themselves our inferiors,
and inferiors feel fear, and nobody can slight any one so long as
he feels afraid of him. that our anger ceases towards those who humble
themselves before us is shown even by dogs, who do not bite people
when they sit down. we also feel calm towards those who are serious
when we are serious, because then we feel that we are treated seriously
and not contemptuously. also towards those who have done us more kindnesses
than we have done them. also towards those who pray to us and beg
for mercy, since they humble themselves by doing so. also towards
those who do not insult or mock at or slight any one at all, or not
any worthy person or any one like ourselves. in general, the things
that make us calm may be inferred by seeing what the opposites are
of those that make us angry. we are not angry with people we fear
or respect, as long as we fear or respect them; you cannot be afraid
of a person and also at the same time angry with him. again, we feel
no anger, or comparatively little, with those who have done what they
did through anger: we do not feel that they have done it from a wish
to slight us, for no one slights people when angry with them, since
slighting is painless, and anger is painful. nor do we grow angry
as to the frame of mind that makes people calm, it is plainly the
opposite to that which makes them angry, as when they are amusing
themselves or laughing or feasting; when they are feeling prosperous
or successful or satisfied; when, in fine, they are enjoying freedom
from pain, or inoffensive pleasure, or justifiable hope. also when
time has passed and their anger is no longer fresh, for time puts
an end to anger. and vengeance previously taken on one person puts
an end to even greater anger felt against another person. hence philocrates,
being asked by some one, at a time when the public was angry with
him, 'why don't you defend yourself?' did right to reply, 'the time
is not yet.' 'why, when is the time?' 'when i see someone else calumniated.'
for men become calm when they have spent their anger on somebody else.
this happened in the case of ergophilus: though the people were more
irritated against him than against callisthenes, they acquitted him
because they had condemned callisthenes to death the day before. again,
men become calm if they have convicted the offender; or if he has
already suffered worse things than they in their anger would have
themselves inflicted upon him; for they feel as if they were already
avenged. or if they feel that they themselves are in the wrong and
are suffering justly (for anger is not excited by what is just), since
men no longer think then that they are suffering without justification;
and anger, as we have seen, means this. hence we ought always to inflict
a preliminary punishment in words: if that is done, even slaves are
less aggrieved by the actual punishment. we also feel calm if we think
that the offender will not see that he is punished on our account
and because of the way he has treated us. for anger has to do with
individuals. this is plain from the definition. hence the poet has
"say that it was odysseus, sacker of cities, "
implying that odysseus would not have considered himself avenged
unless the cyclops perceived both by whom and for what he had been
blinded. consequently we do not get angry with any one who cannot
be aware of our anger, and in particular we cease to be angry with
people once they are dead, for we feel that the worst has been done
to them, and that they will neither feel pain nor anything else that
we in our anger aim at making them feel. and therefore the poet has
well made apollo say, in order to put a stop to the anger of achilles
"for behold in his fury he doeth despite to the senseless clay.
it is now plain that when you wish to calm others you must draw upon
these lines of argument; you must put your hearers into the corresponding
frame of mind, and represent those with whom they are angry as formidable,
or as worthy of reverence, or as benefactors, or as involuntary agents,
or as much distressed at what they have done.
let us now turn to friendship and enmity, and ask towards whom these
feelings are entertained, and why. we will begin by defining and friendly
feeling. we may describe friendly feeling towards any one as wishing
for him what you believe to be good things, not for your own sake
but for his, and being inclined, so far as you can, to bring these
things about. a friend is one who feels thus and excites these feelings
in return: those who think they feel thus towards each other think
themselves friends. this being assumed, it follows that your friend
is the sort of man who shares your pleasure in what is good and your
pain in what is unpleasant, for your sake and for no other reason.
this pleasure and pain of his will be the token of his good wishes
for you, since we all feel glad at getting what we wish for, and pained
at getting what we do not. those, then, are friends to whom the same
things are good and evil; and those who are, moreover, friendly or
unfriendly to the same people; for in that case they must have the
same wishes, and thus by wishing for each other what they wish for
themselves, they show themselves each other's friends. again, we feel
friendly to those who have treated us well, either ourselves or those
we care for, whether on a large scale, or readily, or at some particular
crisis; provided it was for our own sake. and also to those who we
think wish to treat us well. and also to our friends' friends, and
to those who like, or are liked by, those whom we like ourselves.
and also to those who are enemies to those whose enemies we are, and
dislike, or are disliked by, those whom we dislike. for all such persons
think the things good which we think good, so that they wish what
is good for us; and this, as we saw, is what friends must do. and
also to those who are willing to treat us well where money or our
personal safety is concerned: and therefore we value those who are
liberal, brave, or just. the just we consider to be those who do not
live on others; which means those who work for their living, especially
farmers and others who work with their own hands. we also like temperate
men, because they are not unjust to others; and, for the same reason,
those who mind their own business. and also those whose friends we
wish to be, if it is plain that they wish to be our friends: such
are the morally good, and those well thought of by every one, by the
best men, or by those whom we admire or who admire us. and also those
with whom it is pleasant to live and spend our days: such are the
good-tempered, and those who are not too ready to show us our mistakes,
and those who are not cantankerous or quarrelsome-such people are
always wanting to fight us, and those who fight us we feel wish for
the opposite of what we wish for ourselves-and those who have the
tact to make and take a joke; here both parties have the same object
in view, when they can stand being made fun of as well as do it prettily
themselves. and we also feel friendly towards those who praise such
good qualities as we possess, and especially if they praise the good
qualities that we are not too sure we do possess. and towards those
who are cleanly in their person, their dress, and all their way of
life. and towards those who do not reproach us with what we have done
amiss to them or they have done to help us, for both actions show
a tendency to criticize us. and towards those who do not nurse grudges
or store up grievances, but are always ready to make friends again;
for we take it that they will behave to us just as we find them behaving
to every one else. and towards those who are not evil speakers and
who are aware of neither their neighbours' bad points nor our own,
but of our good ones only, as a good man always will be. and towards
those who do not try to thwart us when we are angry or in earnest,
which would mean being ready to fight us. and towards those who have
some serious feeling towards us, such as admiration for us, or belief
in our goodness, or pleasure in our company; especially if they feel
like this about qualities in us for which we especially wish to be
admired, esteemed, or liked. and towards those who are like ourselves
in character and occupation, provided they do not get in our way or
gain their living from the same source as we do-for then it will be
"potter to potter and builder to builder begrudge their reward.
and those who desire the same things as we desire, if it is possible
for us both to share them together; otherwise the same trouble arises
here too. and towards those with whom we are on such terms that, while
we respect their opinions, we need not blush before them for doing
what is conventionally wrong: as well as towards those before whom
we should be ashamed to do anything really wrong. again, our rivals,
and those whom we should like to envy us--though without ill-feeling--either
we like these people or at least we wish them to like us. and we feel
friendly towards those whom we help to secure good for themselves,
provided we are not likely to suffer heavily by it ourselves. and
those who feel as friendly to us when we are not with them as when
we are-which is why all men feel friendly towards those who are faithful
to their dead friends. and, speaking generally, towards those who
are really fond of their friends and do not desert them in trouble;
of all good men, we feel most friendly to those who show their goodness
as friends. also towards those who are honest with us, including those
who will tell us of their own weak points: it has just said that with
our friends we are not ashamed of what is conventionally wrong, and
if we do have this feeling, we do not love them; if therefore we do
not have it, it looks as if we did love them. we also like those with
whom we do not feel frightened or uncomfortable-nobody can like a
man of whom he feels frightened. friendship has various forms-comradeship,
things that cause friendship are: doing kindnesses; doing them unasked;
and not proclaiming the fact when they are done, which shows that
they were done for our own sake and not for some other reason.
enmity and hatred should clearly be studied by reference to their
opposites. enmity may be produced by anger or spite or calumny. now
whereas anger arises from offences against oneself, enmity may arise
even without that; we may hate people merely because of what we take
to be their character. anger is always concerned with individuals-a
callias or a socrates-whereas hatred is directed also against classes:
we all hate any thief and any informer. moreover, anger can be cured
by time; but hatred cannot. the one aims at giving pain to its object,
the other at doing him harm; the angry man wants his victims to feel;
the hater does not mind whether they feel or not. all painful things
are felt; but the greatest evils, injustice and folly, are the least
felt, since their presence causes no pain. and anger is accompanied
by pain, hatred is not; the angry man feels pain, but the hater does
not. much may happen to make the angry man pity those who offend him,
but the hater under no circumstances wishes to pity a man whom he
has once hated: for the one would have the offenders suffer for what
they have done; the other would have them cease to exist.
it is plain from all this that we can prove people to be friends or
enemies; if they are not, we can make them out to be so; if they claim
to be so, we can refute their claim; and if it is disputed whether
an action was due to anger or to hatred, we can attribute it to whichever
to turn next to fear, what follows will show things and persons of
which, and the states of mind in which, we feel afraid. fear may be
defined as a pain or disturbance due to a mental picture of some destructive
or painful evil in the future. of destructive or painful evils only;
for there are some evils, e.g. wickedness or stupidity, the prospect
of which does not frighten us: i mean only such as amount to great
pains or losses. and even these only if they appear not remote but
so near as to be imminent: we do not fear things that are a very long
way off: for instance, we all know we shall die, but we are not troubled
thereby, because death is not close at hand. from this definition
it will follow that fear is caused by whatever we feel has great power
of destroying or of harming us in ways that tend to cause us great
pain. hence the very indications of such things are terrible, making
us feel that the terrible thing itself is close at hand; the approach
of what is terrible is just what we mean by 'danger'. such indications
are the enmity and anger of people who have power to do something
to us; for it is plain that they have the will to do it, and so they
are on the point of doing it. also injustice in possession of power;
for it is the unjust man's will to do evil that makes him unjust.
also outraged virtue in possession of power; for it is plain that,
when outraged, it always has the will to retaliate, and now it has
the power to do so. also fear felt by those who have the power to
do something to us, since such persons are sure to be ready to do
it. and since most men tend to be bad-slaves to greed, and cowards
in danger-it is, as a rule, a terrible thing to be at another man's
mercy; and therefore, if we have done anything horrible, those in
the secret terrify us with the thought that they may betray or desert
us. and those who can do us wrong are terrible to us when we are liable
to be wronged; for as a rule men do wrong to others whenever they
have the power to do it. and those who have been wronged, or believe
themselves to be wronged, are terrible; for they are always looking
out for their opportunity. also those who have done people wrong,
if they possess power, since they stand in fear of retaliation: we
have already said that wickedness possessing power is terrible. again,
our rivals for a thing cause us fear when we cannot both have it at
once; for we are always at war with such men. we also fear those who
are to be feared by stronger people than ourselves: if they can hurt
those stronger people, still more can they hurt us; and, for the same
reason, we fear those whom those stronger people are actually afraid
of. also those who have destroyed people stronger than we are. also
those who are attacking people weaker than we are: either they are
already formidable, or they will be so when they have thus grown stronger.
of those we have wronged, and of our enemies or rivals, it is not
the passionate and outspoken whom we have to fear, but the quiet,
dissembling, unscrupulous; since we never know when they are upon
us, we can never be sure they are at a safe distance. all terrible
things are more terrible if they give us no chance of retrieving a
blunder either no chance at all, or only one that depends on our enemies
and not ourselves. those things are also worse which we cannot, or
cannot easily, help. speaking generally, anything causes us to feel
fear that when it happens to, or threatens, others cause us to feel
the above are, roughly, the chief things that are terrible and are
feared. let us now describe the conditions under which we ourselves
feel fear. if fear is associated with the expectation that something
destructive will happen to us, plainly nobody will be afraid who believes
nothing can happen to him; we shall not fear things that we believe
cannot happen to us, nor people who we believe cannot inflict them
upon us; nor shall we be afraid at times when we think ourselves safe
from them. it follows therefore that fear is felt by those who believe
something to be likely to happen to them, at the hands of particular
persons, in a particular form, and at a particular time. people do
not believe this when they are, or think they a are, in the midst
of great prosperity, and are in consequence insolent, contemptuous,
and reckless-the kind of character produced by wealth, physical strength,
abundance of friends, power: nor yet when they feel they have experienced
every kind of horror already and have grown callous about the future,
like men who are being flogged and are already nearly dead-if they
are to feel the anguish of uncertainty, there must be some faint expectation
of escape. this appears from the fact that fear sets us thinking what
can be done, which of course nobody does when things are hopeless.
consequently, when it is advisable that the audience should be frightened,
the orator must make them feel that they really are in danger of something,
pointing out that it has happened to others who were stronger than
they are, and is happening, or has happened, to people like themselves,
at the hands of unexpected people, in an unexpected form, and at an
having now seen the nature of fear, and of the things that cause it,
and the various states of mind in which it is felt, we can also see
what confidence is, about what things we feel it, and under what conditions.
it is the opposite of fear, and what causes it is the opposite of
what causes fear; it is, therefore, the expectation associated with
a mental picture of the nearness of what keeps us safe and the absence
or remoteness of what is terrible: it may be due either to the near
presence of what inspires confidence or to the absence of what causes
alarm. we feel it if we can take steps-many, or important, or both-to
cure or prevent trouble; if we have neither wronged others nor been
wronged by them; if we have either no rivals at all or no strong ones;
if our rivals who are strong are our friends or have treated us well
or been treated well by us; or if those whose interest is the same
as ours are the more numerous party, or the stronger, or both.
as for our own state of mind, we feel confidence if we believe we
have often succeeded and never suffered reverses, or have often met
danger and escaped it safely. for there are two reasons why human
beings face danger calmly: they may have no experience of it, or they
may have means to deal with it: thus when in danger at sea people
may feel confident about what will happen either because they have
no experience of bad weather, or because their experience gives them
the means of dealing with it. we also feel confident whenever there
is nothing to terrify other people like ourselves, or people weaker
than ourselves, or people than whom we believe ourselves to be stronger-and
we believe this if we have conquered them, or conquered others who
are as strong as they are, or stronger. also if we believe ourselves
superior to our rivals in the number and importance of the advantages
that make men formidable-wealth, physical strength, strong bodies
of supporters, extensive territory, and the possession of all, or
the most important, appliances of war. also if we have wronged no
one, or not many, or not those of whom we are afraid; and generally,
if our relations with the gods are satisfactory, as will be shown
especially by signs and oracles. the fact is that anger makes us confident-that
anger is excited by our knowledge that we are not the wrongers but
the wronged, and that the divine power is always supposed to be on
the side of the wronged. also when, at the outset of an enterprise,
we believe that we cannot and shall not fail, or that we shall succeed
completely.-so much for the causes of fear and confidence.
we now turn to shame and shamelessness; what follows will explain
the things that cause these feelings, and the persons before whom,
and the states of mind under which, they are felt. shame may be defined
as pain or disturbance in regard to bad things, whether present, past,
or future, which seem likely to involve us in discredit; and shamelessness
as contempt or indifference in regard to these same bad things. if
this definition be granted, it follows that we feel shame at such
bad things as we think are disgraceful to ourselves or to those we
care for. these evils are, in the first place, those due to moral
badness. such are throwing away one's shield or taking to flight;
for these bad things are due to cowardice. also, withholding a deposit
or otherwise wronging people about money; for these acts are due to
injustice. also, having carnal intercourse with forbidden persons,
at wrong times, or in wrong places; for these things are due to licentiousness.
also, making profit in petty or disgraceful ways, or out of helpless
persons, e.g. the poor, or the dead-whence the proverb 'he would pick
a corpse's pocket'; for all this is due to low greed and meanness.
also, in money matters, giving less help than you might, or none at
all, or accepting help from those worse off than yourself; so also
borrowing when it will seem like begging; begging when it will seem
like asking the return of a favour; asking such a return when it will
seem like begging; praising a man in order that it may seem like begging;
and going on begging in spite of failure: all such actions are tokens
of meanness. also, praising people to their face, and praising extravagantly
a man's good points and glozing over his weaknesses, and showing extravagant
sympathy with his grief when you are in his presence, and all that
sort of thing; all this shows the disposition of a flatterer. also,
refusing to endure hardships that are endured by people who are older,
more delicately brought up, of higher rank, or generally less capable
of endurance than ourselves: for all this shows effeminacy. also,
accepting benefits, especially accepting them often, from another
man, and then abusing him for conferring them: all this shows a mean,
ignoble disposition. also, talking incessantly about yourself, making
loud professions, and appropriating the merits of others; for this
is due to boastfulness. the same is true of the actions due to any
of the other forms of badness of moral character, of the tokens of
such badness, c.: they are all disgraceful and shameless. another
sort of bad thing at which we feel shame is, lacking a share in the
honourable things shared by every one else, or by all or nearly all
who are like ourselves. by 'those like ourselves' i mean those of
our own race or country or age or family, and generally those who
are on our own level. once we are on a level with others, it is a
disgrace to be, say, less well educated than they are; and so with
other advantages: all the more so, in each case, if it is seen to
be our own fault: wherever we are ourselves to blame for our present,
past, or future circumstances, it follows at once that this is to
a greater extent due to our moral badness. we are moreover ashamed
of having done to us, having had done, or being about to have done
to us acts that involve us in dishonour and reproach; as when we surrender
our persons, or lend ourselves to vile deeds, e.g. when we submit
to outrage. and acts of yielding to the lust of others are shameful
whether willing or unwilling (yielding to force being an instance
of unwillingness), since unresisting submission to them is due to
these things, and others like them, are what cause the feeling of
shame. now since shame is a mental picture of disgrace, in which we
shrink from the disgrace itself and not from its consequences, and
we only care what opinion is held of us because of the people who
form that opinion, it follows that the people before whom we feel
shame are those whose opinion of us matters to us. such persons are:
those who admire us, those whom we admire, those by whom we wish to
be admired, those with whom we are competing, and those whose opinion
of us we respect. we admire those, and wish those to admire us, who
possess any good thing that is highly esteemed; or from whom we are
very anxious to get something that they are able to give us-as a lover
feels. we compete with our equals. we respect, as true, the views
of sensible people, such as our elders and those who have been well
educated. and we feel more shame about a thing if it is done openly,
before all men's eyes. hence the proverb, 'shame dwells in the eyes'.
for this reason we feel most shame before those who will always be
with us and those who notice what we do, since in both cases eyes
are upon us. we also feel it before those not open to the same imputation
as ourselves: for it is plain that their opinions about it are the
opposite of ours. also before those who are hard on any one whose
conduct they think wrong; for what a man does himself, he is said
not to resent when his neighbours do it: so that of course he does
resent their doing what he does not do himself. and before those who
are likely to tell everybody about you; not telling others is as good
as not be lieving you wrong. people are likely to tell others about
you if you have wronged them, since they are on the look out to harm
you; or if they speak evil of everybody, for those who attack the
innocent will be still more ready to attack the guilty. and before
those whose main occupation is with their neighbours' failings-people
like satirists and writers of comedy; these are really a kind of evil-speakers
and tell-tales. and before those who have never yet known us come
to grief, since their attitude to us has amounted to admiration so
far: that is why we feel ashamed to refuse those a favour who ask
one for the first time-we have not as yet lost credit with them. such
are those who are just beginning to wish to be our friends; for they
have seen our best side only (hence the appropriateness of euripides'
reply to the syracusans): and such also are those among our old acquaintances
who know nothing to our discredit. and we are ashamed not merely of
the actual shameful conduct mentioned, but also of the evidences of
it: not merely, for example, of actual sexual intercourse, but also
of its evidences; and not merely of disgraceful acts but also of disgraceful
talk. similarly we feel shame not merely in presence of the persons
mentioned but also of those who will tell them what we have done,
such as their servants or friends. and, generally, we feel no shame
before those upon whose opinions we quite look down as untrustworthy
(no one feels shame before small children or animals); nor are we
ashamed of the same things before intimates as before strangers, but
before the former of what seem genuine faults, before the latter of
the conditions under which we shall feel shame are these: first, having
people related to us like those before whom, as has been said, we
feel shame. these are, as was stated, persons whom we admire, or who
admire us, or by whom we wish to be admired, or from whom we desire
some service that we shall not obtain if we forfeit their good opinion.
these persons may be actually looking on (as cydias represented them
in his speech on land assignments in samos, when he told the athenians
to imagine the greeks to be standing all around them, actually seeing
the way they voted and not merely going to hear about it afterwards):
or again they may be near at hand, or may be likely to find out about
what we do. this is why in misfortune we do not wish to be seen by
those who once wished themselves like us; for such a feeling implies
admiration. and men feel shame when they have acts or exploits to
their credit on which they are bringing dishonour, whether these are
their own, or those of their ancestors, or those of other persons
with whom they have some close connexion. generally, we feel shame
before those for whose own misconduct we should also feel it-those
already mentioned; those who take us as their models; those whose
teachers or advisers we have been; or other people, it may be, like
ourselves, whose rivals we are. for there are many things that shame
before such people makes us do or leave undone. and we feel more shame
when we are likely to be continually seen by, and go about under the
eyes of, those who know of our disgrace. hence, when antiphon the
poet was to be cudgelled to death by order of dionysius, and saw those
who were to perish with him covering their faces as they went through
the gates, he said, 'why do you cover your faces? is it lest some
of these spectators should see you to-morrow?'
so much for shame; to understand shamelessness, we need only consider
the converse cases, and plainly we shall have all we need.
to take kindness next: the definition of it will show us towards whom
it is felt, why, and in what frames of mind. kindness-under the influence
of which a man is said to 'be kind' may be defined as helpfulness
towards some one in need, not in return for anything, nor for the
advantage of the helper himself, but for that of the person helped.
kindness is great if shown to one who is in great need, or who needs
what is important and hard to get, or who needs it at an important
and difficult crisis; or if the helper is the only, the first, or
the chief person to give the help. natural cravings constitute such
needs; and in particular cravings, accompanied by pain, for what is
not being attained. the appetites are cravings for this kind: sexual
desire, for instance, and those which arise during bodily injuries
and in dangers; for appetite is active both in danger and in pain.
hence those who stand by us in poverty or in banishment, even if they
do not help us much, are yet really kind to us, because our need is
great and the occasion pressing; for instance, the man who gave the
mat in the lyceum. the helpfulness must therefore meet, preferably,
just this kind of need; and failing just this kind, some other kind
as great or greater. we now see to whom, why, and under what conditions
kindness is shown; and these facts must form the basis of our arguments.
we must show that the persons helped are, or have been, in such pain
and need as has been described, and that their helpers gave, or are
giving, the kind of help described, in the kind of need described.
we can also see how to eliminate the idea of kindness and make our
opponents appear unkind: we may maintain that they are being or have
been helpful simply to promote their own interest-this, as has been
stated, is not kindness; or that their action was accidental, or was
forced upon them; or that they were not doing a favour, but merely
returning one, whether they know this or not-in either case the action
is a mere return, and is therefore not a kindness even if the doer
does not know how the case stands. in considering this subject we
must look at all the categories: an act may be an act of kindness
because (1) it is a particular thing, (2) it has a particular magnitude
or (3) quality, or (4) is done at a particular time or (5) place.
as evidence of the want of kindness, we may point out that a smaller
service had been refused to the man in need; or that the same service,
or an equal or greater one, has been given to his enemies; these facts
show that the service in question was not done for the sake of the
person helped. or we may point out that the thing desired was worthless
and that the helper knew it: no one will admit that he is in need
so much for kindness and unkindness. let us now consider pity, asking
ourselves what things excite pity, and for what persons, and in what
states of our mind pity is felt. pity may be defined as a feeling
of pain caused by the sight of some evil, destructive or painful,
which befalls one who does not deserve it, and which we might expect
to befall ourselves or some friend of ours, and moreover to befall
us soon. in order to feel pity, we must obviously be capable of supposing
that some evil may happen to us or some friend of ours, and moreover
some such evil as is stated in our definition or is more or less of
that kind. it is therefore not felt by those completely ruined, who
suppose that no further evil can befall them, since the worst has
befallen them already; nor by those who imagine themselves immensely
fortunate-their feeling is rather presumptuous insolence, for when
they think they possess all the good things of life, it is clear that
the impossibility of evil befalling them will be included, this being
one of the good things in question. those who think evil may befall
them are such as have already had it befall them and have safely escaped
from it; elderly men, owing to their good sense and their experience;
weak men, especially men inclined to cowardice; and also educated
people, since these can take long views. also those who have parents
living, or children, or wives; for these are our own, and the evils
mentioned above may easily befall them. and those who neither moved
by any courageous emotion such as anger or confidence (these emotions
take no account of the future), nor by a disposition to presumptuous
insolence (insolent men, too, take no account of the possibility that
something evil will happen to them), nor yet by great fear (panic-stricken
people do not feel pity, because they are taken up with what is happening
to themselves); only those feel pity who are between these two extremes.
in order to feel pity we must also believe in the goodness of at least
some people; if you think nobody good, you will believe that everybody
deserves evil fortune. and, generally, we feel pity whenever we are
in the condition of remembering that similar misfortunes have happened
to us or ours, or expecting them to happen in the future.
so much for the mental conditions under which we feel pity. what we
pity is stated clearly in the definition. all unpleasant and painful
things excite pity if they tend to destroy pain and annihilate; and
all such evils as are due to chance, if they are serious. the painful
and destructive evils are: death in its various forms, bodily injuries
and afflictions, old age, diseases, lack of food. the evils due to
chance are: friendlessness, scarcity of friends (it is a pitiful thing
to be torn away from friends and companions), deformity, weakness,
mutilation; evil coming from a source from which good ought to have
come; and the frequent repetition of such misfortunes. also the coming
of good when the worst has happened: e.g. the arrival of the great
king's gifts for diopeithes after his death. also that either no good
should have befallen a man at all, or that he should not be able to
the grounds, then, on which we feel pity are these or like these.
the people we pity are: those whom we know, if only they are not very
closely related to us-in that case we feel about them as if we were
in danger ourselves. for this reason amasis did not weep, they say,
at the sight of his son being led to death, but did weep when he saw
his friend begging: the latter sight was pitiful, the former terrible,
and the terrible is different from the pitiful; it tends to cast out
pity, and often helps to produce the opposite of pity. again, we feel
pity when the danger is near ourselves. also we pity those who are
like us in age, character, disposition, social standing, or birth;
for in all these cases it appears more likely that the same misfortune
may befall us also. here too we have to remember the general principle
that what we fear for ourselves excites our pity when it happens to
others. further, since it is when the sufferings of others are close
to us that they excite our pity (we cannot remember what disasters
happened a hundred centuries ago, nor look forward to what will happen
a hundred centuries hereafter, and therefore feel little pity, if
any, for such things): it follows that those who heighten the effect
of their words with suitable gestures, tones, dress, and dramatic
action generally, are especially successful in exciting pity: they
thus put the disasters before our eyes, and make them seem close to
us, just coming or just past. anything that has just happened, or
is going to happen soon, is particularly piteous: so too therefore
are the tokens and the actions of sufferers-the garments and the like
of those who have already suffered; the words and the like of those
actually suffering-of those, for instance, who are on the point of
death. most piteous of all is it when, in such times of trial, the
victims are persons of noble character: whenever they are so, our
pity is especially excited, because their innocence, as well as the
setting of their misfortunes before our eyes, makes their misfortunes
most directly opposed to pity is the feeling called indignation. pain
at unmerited good fortune is, in one sense, opposite to pain at unmerited
bad fortune, and is due to the same moral qualities. both feelings
are associated with good moral character; it is our duty both to feel
sympathy and pity for unmerited distress, and to feel indignation
at unmerited prosperity; for whatever is undeserved is unjust, and
that is why we ascribe indignation even to the gods. it might indeed
be thought that envy is similarly opposed to pity, on the ground that
envy it closely akin to indignation, or even the same thing. but it
is not the same. it is true that it also is a disturbing pain excited
by the prosperity of others. but it is excited not by the prosperity
of the undeserving but by that of people who are like us or equal
with us. the two feelings have this in common, that they must be due
not to some untoward thing being likely to befall ourselves, but only
to what is happening to our neighbour. the feeling ceases to be envy
in the one case and indignation in the other, and becomes fear, if
the pain and disturbance are due to the prospect of something bad
for ourselves as the result of the other man's good fortune. the feelings
of pity and indignation will obviously be attended by the converse
feelings of satisfaction. if you are pained by the unmerited distress
of others, you will be pleased, or at least not pained, by their merited
distress. thus no good man can be pained by the punishment of parricides
or murderers. these are things we are bound to rejoice at, as we must
at the prosperity of the deserving; both these things are just, and
both give pleasure to any honest man, since he cannot help expecting
that what has happened to a man like him will happen to him too. all
these feelings are associated with the same type of moral character.
and their contraries are associated with the contrary type; the man
who is delighted by others' misfortunes is identical with the man
who envies others' prosperity. for any one who is pained by the occurrence
or existence of a given thing must be pleased by that thing's non-existence
or destruction. we can now see that all these feelings tend to prevent
pity (though they differ among themselves, for the reasons given),
so that all are equally useful for neutralizing an appeal to pity.
we will first consider indignation-reserving the other emotions for
subsequent discussion-and ask with whom, on what grounds, and in what
states of mind we may be indignant. these questions are really answered
by what has been said already. indignation is pain caused by the sight
of undeserved good fortune. it is, then, plain to begin with that
there are some forms of good the sight of which cannot cause it. thus
a man may be just or brave, or acquire moral goodness: but we shall
not be indignant with him for that reason, any more than we shall
pity him for the contrary reason. indignation is roused by the sight
of wealth, power, and the like-by all those things, roughly speaking,
which are deserved by good men and by those who possess the goods
of nature-noble birth, beauty, and so on. again, what is long established
seems akin to what exists by nature; and therefore we feel more indignation
at those possessing a given good if they have as a matter of fact
only just got it and the prosperity it brings with it. the newly rich
give more offence than those whose wealth is of long standing and
inherited. the same is true of those who have office or power, plenty
of friends, a fine family, c. we feel the same when these advantages
of theirs secure them others. for here again, the newly rich give
us more offence by obtaining office through their riches than do those
whose wealth is of long standing; and so in all other cases. the reason
is that what the latter have is felt to be really their own, but what
the others have is not; what appears to have been always what it is
is regarded as real, and so the possessions of the newly rich do not
seem to be really their own. further, it is not any and every man
that deserves any given kind of good; there is a certain correspondence
and appropriateness in such things; thus it is appropriate for brave
men, not for just men, to have fine weapons, and for men of family,
not for parvenus, to make distinguished marriages. indignation may
therefore properly be felt when any one gets what is not appropriate
for him, though he may be a good man enough. it may also be felt when
any one sets himself up against his superior, especially against his
superior in some particular respect-whence the lines
"only from battle he shrank with aias telamon's son;
but also, even apart from that, when the inferior in any sense contends
with his superior; a musician, for instance, with a just man, for
enough has been said to make clear the grounds on which, and the persons
against whom, indignation is felt-they are those mentioned, and others
like him. as for the people who feel it; we feel it if we do ourselves
deserve the greatest possible goods and moreover have them, for it
is an injustice that those who are not our equals should have been
held to deserve as much as we have. or, secondly, we feel it if we
are really good and honest people; our judgement is then sound, and
we loathe any kind of injustice. also if we are ambitious and eager
to gain particular ends, especially if we are ambitious for what others
are getting without deserving to get it. and, generally, if we think
that we ourselves deserve a thing and that others do not, we are disposed
to be indignant with those others so far as that thing is concerned.
hence servile, worthless, unambitious persons are not inclined to
indignation, since there is nothing they can believe themselves to
from all this it is plain what sort of men those are at whose misfortunes,
distresses, or failures we ought to feel pleased, or at least not
pained: by considering the facts described we see at once what their
contraries are. if therefore our speech puts the judges in such a
frame of mind as that indicated and shows that those who claim pity
on certain definite grounds do not deserve to secure pity but do deserve
not to secure it, it will be impossible for the judges to feel pity.
to take envy next: we can see on what grounds, against what persons,
and in what states of mind we feel it. envy is pain at the sight of
such good fortune as consists of the good things already mentioned;
we feel it towards our equals; not with the idea of getting something
for ourselves, but because the other people have it. we shall feel
it if we have, or think we have, equals; and by 'equals' i mean equals
in birth, relationship, age, disposition, distinction, or wealth.
we feel envy also if we fall but a little short of having everything;
which is why people in high place and prosperity feel it-they think
every one else is taking what belongs to themselves. also if we are
exceptionally distinguished for some particular thing, and especially
if that thing is wisdom or good fortune. ambitious men are more envious
than those who are not. so also those who profess wisdom; they are
ambitious to be thought wise. indeed, generally, those who aim at
a reputation for anything are envious on this particular point. and
small-minded men are envious, for everything seems great to them.
the good things which excite envy have already been mentioned. the
deeds or possessions which arouse the love of reputation and honour
and the desire for fame, and the various gifts of fortune, are almost
all subject to envy; and particularly if we desire the thing ourselves,
or think we are entitled to it, or if having it puts us a little above
others, or not having it a little below them. it is clear also what
kind of people we envy; that was included in what has been said already:
we envy those who are near us in time, place, age, or reputation.
"ay, kin can even be jealous of their kin. "
also our fellow-competitors, who are indeed the people just mentioned-we
do not compete with men who lived a hundred centuries ago, or those
not yet born, or the dead, or those who dwell near the pillars of
hercules, or those whom, in our opinion or that of others, we take
to be far below us or far above us. so too we compete with those who
follow the same ends as ourselves: we compete with our rivals in sport
or in love, and generally with those who are after the same things;
and it is therefore these whom we are bound to envy beyond all others.
we also envy those whose possession of or success in a thing is a
reproach to us: these are our neighbours and equals; for it is clear
that it is our own fault we have missed the good thing in question;
this annoys us, and excites envy in us. we also envy those who have
what we ought to have, or have got what we did have once. hence old
men envy younger men, and those who have spent much envy those who
have spent little on the same thing. and men who have not got a thing,
or not got it yet, envy those who have got it quickly. we can also
see what things and what persons give pleasure to envious people,
and in what states of mind they feel it: the states of mind in which
they feel pain are those under which they will feel pleasure in the
contrary things. if therefore we ourselves with whom the decision
rests are put into an envious state of mind, and those for whom our
pity, or the award of something desirable, is claimed are such as
have been described, it is obvious that they will win no pity from
we will next consider emulation, showing in what follows its causes
and objects, and the state of mind in which it is felt. emulation
is pain caused by seeing the presence, in persons whose nature is
like our own, of good things that are highly valued and are possible
for ourselves to acquire; but it is felt not because others have these
goods, but because we have not got them ourselves. it is therefore
a good feeling felt by good persons, whereas envy is a bad feeling
felt by bad persons. emulation makes us take steps to secure the good
things in question, envy makes us take steps to stop our neighbour
having them. emulation must therefore tend to be felt by persons who
believe themselves to deserve certain good things that they have not
got, it being understood that no one aspires to things which appear
impossible. it is accordingly felt by the young and by persons of
lofty disposition. also by those who possess such good things as are
deserved by men held in honour-these are wealth, abundance of friends,
public office, and the like; on the assumption that they ought to
be good men, they are emulous to gain such goods because they ought,
in their belief, to belong to men whose state of mind is good. also
by those whom all others think deserving. we also feel it about anything
for which our ancestors, relatives, personal friends, race, or country
are specially honoured, looking upon that thing as really our own,
and therefore feeling that we deserve to have it. further, since all
good things that are highly honoured are objects of emulation, moral
goodness in its various forms must be such an object, and also all
those good things that are useful and serviceable to others: for men
honour those who are morally good, and also those who do them service.
so with those good things our possession of which can give enjoyment
to our neighbours-wealth and beauty rather than health. we can see,
too, what persons are the objects of the feeling. they are those who
have these and similar things-those already mentioned, as courage,
wisdom, public office. holders of public office-generals, orators,
and all who possess such powers-can do many people a good turn. also
those whom many people wish to be like; those who have many acquaintances
or friends; those whom admire, or whom we ourselves admire; and those
who have been praised and eulogized by poets or prose-writers. persons
of the contrary sort are objects of contempt: for the feeling and
notion of contempt are opposite to those of emulation. those who are
such as to emulate or be emulated by others are inevitably disposed
to be contemptuous of all such persons as are subject to those bad
things which are contrary to the good things that are the objects
of emulation: despising them for just that reason. hence we often
despise the fortunate, when luck comes to them without their having
those good things which are held in honour.
this completes our discussion of the means by which the several emotions
may be produced or dissipated, and upon which depend the persuasive
let us now consider the various types of human character, in relation
to the emotions and moral qualities, showing how they correspond to
our various ages and fortunes. by emotions i mean anger, desire, and
the like; these we have discussed already. by moral qualities i mean
virtues and vices; these also have been discussed already, as well
as the various things that various types of men tend to will and to
do. by ages i mean youth, the prime of life, and old age. by fortune
i mean birth, wealth, power, and their opposites-in fact, good fortune
to begin with the youthful type of character. young men have strong
passions, and tend to gratify them indiscriminately. of the bodily
desires, it is the sexual by which they are most swayed and in which
they show absence of self-control. they are changeable and fickle
in their desires, which are violent while they last, but quickly over:
their impulses are keen but not deep-rooted, and are like sick people's
attacks of hunger and thirst. they are hot-tempered, and quick-tempered,
and apt to give way to their anger; bad temper often gets the better
of them, for owing to their love of honour they cannot bear being
slighted, and are indignant if they imagine themselves unfairly treated.
while they love honour, they love victory still more; for youth is
eager for superiority over others, and victory is one form of this.
they love both more than they love money, which indeed they love very
little, not having yet learnt what it means to be without it-this
is the point of pittacus' remark about amphiaraus. they look at the
good side rather than the bad, not having yet witnessed many instances
of wickedness. they trust others readily, because they have not yet
often been cheated. they are sanguine; nature warms their blood as
though with excess of wine; and besides that, they have as yet met
with few disappointments. their lives are mainly spent not in memory
but in expectation; for expectation refers to the future, memory to
the past, and youth has a long future before it and a short past behind
it: on the first day of one's life one has nothing at all to remember,
and can only look forward. they are easily cheated, owing to the sanguine
disposition just mentioned. their hot tempers and hopeful dispositions
make them more courageous than older men are; the hot temper prevents
fear, and the hopeful disposition creates confidence; we cannot feel
fear so long as we are feeling angry, and any expectation of good
makes us confident. they are shy, accepting the rules of society in
which they have been trained, and not yet believing in any other standard
of honour. they have exalted notions, because they have not yet been
humbled by life or learnt its necessary limitations; moreover, their
hopeful disposition makes them think themselves equal to great things-and
that means having exalted notions. they would always rather do noble
deeds than useful ones: their lives are regulated more by moral feeling
than by reasoning; and whereas reasoning leads us to choose what is
useful, moral goodness leads us to choose what is noble. they are
fonder of their friends, intimates, and companions than older men
are, because they like spending their days in the company of others,
and have not yet come to value either their friends or anything else
by their usefulness to themselves. all their mistakes are in the direction
of doing things excessively and vehemently. they disobey chilon's
precept by overdoing everything, they love too much and hate too much,
and the same thing with everything else. they think they know everything,
and are always quite sure about it; this, in fact, is why they overdo
everything. if they do wrong to others, it is because they mean to
insult them, not to do them actual harm. they are ready to pity others,
because they think every one an honest man, or anyhow better than
he is: they judge their neighbour by their own harmless natures, and
so cannot think he deserves to be treated in that way. they are fond
of fun and therefore witty, wit being well-bred insolence.
such, then is the character of the young. the character of elderly
men-men who are past their prime-may be said to be formed for the
most part of elements that are the contrary of all these. they have
lived many years; they have often been taken in, and often made mistakes;
and life on the whole is a bad business. the result is that they are
sure about nothing and under-do everything. they 'think', but they
never 'know'; and because of their hesitation they always add a 'possibly'or
a 'perhaps', putting everything this way and nothing positively. they
are cynical; that is, they tend to put the worse construction on everything.
further, their experience makes them distrustful and therefore suspicious
of evil. consequently they neither love warmly nor hate bitterly,
but following the hint of bias they love as though they will some
day hate and hate as though they will some day love. they are small-minded,
because they have been humbled by life: their desires are set upon
nothing more exalted or unusual than what will help them to keep alive.
they are not generous, because money is one of the things they must
have, and at the same time their experience has taught them how hard
it is to get and how easy to lose. they are cowardly, and are always
anticipating danger; unlike that of the young, who are warm-blooded,
their temperament is chilly; old age has paved the way for cowardice;
fear is, in fact, a form of chill. they love life; and all the more
when their last day has come, because the object of all desire is
something we have not got, and also because we desire most strongly
that which we need most urgently. they are too fond of themselves;
this is one form that small-mindedness takes. because of this, they
guide their lives too much by considerations of what is useful and
too little by what is noble-for the useful is what is good for oneself,
and the noble what is good absolutely. they are not shy, but shameless
rather; caring less for what is noble than for what is useful, they
feel contempt for what people may think of them. they lack confidence
in the future; partly through experience-for most things go wrong,
or anyhow turn out worse than one expects; and partly because of their
cowardice. they live by memory rather than by hope; for what is left
to them of life is but little as compared with the long past; and
hope is of the future, memory of the past. this, again, is the cause
of their loquacity; they are continually talking of the past, because
they enjoy remembering it. their fits of anger are sudden but feeble.
their sensual passions have either altogether gone or have lost their
vigour: consequently they do not feel their passions much, and their
actions are inspired less by what they do feel than by the love of
gain. hence men at this time of life are often supposed to have a
self-controlled character; the fact is that their passions have slackened,
and they are slaves to the love of gain. they guide their lives by
reasoning more than by moral feeling; reasoning being directed to
utility and moral feeling to moral goodness. if they wrong others,
they mean to injure them, not to insult them. old men may feel pity,
as well as young men, but not for the same reason. young men feel
it out of kindness; old men out of weakness, imagining that anything
that befalls any one else might easily happen to them, which, as we
saw, is a thought that excites pity. hence they are querulous, and
not disposed to jesting or laughter-the love of laughter being the
such are the characters of young men and elderly men. people always
think well of speeches adapted to, and reflecting, their own character:
and we can now see how to compose our speeches so as to adapt both
as for men in their prime, clearly we shall find that they have a
character between that of the young and that of the old, free from
the extremes of either. they have neither that excess of confidence
which amounts to rashness, nor too much timidity, but the right amount
of each. they neither trust everybody nor distrust everybody, but
judge people correctly. their lives will be guided not by the sole
consideration either of what is noble or of what is useful, but by
both; neither by parsimony nor by prodigality, but by what is fit
and proper. so, too, in regard to anger and desire; they will be brave
as well as temperate, and temperate as well as brave; these virtues
are divided between the young and the old; the young are brave but
intemperate, the old temperate but cowardly. to put it generally,
all the valuable qualities that youth and age divide between them
are united in the prime of life, while all their excesses or defects
are replaced by moderation and fitness. the body is in its prime from
thirty to five-and-thirty; the mind about forty-nine.
so much for the types of character that distinguish youth, old age,
and the prime of life. we will now turn to those gifts of fortune
by which human character is affected. first let us consider good birth.
its effect on character is to make those who have it more ambitious;
it is the way of all men who have something to start with to add to
the pile, and good birth implies ancestral distinction. the well-born
man will look down even on those who are as good as his own ancestors,
because any far-off distinction is greater than the same thing close
to us, and better to boast about. being well-born, which means coming
of a fine stock, must be distinguished from nobility, which means
being true to the family nature-a quality not usually found in the
well-born, most of whom are poor creatures. in the generations of
men as in the fruits of the earth, there is a varying yield; now and
then, where the stock is good, exceptional men are produced for a
while, and then decadence sets in. a clever stock will degenerate
towards the insane type of character, like the descendants of alcibiades
or of the elder dionysius; a steady stock towards the fatuous and
torpid type, like the descendants of cimon, pericles, and socrates.
the type of character produced by wealth lies on the surface for all
to see. wealthy men are insolent and arrogant; their possession of
wealth affects their understanding; they feel as if they had every
good thing that exists; wealth becomes a sort of standard of value
for everything else, and therefore they imagine there is nothing it
cannot buy. they are luxurious and ostentatious; luxurious, because
of the luxury in which they live and the prosperity which they display;
ostentatious and vulgar, because, like other people's, their minds
are regularly occupied with the object of their love and admiration,
and also because they think that other people's idea of happiness
is the same as their own. it is indeed quite natural that they should
be affected thus; for if you have money, there are always plenty of
people who come begging from you. hence the saying of simonides about
wise men and rich men, in answer to hiero's wife, who asked him whether
it was better to grow rich or wise. 'why, rich,' he said; 'for i see
the wise men spending their days at the rich men's doors.' rich men
also consider themselves worthy to hold public office; for they consider
they already have the things that give a claim to office. in a word,
the type of character produced by wealth is that of a prosperous fool.
there is indeed one difference between the type of the newly-enriched
and those who have long been rich: the newly-enriched have all the
bad qualities mentioned in an exaggerated and worse form--to be newly-enriched
means, so to speak, no education in riches. the wrongs they do others
are not meant to injure their victims, but spring from insolence or
self-indulgence, e.g. those that end in assault or in adultery.
as to power: here too it may fairly be said that the type of character
it produces is mostly obvious enough. some elements in this type it
shares with the wealthy type, others are better. those in power are
more ambitious and more manly in character than the wealthy, because
they aspire to do the great deeds that their power permits them to
do. responsibility makes them more serious: they have to keep paying
attention to the duties their position involves. they are dignified
rather than arrogant, for the respect in which they are held inspires
them with dignity and therefore with moderation-dignity being a mild
and becoming form of arrogance. if they wrong others, they wrong them
good fortune in certain of its branches produces the types of character
belonging to the conditions just described, since these conditions
are in fact more or less the kinds of good fortune that are regarded
as most important. it may be added that good fortune leads us to gain
all we can in the way of family happiness and bodily advantages. it
does indeed make men more supercilious and more reckless; but there
is one excellent quality that goes with it-piety, and respect for
the divine power, in which they believe because of events which are
this account of the types of character that correspond to differences
of age or fortune may end here; for to arrive at the opposite types
to those described, namely, those of the poor, the unfortunate, and
the powerless, we have only to ask what the opposite qualities are.
the use of persuasive speech is to lead to decisions. (when we know
a thing, and have decided about it, there is no further use in speaking
about it.) this is so even if one is addressing a single person and
urging him to do or not to do something, as when we scold a man for
his conduct or try to change his views: the single person is as much
your 'judge' as if he were one of many; we may say, without qualification,
that any one is your judge whom you have to persuade. nor does it
matter whether we are arguing against an actual opponent or against
a mere proposition; in the latter case we still have to use speech
and overthrow the opposing arguments, and we attack these as we should
attack an actual opponent. our principle holds good of ceremonial
speeches also; the 'onlookers' for whom such a speech is put together
are treated as the judges of it. broadly speaking, however, the only
sort of person who can strictly be called a judge is the man who decides
the issue in some matter of public controversy; that is, in law suits
and in political debates, in both of which there are issues to be
decided. in the section on political oratory an account has already
been given of the types of character that mark the different constitutions.
the manner and means of investing speeches with moral character may
each of the main divisions of oratory has, we have seen, its own distinct
purpose. with regard to each division, we have noted the accepted
views and propositions upon which we may base our arguments-for political,
for ceremonial, and for forensic speaking. we have further determined
completely by what means speeches may be invested with the required
moral character. we are now to proceed to discuss the arguments common
to all oratory. all orators, besides their special lines of argument,
are bound to use, for instance, the topic of the possible and impossible;
and to try to show that a thing has happened, or will happen in future.
again, the topic of size is common to all oratory; all of us have
to argue that things are bigger or smaller than they seem, whether
we are making political speeches, speeches of eulogy or attack, or
prosecuting or defending in the law-courts. having analysed these
subjects, we will try to say what we can about the general principles
of arguing by 'enthymeme' and 'example', by the addition of which
we may hope to complete the project with which we set out. of the
above-mentioned general lines of argument, that concerned with amplification
is-as has been already said-most appropriate to ceremonial speeches;
that concerned with the past, to forensic speeches, where the required
decision is always about the past; that concerned with possibility
let us first speak of the possible and impossible. it may plausibly
be argued: that if it is possible for one of a pair of contraries
to be or happen, then it is possible for the other: e.g. if a man
can be cured, he can also fall ill; for any two contraries are equally
possible, in so far as they are contraries. that if of two similar
things one is possible, so is the other. that if the harder of two
things is possible, so is the easier. that if a thing can come into
existence in a good and beautiful form, then it can come into existence
generally; thus a house can exist more easily than a beautiful house.
that if the beginning of a thing can occur, so can the end; for nothing
impossible occurs or begins to occur; thus the commensurability of
the diagonal of a square with its side neither occurs nor can begin
to occur. that if the end is possible, so is the beginning; for all
things that occur have a beginning. that if that which is posterior
in essence or in order of generation can come into being, so can that
which is prior: thus if a man can come into being, so can a boy, since
the boy comes first in order of generation; and if a boy can, so can
a man, for the man also is first. that those things are possible of
which the love or desire is natural; for no one, as a rule, loves
or desires impossibilities. that things which are the object of any
kind of science or art are possible and exist or come into existence.
that anything is possible the first step in whose production depends
on men or things which we can compel or persuade to produce it, by
our greater strength, our control of them, or our friendship with
them. that where the parts are possible, the whole is possible; and
where the whole is possible, the parts are usually possible. for if
the slit in front, the toe-piece, and the upper leather can be made,
then shoes can be made; and if shoes, then also the front slit and
toe-piece. that if a whole genus is a thing that can occur, so can
the species; and if the species can occur, so can the genus: thus,
if a sailing vessel can be made, so also can a trireme; and if a trireme,
then a sailing vessel also. that if one of two things whose existence
depends on each other is possible, so is the other; for instance,
if 'double', then 'half', and if 'half', then 'double'. that if a
thing can be produced without art or preparation, it can be produced
still more certainly by the careful application of art to it. hence
"to some things we by art must needs attain,
that if anything is possible to inferior, weaker, and stupider people,
it is more so for their opposites; thus isocrates said that it would
be a strange thing if he could not discover a thing that euthynus
had found out. as for impossibility, we can clearly get what we want
by taking the contraries of the arguments stated above.
questions of past fact may be looked at in the following ways: first,
that if the less likely of two things has occurred, the more likely
must have occurred also. that if one thing that usually follows another
has happened, then that other thing has happened; that, for instance,
if a man has forgotten a thing, he has also once learnt it. that if
a man had the power and the wish to do a thing, he has done it; for
every one does do whatever he intends to do whenever he can do it,
there being nothing to stop him. that, further, he has done the thing
in question either if he intended it and nothing external prevented
him; or if he had the power to do it and was angry at the time; or
if he had the power to do it and his heart was set upon it-for people
as a rule do what they long to do, if they can; bad people through
lack of self-control; good people, because their hearts are set upon
good things. again, that if a thing was 'going to happen', it has
happened; if a man was 'going to do something', he has done it, for
it is likely that the intention was carried out. that if one thing
has happened which naturally happens before another or with a view
to it, the other has happened; for instance, if it has lightened,
it has also thundered; and if an action has been attempted, it has
been done. that if one thing has happened which naturally happens
after another, or with a view to which that other happens, then that
other (that which happens first, or happens with a view to this thing)
has also happened; thus, if it has thundered it has lightened, and
if an action has been done it has been attempted. of all these sequences
some are inevitable and some merely usual. the arguments for the non-occurrence
of anything can obviously be found by considering the opposites of
how questions of future fact should be argued is clear from the same
considerations: that a thing will be done if there is both the power
and the wish to do it; or if along with the power to do it there is
a craving for the result, or anger, or calculation, prompting it.
that the thing will be done, in these cases, if the man is actually
setting about it, or even if he means to do it later-for usually what
we mean to do happens rather than what we do not mean to do. that
a thing will happen if another thing which naturally happens before
it has already happened; thus, if it is clouding over, it is likely
to rain. that if the means to an end have occurred, then the end is
likely to occur; thus, if there is a foundation, there will be a house.
for arguments about the greatness and smallness of things, the greater
and the lesser, and generally great things and small, what we have
already said will show the line to take. in discussing deliberative
oratory we have spoken about the relative greatness of various goods,
and about the greater and lesser in general. since therefore in each
type oratory the object under discussion is some kind of good-whether
it is utility, nobleness, or justice-it is clear that every orator
must obtain the materials of amplification through these channels.
to go further than this, and try to establish abstract laws of greatness
and superiority, is to argue without an object; in practical life,
particular facts count more than generalizations.
enough has now been said about these questions of possibility and
the reverse, of past or future fact, and of the relative greatness
the special forms of oratorical argument having now been discussed,
we have next to treat of those which are common to all kinds of oratory.
these are of two main kinds, 'example' and 'enthymeme'; for the 'maxim'
we will first treat of argument by example, for it has the nature
of induction, which is the foundation of reasoning. this form of argument
has two varieties; one consisting in the mention of actual past facts,
the other in the invention of facts by the speaker. of the latter,
again, there are two varieties, the illustrative parallel and the
fable (e.g. the fables of aesop, those from libya). as an instance
of the mention of actual facts, take the following. the speaker may
argue thus: 'we must prepare for war against the king of persia and
not let him subdue egypt. for darius of old did not cross the aegean
until he had seized egypt; but once he had seized it, he did cross.
and xerxes, again, did not attack us until he had seized egypt; but
once he had seized it, he did cross. if therefore the present king
seizes egypt, he also will cross, and therefore we must not let him.'
the illustrative parallel is the sort of argument socrates used: e.g.
'public officials ought not to be selected by lot. that is like using
the lot to select athletes, instead of choosing those who are fit
for the contest; or using the lot to select a steersman from among
a ship's crew, as if we ought to take the man on whom the lot falls,
and not the man who knows most about it.'
instances of the fable are that of stesichorus about phalaris, and
that of aesop in defence of the popular leader. when the people of
himera had made phalaris military dictator, and were going to give
him a bodyguard, stesichorus wound up a long talk by telling them
the fable of the horse who had a field all to himself. presently there
came a stag and began to spoil his pasturage. the horse, wishing to
revenge himself on the stag, asked a man if he could help him to do
so. the man said, 'yes, if you will let me bridle you and get on to
your back with javelins in my hand'. the horse agreed, and the man
mounted; but instead of getting his revenge on the stag, the horse
found himself the slave of the man. 'you too', said stesichorus, 'take
care lest your desire for revenge on your enemies, you meet the same
fate as the horse. by making phalaris military dictator, you have
already let yourselves be bridled. if you let him get on to your backs
by giving him a bodyguard, from that moment you will be his slaves.'
aesop, defending before the assembly at samos a poular leader who
was being tried for his life, told this story: a fox, in crossing
a river, was swept into a hole in the rocks; and, not being able to
get out, suffered miseries for a long time through the swarms of fleas
that fastened on her. a hedgehog, while roaming around, noticed the
fox; and feeling sorry for her asked if he might remove the fleas.
but the fox declined the offer; and when the hedgehog asked why, she
replied, 'these fleas are by this time full of me and not sucking
much blood; if you take them away, others will come with fresh appetites
and drink up all the blood i have left.' 'so, men of samos', said
aesop, 'my client will do you no further harm; he is wealthy already.
but if you put him to death, others will come along who are not rich,
and their peculations will empty your treasury completely.'
fables are suitable for addresses to popular assemblies; and they
have one advantage-they are comparatively easy to invent, whereas
it is hard to find parallels among actual past events. you will in
fact frame them just as you frame illustrative parallels: all you
require is the power of thinking out your analogy, a power developed
by intellectual training. but while it is easier to supply parallels
by inventing fables, it is more valuable for the political speaker
to supply them by quoting what has actually happened, since in most
respects the future will be like what the past has been.
where we are unable to argue by enthymeme, we must try to demonstrate
our point by this method of example, and to convince our hearers thereby.
if we can argue by enthymeme, we should use our examples as subsequent
supplementary evidence. they should not precede the enthymemes: that
will give the argument an inductive air, which only rarely suits the
conditions of speech-making. if they follow the enthymemes, they have
the effect of witnesses giving evidence, and this alway tells. for
the same reason, if you put your examples first you must give a large
number of them; if you put them last, a single one is sufficient;
even a single witness will serve if he is a good one. it has now been
stated how many varieties of argument by example there are, and how
we now turn to the use of maxims, in order to see upon what subjects
and occasions, and for what kind of speaker, they will appropriately
form part of a speech. this will appear most clearly when we have
defined a maxim. it is a statement; not a particular fact, such as
the character of lphicrates, but of a general kind; nor is it about
any and every subject--e.g. 'straight is the contrary of curved' is
not a maxim--but only about questions of practical conduct, courses
of conduct to be chosen or avoided. now an enthymeme is a syllogism
dealing with such practical subjects. it is therefore roughly true
that the premisses or conclusions of enthymemes, considered apart
from the rest of the argument, are maxims: e.g.
"never should any man whose wits are sound
"have his sons taught more wisdom than their fellows. "
here we have a maxim; add the reason or explanation, and the whole
"it makes them idle; and therewith they earn
"ill-will and jealousy throughout the city. "
"there is no man in all things prosperous, "
"there is no man among us all is free, "
are maxims; but the latter, taken with what follows it, is an enthymeme-
"for all are slaves of money or of chance. "
from this definition of a maxim it follows that there are four kinds
of maxims. in the first place, the maxim may or may not have a supplement.
proof is needed where the statement is paradoxical or disputable;
no supplement is wanted where the statement contains nothing paradoxical,
either because the view expressed is already a known truth, e.g.
"chiefest of blessings is health for a man, as it seemeth to me,
this being the general opinion: or because, as soon as the view is
"no love is true save that which loves for ever. "
of the maxims that do have a supplement attached, some are part of
"never should any man whose wits are sound, c. "
others have the essential character of enthymemes, but are not stated
as parts of enthymemes; these latter are reckoned the best; they are
those in which the reason for the view expressed is simply implied,
"o mortal man, nurse not immortal wrath. "
to say 'it is not right to nurse immortal wrath' is a maxim; the added
words 'mortal man' give the reason. similarly, with the words mortal
creatures ought to cherish mortal, not immortal thoughts.
what has been said has shown us how many kinds of maxims there are,
and to what subjects the various kinds are appropriate. they must
not be given without supplement if they express disputed or paradoxical
views: we must, in that case, either put the supplement first and
make a maxim of the conclusion, e.g. you might say, 'for my part,
since both unpopularity and idleness are undesirable, i hold that
it is better not to be educated'; or you may say this first, and then
add the previous clause. where a statement, without being paradoxical,
is not obviously true, the reason should be added as concisely as
possible. in such cases both laconic and enigmatic sayings are suitable:
thus one might say what stesichorus said to the locrians, 'insolence
is better avoided, lest the cicalas chirp on the ground'.
the use of maxims is appropriate only to elderly men, and in handling
subjects in which the speaker is experienced. for a young man to use
them is-like telling stories-unbecoming; to use them in handling things
in which one has no experience is silly and ill-bred: a fact sufficiently
proved by the special fondness of country fellows for striking out
maxims, and their readiness to air them.
to declare a thing to be universally true when it is not is most appropriate
when working up feelings of horror and indignation in our hearers;
especially by way of preface, or after the facts have been proved.
even hackneyed and commonplace maxims are to be used, if they suit
one's purpose: just because they are commonplace, every one seems
to agree with them, and therefore they are taken for truth. thus,
any one who is calling on his men to risk an engagement without obtaining
"one omen of all is hest, that we fight for our fatherland.
or, if he is calling on them to attack a stronger force-
or, if he is urging people to destroy the innocent children of their
"fool, who slayeth the father and leaveth his sons to avenge him.
some proverbs are also maxims, e.g. the proverb 'an attic neighbour'.
you are not to avoid uttering maxims that contradict such sayings
as have become public property (i mean such sayings as 'know thyself'
and 'nothing in excess') if doing so will raise your hearers' opinion
of your character, or convey an effect of strong emotion--e.g. an
angry speaker might well say, 'it is not true that we ought to know
ourselves: anyhow, if this man had known himself, he would never have
thought himself fit for an army command.' it will raise people's opinion
of our character to say, for instance, 'we ought not to follow the
saying that bids us treat our friends as future enemies: much better
to treat our enemies as future friends.' the moral purpose should
be implied partly by the very wording of our maxim. failing this,
we should add our reason: e.g. having said 'we should treat our friends,
not as the saying advises, but as if they were going to be our friends
always', we should add 'for the other behaviour is that of a traitor':
or we might put it, i disapprove of that saying. a true friend will
treat his friend as if he were going to be his friend for ever'; and
again, 'nor do i approve of the saying "nothing in excess": we are
bound to hate bad men excessively.' one great advantage of maxims
to a speaker is due to the want of intelligence in his hearers, who
love to hear him succeed in expressing as a universal truth the opinions
which they hold themselves about particular cases. i will explain
what i mean by this, indicating at the same time how we are to hunt
down the maxims required. the maxim, as has been already said, a general
statement and people love to hear stated in general terms what they
already believe in some particular connexion: e.g. if a man happens
to have bad neighbours or bad children, he will agree with any one
who tells him, 'nothing is more annoying than having neighbours',
or, 'nothing is more foolish than to be the parent of children.' the
orator has therefore to guess the subjects on which his hearers really
hold views already, and what those views are, and then must express,
as general truths, these same views on these same subjects. this is
one advantage of using maxims. there is another which is more important-it
invests a speech with moral character. there is moral character in
every speech in which the moral purpose is conspicuous: and maxims
always produce this effect, because the utterance of them amounts
to a general declaration of moral principles: so that, if the maxims
are sound, they display the speaker as a man of sound moral character.
so much for the maxim-its nature, varieties, proper use, and advantages.
we now come to the enthymemes, and will begin the subject with some
general consideration of the proper way of looking for them, and then
proceed to what is a distinct question, the lines of argument to be
embodied in them. it has already been pointed out that the enthymeme
is a syllogism, and in what sense it is so. we have also noted the
differences between it and the syllogism of dialectic. thus we must
not carry its reasoning too far back, or the length of our argument
will cause obscurity: nor must we put in all the steps that lead to
our conclusion, or we shall waste words in saying what is manifest.
it is this simplicity that makes the uneducated more effective than
the educated when addressing popular audiences-makes them, as the
poets tell us, 'charm the crowd's ears more finely'. educated men
lay down broad general principles; uneducated men argue from common
knowledge and draw obvious conclusions. we must not, therefore, start
from any and every accepted opinion, but only from those we have defined-those
accepted by our judges or by those whose authority they recognize:
and there must, moreover, be no doubt in the minds of most, if not
all, of our judges that the opinions put forward really are of this
sort. we should also base our arguments upon probabilities as well
the first thing we have to remember is this. whether our argument
concerns public affairs or some other subject, we must know some,
if not all, of the facts about the subject on which we are to speak
and argue. otherwise we can have no materials out of which to construct
arguments. i mean, for instance, how could we advise the athenians
whether they should go to war or not, if we did not know their strength,
whether it was naval or military or both, and how great it is; what
their revenues amount to; who their friends and enemies are; what
wars, too, they have waged, and with what success; and so on? or how
could we eulogize them if we knew nothing about the sea-fight at salamis,
or the battle of marathon, or what they did for the heracleidae, or
any other facts like that? all eulogy is based upon the noble deeds--real
or imaginary--that stand to the credit of those eulogized. on the
same principle, invectives are based on facts of the opposite kind:
the orator looks to see what base deeds--real or imaginary--stand
to the discredit of those he is attacking, such as treachery to the
cause of hellenic freedom, or the enslavement of their gallant allies
against the barbarians (aegina, potidaea, c.), or any other misdeeds
of this kind that are recorded against them. so, too, in a court of
law: whether we are prosecuting or defending, we must pay attention
to the existing facts of the case. it makes no difference whether
the subject is the lacedaemonians or the athenians, a man or a god;
we must do the same thing. suppose it to be achilles whom we are to
advise, to praise or blame, to accuse or defend; here too we must
take the facts, real or imaginary; these must be our material, whether
we are to praise or blame him for the noble or base deeds he has done,
to accuse or defend him for his just or unjust treatment of others,
or to advise him about what is or is not to his interest. the same
thing applies to any subject whatever. thus, in handling the question
whether justice is or is not a good, we must start with the real facts
about justice and goodness. we see, then, that this is the only way
in which any one ever proves anything, whether his arguments are strictly
cogent or not: not all facts can form his basis, but only those that
bear on the matter in hand: nor, plainly, can proof be effected otherwise
by means of the speech. consequently, as appears in the topics, we
must first of all have by us a selection of arguments about questions
that may arise and are suitable for us to handle; and then we must
try to think out arguments of the same type for special needs as they
emerge; not vaguely and indefinitely, but by keeping our eyes on the
actual facts of the subject we have to speak on, and gathering in
as many of them as we can that bear closely upon it: for the more
actual facts we have at our command, the more easily we prove our
case; and the more closely they bear on the subject, the more they
will seem to belong to that speech only instead of being commonplaces.
by 'commonplaces' i mean, for example, eulogy of achilles because
he is a human being or a demi-god, or because he joined the expedition
against troy: these things are true of many others, so that this kind
of eulogy applies no better to achilles than to diomede. the special
facts here needed are those that are true of achilles alone; such
facts as that he slew hector, the bravest of the trojans, and cycnus
the invulnerable, who prevented all the greeks from landing, and again
that he was the youngest man who joined the expedition, and was not
here, again, we have our first principle of selection of enthymemes-that
which refers to the lines of argument selected. we will now consider
the various elementary classes of enthymemes. (by an 'elementary class'
of enthymeme i mean the same thing as a 'line of argument'.) we will
begin, as we must begin, by observing that there are two kinds of
enthymemes. one kind proves some affirmative or negative proposition;
the other kind disproves one. the difference between the two kinds
is the same as that between syllogistic proof and disproof in dialectic.
the demonstrative enthymeme is formed by the conjunction of compatible
propositions; the refutative, by the conjunction of incompatible propositions.
we may now be said to have in our hands the lines of argument for
the various special subjects that it is useful or necessary to handle,
having selected the propositions suitable in various cases. we have,
in fact, already ascertained the lines of argument applicable to enthymemes
about good and evil, the noble and the base, justice and injustice,
and also to those about types of character, emotions, and moral qualities.
let us now lay hold of certain facts about the whole subject, considered
from a different and more general point of view. in the course of
our discussion we will take note of the distinction between lines
of proof and lines of disproof: and also of those lines of argument
used in what seems to be enthymemes, but are not, since they do not
represent valid syllogisms. having made all this clear, we will proceed
to classify objections and refutations, showing how they can be brought
1. one line of positive proof is based upon consideration of the opposite
of the thing in question. observe whether that opposite has the opposite
quality. if it has not, you refute the original proposition; if it
has, you establish it. e.g. 'temperance is beneficial; for licentiousness
is hurtful'. or, as in the messenian speech, 'if war is the cause
of our present troubles, peace is what we need to put things right
"anger us if they meant not what they did,
"as were constrained to do the good they did us. "
"since in this world liars may win belief,
"be sure of the opposite likewise-that this world
"hears many a true word and believes it not. "
2. another line of proof is got by considering some modification of
the key-word, and arguing that what can or cannot be said of the one,
can or cannot be said of the other: e.g. 'just' does not always mean
'beneficial', or 'justly' would always mean 'beneficially', whereas
it is not desirable to be justly put to death.
3. another line of proof is based upon correlative ideas. if it is
true that one man noble or just treatment to another, you argue that
the other must have received noble or just treatment; or that where
it is right to command obedience, it must have been right to obey
the command. thus diomedon, the tax-farmer, said of the taxes: 'if
it is no disgrace for you to sell them, it is no disgrace for us to
buy them'. further, if 'well' or 'justly' is true of the person to
whom a thing is done, you argue that it is true of the doer. but it
is possible to draw a false conclusion here. it may be just that a
should be treated in a certain way, and yet not just that he should
be so treated by b. hence you must ask yourself two distinct questions:
(1) is it right that a should be thus treated? (2) is it right that
b should thus treat him? and apply your results properly, according
as your answers are yes or no. sometimes in such a case the two answers
differ: you may quite easily have a position like that in the alcmaeon
"and was there none to loathe thy mother's crime? "
to which question alcmaeon in reply says,
"why, there are two things to examine here. "
and when alphesiboea asks what he means, he rejoins:
"they judged her fit to die, not me to slay her. "
again there is the lawsuit about demosthenes and the men who killed
nicanor; as they were judged to have killed him justly, it was thought
that he was killed justly. and in the case of the man who was killed
at thebes, the judges were requested to decide whether it was unjust
that he should be killed, since if it was not, it was argued that
it could not have been unjust to kill him.
4. another line of proof is the 'a fortiori'. thus it may be argued
that if even the gods are not omniscient, certainly human beings are
not. the principle here is that, if a quality does not in fact exist
where it is more likely to exist, it clearly does not exist where
it is less likely. again, the argument that a man who strikes his
father also strikes his neighbours follows from the principle that,
if the less likely thing is true, the more likely thing is true also;
for a man is less likely to strike his father than to strike his neighbours.
the argument, then, may run thus. or it may be urged that, if a thing
is not true where it is more likely, it is not true where it is less
likely; or that, if it is true where it is less likely, it is true
where it is more likely: according as we have to show that a thing
is or is not true. this argument might also be used in a case of parity,
"thou hast pity for thy sire, who has lost his sons:
"hast none for oeneus, whose brave son is dead? "
and, again, 'if theseus did no wrong, neither did paris'; or 'the
sons of tyndareus did no wrong, neither did paris'; or 'if hector
did well to slay patroclus, paris did well to slay achilles'. and
'if other followers of an art are not bad men, neither are philosophers'.
and 'if generals are not bad men because it often happens that they
are condemned to death, neither are sophists'. and the remark that
'if each individual among you ought to think of his own city's reputation,
you ought all to think of the reputation of greece as a whole'.
5. another line of argument is based on considerations of time. thus
iphicrates, in the case against harmodius, said, 'if before doing
the deed i had bargained that, if i did it, i should have a statue,
you would have given me one. will you not give me one now that i have
done the deed? you must not make promises when you are expecting a
thing to be done for you, and refuse to fulfil them when the thing
has been done.' and, again, to induce the thebans to let philip pass
through their territory into attica, it was argued that 'if he had
insisted on this before he helped them against the phocians, they
would have promised to do it. it is monstrous, therefore, that just
because he threw away his advantage then, and trusted their honour,
they should not let him pass through now'.
6. another line is to apply to the other speaker what he has said
against yourself. it is an excellent turn to give to a debate, as
may be seen in the teucer. it was employed by iphicrates in his reply
to aristophon. 'would you', he asked, 'take a bribe to betray the
fleet?' 'no', said aristophon; and iphicrates replied, 'very good:
if you, who are aristophon, would not betray the fleet, would i, who
am iphicrates?' only, it must be recognized beforehand that the other
man is more likely than you are to commit the crime in question. otherwise
you will make yourself ridiculous; it is aristeides who is prosecuting,
you cannot say that sort of thing to him. the purpose is to discredit
the prosecutor, who as a rule would have it appear that his character
is better than that of the defendant, a pretension which it is desirable
to upset. but the use of such an argument is in all cases ridiculous
if you are attacking others for what you do or would do yourself,
or are urging others to do what you neither do nor would do yourself.
7. another line of proof is secured by defining your terms. thus,
'what is the supernatural? surely it is either a god or the work of
a god. well, any one who believes that the work of a god exists, cannot
help also believing that gods exist.' or take the argument of iphicrates,
'goodness is true nobility; neither harmodius nor aristogeiton had
any nobility before they did a noble deed'. he also argued that he
himself was more akin to harmodius and aristogeiton than his opponent
was. 'at any rate, my deeds are more akin to those of harmodius and
aristogeiton than yours are'. another example may be found in the
alexander. 'every one will agree that by incontinent people we mean
those who are not satisfied with the enjoyment of one love.' a further
example is to be found in the reason given by socrates for not going
to the court of archelaus. he said that 'one is insulted by being
unable to requite benefits, as well as by being unable to requite
injuries'. all the persons mentioned define their term and get at
its essential meaning, and then use the result when reasoning on the
8. another line of argument is founded upon the various senses of
a word. such a word is 'rightly', as has been explained in the topics.
another line is based upon logical division. thus, 'all men do wrong
from one of three motives, a, b, or c: in my case a and b are out
of the question, and even the accusers do not allege c'.
10. another line is based upon induction. thus from the case of the
woman of peparethus it might be argued that women everywhere can settle
correctly the facts about their children. another example of this
occurred at athens in the case between the orator mantias and his
son, when the boy's mother revealed the true facts: and yet another
at thebes, in the case between ismenias and stilbon, when dodonis
proved that it was ismenias who was the father of her son thettaliscus,
and he was in consequence always regarded as being so. a further instance
of induction may be taken from the law of theodectes: 'if we do not
hand over our horses to the care of men who have mishandled other
people's horses, nor ships to those who have wrecked other people's
ships, and if this is true of everything else alike, then men who
have failed to secure other people's safety are not to be employed
to secure our own.' another instance is the argument of alcidamas:
'every one honours the wise'. thus the parians have honoured archilochus,
in spite of his bitter tongue; the chians homer, though he was not
their countryman; the mytilenaeans sappho, though she was a woman;
the lacedaemonians actually made chilon a member of their senate,
though they are the least literary of men; the italian greeks honoured
pythagoras; the inhabitants of lampsacus gave public burial to anaxagoras,
though he was an alien, and honour him even to this day. (it may be
argued that peoples for whom philosophers legislate are always prosperous)
on the ground that the athenians became prosperous under solon's laws
and the lacedaemonians under those of lycurgus, while at thebes no
sooner did the leading men become philosophers than the country began
11. another line of argument is founded upon some decision already
pronounced, whether on the same subject or on one like it or contrary
to it. such a proof is most effective if every one has always decided
thus; but if not every one, then at any rate most people; or if all,
or most, wise or good men have thus decided, or the actual judges
of the present question, or those whose authority they accept, or
any one whose decision they cannot gainsay because he has complete
control over them, or those whom it is not seemly to gainsay, as the
gods, or one's father, or one's teachers. thus autocles said, when
attacking mixidemides, that it was a strange thing that the dread
goddesses could without loss of dignity submit to the judgement of
the areopagus, and yet mixidemides could not. or as sappho said, 'death
is an evil thing; the gods have so judged it, or they would die'.
or again as aristippus said in reply to plato when he spoke somewhat
too dogmatically, as aristippus thought: 'well, anyhow, our friend',
meaning socrates, 'never spoke like that'. and hegesippus, having
previously consulted zeus at olympia, asked apollo at delphi 'whether
his opinion was the same as his father's', implying that it would
be shameful for him to contradict his father. thus too isocrates argued
that helen must have been a good woman, because theseus decided that
she was; and paris a good man, because the goddesses chose him before
all others; and evagoras also, says isocrates, was good, since when
conon met with his misfortune he betook himself to evagoras without
12. another line of argument consists in taking separately the parts
of a subject. such is that given in the topics: 'what sort of motion
is the soul? for it must be this or that.' the socrates of theodectes
provides an example: 'what temple has he profaned? what gods recognized
13. since it happens that any given thing usually has both good and
bad consequences, another line of argument consists in using those
consequences as a reason for urging that a thing should or should
not be done, for prosecuting or defending any one, for eulogy or censure.
e.g. education leads both to unpopularity, which is bad, and to wisdom,
which is good. hence you either argue, 'it is therefore not well to
be educated, since it is not well to be unpopular': or you answer,
'no, it is well to be educated, since it is well to be wise'. the
art of rhetoric of callippus is made up of this line of argument,
with the addition of those of possibility and the others of that kind
14. another line of argument is used when we have to urge or discourage
a course of action that may be done in either of two opposite ways,
and have to apply the method just mentioned to both. the difference
between this one and the last is that, whereas in the last any two
things are contrasted, here the things contrasted are opposites. for
instance, the priestess enjoined upon her son not to take to public
speaking: 'for', she said, 'if you say what is right, men will hate
you; if you say what is wrong, the gods will hate you.' the reply
might be, 'on the contrary, you ought to take to public speaking:
for if you say what is right the gods will love you; if you say what
is wrong, men will love you.' this amounts to the proverbial 'buying
the marsh with the salt'. it is just this situation, viz. when each
of two opposites has both a good and a bad consequence opposite respectively
to each other, that has been termed divarication.
15. another line of argument is this: the things people approve of
openly are not those which they approve of secretly: openly, their
chief praise is given to justice and nobleness; but in their hearts
they prefer their own advantage. try, in face of this, to establish
the point of view which your opponent has not adopted. this is the
most effective of the forms of argument that contradict common opinion.
16. another line is that of rational correspondence. e.g. iphicrates,
when they were trying to compel his son, a youth under the prescribed
age, to perform one of the state duties because he was tall, said
'if you count tall boys men, you will next be voting short men boys'.
and theodectes in his law said, 'you make citizens of such mercenaries
as strabax and charidemus, as a reward of their merits; will you not
make exiles of such citizens as those who have done irreparable harm
17. another line is the argument that if two results are the same
their antecedents are also the same. for instance, it was a saying
of xenophanes that to assert that the gods had birth is as impious
as to say that they die; the consequence of both statements is that
there is a time when the gods do not exist. this line of proof assumes
generally that the result of any given thing is always the same: e.g.
'you are going to decide not about isocrates, but about the value
of the whole profession of philosophy.' or, 'to give earth and water'
means slavery; or, 'to share in the common peace' means obeying orders.
we are to make either such assumptions or their opposite, as suits
18. another line of argument is based on the fact that men do not
always make the same choice on a later as on an earlier occasion,
but reverse their previous choice. e.g. the following enthymeme: 'when
we were exiles, we fought in order to return; now we have returned,
it would be strange to choose exile in order not to have to fight.'
one occasion, that is, they chose to be true to their homes at the
cost of fighting, and on the other to avoid fighting at the cost of
19. another line of argument is the assertion that some possible motive
for an event or state of things is the real one: e.g. that a gift
was given in order to cause pain by its withdrawal. this notion underlies
"not of good god towards them, but to make
or take the passage from the meleager of antiphon:
or the argument in the ajax of theodectes, that diomede chose out
odysseus not to do him honour, but in order that his companion might
be a lesser man than himself-such a motive for doing so is quite possible.
20. another line of argument is common to forensic and deliberative
oratory, namely, to consider inducements and deterrents, and the motives
people have for doing or avoiding the actions in question. these are
the conditions which make us bound to act if they are for us, and
to refrain from action if they are against us: that is, we are bound
to act if the action is possible, easy, and useful to ourselves or
our friends or hurtful to our enemies; this is true even if the action
entails loss, provided the loss is outweighed by the solid advantage.
a speaker will urge action by pointing to such conditions, and discourage
it by pointing to the opposite. these same arguments also form the
materials for accusation or defence-the deterrents being pointed out
by the defence, and the inducements by the prosecution. as for the
defence,...this topic forms the whole art of rhetoric both of pamphilus
21. another line of argument refers to things which are supposed to
happen and yet seem incredible. we may argue that people could not
have believed them, if they had not been true or nearly true: even
that they are the more likely to be true because they are incredible.
for the things which men believe are either facts or probabilities:
if, therefore, a thing that is believed is improbable and even incredible,
it must be true, since it is certainly not believed because it is
at all probable or credible. an example is what androcles of the deme
pitthus said in his well-known arraignment of the law. the audience
tried to shout him down when he observed that the laws required a
law to set them right. 'why', he went on, 'fish need salt, improbable
and incredible as this might seem for creatures reared in salt water;
and olive-cakes need oil, incredible as it is that what produces oil
22. another line of argument is to refute our opponent's case by noting
any contrasts or contradictions of dates, acts, or words that it anywhere
displays; and this in any of the three following connexions. (1) referring
to our opponent's conduct, e.g. 'he says he is devoted to you, yet
he conspired with the thirty.' (2) referring to our own conduct, e.g.
'he says i am litigious, and yet he cannot prove that i have been
engaged in a single lawsuit.' (3) referring to both of us together,
e.g. 'he has never even lent any one a penny, but i have ransomed
23. another line that is useful for men and causes that have been
really or seemingly slandered, is to show why the facts are not as
supposed; pointing out that there is a reason for the false impression
given. thus a woman, who had palmed off her son on another woman,
was thought to be the lad's mistress because she embraced him; but
when her action was explained the charge was shown to be groundless.
another example is from the ajax of theodectes, where odysseus tells
ajax the reason why, though he is really braver than ajax, he is not
24. another line of argument is to show that if the cause is present,
the effect is present, and if absent, absent. for by proving the cause
you at once prove the effect, and conversely nothing can exist without
its cause. thus thrasybulus accused leodamas of having had his name
recorded as a criminal on the slab in the acropolis, and of erasing
the record in the time of the thirty tyrants: to which leodamas replied,
'impossible: for the thirty would have trusted me all the more if
my quarrel with the commons had been inscribed on the slab.'
25. another line is to consider whether the accused person can take
or could have taken a better course than that which he is recommending
or taking, or has taken. if he has not taken this better course, it
is clear that he is not guilty, since no one deliberately and consciously
chooses what is bad. this argument is, however, fallacious, for it
often becomes clear after the event how the action could have been
done better, though before the event this was far from clear.
26. another line is, when a contemplated action is inconsistent with
any past action, to examine them both together. thus, when the people
of elea asked xenophanes if they should or should not sacrifice to
leucothea and mourn for her, he advised them not to mourn for her
if they thought her a goddess, and not to sacrifice to her if they
27. another line is to make previous mistakes the grounds of accusation
or defence. thus, in the medea of carcinus the accusers allege that
medea has slain her children; 'at all events', they say, 'they are
not to be seen'-medea having made the mistake of sending her children
away. in defence she argues that it is not her children, but jason,
whom she would have slain; for it would have been a mistake on her
part not to do this if she had done the other. this special line of
argument for enthymeme forms the whole of the art of rhetoric in use
"another line is to draw meanings from names. sophocles, for instance,
"o steel in heart as thou art steel in name. "
this line of argument is common in praises of the gods. thus, too,
conon called thrasybulus rash in counsel. and herodicus said of thrasymachus,
'you are always bold in battle'; of polus, 'you are always a colt';
and of the legislator draco that his laws were those not of a human
being but of a dragon, so savage were they. and, in euripides, hecuba
"her name and folly's (aphrosuns) lightly begin alike, "
"pentheus-a name foreshadowing grief (penthos) to come. "
the refutative enthymeme has a greater reputation than the demonstrative,
because within a small space it works out two opposing arguments,
and arguments put side by side are clearer to the audience. but of
all syllogisms, whether refutative or demonstrative, those are most
applauded of which we foresee the conclusions from the beginning,
so long as they are not obvious at first sight-for part of the pleasure
we feel is at our own intelligent anticipation; or those which we
follow well enough to see the point of them as soon as the last word
besides genuine syllogisms, there may be syllogisms that look genuine
but are not; and since an enthymeme is merely a syllogism of a particular
kind, it follows that, besides genuine enthymemes, there may be those
1. among the lines of argument that form the spurious enthymeme the
first is that which arises from the particular words employed.
(a) one variety of this is when-as in dialectic, without having gone
through any reasoning process, we make a final statement as if it
were the conclusion of such a process, 'therefore so-and-so is not
true', 'therefore also so-and-so must be true'-so too in rhetoric
a compact and antithetical utterance passes for an enthymeme, such
language being the proper province of enthymeme, so that it is seemingly
the form of wording here that causes the illusion mentioned. in order
to produce the effect of genuine reasoning by our form of wording
it is useful to summarize the results of a number of previous reasonings:
as 'some he saved-others he avenged-the greeks he freed'. each of
these statements has been previously proved from other facts; but
the mere collocation of them gives the impression of establishing
(b) another variety is based on the use of similar words for different
things; e.g. the argument that the mouse must be a noble creature,
since it gives its name to the most august of all religious rites-for
such the mysteries are. or one may introduce, into a eulogy of the
dog, the dog-star; or pan, because pindar said:
or we may argue that, because there is much disgrace in there not
being a dog about, there is honour in being a dog. or that hermes
is readier than any other god to go shares, since we never say 'shares
all round' except of him. or that speech is a very excellent thing,
since good men are not said to be worth money but to be worthy of
esteem-the phrase 'worthy of esteem' also having the meaning of 'worth
2. another line is to assert of the whole what is true of the parts,
or of the parts what is true of the whole. a whole and its parts are
supposed to be identical, though often they are not. you have therefore
to adopt whichever of these two lines better suits your purpose. that
is how euthydemus argues: e.g. that any one knows that there is a
trireme in the peiraeus, since he knows the separate details that
make up this statement. there is also the argument that one who knows
the letters knows the whole word, since the word is the same thing
as the letters which compose it; or that, if a double portion of a
certain thing is harmful to health, then a single portion must not
be called wholesome, since it is absurd that two good things should
make one bad thing. put thus, the enthymeme is refutative; put as
follows; demonstrative: 'for one good thing cannot be made up of two
bad things.' the whole line of argument is fallacious. again, there
is polycrates' saying that thrasybulus put down thirty tyrants, where
the speaker adds them up one by one. or the argument in the orestes
of theodectes, where the argument is from part to whole:
"'tis right that she who slays her lord should die. "
'it is right, too, that the son should avenge his father. very good:
these two things are what orestes has done.' still, perhaps the two
things, once they are put together, do not form a right act. the fallacy
might also be said to be due to omission, since the speaker fails
to say by whose hand a husband-slayer should die.
3. another line is the use of indignant language, whether to support
your own case or to overthrow your opponent's. we do this when we
paint a highly-coloured picture of the situation without having proved
the facts of it: if the defendant does so, he produces an impression
of his innocence; and if the prosecutor goes into a passion, he produces
an impression of the defendant's guilt. here there is no genuine enthymeme:
the hearer infers guilt or innocence, but no proof is given, and the
4. another line is to use a 'sign', or single instance, as certain
evidence; which, again, yields no valid proof. thus, it might be said
that lovers are useful to their countries, since the love of harmodius
and aristogeiton caused the downfall of the tyrant hipparchus. or,
again, that dionysius is a thief, since he is a vicious man-there
is, of course, no valid proof here; not every vicious man is a thief,
5. another line represents the accidental as essential. an instance
is what polycrates says of the mice, that they 'came to the rescue'
because they gnawed through the bowstrings. or it might be maintained
that an invitation to dinner is a great honour, for it was because
he was not invited that achilles was 'angered' with the greeks at
tenedos? as a fact, what angered him was the insult involved; it was
a mere accident that this was the particular form that the insult
6. another is the argument from consequence. in the alexander, for
instance, it is argued that paris must have had a lofty disposition,
since he despised society and lived by himself on mount ida: because
lofty people do this kind of thing, therefore paris too, we are to
suppose, had a lofty soul. or, if a man dresses fashionably and roams
around at night, he is a rake, since that is the way rakes behave.
another similar argument points out that beggars sing and dance in
temples, and that exiles can live wherever they please, and that such
privileges are at the disposal of those we account happy and therefore
every one might be regarded as happy if only he has those privileges.
what matters, however, is the circumstances under which the privileges
are enjoyed. hence this line too falls under the head of fallacies
7. another line consists in representing as causes things which are
not causes, on the ground that they happened along with or before
the event in question. they assume that, because b happens after a,
it happens because of a. politicians are especially fond of taking
this line. thus demades said that the policy of demosthenes was the
cause of all the mischief, 'for after it the war occurred'.
8. another line consists in leaving out any mention of time and circumstances.
e.g. the argument that paris was justified in taking helen, since
her father left her free to choose: here the freedom was presumably
not perpetual; it could only refer to her first choice, beyond which
her father's authority could not go. or again, one might say that
to strike a free man is an act of wanton outrage; but it is not so
in every case-only when it is unprovoked.
9. again, a spurious syllogism may, as in 'eristical' discussions,
be based on the confusion of the absolute with that which is not absolute
but particular. as, in dialectic, for instance, it may be argued that
what-is-not is, on the ground that what-is-not is what-is-not: or
that the unknown can be known, on the ground that it can be known
to he unknown: so also in rhetoric a spurious enthymeme may be based
on the confusion of some particular probability with absolute probability.
now no particular probability is universally probable: as agathon
"one might perchance say that was probable-
"that things improbable oft will hap to men. "
for what is improbable does happen, and therefore it is probable
that improbable things will happen. granted this, one might argue
that 'what is improbable is probable'. but this is not true absolutely.
as, in eristic, the imposture comes from not adding any clause specifying
relationship or reference or manner; so here it arises because the
probability in question is not general but specific. it is of this
line of argument that corax's art of rhetoric is composed. if the
accused is not open to the charge-for instance if a weakling be tried
for violent assault-the defence is that he was not likely to do such
a thing. but if he is open to the charge-i.e. if he is a strong man-the
defence is still that he was not likely to do such a thing, since
he could be sure that people would think he was likely to do it. and
so with any other charge: the accused must be either open or not open
to it: there is in either case an appearance of probable innocence,
but whereas in the latter case the probability is genuine, in the
former it can only be asserted in the special sense mentioned. this
sort of argument illustrates what is meant by making the worse argument
seem the better. hence people were right in objecting to the training
protagoras undertook to give them. it was a fraud; the probability
it handled was not genuine but spurious, and has a place in no art
enthymemes, genuine and apparent, have now been described; the next
an argument may be refuted either by a counter-syllogism or by bringing
an objection. it is clear that counter-syllogisms can be built up
from the same lines of arguments as the original syllogisms: for the
materials of syllogisms are the ordinary opinions of men, and such
opinions often contradict each other. objections, as appears in the
topics, may be raised in four ways-either by directly attacking your
opponent's own statement, or by putting forward another statement
like it, or by putting forward a statement contrary to it, or by quoting
1. by 'attacking your opponent's own statement' i mean, for instance,
this: if his enthymeme should assert that love is always good, the
objection can be brought in two ways, either by making the general
statement that 'all want is an evil', or by making the particular
one that there would be no talk of 'caunian love' if there were not
2. an objection 'from a contrary statement' is raised when, for instance,
the opponent's enthymeme having concluded that a good man does good
to all his friends, you object, 'that proves nothing, for a bad man
3. an example of an objection 'from a like statement' is, the enthymeme
having shown that ill-used men always hate their ill-users, to reply,
'that proves nothing, for well-used men do not always love those who
4. the 'decisions' mentioned are those proceeding from well-known
men; for instance, if the enthymeme employed has concluded that 'that
allowance ought to be made for drunken offenders, since they did not
know what they were doing', the objection will be, 'pittacus, then,
deserves no approval, or he would not have prescribed specially severe
penalties for offences due to drunkenness'.
enthymemes are based upon one or other of four kinds of alleged fact:
(1) probabilities, (2) examples, (3) infallible signs, (4) ordinary
signs. (1) enthymemes based upon probabilities are those which argue
from what is, or is supposed to be, usually true. (2) enthymemes based
upon example are those which proceed by induction from one or more
similar cases, arrive at a general proposition, and then argue deductively
to a particular inference. (3) enthymemes based upon infallible signs
are those which argue from the inevitable and invariable. (4) enthymemes
based upon ordinary signs are those which argue from some universal
or particular proposition, true or false.
now (1) as a probability is that which happens usually but not always,
enthymemes founded upon probabilities can, it is clear, always be
refuted by raising some objection. the refutation is not always genuine:
it may be spurious: for it consists in showing not that your opponent's
premiss is not probable, but only in showing that it is not inevitably
true. hence it is always in defence rather than in accusation that
it is possible to gain an advantage by using this fallacy. for the
accuser uses probabilities to prove his case: and to refute a conclusion
as improbable is not the same thing as to refute it as not inevitable.
any argument based upon what usually happens is always open to objection:
otherwise it would not be a probability but an invariable and necessary
truth. but the judges think, if the refutation takes this form, either
that the accuser's case is not probable or that they must not decide
it; which, as we said, is a false piece of reasoning. for they ought
to decide by considering not merely what must be true but also what
is likely to be true: this is, indeed, the meaning of 'giving a verdict
in accordance with one's honest opinion'. therefore it is not enough
for the defendant to refute the accusation by proving that the charge
is not hound to be true: he must do so by showing that it is not likely
to be true. for this purpose his objection must state what is more
usually true than the statement attacked. it may do so in either of
two ways: either in respect of frequency or in respect of exactness.
it will be most convincing if it does so in both respects; for if
the thing in question both happens oftener as we represent it and
happens more as we represent it, the probability is particularly great.
(2) fallible signs, and enthymemes based upon them, can be refuted
even if the facts are correct, as was said at the outset. for we have
shown in the analytics that no fallible sign can form part of a valid
(3) enthymemes depending on examples may be refuted in the same way
as probabilities. if we have a negative instance, the argument is
refuted, in so far as it is proved not inevitable, even though the
positive examples are more similar and more frequent. and if the positive
examples are more numerous and more frequent, we must contend that
the present case is dissimilar, or that its conditions are dissimilar,
or that it is different in some way or other.
(4) it will be impossible to refute infallible signs, and enthymemes
resting on them, by showing in any way that they do not form a valid
logical proof: this, too, we see from the analytics. all we can do
is to show that the fact alleged does not exist. if there is no doubt
that it does, and that it is an infallible sign, refutation now becomes
impossible: for this is equivalent to a demonstration which is clear
amplification and depreciation are not an element of enthymeme. by
'an element of enthymeme' i mean the same thing as a line of enthymematic
argument-a general class embracing a large number of particular kinds
of enthymeme. amplification and depreciation are one kind of enthymeme,
viz. the kind used to show that a thing is great or small; just as
there are other kinds used to show that a thing is good or bad, just
or unjust, and anything else of the sort. all these things are the
subject-matter of syllogisms and enthymemes; none of these is the
line of argument of an enthymeme; no more, therefore, are amplification
and depreciation. nor are refutative enthymemes a different species
from constructive. for it is clear that refutation consists either
in offering positive proof or in raising an objection. in the first
case we prove the opposite of our adversary's statements. thus, if
he shows that a thing has happened, we show that it has not; if he
shows that it has not happened, we show that it has. this, then, could
not be the distinction if there were one, since the same means are
employed by both parties, enthymemes being adduced to show that the
fact is or is not so-and-so. an objection, on the other hand, is not
an enthymeme at all, as was said in the topics, consists in stating
some accepted opinion from which it will be clear that our opponent
has not reasoned correctly or has made a false assumption.
three points must be studied in making a speech; and we have now completed
the account of (1) examples, maxims, enthymemes, and in general the
thought-element the way to invent and refute arguments. we have next
to discuss (2) style, and (3) arrangement.
in making a speech one must study three points: first, the means
of producing persuasion; second, the style, or language, to be used;
third, the proper arrangement of the various parts of the speech.
we have already specified the sources of persuasion. we have shown
that these are three in number; what they are; and why there are only
these three: for we have shown that persuasion must in every case
be effected either (1) by working on the emotions of the judges themselves,
(2) by giving them the right impression of the speakers' character,
or (3) by proving the truth of the statements made.
enthymemes also have been described, and the sources from which they
should be derived; there being both special and general lines of argument
our next subject will be the style of expression. for it is not enough
to know what we ought to say; we must also say it as we ought; much
help is thus afforded towards producing the right impression of a
speech. the first question to receive attention was naturally the
one that comes first naturally-how persuasion can be produced from
the facts themselves. the second is how to set these facts out in
language. a third would be the proper method of delivery; this is
a thing that affects the success of a speech greatly; but hitherto
the subject has been neglected. indeed, it was long before it found
a way into the arts of tragic drama and epic recitation: at first
poets acted their tragedies themselves. it is plain that delivery
has just as much to do with oratory as with poetry. (in connexion
with poetry, it has been studied by glaucon of teos among others.)
it is, essentially, a matter of the right management of the voice
to express the various emotions-of speaking loudly, softly, or between
the two; of high, low, or intermediate pitch; of the various rhythms
that suit various subjects. these are the three things-volume of sound,
modulation of pitch, and rhythm-that a speaker bears in mind. it is
those who do bear them in mind who usually win prizes in the dramatic
contests; and just as in drama the actors now count for more than
the poets, so it is in the contests of public life, owing to the defects
of our political institutions. no systematic treatise upon the rules
of delivery has yet been composed; indeed, even the study of language
made no progress till late in the day. besides, delivery is-very properly-not
regarded as an elevated subject of inquiry. still, the whole business
of rhetoric being concerned with appearances, we must pay attention
to the subject of delivery, unworthy though it is, because we cannot
do without it. the right thing in speaking really is that we should
be satisfied not to annoy our hearers, without trying to delight them:
we ought in fairness to fight our case with no help beyond the bare
facts: nothing, therefore, should matter except the proof of those
facts. still, as has been already said, other things affect the result
considerably, owing to the defects of our hearers. the arts of language
cannot help having a small but real importance, whatever it is we
have to expound to others: the way in which a thing is said does affect
its intelligibility. not, however, so much importance as people think.
all such arts are fanciful and meant to charm the hearer. nobody uses
when the principles of delivery have been worked out, they will produce
the same effect as on the stage. but only very slight attempts to
deal with them have been made and by a few people, as by thrasymachus
in his 'appeals to pity'. dramatic ability is a natural gift, and
can hardly be systematically taught. the principles of good diction
can be so taught, and therefore we have men of ability in this direction
too, who win prizes in their turn, as well as those speakers who excel
in delivery-speeches of the written or literary kind owe more of their
effect to their direction than to their thought.
it was naturally the poets who first set the movement going; for words
represent things, and they had also the human voice at their disposal,
which of all our organs can best represent other things. thus the
arts of recitation and acting were formed, and others as well. now
it was because poets seemed to win fame through their fine language
when their thoughts were simple enough, that the language of oratorical
prose at first took a poetical colour, e.g. that of gorgias. even
now most uneducated people think that poetical language makes the
finest discourses. that is not true: the language of prose is distinct
from that of poetry. this is shown by the state of things to-day,
when even the language of tragedy has altered its character. just
as iambics were adopted, instead of tetrameters, because they are
the most prose-like of all metres, so tragedy has given up all those
words, not used in ordinary talk, which decorated the early drama
and are still used by the writers of hexameter poems. it is therefore
ridiculous to imitate a poetical manner which the poets themselves
have dropped; and it is now plain that we have not to treat in detail
the whole question of style, but may confine ourselves to that part
of it which concerns our present subject, rhetoric. the other--the
poetical--part of it has been discussed in the treatise on the art
we may, then, start from the observations there made, including the
definition of style. style to be good must be clear, as is proved
by the fact that speech which fails to convey a plain meaning will
fail to do just what speech has to do. it must also be appropriate,
avoiding both meanness and undue elevation; poetical language is certainly
free from meanness, but it is not appropriate to prose. clearness
is secured by using the words (nouns and verbs alike) that are current
and ordinary. freedom from meanness, and positive adornment too, are
secured by using the other words mentioned in the art of poetry. such
variation from what is usual makes the language appear more stately.
people do not feel towards strangers as they do towards their own
countrymen, and the same thing is true of their feeling for language.
it is therefore well to give to everyday speech an unfamiliar air:
people like what strikes them, and are struck by what is out of the
way. in verse such effects are common, and there they are fitting:
the persons and things there spoken of are comparatively remote from
ordinary life. in prose passages they are far less often fitting because
the subject-matter is less exalted. even in poetry, it is not quite
appropriate that fine language should be used by a slave or a very
young man, or about very trivial subjects: even in poetry the style,
to be appropriate, must sometimes be toned down, though at other times
heightened. we can now see that a writer must disguise his art and
give the impression of speaking naturally and not artificially. naturalness
is persuasive, artificiality is the contrary; for our hearers are
prejudiced and think we have some design against them, as if we were
mixing their wines for them. it is like the difference between the
quality of theodorus' voice and the voices of all other actors: his
really seems to be that of the character who is speaking, theirs do
not. we can hide our purpose successfully by taking the single words
of our composition from the speech of ordinary life. this is done
in poetry by euripides, who was the first to show the way to his successors.
language is composed of nouns and verbs. nouns are of the various
kinds considered in the treatise on poetry. strange words, compound
words, and invented words must be used sparingly and on few occasions:
on what occasions we shall state later. the reason for this restriction
has been already indicated: they depart from what is suitable, in
the direction of excess. in the language of prose, besides the regular
and proper terms for things, metaphorical terms only can be used with
advantage. this we gather from the fact that these two classes of
terms, the proper or regular and the metaphorical-these and no others-are
used by everybody in conversation. we can now see that a good writer
can produce a style that is distinguished without being obtrusive,
and is at the same time clear, thus satisfying our definition of good
oratorical prose. words of ambiguous meaning are chiefly useful to
enable the sophist to mislead his hearers. synonyms are useful to
the poet, by which i mean words whose ordinary meaning is the same,
e.g. 'porheueseai' (advancing) and 'badizein' (proceeding); these
two are ordinary words and have the same meaning.
in the art of poetry, as we have already said, will be found definitions
of these kinds of words; a classification of metaphors; and mention
of the fact that metaphor is of great value both in poetry and in
prose. prose-writers must, however, pay specially careful attention
to metaphor, because their other resources are scantier than those
of poets. metaphor, moreover, gives style clearness, charm, and distinction
as nothing else can: and it is not a thing whose use can be taught
by one man to another. metaphors, like epithets, must be fitting,
which means that they must fairly correspond to the thing signified:
failing this, their inappropriateness will be conspicuous: the want
of harmony between two things is emphasized by their being placed
side by side. it is like having to ask ourselves what dress will suit
an old man; certainly not the crimson cloak that suits a young man.
and if you wish to pay a compliment, you must take your metaphor from
something better in the same line; if to disparage, from something
worse. to illustrate my meaning: since opposites are in the same class,
you do what i have suggested if you say that a man who begs 'prays',
and a man who prays 'begs'; for praying and begging are both varieties
of asking. so iphicrates called callias a 'mendicant priest' instead
of a 'torch-bearer', and callias replied that iphicrates must be uninitiated
or he would have called him not a 'mendicant priest' but a 'torch-bearer'.
both are religious titles, but one is honourable and the other is
not. again, somebody calls actors 'hangers-on of dionysus', but they
call themselves 'artists': each of these terms is a metaphor, the
one intended to throw dirt at the actor, the other to dignify him.
and pirates now call themselves 'purveyors'. we can thus call a crime
a mistake, or a mistake a crime. we can say that a thief 'took' a
thing, or that he 'plundered' his victim. an expression like that
"king of the oar, on mysia's coast he landed, "
is inappropriate; the word 'king' goes beyond the dignity of the
subject, and so the art is not concealed. a metaphor may be amiss
because the very syllables of the words conveying it fail to indicate
sweetness of vocal utterance. thus dionysius the brazen in his elegies
calls poetry 'calliope's screech'. poetry and screeching are both,
to be sure, vocal utterances. but the metaphor is bad, because the
sounds of 'screeching', unlike those of poetry, are discordant and
unmeaning. further, in using metaphors to give names to nameless things,
we must draw them not from remote but from kindred and similar things,
so that the kinship is clearly perceived as soon as the words are
"i marked how a man glued bronze with fire to another man's body,
the process is nameless; but both it and gluing are a kind of application,
and that is why the application of the cupping-glass is here called
a 'gluing'. good riddles do, in general, provide us with satisfactory
metaphors: for metaphors imply riddles, and therefore a good riddle
can furnish a good metaphor. further, the materials of metaphors must
be beautiful; and the beauty, like the ugliness, of all words may,
as licymnius says, lie in their sound or in their meaning. further,
there is a third consideration-one that upsets the fallacious argument
of the sophist bryson, that there is no such thing as foul language,
because in whatever words you put a given thing your meaning is the
same. this is untrue. one term may describe a thing more truly than
another, may be more like it, and set it more intimately before our
eyes. besides, two different words will represent a thing in two different
lights; so on this ground also one term must be held fairer or fouler
than another. for both of two terms will indicate what is fair, or
what is foul, but not simply their fairness or their foulness, or
if so, at any rate not in an equal degree. the materials of metaphor
must be beautiful to the ear, to the understanding, to the eye or
some other physical sense. it is better, for instance, to say 'rosy-fingered
morn', than 'crimson-fingered' or, worse still, 'red-fingered morn'.
the epithets that we apply, too, may have a bad and ugly aspect, as
when orestes is called a 'mother-slayer'; or a better one, as when
he is called his 'father's avenger'. simonides, when the victor in
the mule-race offered him a small fee, refused to write him an ode,
because, he said, it was so unpleasant to write odes to half-asses:
but on receiving an adequate fee, he wrote
"hail to you, daughters of storm-footed steeds? "
though of course they were daughters of asses too. the same effect
is attained by the use of diminutives, which make a bad thing less
bad and a good thing less good. take, for instance, the banter of
aristophanes in the babylonians where he uses 'goldlet' for 'gold',
'cloaklet' for 'cloak', 'scoffiet' for 'scoff, and 'plaguelet'. but
alike in using epithets and in using diminutives we must be wary and
bad taste in language may take any of four forms:
(1) the misuse of compound words. lycophron, for instance, talks of
the 'many visaged heaven' above the 'giant-crested earth', and again
the 'strait-pathed shore'; and gorgias of the 'pauper-poet flatterer'
and 'oath-breaking and over-oath-keeping'. alcidamas uses such expressions
as 'the soul filling with rage and face becoming flame-flushed', and
'he thought their enthusiasm would be issue-fraught' and 'issue-fraught
he made the persuasion of his words', and 'sombre-hued is the floor
of the sea'.the way all these words are compounded makes them, we
feel, fit for verse only. this, then, is one form in which bad taste
(2) another is the employment of strange words. for instance, lycophron
talks of 'the prodigious xerxes' and 'spoliative sciron'; alcidamas
of 'a toy for poetry' and 'the witlessness of nature', and says 'whetted
with the unmitigated temper of his spirit'.
(3) a third form is the use of long, unseasonable, or frequent epithets.
it is appropriate enough for a poet to talk of 'white milk', in prose
such epithets are sometimes lacking in appropriateness or, when spread
too thickly, plainly reveal the author turning his prose into poetry.
of course we must use some epithets, since they lift our style above
the usual level and give it an air of distinction. but we must aim
at the due mean, or the result will be worse than if we took no trouble
at all; we shall get something actually bad instead of something merely
not good. that is why the epithets of alcidamas seem so tasteless;
he does not use them as the seasoning of the meat, but as the meat
itself, so numerous and swollen and aggressive are they. for instance,
he does not say 'sweat', but 'the moist sweat'; not 'to the isthmian
games', but 'to the world-concourse of the isthmian games'; not 'laws',
but 'the laws that are monarchs of states'; not 'at a run', but 'his
heart impelling him to speed of foot'; not 'a school of the muses',
but 'nature's school of the muses had he inherited'; and so 'frowning
care of heart', and 'achiever' not of 'popularity' but of 'universal
popularity', and 'dispenser of pleasure to his audience', and 'he
concealed it' not 'with boughs' but 'with boughs of the forest trees',
and 'he clothed' not 'his body' but 'his body's nakedness', and 'his
soul's desire was counter imitative' (this's at one and the same time
a compound and an epithet, so that it seems a poet's effort), and
'so extravagant the excess of his wickedness'. we thus see how the
inappropriateness of such poetical language imports absurdity and
tastelessness into speeches, as well as the obscurity that comes from
all this verbosity-for when the sense is plain, you only obscure and
the ordinary use of compound words is where there is no term for a
thing and some compound can be easily formed, like 'pastime' (chronotribein);
but if this is much done, the prose character disappears entirely.
we now see why the language of compounds is just the thing for writers
of dithyrambs, who love sonorous noises; strange words for writers
of epic poetry, which is a proud and stately affair; and metaphor
for iambic verse, the metre which (as has been already' said) is widely
(4) there remains the fourth region in which bad taste may be shown,
metaphor. metaphors like other things may be inappropriate. some are
so because they are ridiculous; they are indeed used by comic as well
as tragic poets. others are too grand and theatrical; and these, if
they are far-fetched, may also be obscure. for instance, gorgias talks
of 'events that are green and full of sap', and says 'foul was the
deed you sowed and evil the harvest you reaped'. that is too much
like poetry. alcidamas, again, called philosophy 'a fortress that
threatens the power of law', and the odyssey 'a goodly looking-glass
of human life',' talked about 'offering no such toy to poetry': all
these expressions fail, for the reasons given, to carry the hearer
with them. the address of gorgias to the swallow, when she had let
her droppings fall on him as she flew overhead, is in the best tragic
manner. he said, 'nay, shame, o philomela'. considering her as a bird,
you could not call her act shameful; considering her as a girl, you
could; and so it was a good gibe to address her as what she was once
the simile also is a metaphor; the difference is but slight. when
this is a simile; when he says of him 'the lion leapt', it is a metaphor-here,
since both are courageous, he has transferred to achilles the name
of 'lion'. similes are useful in prose as well as in verse; but not
often, since they are of the nature of poetry. they are to be employed
just as metaphors are employed, since they are really the same thing
the following are examples of similes. androtion said of idrieus that
he was like a terrier let off the chain, that flies at you and bites
you-idrieus too was savage now that he was let out of his chains.
theodamas compared archidamus to an euxenus who could not do geometry-a
proportional simile, implying that euxenus is an archidamus who can
do geometry. in plato's republic those who strip the dead are compared
to curs which bite the stones thrown at them but do not touch the
thrower, and there is the simile about the athenian people, who are
compared to a ship's captain who is strong but a little deaf; and
the one about poets' verses, which are likened to persons who lack
beauty but possess youthful freshness-when the freshness has faded
the charm perishes, and so with verses when broken up into prose.
pericles compared the samians to children who take their pap but go
on crying; and the boeotians to holm-oaks, because they were ruining
one another by civil wars just as one oak causes another oak's fall.
demosthenes said that the athenian people were like sea-sick men on
board ship. again, demosthenes compared the political orators to nurses
who swallow the bit of food themselves and then smear the children's
lips with the spittle. antisthenes compared the lean cephisodotus
to frankincense, because it was his consumption that gave one pleasure.
all these ideas may be expressed either as similes or as metaphors;
those which succeed as metaphors will obviously do well also as similes,
and similes, with the explanation omitted, will appear as metaphors.
but the proportional metaphor must always apply reciprocally to either
of its co-ordinate terms. for instance, if a drinking-bowl is the
shield of dionysus, a shield may fittingly be called the drinking-bowl
such, then, are the ingredients of which speech is composed. the foundation
of good style is correctness of language, which falls under five heads.
(1) first, the proper use of connecting words, and the arrangement
of them in the natural sequence which some of them require. for instance,
the connective 'men' (e.g. ego men) requires the correlative de (e.g.
o de). the answering word must be brought in before the first has
been forgotten, and not be widely separated from it; nor, except in
the few cases where this is appropriate, is another connective to
be introduced before the one required. consider the sentence, 'but
as soon as he told me (for cleon had come begging and praying), took
them along and set out.' in this sentence many connecting words are
inserted in front of the one required to complete the sense; and if
there is a long interval before 'set out', the result is obscurity.
one merit, then, of good style lies in the right use of connecting
words. (2) the second lies in calling things by their own special
names and not by vague general ones. (3) the third is to avoid ambiguities;
unless, indeed, you definitely desire to be ambiguous, as those do
who have nothing to say but are pretending to mean something. such
people are apt to put that sort of thing into verse. empedocles, for
instance, by his long circumlocutions imposes on his hearers; these
are affected in the same way as most people are when they listen to
diviners, whose ambiguous utterances are received with nods of acquiescence-
"croesus by crossing the halys will ruin a mighty realm.
diviners use these vague generalities about the matter in hand because
their predictions are thus, as a rule, less likely to be falsified.
we are more likely to be right, in the game of 'odd and even', if
we simply guess 'even' or 'odd' than if we guess at the actual number;
and the oracle-monger is more likely to be right if he simply says
that a thing will happen than if he says when it will happen, and
therefore he refuses to add a definite date. all these ambiguities
have the same sort of effect, and are to be avoided unless we have
some such object as that mentioned. (4) a fourth rule is to observe
protagoras' classification of nouns into male, female, and inanimate;
for these distinctions also must be correctly given. 'upon her arrival
she said her say and departed (e d elthousa kai dialechtheisa ocheto).'
(5) a fifth rule is to express plurality, fewness, and unity by the
correct wording, e.g. 'having come, they struck me (oi d elthontes
it is a general rule that a written composition should be easy to
read and therefore easy to deliver. this cannot be so where there
are many connecting words or clauses, or where punctuation is hard,
as in the writings of heracleitus. to punctuate heracleitus is no
easy task, because we often cannot tell whether a particular word
belongs to what precedes or what follows it. thus, at the outset of
his treatise he says, 'though this truth is always men understand
it not', where it is not clear with which of the two clauses the word
'always' should be joined by the punctuation. further, the following
fact leads to solecism, viz. that the sentence does not work out properly
if you annex to two terms a third which does not suit them both. thus
either 'sound' or 'colour' will fail to work out properly with some
verbs: 'perceive' will apply to both, 'see' will not. obscurity is
also caused if, when you intend to insert a number of details, you
do not first make your meaning clear; for instance, if you say, 'i
meant, after telling him this, that and the other thing, to set out',
rather than something of this kind 'i meant to set out after telling
him; then this, that, and the other thing occurred.'
the following suggestions will help to give your language impressiveness.
(1) describe a thing instead of naming it: do not say 'circle', but
'that surface which extends equally from the middle every way'. to
achieve conciseness, do the opposite-put the name instead of the description.
when mentioning anything ugly or unseemly, use its name if it is the
description that is ugly, and describe it if it is the name that is
ugly. (2) represent things with the help of metaphors and epithets,
being careful to avoid poetical effects. (3) use plural for singular,
"here are my letter's many-leaved folds. "
(4) do not bracket two words under one article, but put one article
with each; e.g. 'that wife of ours.' the reverse to secure conciseness;
e.g. 'our wife.' use plenty of connecting words; conversely, to secure
conciseness, dispense with connectives, while still preserving connexion;
e.g. 'having gone and spoken', and 'having gone, i spoke', respectively.
(6) and the practice of antimachus, too, is useful-to describe a thing
by mentioning attributes it does not possess; as he does in talking
"there is a little wind-swept knoll... "
a subject can be developed indefinitely along these lines. you may
apply this method of treatment by negation either to good or to bad
qualities, according to which your subject requires. it is from this
source that the poets draw expressions such as the 'stringless' or
'lyreless' melody, thus forming epithets out of negations. this device
is popular in proportional metaphors, as when the trumpet's note is
your language will be appropriate if it expresses emotion and character,
and if it corresponds to its subject. 'correspondence to subject'
means that we must neither speak casually about weighty matters, nor
solemnly about trivial ones; nor must we add ornamental epithets to
commonplace nouns, or the effect will be comic, as in the works of
cleophon, who can use phrases as absurd as 'o queenly fig-tree'. to
express emotion, you will employ the language of anger in speaking
of outrage; the language of disgust and discreet reluctance to utter
a word when speaking of impiety or foulness; the language of exultation
for a tale of glory, and that of humiliation for a tale of and so
this aptness of language is one thing that makes people believe in
the truth of your story: their minds draw the false conclusion that
you are to be trusted from the fact that others behave as you do when
things are as you describe them; and therefore they take your story
to be true, whether it is so or not. besides, an emotional speaker
always makes his audience feel with him, even when there is nothing
in his arguments; which is why many speakers try to overwhelm their
furthermore, this way of proving your story by displaying these signs
of its genuineness expresses your personal character. each class of
men, each type of disposition, will have its own appropriate way of
letting the truth appear. under 'class' i include differences of age,
as boy, man, or old man; of sex, as man or woman; of nationality,
as spartan or thessalian. by 'dispositions' i here mean those dispositions
only which determine the character of a man's for it is not every
disposition that does this. if, then, a speaker uses the very words
which are in keeping with a particular disposition, he will reproduce
the corresponding character; for a rustic and an educated man will
not say the same things nor speak in the same way. again, some impression
is made upon an audience by a device which speech-writers employ to
nauseous excess, when they say 'who does not know this?' or 'it is
known to everybody.' the hearer is ashamed of his ignorance, and agrees
with the speaker, so as to have a share of the knowledge that everybody
all the variations of oratorical style are capable of being used in
season or out of season. the best way to counteract any exaggeration
is the well-worn device by which the speaker puts in some criticism
of himself; for then people feel it must be all right for him to talk
thus, since he certainly knows what he is doing. further, it is better
not to have everything always just corresponding to everything else-your
hearers will see through you less easily thus. i mean for instance,
if your words are harsh, you should not extend this harshness to your
voice and your countenance and have everything else in keeping. if
you do, the artificial character of each detail becomes apparent;
whereas if you adopt one device and not another, you are using art
all the same and yet nobody notices it. (to be sure, if mild sentiments
are expressed in harsh tones and harsh sentiments in mild tones, you
become comparatively unconvincing.) compound words, fairly plentiful
epithets, and strange words best suit an emotional speech. we forgive
an angry man for talking about a wrong as 'heaven-high' or 'colossal';
and we excuse such language when the speaker has his hearers already
in his hands and has stirred them deeply either by praise or blame
or anger or affection, as isocrates, for instance, does at the end
of his panegyric, with his 'name and fame' and 'in that they brooked'.
men do speak in this strain when they are deeply stirred, and so,
once the audience is in a like state of feeling, approval of course
follows. this is why such language is fitting in poetry, which is
an inspired thing. this language, then, should be used either under
stress of emotion, or ironically, after the manner of gorgias and
the form of a prose composition should be neither metrical nor destitute
of rhythm. the metrical form destroys the hearer's trust by its artificial
appearance, and at the same time it diverts his attention, making
him watch for metrical recurrences, just as children catch up the
herald's question, 'whom does the freedman choose as his advocate?',
with the answer 'cleon!' on the other hand, unrhythmical language
is too unlimited; we do not want the limitations of metre, but some
limitation we must have, or the effect will be vague and unsatisfactory.
now it is number that limits all things; and it is the numerical limitation
of the forms of a composition that constitutes rhythm, of which metres
are definite sections. prose, then, is to be rhythmical, but not metrical,
or it will become not prose but verse. it should not even have too
precise a prose rhythm, and therefore should only be rhythmical to
of the various rhythms, the heroic has dignity, but lacks the tones
of the spoken language. the iambic is the very language of ordinary
people, so that in common talk iambic lines occur oftener than any
others: but in a speech we need dignity and the power of taking the
hearer out of his ordinary self. the trochee is too much akin to wild
dancing: we can see this in tetrameter verse, which is one of the
there remains the paean, which speakers began to use in the time of
thrasymachus, though they had then no name to give it. the paean is
a third class of rhythm, closely akin to both the two already mentioned;
it has in it the ratio of three to two, whereas the other two kinds
have the ratio of one to one, and two to one respectively. between
the two last ratios comes the ratio of one-and-a-half to one, which
now the other two kinds of rhythm must be rejected in writing prose,
partly for the reasons given, and partly because they are too metrical;
and the paean must be adopted, since from this alone of the rhythms
mentioned no definite metre arises, and therefore it is the least
obtrusive of them. at present the same form of paean is employed at
the beginning a at the end of sentences, whereas the end should differ
from the beginning. there are two opposite kinds of paean, one of
which is suitable to the beginning of a sentence, where it is indeed
actually used; this is the kind that begins with a long syllable and
the other paean begins, conversely, with three short syllables and
"meta de lan udata t ok eanon e oanise nux. "
this kind of paean makes a real close: a short syllable can give
no effect of finality, and therefore makes the rhythm appear truncated.
a sentence should break off with the long syllable: the fact that
it is over should be indicated not by the scribe, or by his period-mark
in the margin, but by the rhythm itself.
we have now seen that our language must be rhythmical and not destitute
of rhythm, and what rhythms, in what particular shape, make it so.
the language of prose must be either free-running, with its parts
united by nothing except the connecting words, like the preludes in
dithyrambs; or compact and antithetical, like the strophes of the
old poets. the free-running style is the ancient one, e.g. 'herein
is set forth the inquiry of herodotus the thurian.' every one used
this method formerly; not many do so now. by 'free-running' style
i mean the kind that has no natural stopping-places, and comes to
a stop only because there is no more to say of that subject. this
style is unsatisfying just because it goes on indefinitely-one always
likes to sight a stopping-place in front of one: it is only at the
goal that men in a race faint and collapse; while they see the end
of the course before them, they can keep on going. such, then, is
the free-running kind of style; the compact is that which is in periods.
by a period i mean a portion of speech that has in itself a beginning
and an end, being at the same time not too big to be taken in at a
glance. language of this kind is satisfying and easy to follow. it
is satisfying, because it is just the reverse of indefinite; and moreover,
the hearer always feels that he is grasping something and has reached
some definite conclusion; whereas it is unsatisfactory to see nothing
in front of you and get nowhere. it is easy to follow, because it
can easily be remembered; and this because language when in periodic
form can be numbered, and number is the easiest of all things to remember.
that is why verse, which is measured, is always more easily remembered
than prose, which is not: the measures of verse can be numbered. the
period must, further, not be completed until the sense is complete:
it must not be capable of breaking off abruptly, as may happen with
the following iambic lines of sophocles-
"calydon's soil is this; of pelops' land
"(the smiling plains face us across the strait.) "
by a wrong division of the words the hearer may take the meaning to
be the reverse of what it is: for instance, in the passage quoted,
one might imagine that calydon is in the peloponnesus.
a period may be either divided into several members or simple. the
period of several members is a portion of speech (1) complete in itself,
(2) divided into parts, and (3) easily delivered at a single breath-as
a whole, that is; not by fresh breath being taken at the division.
a member is one of the two parts of such a period. by a 'simple' period,
i mean that which has only one member. the members, and the whole
periods, should be neither curt nor long. a member which is too short
often makes the listener stumble; he is still expecting the rhythm
to go on to the limit his mind has fixed for it; and if meanwhile
he is pulled back by the speaker's stopping, the shock is bound to
make him, so to speak, stumble. if, on the other hand, you go on too
long, you make him feel left behind, just as people who when walking
pass beyond the boundary before turning back leave their companions
behind so too if a period is too long you turn it into a speech, or
something like a dithyrambic prelude. the result is much like the
preludes that democritus of chios jeered at melanippides for writing
"he that sets traps for another man's feet
"and long-winded preludes do harm to us all,
which applies likewise to long-membered orators. periods whose members
are altogether too short are not periods at all; and the result is
the periodic style which is divided into members is of two kinds.
it is either simply divided, as in 'i have often wondered at the conveners
of national gatherings and the founders of athletic contests'; or
it is antithetical, where, in each of the two members, one of one
pair of opposites is put along with one of another pair, or the same
word is used to bracket two opposites, as 'they aided both parties-not
only those who stayed behind but those who accompanied them: for the
latter they acquired new territory larger than that at home, and to
the former they left territory at home that was large enough'. here
the contrasted words are 'staying behind' and 'accompanying', 'enough'
and 'larger'. so in the example, 'both to those who want to get property
and to those who desire to enjoy it' where 'enjoyment' is contrasted
with 'getting'. again, 'it often happens in such enterprises that
the wise men fail and the fools succeed'; 'they were awarded the prize
of valour immediately, and won the command of the sea not long afterwards';
'to sail through the mainland and march through the sea, by bridging
the hellespont and cutting through athos'; 'nature gave them their
country and law took it away again'; 'of them perished in misery,
others were saved in disgrace'; 'athenian citizens keep foreigners
in their houses as servants, while the city of athens allows her allies
by thousands to live as the foreigner's slaves'; and 'to possess in
life or to bequeath at death'. there is also what some one said about
peitholaus and lycophron in a law-court, 'these men used to sell you
when they were at home, and now they have come to you here and bought
you'. all these passages have the structure described above. such
a form of speech is satisfying, because the significance of contrasted
ideas is easily felt, especially when they are thus put side by side,
and also because it has the effect of a logical argument; it is by
putting two opposing conclusions side by side that you prove one of
such, then, is the nature of antithesis. parisosis is making the two
members of a period equal in length. paromoeosis is making the extreme
words of both members like each other. this must happen either at
the beginning or at the end of each member. if at the beginning, the
resemblance must always be between whole words; at the end, between
final syllables or inflexions of the same word or the same word repeated.
"dorhetoi t epelonto pararretoi t epeessin "
"en pleiotals de opontisi kai en elachistais elpisin "
an example of inflexions of the same word is
"axios de staoenai chalkous ouk axios on chalkou; "
"su d' auton kai zonta eleges kakos kai nun grafeis kakos.
"ti d' an epaoes deinon, ei andrh' eides arhgon; "
it is possible for the same sentence to have all these features together-antithesis,
parison, and homoeoteleuton. (the possible beginnings of periods have
been pretty fully enumerated in the theodectea.) there are also spurious
"there one time i as their guest did stay,
"and they were my hosts on another day. "
we may now consider the above points settled, and pass on to say something
about the way to devise lively and taking sayings. their actual invention
can only come through natural talent or long practice; but this treatise
may indicate the way it is done. we may deal with them by enumerating
the different kinds of them. we will begin by remarking that we all
naturally find it agreeable to get hold of new ideas easily: words
express ideas, and therefore those words are the most agreeable that
enable us to get hold of new ideas. now strange words simply puzzle
us; ordinary words convey only what we know already; it is from metaphor
that we can best get hold of something fresh. when the poet calls
'old age a withered stalk', he conveys a new idea, a new fact, to
us by means of the general notion of bloom, which is common to both
things. the similes of the poets do the same, and therefore, if they
are good similes, give an effect of brilliance. the simile, as has
been said before, is a metaphor, differing from it only in the way
it is put; and just because it is longer it is less attractive. besides,
it does not say outright that 'this' is 'that', and therefore the
hearer is less interested in the idea. we see, then, that both speech
and reasoning are lively in proportion as they make us seize a new
idea promptly. for this reason people are not much taken either by
obvious arguments (using the word 'obvious' to mean what is plain
to everybody and needs no investigation), nor by those which puzzle
us when we hear them stated, but only by those which convey their
information to us as soon as we hear them, provided we had not the
information already; or which the mind only just fails to keep up
with. these two kinds do convey to us a sort of information: but the
obvious and the obscure kinds convey nothing, either at once or later
on. it is these qualities, then, that, so far as the meaning of what
is said is concerned, make an argument acceptable. so far as the style
is concerned, it is the antithetical form that appeals to us, e.g.
'judging that the peace common to all the rest was a war upon their
own private interests', where there is an antithesis between war and
peace. it is also good to use metaphorical words; but the metaphors
must not be far-fetched, or they will be difficult to grasp, nor obvious,
or they will have no effect. the words, too, ought to set the scene
before our eyes; for events ought to be seen in progress rather than
in prospect. so we must aim at these three points: antithesis, metaphor,
of the four kinds of metaphor the most taking is the proportional
kind. thus pericles, for instance, said that the vanishing from their
country of the young men who had fallen in the war was 'as if the
spring were taken out of the year'. leptines, speaking of the lacedaemonians,
said that he would not have the athenians let greece 'lose one of
her two eyes'. when chares was pressing for leave to be examined upon
his share in the olynthiac war, cephisodotus was indignant, saying
that he wanted his examination to take place 'while he had his fingers
upon the people's throat'. the same speaker once urged the athenians
to march to euboea, 'with miltiades' decree as their rations'. iphicrates,
indignant at the truce made by the athenians with epidaurus and the
neighbouring sea-board, said that they had stripped themselves of
their travelling money for the journey of war. peitholaus called the
state-galley 'the people's big stick', and sestos 'the corn-bin of
the peiraeus'. pericles bade his countrymen remove aegina, 'that eyesore
of the peiraeus.' and moerocles said he was no more a rascal than
was a certain respectable citizen he named, 'whose rascality was worth
over thirty per cent per annum to him, instead of a mere ten like
his own'.there is also the iambic line of anaxandrides about the way
"my daughters' marriage-bonds are overdue. "
polyeuctus said of a paralytic man named speusippus that he could
not keep quiet, 'though fortune had fastened him in the pillory of
disease'. cephisodotus called warships 'painted millstones'. diogenes
the dog called taverns 'the mess-rooms of attica'. aesion said that
the athenians had 'emptied' their town into sicily: this is a graphic
metaphor. 'till all hellas shouted aloud' may be regarded as a metaphor,
and a graphic one again. cephisodotus bade the athenians take care
not to hold too many 'parades'. isocrates used the same word of those
who 'parade at the national festivals.' another example occurs in
the funeral speech: 'it is fitting that greece should cut off her
hair beside the tomb of those who fell at salamis, since her freedom
and their valour are buried in the same grave.' even if the speaker
here had only said that it was right to weep when valour was being
buried in their grave, it would have been a metaphor, and a graphic
one; but the coupling of 'their valour' and 'her freedom' presents
a kind of antithesis as well. 'the course of my words', said iphicrates,
'lies straight through the middle of chares' deeds': this is a proportional
metaphor, and the phrase 'straight through the middle' makes it graphic.
the expression 'to call in one danger to rescue us from another' is
a graphic metaphor. lycoleon said, defending chabrias, 'they did not
respect even that bronze statue of his that intercedes for him yonder'.this
was a metaphor for the moment, though it would not always apply; a
vivid metaphor, however; chabrias is in danger, and his statue intercedes
for him-that lifeless yet living thing which records his services
to his country. 'practising in every way littleness of mind' is metaphorical,
for practising a quality implies increasing it. so is 'god kindled
our reason to be a lamp within our soul', for both reason and light
reveal things. so is 'we are not putting an end to our wars, but only
postponing them', for both literal postponement and the making of
such a peace as this apply to future action. so is such a saying as
'this treaty is a far nobler trophy than those we set up on fields
of battle; they celebrate small gains and single successes; it celebrates
our triumph in the war as a whole'; for both trophy and treaty are
signs of victory. so is 'a country pays a heavy reckoning in being
condemned by the judgement of mankind', for a reckoning is damage
it has already been mentioned that liveliness is got by using the
proportional type of metaphor and being making (ie. making your hearers
see things). we have still to explain what we mean by their 'seeing
things', and what must be done to effect this. by 'making them see
things' i mean using expressions that represent things as in a state
of activity. thus, to say that a good man is 'four-square' is certainly
a metaphor; both the good man and the square are perfect; but the
metaphor does not suggest activity. on the other hand, in the expression
'with his vigour in full bloom' there is a notion of activity; and
so in 'but you must roam as free as a sacred victim'; and in
"thereas up sprang the hellenes to their feet, "
where 'up sprang' gives us activity as well as metaphor, for it at
once suggests swiftness. so with homer's common practice of giving
metaphorical life to lifeless things: all such passages are distinguished
by the effect of activity they convey. thus,
"downward anon to the valley rebounded the boulder remorseless; and
stuck in the earth, still panting to feed on the flesh of the heroes;
"and the point of the spear in its fury drove
in all these examples the things have the effect of being active
because they are made into living beings; shameless behaviour and
fury and so on are all forms of activity. and the poet has attached
these ideas to the things by means of proportional metaphors: as the
stone is to sisyphus, so is the shameless man to his victim. in his
famous similes, too, he treats inanimate things in the same way:
"curving and crested with white, host following
here he represents everything as moving and living; and activity
metaphors must be drawn, as has been said already, from things that
are related to the original thing, and yet not obviously so related-just
as in philosophy also an acute mind will perceive resemblances even
in things far apart. thus archytas said that an arbitrator and an
altar were the same, since the injured fly to both for refuge. or
you might say that an anchor and an overhead hook were the same, since
both are in a way the same, only the one secures things from below
and the other from above. and to speak of states as 'levelled' is
to identify two widely different things, the equality of a physical
surface and the equality of political powers.
liveliness is specially conveyed by metaphor, and by the further power
of surprising the hearer; because the hearer expected something different,
his acquisition of the new idea impresses him all the more. his mind
seems to say, 'yes, to be sure; i never thought of that'. the liveliness
of epigrammatic remarks is due to the meaning not being just what
the words say: as in the saying of stesichorus that 'the cicalas will
chirp to themselves on the ground'. well-constructed riddles are attractive
for the same reason; a new idea is conveyed, and there is metaphorical
expression. so with the 'novelties' of theodorus. in these the thought
is startling, and, as theodorus puts it, does not fit in with the
ideas you already have. they are like the burlesque words that one
finds in the comic writers. the effect is produced even by jokes depending
upon changes of the letters of a word; this too is a surprise. you
find this in verse as well as in prose. the word which comes is not
"onward he came, and his feet were shod with his-chilblains,
where one imagined the word would be 'sandals'. but the point should
be clear the moment the words are uttered. jokes made by altering
the letters of a word consist in meaning, not just what you say, but
something that gives a twist to the word used; e.g. the remark of
theodorus about nicon the harpist thratt' ei su ('you thracian slavey'),
where he pretends to mean thratteis su ('you harpplayer'), and surprises
us when we find he means something else. so you enjoy the point when
you see it, though the remark will fall flat unless you are aware
that nicon is thracian. or again: boulei auton persai. in both these
cases the saying must fit the facts. this is also true of such lively
remarks as the one to the effect that to the athenians their empire
(arche) of the sea was not the beginning (arche) of their troubles,
since they gained by it. or the opposite one of isocrates, that their
empire (arche) was the beginning (arche) of their troubles. either
way, the speaker says something unexpected, the soundness of which
is thereupon recognized. there would be nothing clever is saying 'empire
is empire'. isocrates means more than that, and uses the word with
a new meaning. so too with the former saying, which denies that arche
in one sense was arche in another sense. in all these jokes, whether
a word is used in a second sense or metaphorically, the joke is good
if it fits the facts. for instance, anaschetos (proper name) ouk anaschetos:
where you say that what is so-and-so in one sense is not so-and-so
in another; well, if the man is unpleasant, the joke fits the facts.
"thou must not be a stranger stranger than thou should'st.
do not the words 'thou must not be', c., amount to saying that the
stranger must not always be strange? here again is the use of one
word in different senses. of the same kind also is the much-praised
"deeds that would make death fit for you. "
this amounts to saying 'it is a fit thing to die when you are not
fit to die', or 'it is a fit thing to die when death is not fit for
you', i.e. when death is not the fit return for what you are doing.
the type of language employed-is the same in all these examples; but
the more briefly and antithetically such sayings can be expressed,
the more taking they are, for antithesis impresses the new idea more
firmly and brevity more quickly. they should always have either some
personal application or some merit of expression, if they are to be
true without being commonplace-two requirements not always satisfied
simultaneously. thus 'a man should die having done no wrong' is true
but dull: 'the right man should marry the right woman' is also true
but dull. no, there must be both good qualities together, as in 'it
is fitting to die when you are not fit for death'. the more a saying
has these qualitis, the livelier it appears: if, for instance, its
wording is metaphorical, metaphorical in the right way, antithetical,
and balanced, and at the same time it gives an idea of activity.
successful similes also, as has been said above, are in a sense metaphors,
since they always involve two relations like the proportional metaphor.
thus: a shield, we say, is the 'drinking-bowl of ares', and a bow
is the 'chordless lyre'. this way of putting a metaphor is not 'simple',
as it would be if we called the bow a lyre or the shield a drinking-bowl.
there are 'simple' similes also: we may say that a flute-player is
like a monkey, or that a short-sighted man's eyes are like a lamp-flame
with water dropping on it, since both eyes and flame keep winking.
a simile succeeds best when it is a converted metaphor, for it is
possible to say that a shield is like the drinking-bowl of ares, or
that a ruin is like a house in rags, and to say that niceratus is
like a philoctetes stung by pratys-the simile made by thrasyniachus
when he saw niceratus, who had been beaten by pratys in a recitation
competition, still going about unkempt and unwashed. it is in these
respects that poets fail worst when they fail, and succeed best when
they succeed, i.e. when they give the resemblance pat, as in
"those legs of his curl just like parsley leaves; "
"just like philammon struggling with his punchball. "
these are all similes; and that similes are metaphors has been stated
proverbs, again, are metaphors from one species to another. suppose,
for instance, a man to start some undertaking in hope of gain and
then to lose by it later on, 'here we have once more the man of carpathus
and his hare', says he. for both alike went through the said experience.
it has now been explained fairly completely how liveliness is secured
and why it has the effect it has. successful hyperboles are also metaphors,
e.g. the one about the man with a black eye, 'you would have thought
he was a basket of mulberries'; here the 'black eye' is compared to
a mulberry because of its colour, the exaggeration lying in the quantity
of mulberries suggested. the phrase 'like so-and-so' may introduce
a hyperbole under the form of a simile. thus
"just like philammon struggling with his punchball "
is equivalent to 'you would have thought he was philammon struggling
"those legs of his curl just like parsley leaves "
is equivalent to 'his legs are so curly that you would have thought
they were not legs but parsley leaves'. hyperboles are for young men
to use; they show vehemence of character; and this is why angry people
"not though he gave me as much as the dust
"but her, the daughter of atreus' son, i never will marry,
"nay, not though she were fairer than aphrodite the golden,
(the attic orators are particularly fond of this method of speech.)
consequently it does not suit an elderly speaker.
it should be observed that each kind of rhetoric has its own appropriate
style. the style of written prose is not that of spoken oratory, nor
are those of political and forensic speaking the same. both written
and spoken have to be known. to know the latter is to know how to
speak good greek. to know the former means that you are not obliged,
as otherwise you are, to hold your tongue when you wish to communicate
the written style is the more finished: the spoken better admits of
dramatic delivery-like the kind of oratory that reflects character
and the kind that reflects emotion. hence actors look out for plays
written in the latter style, and poets for actors competent to act
in such plays. yet poets whose plays are meant to be read are read
and circulated: chaeremon, for instance, who is as finished as a professional
speech-writer; and licymnius among the dithyrambic poets. compared
with those of others, the speeches of professional writers sound thin
in actual contests. those of the orators, on the other hand, are good
to hear spoken, but look amateurish enough when they pass into the
hands of a reader. this is just because they are so well suited for
an actual tussle, and therefore contain many dramatic touches, which,
being robbed of all dramatic rendering, fail to do their own proper
work, and consequently look silly. thus strings of unconnected words,
and constant repetitions of words and phrases, are very properly condemned
in written speeches: but not in spoken speeches-speakers use them
freely, for they have a dramatic effect. in this repetition there
must be variety of tone, paving the way, as it were, to dramatic effect;
e.g. 'this is the villain among you who deceived you, who cheated
you, who meant to betray you completely'. this is the sort of thing
that philemon the actor used to do in the old men's madness of anaxandrides
whenever he spoke the words 'rhadamanthus and palamedes', and also
in the prologue to the saints whenever he pronounced the pronoun 'i'.
if one does not deliver such things cleverly, it becomes a case of
'the man who swallowed a poker'. so too with strings of unconnected
words, e.g.'i came to him; i met him; i besought him'. such passages
must be acted, not delivered with the same quality and pitch of voice,
as though they had only one idea in them. they have the further peculiarity
of suggesting that a number of separate statements have been made
in the time usually occupied by one. just as the use of conjunctions
makes many statements into a single one, so the omission of conjunctions
acts in the reverse way and makes a single one into many. it thus
makes everything more important: e.g. 'i came to him; i talked to
him; i entreated him'-what a lot of facts! the hearer thinks-'he paid
no attention to anything i said'. this is the effect which homer seeks
"nireus likewise from syme (three well-fashioned ships did bring),
"nireus, the son of aglaia (and charopus, bright-faced king),
"nireus, the comeliest man (of all that to ilium's strand).
if many things are said about a man, his name must be mentioned many
times; and therefore people think that, if his name is mentioned many
times, many things have been said about him. so that homer, by means
of this illusion, has made a great deal of though he has mentioned
him only in this one passage, and has preserved his memory, though
he nowhere says a word about him afterwards.
now the style of oratory addressed to public assemblies is really
just like scene-painting. the bigger the throng, the more distant
is the point of view: so that, in the one and the other, high finish
in detail is superfluous and seems better away. the forensic style
is more highly finished; still more so is the style of language addressed
to a single judge, with whom there is very little room for rhetorical
artifices, since he can take the whole thing in better, and judge
of what is to the point and what is not; the struggle is less intense
and so the judgement is undisturbed. this is why the same speakers
do not distinguish themselves in all these branches at once; high
finish is wanted least where dramatic delivery is wanted most, and
here the speaker must have a good voice, and above all, a strong one.
it is ceremonial oratory that is most literary, for it is meant to
be read; and next to it forensic oratory.
to analyse style still further, and add that it must be agreeable
or magnificent, is useless; for why should it have these traits any
more than 'restraint', 'liberality', or any other moral excellence?
obviously agreeableness will be produced by the qualities already
mentioned, if our definition of excellence of style has been correct.
for what other reason should style be 'clear', and 'not mean' but
'appropriate'? if it is prolix, it is not clear; nor yet if it is
curt. plainly the middle way suits best. again, style will be made
agreeable by the elements mentioned, namely by a good blending of
ordinary and unusual words, by the rhythm, and by-the persuasiveness
this concludes our discussion of style, both in its general aspects
and in its special applications to the various branches of rhetoric.
a speech has two parts. you must state your case, and you must prove
it. you cannot either state your case and omit to prove it, or prove
it without having first stated it; since any proof must be a proof
of something, and the only use of a preliminary statement is the proof
that follows it. of these two parts the first part is called the statement
of the case, the second part the argument, just as we distinguish
between enunciation and demonstration. the current division is absurd.
for 'narration' surely is part of a forensic speech only: how in a
political speech or a speech of display can there be 'narration' in
the technical sense? or a reply to a forensic opponent? or an epilogue
in closely-reasoned speeches? again, introduction, comparison of conflicting
arguments, and recapitulation are only found in political speeches
when there is a struggle between two policies. they may occur then;
so may even accusation and defence, often enough; but they form no
essential part of a political speech. even forensic speeches do not
always need epilogues; not, for instance, a short speech, nor one
in which the facts are easy to remember, the effect of an epilogue
being always a reduction in the apparent length. it follows, then,
that the only necessary parts of a speech are the statement and the
argument. these are the essential features of a speech; and it cannot
in any case have more than introduction, statement, argument, and
epilogue. 'refutation of the opponent' is part of the arguments: so
is 'comparison' of the opponent's case with your own, for that process
is a magnifying of your own case and therefore a part of the arguments,
since one who does this proves something. the introduction does nothing
like this; nor does the epilogue-it merely reminds us of what has
been said already. if we make such distinctions we shall end, like
theodorus and his followers, by distinguishing 'narration' proper
from 'post-narration' and 'pre-narration', and 'refutation' from 'final
refutation'. but we ought only to bring in a new name if it indicates
a real species with distinct specific qualities; otherwise the practice
is pointless and silly, like the way licymnius invented names in his
art of rhetoric-'secundation', 'divagation', 'ramification'.
the introduction is the beginning of a speech, corresponding to the
prologue in poetry and the prelude in flute-music; they are all beginnings,
paving the way, as it were, for what is to follow. the musical prelude
resembles the introduction to speeches of display; as flute players
play first some brilliant passage they know well and then fit it on
to the opening notes of the piece itself, so in speeches of display
the writer should proceed in the same way; he should begin with what
best takes his fancy, and then strike up his theme and lead into it;
which is indeed what is always done. (take as an example the introduction
to the helen of isocrates-there is nothing in common between the 'eristics'
and helen.) and here, even if you travel far from your subject, it
is fitting, rather than that there should be sameness in the entire
the usual subject for the introductions to speeches of display is
some piece of praise or censure. thus gorgias writes in his olympic
speech, 'you deserve widespread admiration, men of greece', praising
thus those who start,ed the festival gatherings.' isocrates, on the
other hand, censures them for awarding distinctions to fine athletes
but giving no prize for intellectual ability. or one may begin with
a piece of advice, thus: 'we ought to honour good men and so i myself
am praising aristeides' or 'we ought to honour those who are unpopular
but not bad men, men whose good qualities have never been noticed,
like alexander son of priam.' here the orator gives advice. or we
may begin as speakers do in the law-courts; that is to say, with appeals
to the audience to excuse us if our speech is about something paradoxical,
difficult, or hackneyed; like choerilus in the lines-
"but now when allotment of all has been made... "
introductions to speeches of display, then, may be composed of some
piece of praise or censure, of advice to do or not to do something,
or of appeals to the audience; and you must choose between making
these preliminary passages connected or disconnected with the speech
introductions to forensic speeches, it must be observed, have the
same value as the prologues of dramas and the introductions to epic
poems; the dithyrambic prelude resembling the introduction to a speech
"for thee, and thy gilts, and thy battle-spoils.... "
in prologues, and in epic poetry, a foretaste of the theme is given,
intended to inform the hearers of it in advance instead of keeping
their minds in suspense. anything vague puzzles them: so give them
a grasp of the beginning, and they can hold fast to it and follow
"sing, o goddess of song, of the wrath...
"lead me to tell a new tale, how there came great warfare to europe
the tragic poets, too, let us know the pivot of their play; if not
at the outset like euripides, at least somewhere in the preface to
and so in comedy. this, then, is the most essential function and
distinctive property of the introduction, to show what the aim of
the speech is; and therefore no introduction ought to be employed
where the subject is not long or intricate.
the other kinds of introduction employed are remedial in purpose,
and may be used in any type of speech. they are concerned with the
speaker, the hearer, the subject, or the speaker's opponent. those
concerned with the speaker himself or with his opponent are directed
to removing or exciting prejudice. but whereas the defendant will
begin by dealing with this sort of thing, the prosecutor will take
quite another line and deal with such matters in the closing part
of his speech. the reason for this is not far to seek. the defendant,
when he is going to bring himself on the stage, must clear away any
obstacles, and therefore must begin by removing any prejudice felt
against him. but if you are to excite prejudice, you must do so at
the close, so that the judges may more easily remember what you have
the appeal to the hearer aims at securing his goodwill, or at arousing
his resentment, or sometimes at gaining his serious attention to the
case, or even at distracting it-for gaining it is not always an advantage,
and speakers will often for that reason try to make him laugh.
you may use any means you choose to make your hearer receptive; among
others, giving him a good impression of your character, which always
helps to secure his attention. he will be ready to attend to anything
that touches himself and to anything that is important, surprising,
or agreeable; and you should accordingly convey to him the impression
that what you have to say is of this nature. if you wish to distract
his attention, you should imply that the subject does not affect him,
or is trivial or disagreeable. but observe, all this has nothing to
do with the speech itself. it merely has to do with the weak-minded
tendency of the hearer to listen to what is beside the point. where
this tendency is absent, no introduction wanted beyond a summary statement
of your subject, to put a sort of head on the main body of your speech.
moreover, calls for attention, when required, may come equally well
in any part of a speech; in fact, the beginning of it is just where
there is least slackness of interest; it is therefore ridiculous to
put this kind of thing at the beginning, when every one is listening
with most attention. choose therefore any point in the speech where
such an appeal is needed, and then say 'now i beg you to note this
point-it concerns you quite as much as myself'; or
"i will tell you that whose like you have never yet "
heard for terror, or for wonder. this is what prodicus called 'slipping
in a bit of the fifty-drachma show-lecture for the audience whenever
they began to nod'. it is plain that such introductions are addressed
not to ideal hearers, but to hearers as we find them. the use of introductions
to excite prejudice or to dispel misgivings is universal-
"my lord, i will not say that eagerly... "
introductions are popular with those whose case is weak, or looks
weak; it pays them to dwell on anything rather than the actual facts
of it. that is why slaves, instead of answering the questions put
to them, make indirect replies with long preambles. the means of exciting
in your hearers goodwill and various other feelings of the same kind
have already been described. the poet finely says may i find in phaeacian
hearts, at my coming, goodwill and compassion; and these are the two
things we should aim at. in speeches of display we must make the hearer
feel that the eulogy includes either himself or his family or his
way of life or something or other of the kind. for it is true, as
socrates says in the funeral speech, that 'the difficulty is not to
praise the athenians at athens but at sparta'.
the introductions of political oratory will be made out of the same
materials as those of the forensic kind, though the nature of political
oratory makes them very rare. the subject is known already, and therefore
the facts of the case need no introduction; but you may have to say
something on account of yourself or to your opponents; or those present
may be inclined to treat the matter either more or less seriously
than you wish them to. you may accordingly have to excite or dispel
some prejudice, or to make the matter under discussion seem more or
less important than before: for either of which purposes you will
want an introduction. you may also want one to add elegance to your
remarks, feeling that otherwise they will have a casual air, like
gorgias' eulogy of the eleans, in which, without any preliminary sparring
or fencing, he begins straight off with 'happy city of elis!'
in dealing with prejudice, one class of argument is that whereby you
can dispel objectionable suppositions about yourself. it makes no
practical difference whether such a supposition has been put into
words or not, so that this distinction may be ignored. another way
is to meet any of the issues directly: to deny the alleged fact; or
to say that you have done no harm, or none to him, or not as much
as he says; or that you have done him no injustice, or not much; or
that you have done nothing disgraceful, or nothing disgraceful enough
to matter: these are the sort of questions on which the dispute hinges.
thus iphicrates replying to nausicrates, admitted that he had done
the deed alleged, and that he had done nausicrates harm, but not that
he had done him wrong. or you may admit the wrong, but balance it
with other facts, and say that, if the deed harmed him, at any rate
it was honourable; or that, if it gave him pain, at least it did him
good; or something else like that. another way is to allege that your
action was due to mistake, or bad luck, or necessity as sophocles
said he was not trembling, as his traducer maintained, in order to
make people think him an old man, but because he could not help it;
he would rather not be eighty years old. you may balance your motive
against your actual deed; saying, for instance, that you did not mean
to injure him but to do so-and-so; that you did not do what you are
falsely charged with doing-the damage was accidental-'i should indeed
be a detestable person if i had deliberately intended this result.'
another way is open when your calumniator, or any of his connexions,
is or has been subject to the same grounds for suspicion. yet another,
when others are subject to the same grounds for suspicion but are
admitted to be in fact innocent of the charge: e.g. 'must i be a profligate
because i am well-groomed? then so-and-so must be one too.' another,
if other people have been calumniated by the same man or some one
else, or, without being calumniated, have been suspected, like yourself
now, and yet have been proved innocent. another way is to return calumny
for calumny and say, 'it is monstrous to trust the man's statements
when you cannot trust the man himself.' another is when the question
has been already decided. so with euripides' reply to hygiaenon, who,
in the action for an exchange of properties, accused him of impiety
in having written a line encouraging perjury-
"my tongue hath sworn: no oath is on my soul. "
euripides said that his opponent himself was guilty in bringing into
the law-courts cases whose decision belonged to the dionysiac contests.
'if i have not already answered for my words there, i am ready to
do so if you choose to prosecute me there.' another method is to denounce
calumny, showing what an enormity it is, and in particular that it
raises false issues, and that it means a lack of confidence in the
merits of his case. the argument from evidential circumstances is
available for both parties: thus in the teucer odysseus says that
teucer is closely bound to priam, since his mother hesione was priam's
sister. teucer replies that telamon his father was priam's enemy,
and that he himself did not betray the spies to priam. another method,
suitable for the calumniator, is to praise some trifling merit at
great length, and then attack some important failing concisely; or
after mentioning a number of good qualities to attack one bad one
that really bears on the question. this is the method of thoroughly
skilful and unscrupulous prosecutors. by mixing up the man's merits
with what is bad, they do their best to make use of them to damage
there is another method open to both calumniator and apologist. since
a given action can be done from many motives, the former must try
to disparage it by selecting the worse motive of two, the latter to
put the better construction on it. thus one might argue that diomedes
chose odysseus as his companion because he supposed odysseus to be
the best man for the purpose; and you might reply to this that it
was, on the contrary, because he was the only hero so worthless that
we may now pass from the subject of calumny to that of narration.
narration in ceremonial oratory is not continuous but intermittent.
there must, of course, be some survey of the actions that form the
subject-matter of the speech. the speech is a composition containing
two parts. one of these is not provided by the orator's art, viz.
the actions themselves, of which the orator is in no sense author.
the other part is provided by his namely, the proof (where proof is
needed) that the actions were done, the description of their quality
or of their extent, or even all these three things together. now the
reason why sometimes it is not desirable to make the whole narrative
continuous is that the case thus expounded is hard to keep in mind.
show, therefore, from one set of facts that your hero is, e.g. brave,
and from other sets of facts that he is able, just, c. a speech thus
arranged is comparatively simple, instead of being complicated and
elaborate. you will have to recall well-known deeds among others;
and because they are well-known, the hearer usually needs no narration
of them; none, for instance, if your object is the praise of achilles;
we all know the facts of his life-what you have to do is to apply
those facts. but if your object is the praise of critias, you must
narrate his deeds, which not many people know of...
nowadays it is said, absurdly enough, that the narration should be
rapid. remember what the man said to the baker who asked whether he
was to make the cake hard or soft: 'what, can't you make it right?'
just so here. we are not to make long narrations, just as we are not
to make long introductions or long arguments. here, again, rightness
does not consist either in rapidity or in conciseness, but in the
happy mean; that is, in saying just so much as will make the facts
plain, or will lead the hearer to believe that the thing has happened,
or that the man has caused injury or wrong to some one, or that the
facts are really as important as you wish them to be thought: or the
opposite facts to establish the opposite arguments.
you may also narrate as you go anything that does credit to yourself,
e.g. 'i kept telling him to do his duty and not abandon his children';
or discredit to your adversary, e.g. 'but he answered me that, wherever
he might find himself, there he would find other children', the answer
herodotus' records of the egyptian mutineers. slip in anything else
the defendant will make less of the narration. he has to maintain
that the thing has not happened, or did no harm, or was not unjust,
or not so bad as is alleged. he must therefor snot waste time about
what is admitted fact, unless this bears on his own contention; e.g.
that the thing was done, but was not wrong. further, we must speak
of events as past and gone, except where they excite pity or indignation
by being represented as present. the story told to alcinous is an
example of a brief chronicle, when it is repeated to penelope in sixty
lines. another instance is the epic cycle as treated by phayllus,
the narration should depict character; to which end you must know
what makes it do so. one such thing is the indication of moral purpose;
the quality of purpose indicated determines the quality of character
depicted and is itself determined by the end pursued. thus it is that
mathematical discourses depict no character; they have nothing to
do with moral purpose, for they represent nobody as pursuing any end.
on the other hand, the socratic dialogues do depict character, being
concerned with moral questions. this end will also be gained by describing
the manifestations of various types of character, e.g. 'he kept walking
along as he talked', which shows the man's recklessness and rough
manners. do not let your words seem inspired so much by intelligence,
in the manner now current, as by moral purpose: e.g. 'i willed this;
aye, it was my moral purpose; true, i gained nothing by it, still
it is better thus.' for the other way shows good sense, but this shows
good character; good sense making us go after what is useful, and
good character after what is noble. where any detail may appear incredible,
then add the cause of it; of this sophocles provides an example in
the antigone, where antigone says she had cared more for her brother
than for husband or children, since if the latter perished they might
"but since my father and mother in their graves
"lie dead, no brother can be born to me. "
if you have no such cause to suggest, just say that you are aware
that no one will believe your words, but the fact remains that such
is our nature, however hard the world may find it to believe that
a man deliberately does anything except what pays him.
again, you must make use of the emotions. relate the familiar manifestations
of them, and those that distinguish yourself and your opponent; for
instance, 'he went away scowling at me'. so aeschines described cratylus
as 'hissing with fury and shaking his fists'. these details carry
conviction: the audience take the truth of what they know as so much
evidence for the truth of what they do not. plenty of such details
"thus did she say: but the old woman buried her face in her hands:
a true touch-people beginning to cry do put their hands over their
bring yourself on the stage from the first in the right character,
that people may regard you in that light; and the same with your adversary;
but do not let them see what you are about. how easily such impressions
may be conveyed we can see from the way in which we get some inkling
of things we know nothing of by the mere look of the messenger bringing
news of them. have some narrative in many different parts of your
speech; and sometimes let there be none at the beginning of it.
in political oratory there is very little opening for narration; nobody
can 'narrate' what has not yet happened. if there is narration at
all, it will be of past events, the recollection of which is to help
the hearers to make better plans for the future. or it may be employed
to attack some one's character, or to eulogize him-only then you will
not be doing what the political speaker, as such, has to do.
if any statement you make is hard to believe, you must guarantee its
truth, and at once offer an explanation, and then furnish it with
such particulars as will be expected. thus carcinus' jocasta, in his
oedipus, keeps guaranteeing the truth of her answers to the inquiries
of the man who is seeking her son; and so with haemon in sophocles.
the duty of the arguments is to attempt demonstrative proofs. these
proofs must bear directly upon the question in dispute, which must
fall under one of four heads. (1) if you maintain that the act was
not committed, your main task in court is to prove this. (2) if you
maintain that the act did no harm, prove this. if you maintain that
(3) the act was less than is alleged, or (4) justified, prove these
facts, just as you would prove the act not to have been committed
it should be noted that only where the question in dispute falls under
the first of these heads can it be true that one of the two parties
is necessarily a rogue. here ignorance cannot be pleaded, as it might
if the dispute were whether the act was justified or not. this argument
must therefore be used in this case only, not in the others.
in ceremonial speeches you will develop your case mainly by arguing
that what has been done is, e.g., noble and useful. the facts themselves
are to be taken on trust; proof of them is only submitted on those
rare occasions when they are not easily credible or when they have
in political speeches you may maintain that a proposal is impracticable;
or that, though practicable, it is unjust, or will do no good, or
is not so important as its proposer thinks. note any falsehoods about
irrelevant matters-they will look like proof that his other statements
also are false. argument by 'example' is highly suitable for political
oratory, argument by 'enthymeme' better suits forensic. political
oratory deals with future events, of which it can do no more than
quote past events as examples. forensic oratory deals with what is
or is not now true, which can better be demonstrated, because not
contingent-there is no contingency in what has now already happened.
do not use a continuous succession of enthymemes: intersperse them
with other matter, or they will spoil one another's effect. there
friend, you have spoken as much as a sensible man would have spoken.
,as much' says homer, not 'as well'. nor should you try to make enthymemes
on every point; if you do, you will be acting just like some students
of philosophy, whose conclusions are more familiar and believable
than the premisses from which they draw them. and avoid the enthymeme
form when you are trying to rouse feeling; for it will either kill
the feeling or will itself fall flat: all simultaneous motions tend
to cancel each other either completely or partially. nor should you
go after the enthymeme form in a passage where you are depicting character-the
process of demonstration can express neither moral character nor moral
purpose. maxims should be employed in the arguments-and in the narration
too-since these do express character: 'i have given him this, though
i am quite aware that one should "trust no man".' or if you are appealing
to the emotions: 'i do not regret it, though i have been wronged;
if he has the profit on his side, i have justice on mine.'
political oratory is a more difficult task than forensic; and naturally
so, since it deals with the future, whereas the pleader deals with
the past, which, as epimenides of crete said, even the diviners already
know. (epimenides did not practise divination about the future; only
about the obscurities of the past.) besides, in forensic oratory you
have a basis in the law; and once you have a starting-point, you can
prove anything with comparative ease. then again, political oratory
affords few chances for those leisurely digressions in which you may
attack your adversary, talk about yourself, or work on your hearers'
emotions; fewer chances indeed, than any other affords, unless your
set purpose is to divert your hearers' attention. accordingly, if
you find yourself in difficulties, follow the lead of the athenian
speakers, and that of isocrates, who makes regular attacks upon people
in the course of a political speech, e.g. upon the lacedaemonians
in the panegyricus, and upon chares in the speech about the allies.
in ceremonial oratory, intersperse your speech with bits of episodic
eulogy, like isocrates, who is always bringing some one forward for
this purpose. and this is what gorgias meant by saying that he always
found something to talk about. for if he speaks of achilles, he praises
peleus, then aeacus, then zeus; and in like manner the virtue of valour,
describing its good results, and saying what it is like.
now if you have proofs to bring forward, bring them forward, and your
moral discourse as well; if you have no enthymemes, then fall back
upon moral discourse: after all, it is more fitting for a good man
to display himself as an honest fellow than as a subtle reasoner.
refutative enthymemes are more popular than demonstrative ones: their
logical cogency is more striking: the facts about two opposites always
stand out clearly when the two are nut side by side.
the 'reply to the opponent' is not a separate division of the speech;
it is part of the arguments to break down the opponent's case, whether
by objection or by counter-syllogism. both in political speaking and
when pleading in court, if you are the first speaker you should put
your own arguments forward first, and then meet the arguments on the
other side by refuting them and pulling them to pieces beforehand.
if, however, the case for the other side contains a great variety
of arguments, begin with these, like callistratus in the messenian
assembly, when he demolished the arguments likely to be used against
him before giving his own. if you speak later, you must first, by
means of refutation and counter-syllogism, attempt some answer to
your opponent's speech, especially if his arguments have been well
received. for just as our minds refuse a favourable reception to a
person against whom they are prejudiced, so they refuse it to a speech
when they have been favourably impressed by the speech on the other
side. you should, therefore, make room in the minds of the audience
for your coming speech; and this will be done by getting your opponent's
speech out of the way. so attack that first-either the whole of it,
or the most important, successful, or vulnerable points in it, and
thus inspire confidence in what you have to say yourself-
"first, champion will i be of goddesses...
where the speaker has attacked the silliest argument first. so much
with regard to the element of moral character: there are assertions
which, if made about yourself, may excite dislike, appear tedious,
or expose you to the risk of contradiction; and other things which
you cannot say about your opponent without seeming abusive or ill-bred.
put such remarks, therefore, into the mouth of some third person.
this is what isocrates does in the philippus and in the antidosis,
and archilochus in his satires. the latter represents the father himself
as attacking his daughter in the lampoon
"nor swear that it shall not befall... "
and puts into the mouth of charon the carpenter the lampoon which
so too sophocles makes haemon appeal to his father on behalf of antigone
again, sometimes you should restate your enthymemes in the form of
maxims; e.g. 'wise men will come to terms in the hour of success;
for they will gain most if they do'. expressed as an enthymeme, this
would run, 'if we ought to come to terms when doing so will enable
us to gain the greatest advantage, then we ought to come to terms
next as to interrogation. the best moment to a employ this is when
your opponent has so answered one question that the putting of just
one more lands him in absurdity. thus pericles questioned lampon about
the way of celebrating the rites of the saviour goddess. lampon declared
that no uninitiated person could be told of them. pericles then asked,
'do you know them yourself?' 'yes', answered lampon. 'why,' said pericles,
'how can that be, when you are uninitiated?'
another good moment is when one premiss of an argument is obviously
true, and you can see that your opponent must say 'yes' if you ask
him whether the other is true. having first got this answer about
the other, do not go on to ask him about the obviously true one, but
just state the conclusion yourself. thus, when meletus denied that
socrates believed in the existence of gods but admitted that he talked
about a supernatural power, socrates proceeded to to ask whether 'supernatural
beings were not either children of the gods or in some way divine?'
'yes', said meletus. 'then', replied socrates, 'is there any one who
believes in the existence of children of the gods and yet not in the
existence of the gods themselves?' another good occasion is when you
expect to show that your opponent is contradicting either his own
words or what every one believes. a fourth is when it is impossible
for him to meet your question except by an evasive answer. if he answers
'true, and yet not true', or 'partly true and partly not true', or
'true in one sense but not in another', the audience thinks he is
in difficulties, and applauds his discomfiture. in other cases do
not attempt interrogation; for if your opponent gets in an objection,
you are felt to have been worsted. you cannot ask a series of questions
owing to the incapacity of the audience to follow them; and for this
reason you should also make your enthymemes as compact as possible.
in replying, you must meet ambiguous questions by drawing reasonable
distinctions, not by a curt answer. in meeting questions that seem
to involve you in a contradiction, offer the explanation at the outset
of your answer, before your opponent asks the next question or draws
his conclusion. for it is not difficult to see the drift of his argument
in advance. this point, however, as well as the various means of refutation,
may be regarded as known to us from the topics.
when your opponent in drawing his conclusion puts it in the form of
a question, you must justify your answer. thus when sophocles was
asked by peisander whether he had, like the other members of the board
of safety, voted for setting up the four hundred, he said 'yes.'-'why,
did you not think it wicked?'-'yes.'-'so you committed this wickedness?'
'yes', said sophocles, 'for there was nothing better to do.' again,
the lacedaemonian, when he was being examined on his conduct as ephor,
was asked whether he thought that the other ephors had been justly
put to death. 'yes', he said. 'well then', asked his opponent, 'did
not you propose the same measures as they?'-'yes.'-'well then, would
not you too be justly put to death?'-'not at all', said he; 'they
were bribed to do it, and i did it from conviction'. hence you should
not ask any further questions after drawing the conclusion, nor put
the conclusion itself in the form of a further question, unless there
is a large balance of truth on your side.
as to jests. these are supposed to be of some service in controversy.
gorgias said that you should kill your opponents' earnestness with
jesting and their jesting with earnestness; in which he was right.
jests have been classified in the poetics. some are becoming to a
gentleman, others are not; see that you choose such as become you.
irony better befits a gentleman than buffoonery; the ironical man
jokes to amuse himself, the buffoon to amuse other people.
the epilogue has four parts. you must (1) make the audience well-disposed
towards yourself and ill-disposed towards your opponent (2) magnify
or minimize the leading facts, (3) excite the required state of emotion
in your hearers, and (4) refresh their memories.
(1) having shown your own truthfulness and the untruthfulness of your
opponent, the natural thing is to commend yourself, censure him, and
hammer in your points. you must aim at one of two objects-you must
make yourself out a good man and him a bad one either in yourselves
or in relation to your hearers. how this is to be managed-by what
lines of argument you are to represent people as good or bad-this
(2) the facts having been proved, the natural thing to do next is
to magnify or minimize their importance. the facts must be admitted
before you can discuss how important they are; just as the body cannot
grow except from something already present. the proper lines of argument
to be used for this purpose of amplification and depreciation have
(3) next, when the facts and their importance are clearly understood,
you must excite your hearers' emotions. these emotions are pity, indignation,
anger, hatred, envy, emulation, pugnacity. the lines of argument to
be used for these purposes also have been previously mentioned.
(4) finally you have to review what you have already said. here you
may properly do what some wrongly recommend doing in the introduction-repeat
your points frequently so as to make them easily understood. what
you should do in your introduction is to state your subject, in order
that the point to be judged may be quite plain; in the epilogue you
should summarize the arguments by which your case has been proved.
the first step in this reviewing process is to observe that you have
done what you undertook to do. you must, then, state what you have
said and why you have said it. your method may be a comparison of
your own case with that of your opponent; and you may compare either
the ways you have both handled the same point or make your comparison
less direct: 'my opponent said so-and-so on this point; i said so-and-so,
and this is why i said it'. or with modest irony, e.g. 'he certainly
said so-and-so, but i said so-and-so'. or 'how vain he would have
been if he had proved all this instead of that!' or put it in the
form of a question. 'what has not been proved by me?' or 'what has
my opponent proved?' you may proceed then, either in this way by setting
point against point, or by following the natural order of the arguments
as spoken, first giving your own, and then separately, if you wish,
for the conclusion, the disconnected style of language is appropriate,
and will mark the difference between the oration and the peroration.
'i have done. you have heard me. the facts are before you. i ask for
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translation of "the deeds of the divine augustus" by augustus iscornell university library produced this volume to replace the
irreparably deteriorated original. it was scanned using xerox
software and equipment at 600 dots per inch resolution and com-
pressed prior to storage using ccitt group 4 compression. the
digital data were used to create cornell's replacement volume
on paper that meets the ansi standard z39.48-1984. the produc-
tion of this volume was supported in part by the commission on
preservation and access and the xerox corporation. digital
file copyright by cornell university library 1991.
transcriber's note: the index has been regenerated to fit the pagination of this edition. despite the author's stated hope that "few misprints have escaped detection" there were several, which have here been corrected and noted at the end of the text.
.......................................................................
with numerous examples and questions for examination.
third edition, revised and enlarged, price 3s. 6d., cloth.
containing an easy introduction to modern geometry:
third edition, price 4s. 6d.; or in two parts, each 2s. 6d.
the elements of euclid, books i.-vi., and
together with an appendix on the cylinder, sphere,
a treatise on the analytical geometry of
the point, line, circle, conic sections,
containing an account of its most recent extensions,
appendix on the cylinder, sphere, cone, etc.
copious annotations and numerous exercises.
fellow of the royal university of ireland;
member of the mathematical societies of london and france;
and professor of the higher mathematics and of
mathematical physics in the catholic university of ireland.
dublin: hodges, figgis, co., grafton-st.
this edition of the elements of euclid, undertaken at the request of the principals of some of the leading colleges and schools of ireland, is intended to supply a want much felt by teachers at the present day-the production of a work which, while giving
the unrivalled original in all its integrity, would also contain the modern conceptions and developments of the portion of geometry over which the elements extend.
a cursory examination of the work will show that the editor has gone much further in this latter direction than any of his predecessors, for it will be found to contain, not only more actual matter than is given in any of theirs with which he is
acquainted, but also much of a special character, which is not given, so far as he is aware, in any former work on the subject.
the great extension of geometrical methods in recent times has made such a work a necessity for the student, to enable him not only to read with advantage, but even to understand those mathematical writings of modern times which require an accurate
knowledge of elementary geometry, and to which it is in reality the best introduction.
in compiling his work the editor has received invaluable assistance from the late rev. professor townsend, s.f.t.c.d.
the book was rewritten and considerably altered in accordance with his suggestions, and to that distinguished geometer it is largely indebted for whatever merit it possesses.
the questions for examination in the early part of the first book are intended as specimens, which the teacher ought to follow through the entire work.
every person who has had experience in tuition knows well the importance of such examinations in teaching elementary geometry.
the exercises, of which there are over eight hundred, have been all selected with great care. those in the body of each book are intended as applications of euclid's propositions.
they are for the most part of an elementary character, and may be regarded as common property, nearly every one of them having appeared already in previous collections.
the exercises at the end of each book are more advanced; several are due to the late professor townsend, some are original, and a large number have been taken from two important french works-catalan's theoremes et problemes de geometrie elementaire, and
the traite de geometrie, by rouche and de comberousse.
the second edition has been thoroughly revised and greatly enlarged.
the new matter includes several alternative proofs, important examination questions on each of the books, an explanation of the ratio of incommensurable quantities, the first twenty-one propositions of book xi., and an appendix on the properties of the
prism, pyramids, cylinder, sphere, and cone.
the present edition has been very carefully read throughout, and it is hoped that few misprints have escaped detection.
the editor is glad to find from the rapid sale of former editions (each 3000 copies) of his book, and its general adoption in schools, that it is likely to accomplish the double object with which it was written, viz.
to supply students with a manual that will impart a thorough knowledge of the immortal work of the great greek geometer, and introduce them, at the same time, to some of the most important conceptions and developments of the geometry of the present day.
geometry is the science of figured space. figured space is of one, two, or three dimensions, according as it consists of lines, surfaces, or solids. the boundaries of solids are surfaces; of surfaces, lines; and of lines, points.
thus it is the province of geometry to investigate the properties of solids, of surfaces, and of the figures described on surfaces.
the simplest of all surfaces is the plane, and that department of geometry which is occupied with the lines and curves drawn on a plane is called plane geometry; that which demonstrates the properties of solids, of curved surfaces, and the figures
described on curved surfaces, is geometry of three dimensions.
the simplest lines that can be drawn on a plane are the right line and circle, and the study of the properties of the point, the right line, and the circle, is the introduction to geometry, of which it forms an extensive and important department.
this is the part of geometry on which the oldest mathematical book in existence, namely, euclid's elements, is written, and is the subject of the present volume.
the conic sections and other curves that can be described on a plane form special branches, and complete the divisions of this, the most comprehensive of all the sciences.
the student will find in chasles' apercu historique a valuable history of the origin and the development of the methods of geometry.
in the following work, when figures are not drawn, the student should construct them from the given directions. the propositions of euclid will be printed in larger type, and will be referred to by roman numerals enclosed in brackets. thus iii. xxxii.
will denote the 32nd proposition of the 3rd book. the number of the book will be given only when different from that under which the reference occurs. the general and the particular enunciation of every proposition will be given in one.
by omitting the letters enclosed in parentheses we have the general enunciation, and by reading them, the particular. the annotations will be printed in smaller type. the following symbols will be used in them:-
in addition to these we shall employ the usual symbols , , c. of algebra, and also the sign of congruence, namely . this symbol has been introduced by the illustrious gauss.
theory of angles, triangles, parallel lines, and parallelograms.
i. a point is that which has position but not dimensions.
a geometrical magnitude which has three dimensions, that is, length, breadth, and thickness, is a solid; that which has two dimensions, such as length and breadth, is a surface; and that which has but one dimension is a line.
but a point is neither a solid, nor a surface, nor a line; hence it has no dimensions-that is, it has neither length, breadth, nor thickness.
a line is space of one dimension. if it had any breadth, no matter how small, it would be space of two dimensions; and if in addition it had any thickness it would be space of three dimensions; hence a line has neither breadth nor thickness.
iii. the intersections of lines and their extremities are points.
iv. a line which lies evenly between its extreme points is called a straight or right line, such as ab.
if a point move without changing its direction it will describe a right line. the direction in which a point moves is called its "sense."
if the moving point continually changes its direction it will describe a curve; hence it follows that only one right line can be drawn between two points.
the following illustration is due to professor henrici:-"if we suspend a weight by a string, the string becomes stretched, and we say it is straight, by which we mean to express that it has assumed a peculiar definite shape.
if we mentally abstract from this string all thickness, we obtain the notion of the simplest of all lines, which we call a straight line."
v. a surface is that which has length and breadth.
a surface is space of two dimensions. it has no thickness, for if it had any, however small, it would be space of three dimensions.
vi. when a surface is such that the right line joining any two arbitrary points in it lies wholly in the surface, it is called a plane.
a plane is perfectly flat and even, like the surface of still water, or of a smooth floor.-newcomb.
vii. any combination of points, of lines, or of points and lines in a plane, is called a plane figure. if a figure be formed of points only it is called a stigmatic figure; and if of right lines only, a rectilineal figure.
viii. points which lie on the same right line are called collinear points. a figure formed of collinear points is called a row of points.
ix. the inclination of two right lines extending out from one point in different directions is called a rectilineal angle.
x. the two lines are called the legs, and the point the vertex of the angle.
a right line drawn from the vertex and turning about it in the plane of the angle, from the position of coincidence with one leg to that of coincidence with the other, is said to turn through the angle, and the angle is the greater as the quantity of
turning is the greater. again, since the line may turn from one position to the other in either of two ways, two angles are formed by two lines drawn from a point.
thus if ab, ac be the legs, a line may turn from the position ab to the position ac in the two ways indicated by the arrows. the smaller of the angles thus formed is to be understood as the angle contained by the lines.
the larger, called a re-entrant angle, seldom occurs in the "elements."
xi. designation of angles.-a particular angle in a figure is denoted by three letters, as bac, of which the middle one, a, is at the vertex, and the other two along the legs. the angle is then read bac.
xii. the angle formed by joining two or more angles together is called their sum. thus the sum of the two angles abc, pqr is the angle abr,
formed by applying the side qp to the side bc, so that the vertex q shall fall on the vertex b, and the side qr on the opposite side of bc from ba.
xiii. when the sum of two angles bac, cad is such that the legs ba, ad form one right line, they are called supplements of each other.
hence, when one line stands on another, the two angles which it makes on the same side of that on which it stands are supplements of each other.
xiv. when one line stands on another, and makes the adjacent angles at both sides of itself equal, each of the angles is called a right angle, and the line which stands on the other is called a perpendicular to it.
hence a right angle is equal to its supplement.
xv. an acute angle is one which is less than a right angle, as a.
xvi. an obtuse angle is one which is greater than a right angle, as bac.
the supplement of an acute angle is obtuse, and conversely, the supplement of an obtuse angle is acute.
xvii. when the sum of two angles is a right angle, each is called the complement of the other. thus, if the angle bac be right, the angles bad, dac are complements of each other.
xviii. three or more right lines passing through the same point are called concurrent lines.
xix. a system of more than three concurrent lines is called a pencil of lines. each line of a pencil is called a ray, and the common point through which the rays pass is called the vertex.
xx. a triangle is a figure formed by three right lines joined end to end. the three lines are called its sides.
xxi. a triangle whose three sides are unequal is said to be scalene, as a; a triangle having two sides equal, to be isosceles, as b; and and having all its sides equal, to be equilateral, as c.
xxii. a right-angled triangle is one that has one of its angles a right angle, as d. the side which subtends the right angle is called the hypotenuse.
xxiii. an obtuse-angled triangle is one that has one of its angles obtuse, as e.
xxiv. an acute-angled triangle is one that has its three angles acute, as f.
xxv. an exterior angle of a triangle is one that is formed by any side and the continuation of another side.
hence a triangle has six exterior angles; and also each exterior angle is the supplement of the adjacent interior angle.
xxvi. a rectilineal figure bounded by more than three right lines is usually called a polygon.
xxvii. a polygon is said to be convex when it has no re-entrant angle.
xxviii. a polygon of four sides is called a quadrilateral.
xxix. a quadrilateral whose four sides are equal is called a lozenge.
xxx. a lozenge which has a right angle is called a square.
xxxi. a polygon which has five sides is called a pentagon; one which has six sides, a hexagon, and so on.
xxxii. a circle is a plane figure formed by a curved line called the circumference, and is such that all right lines drawn from a certain point within the figure to the circumference are equal to one another. this point is called the centre.
xxxiii. a radius of a circle is any right line drawn from the centre to the circumference, such as cd.
xxxiv. a diameter of a circle is a right line drawn through the centre and terminated both ways by the circumference, such as ab.
from the definition of a circle it follows at once that the path of a movable point in a plane which remains at a constant distance from a fixed point is a circle; also that any point p in the plane is inside, outside, or on the circumference of a circle
according as its distance from the centre is less than, greater than, or equal to, the radius.
i. a right line may be drawn from any one point to any other point.
when we consider a straight line contained between two fixed points which are its ends, such a portion is called a finite straight line.
ii. a terminated right line may be produced to any length in a right line.
every right line may extend without limit in either direction or in both. it is in these cases called an indefinite line. by this postulate a finite right line may be supposed to be produced, whenever we please, into an indefinite right line.
iii. a circle may be described from any centre, and with any distance from that centre as radius.
if there be two points a and b, and if with any instruments, such as a ruler and pen, we draw a line from a to b, this will evidently have some irregularities, and also some breadth and thickness.
hence it will not be a geometrical line no matter how nearly it may approach to one. this is the reason that euclid postulates the drawing of a right line from one point to another.
for if it could be accurately done there would be no need for his asking us to let it be granted. similar observations apply to the other postulates.
it is also worthy of remark that euclid never takes for granted the doing of anything for which a geometrical construction, founded on other problems or on the foregoing postulates, can be given.
i. things which are equal to the same, or to equals, are equal to each other.
thus, if there be three things, and if the first, and the second, be each equal to the third, we infer by this axiom that the first is equal to the second. this axiom relates to all kinds of magnitude.
the same is true of axioms ii., iii., iv., v., vi., vii., ix.; but viii., x., xi., xii., are strictly geometrical.
ii. if equals be added to equals the sums will be equal.
iii. if equals be taken from equals the remainders will be equal.
iv. if equals be added to unequals the sums will be unequal.
v. if equals be taken from unequals the remainders will be unequal.
vi. the doubles of equal magnitudes are equal.
vii. the halves of equal magnitudes are equal.
viii. magnitudes that can be made to coincide are equal.
the placing of one geometrical magnitude on another, such as a line on a line, a triangle on a triangle, or a circle on a circle, c., is called superposition.
the superposition employed in geometry is only mental, that is, we conceive one magnitude placed on the other; and then, if we can prove that they coincide, we infer, by the present axiom, that they are equal.
superposition involves the following principle, of which, without explicitly stating it, euclid makes frequent use:-"any figure may be transferred from one position to another without change of form or size."
this axiom is included in the following, which is a fuller statement:-
ix. the whole is equal to the sum of all its parts.
x. two right lines cannot enclose a space.
this is equivalent to the statement, "if two right lines have two points common to both, they coincide in direction," that is, they form but one line, and this holds true even when one of the points is at infinity.
xi. all right angles are equal to one another.
this can be proved as follows:-let there be two right lines ab, cd, and two perpendiculars to them, namely, ef, gh, then if ab, cd be made to coincide by superposition, so that the point e will coincide with g; then since a right angle is equal to its
supplement, the line ef must coincide with gh. hence the angle aef is equal to cgh.
xii. if two right lines (ab, cd) meet a third line (ac), so as to make the sum of the two interior angles (bac, acd) on the same side less than two right angles, these lines being produced shall meet at some finite distance.
this axiom is the converse of prop. xvii., book i.
axioms.-"elements of human reason," according to dugald stewart, are certain general propositions, the truths of which are self-evident, and which are so fundamental, that they cannot be inferred from any propositions which are more elementary; in other
words, they are incapable of demonstration.
"that two sides of a triangle are greater than the third" is, perhaps, self-evident; but it is not an axiom, inasmuch as it can be inferred by demonstration from other propositions; but we can give no proof of the proposition that "things which are equal
to the same are equal to one another," and, being self-evident, it is an axiom.
propositions which are not axioms are properties of figures obtained by processes of reasoning. they are divided into theorems and problems.
a theorem is the formal statement of a property that may be demonstrated from known propositions. these propositions may themselves be theorems or axioms.
a theorem consists of two parts, the hypothesis, or that which is assumed, and the conclusion, or that which is asserted to follow therefrom. thus, in the typical theorem,
the hypothesis is that x is y , and the conclusion is that z is w.
converse theorems.-two theorems are said to be converse, each of the other, when the hypothesis of either is the conclusion of the other. thus the converse of the theorem (i.) is-
from the two theorems (i.) and (ii.) we may infer two others, called their contrapositives. thus the contrapositive
of (i.) is, if z is not w, then x is not y ; (iii.)
of (ii.) is, if x is not y , then z is not w. (iv.)
the theorem (iv.) is called the obverse of (i.), and (iii.) the obverse of (ii.).
a problem is a proposition in which something is proposed to be done, such as a line to be drawn, or a figure to be constructed, under some given conditions.
the solution of a problem is the method of construction which accomplishes the required end.
the demonstration is the proof, in the case of a theorem, that the conclusion follows from the hypothesis; and in the case of a problem, that the construction accomplishes the object proposed.
the enunciation of a problem consists of two parts, namely, the data, or things supposed to be given, and the quaesita, or things required to be done.
postulates are the elements of geometrical construction, and occupy the same relation with respect to problems as axioms do to theorems.
a corollary is an inference or deduction from a proposition.
a lemma is an auxiliary proposition required in the demonstration of a principal proposition.
a secant or transversal is a line which cuts a system of lines, a circle, or any other geometrical figure.
congruent figures are those that can be made to coincide by superposition. they agree in shape and size, but differ in position.
hence it follows, by axiom viii., that corresponding parts or portions of congruent figures are congruent, and that congruent figures are equal in every respect.
rule of identity.-under this name the following principle will be sometimes referred to:-"if there is but one x and one y , then, from the fact that x is y , it necessarily follows that y is x."-syllabus.
on a given finite right line (ab) to construct an equilateral triangle.
sol.-with a as centre, and ab as radius, describe the circle bcd (post. iii.). with b as centre, and ba as radius, describe the circle ace, cutting the former circle in c. join ca, cb (post. i.). then abc is the equilateral triangle required.
dem.-because a is the centre of the circle bcd, ac is equal to ab (def. xxxii.). again, because b is the centre of the circle ace, bc is equal to ba. hence we have proved.
but things which are equal to the same are equal to one another (axiom i.); therefore ac is equal to bc; therefore the three lines ab, bc, ca are equal to one another. hence the triangle abc is equilateral (def.
xxi.); and it is described on the given line ab, which was required to be done.
1. what is the datum in this proposition?
5. in what part of the construction is the third postulate quoted? and for what purpose? where is the first postulate quoted?
7. what use is made of the definition of a circle? what is a circle?
the following exercises are to be solved when the pupil has mastered the first book:-
1. if the lines af, bf be joined, the figure acbf is a lozenge.
2. if ab be produced to d and e, the triangles cdf and cef are equilateral.
3. if ca, cb be produced to meet the circles again in g and h, the points g, f, h are collinear, and the triangle gch is equilateral.
5. describe a circle in the space acb, bounded by the line ab and the two circles.
from a given point (a) to draw a right line equal to a given finite right line (bc).
sol.-join ab (post. i.); on ab describe the equilateral triangle abd i. . with b as centre, and bc as radius, describe the circle ech (post iii.). produce db to meet the circle ech in e (post. ii.).
with d as centre, and de as radius, describe the circle efg (post. iii.). produce da to meet this circle in f. af is equal to bc.
dem.-because d is the centre of the circle efg, df is equal to de (def. xxxii.). and because dab is an equilateral triangle, da is equal to db (def. xxi.). hence we have
and taking the latter from the former, the remainder af is equal to the remainder be (axiom iii.).
again, because b is the centre of the circle ech, bc is equal to be; and we have proved that af is equal to be; and things which are equal to the same thing are equal to one another (axiom i.). hence af is equal to bc.
therefore from the given point a the line af has been drawn equal to bc.
it is usual with commentators on euclid to say that he allows the use of the rule and compass. were such the case this proposition would have been unnecessary.
the fact is, euclid's object was to teach theoretical and not practical geometry, and the only things he postulates are the drawing of right lines and the describing of circles.
if he allowed the mechanical use of the rule and compass he could give methods of solving many problems that go beyond the limits of the "geometry of the point, line, and circle."-see notes d, f at the end of this work.
1. solve the problem when the point a is in the line bc itself.
2. inflect from a given point a to a given line bc a line equal to a given line. state the number of solutions.
from the greater (ab) of two given right lines to cut off a part equal to (c) the less.
sol.-from a, one of the extremities of ab, draw the right line ad equal to c ii. ; and with a as centre, and ad as radius, describe the circle edf (post. iii.) cutting ab in e. ae shall be equal to c.
dem.-because a is the centre of the circle edf, ae is equal to ad (def. xxxii.), and c is equal to ad (const.); and things which are equal to the same are equal to one another (axiom i.); therefore ae is equal to c.
wherefore from ab, the greater of the two given lines, a part, ae, has been out off equal to c, the less.
1. what previous problem is employed in the solution of this?
4. show how to produce the less of two given lines until the whole produced line becomes equal to the greater.
if two triangles (bac, edf) have two sides (ba, ac) of one equal respectively to two sides (ed, df) of the other, and have also the angles (a, d) included by those sides equal, the triangles shall be equal in every respect-that is, their bases or third
sides (bc, ef) shall be equal, and the angles (b, c) at the base of one shall be respectively equal to the angles (e, f) at the base of the other; namely, those shall be equal to which the equal sides are opposite.
dem.-let us conceive the triangle bac to be applied to edf, so that the point a shall coincide with d, and the line ab with de, and that the point c shall be on the same side of de as f; then because ab is equal to de, the point b shall coincide with e.
again, because the angle bac is equal to the angle edf, the line ac shall coincide with df; and since ac is equal to df (hyp.), the point c shall coincide with f; and we have proved that the point b coincides with e.
hence two points of the line bc coincide with two points of the line ef; and since two right lines cannot enclose a space, bc must coincide with ef.
hence the triangles agree in every respect; therefore bc is equal to ef, the angle b is equal to the angle e, the angle c to the angle f, and the triangle bac to the triangle edf.
1. how many parts in the hypothesis of this proposition? ans. three. name them.
2. how many in the conclusion? name them.
3. what technical term is applied to figures which agree in everything but position? ans. they are said to be congruent.
5. what axiom is made use of in superposition?
6. how many parts in a triangle? ans. six; namely, three sides and three angles.
7. when it is required to prove that two triangles are congruent, how many parts of one must be given equal to corresponding parts of the other? ans. in general, any three except the three angles. this will be established in props. viii.
8. what property of two lines having two common points is quoted in this proposition? they must coincide.
1. the line that bisects the vertical angle of an isosceles triangle bisects the base perpendicularly.
2. if two adjacent sides of a quadrilateral be equal, and the diagonal bisects the angle between them, their other sides are equal.
3. if two lines be at right angles, and if each bisect the other, then any point in either is equally distant from the extremities of the other.
4. if equilateral triangles be described on the sides of any triangle, the distances between the vertices of the original triangle and the opposite vertices of the equilateral triangles are equal.
(this proposition should be proved after the student has read prop. xxxii.)
the angles (abc, acb) at the base (bc) of an isosceles triangle are equal to one another, and if the equal sides (ab, ac) be produced, the external angles (dbc, ecb) below the base shall be equal.
dem.-in bd take any point f, and from ae, the greater, cut off ag equal to af iii . join bg, cf (post. i.).
because af is equal to ag (const.), and ac is equal to ab (hyp.), the two triangles fac, gab have the sides fa, ac in one respectively equal to the sides ga, ab in the other; and the included angle a is common to both triangles. hence iv.
the base fc is equal to gb, the angle afc is equal to agb, and the angle acf is equal to the angle abg.
again, because af is equal to ag (const.), and ab to ac (hyp.), the remainder, bf, is equal to cg (axiom iii); and we have proved that fc is equal to gb, and the angle bfc equal to the angle cgb.
hence the two triangles bfc, cgb have the two sides bf, fc in one equal to the two sides cg, gb in the other; and the angle bfc contained by the two sides of one equal to the angle cgb contained by the two sides of the other. therefore iv.
these triangles have the angle fbc equal to the angle gcb, and these are the angles below the base. also the angle fcb equal to gbc; but the whole angle fca has been proved equal to the whole angle gba.
hence the remaining angle acb is equal to the remaining angle abc, and these are the angles at the base.
observation.-the great difficulty which beginners find in this proposition is due to the fact that the two triangles acf, abg overlap each other.
the teacher should make these triangles separate, as in the annexed diagram, and point out the corresponding parts thus:-
the student should also be shown how to apply one of the triangles to the other, so as to bring them into coincidence. similar illustrations may be given of the triangles bfc, cgb.
the following is a very easy proof of this proposition. conceive the acb to be turned, without alteration, round the line ac, until it falls on the other side.
let acd be its new position; then the angle adc of the displaced triangle is evidently equal to the angle abc, with which it originally coincided.
again, the two s bac, cad have the sides ba, ac of one respectively equal to the sides ac, ad of the other, and the included angles equal; therefore iv.
the angle acb opposite to the side ab is equal to the angle adc opposite to the side ac; but the angle adc is equal to abc; therefore acb is equal to abc.
cor.-every equilateral triangle is equiangular.
def.-a line in any figure, such as ac in the preceding diagram, which is such that, by folding the plane of the figure round it, one part of the diagram will coincide with the other, is called an axis of symmetry of the figure.
1. prove that the angles at the base are equal without producing the sides. also by producing the sides through the vertex.
2. prove that the line joining the point a to the intersection of the lines cf and bg is an axis of symmetry of the figure.
3. if two isosceles triangles be on the same base, and be either at the same or at opposite sides of it, the line joining their vertices is an axis of symmetry of the figure formed by them.
4. show how to prove this proposition by assuming as an axiom that every angle has a bisector.
5. each diagonal of a lozenge is an axis of symmetry of the lozenge.
6. if three points be taken on the sides of an equilateral triangle, namely, one on each side, at equal distances from the angles, the lines joining them form a new equilateral triangle.
if two angles (b, c) of a triangle be equal, the sides (ac, ab) opposite to them are also equal.
dem.-if ab, ac are not equal, one must be greater than the other. suppose ab is the greater, and that the part bd is equal to ac. join cd (post. i.). then the two triangles dbc, acb have bd equal to ac, and bc common to both.
therefore the two sides db, bc in one are equal to the two sides ac, cb in the other; and the angle dbc in one is equal to the angle acb in the other (hyp). therefore iv.
the triangle dbc is equal to the triangle acb-the less to the greater, which is absurd; hence ac, ab are not unequal, that is, they are equal.
1. what is the hypothesis in this proposition?
2. what proposition is this the converse of?
3. what is the obverse of this proposition?
6. how does euclid generally prove converse propositions?
7. what false assumption is made in the demonstration?
if two triangles (acb, adb) on the same base (ab) and on the same side of it have one pair of conterminous sides (ac, ad) equal to one another, the other pair of conterminous sides (bc, bd) must be unequal.
dem.-1. let the vertex of each triangle be without the other. join cd. then because ad is equal to ac (hyp.), the triangle acd is isosceles; therefore v.
the angle acd is equal to the angle adc; but adc is greater than bdc (axiom ix.); therefore acd is greater than bdc: much, more is bcd greater than bdc. now if the side bd were equal to bc, the angle bcd would be equal to bdc v.
; but it has been proved to be greater. hence bd is not equal to bc.
2. let the vertex of one triangle adb fall within the other triangle acb. produce the sides ac, ad to e and f. then because ac is equal to ad (hyp.), the triangle acd is isosceles, and v.
the external angles ecd, fdc at the other side of the base cd are equal; but ecd is greater than bcd (axiom ix.). therefore fdc is greater than bcd: much more is bdc greater than bcd; but if bc were equal to bd, the angle bdc would be equal to bcd v.
3. if the vertex d of the second triangle fall on the line bc, it is evident that bc and bd are unequal.
1. what use is made of prop. vii.? ans. as a lemma to prop. viii.
2. in the demonstration of prop. vii. the contrapositive of prop. v. occurs; show where.
3. show that two circles can intersect each other only in one point on the same side of the line joining their centres, and hence that two circles cannot have more than two points of intersection.
if two triangles (abc, def) have two sides (ab, ac) of one respectively equal to two sides (de, df) of the other, and have also the base (bc) of one equal to the base (ef) of the other; then the two triangles shall be equal, and the angles of one shall be
respectively equal to the angles of the other-namely, those shall be equal to which the equal sides are opposite.
dem.-let the triangle abc be applied to def, so that the point b will coincide with e, and the line bc with the line ef; then because bc is equal to ef, the point c shall coincide with f.
then if the vertex a fall on the same side of ef as the vertex d, the point a must coincide with d; for if not, let it take a different position g; then we have eg equal to ba, and ba is equal to ed (hyp.).
hence (axiom i.) eg is equal to ed: in like manner, fg is equal to fd, and this is impossible vii. .
hence the point a must coincide with d, and the triangle abc agrees in every respect with the triangle def; and therefore the three angles of one are respectively equal to the three angles of the other-namely, a to d, b to e, and c to f, and the two
this proposition is the converse of iv., and is the second case of the congruence of triangles in the elements.
philo's proof.-let the equal bases be applied as in the foregoing proof, but let the vertices be on the opposite sides; then let bgc be the position which edf takes. join ag. then because bg ba, the angle bag bga. in like manner the angle cag cga.
hence the whole angle bac bgc; but bgc edf therefore bac edf.
to bisect a given rectilineal angle (bac).
sol.-in ab take any point d, and cut off iii. ae equal to ad. join de (post. i.), and upon it, on the side remote from a, describe the equilateral triangle def i. join af. af bisects the given angle bac.
dem.-the triangles daf, eaf have the side ad equal to ae (const.) and af common; therefore the two sides da, af are respectively equal to ea, af, and the base df is equal to the base ef, because they are the sides of an equilateral triangle (def. xxi.).
therefore viii. the angle daf is equal to the angle eaf; hence the angle bac is bisected by the line af.
cor.-the line af is an axis of symmetry of the figure.
1. why does euclid describe the equilateral triangle on the side remote from a?
2. in what case would the construction fail, if the equilateral triangle were described on the other side of de?
1. prove this proposition without using prop. viii.
2. prove that af is perpendicular to de.
3. prove that any point in af is equally distant from the points d and e.
4. prove that any point in af is equally distant from the lines ab, ac.
to bisect a given finite right line (ab).
sol.-upon ab describe an equilateral triangle acb i. . bisect the angle acb by the line cd ix. , meeting ab in d, then ab is bisected in d.
dem.-the two triangles acd, bcd, have the side ac equal to bc, being the sides of an equilateral triangle, and cd common.
therefore the two sides ac, cd in one are equal to the two sides bc, cd in the other; and the angle acd is equal to the angle bcd (const.). therefore the base ad is equal to the base db iv. . hence ab is bisected in d.
1. show how to bisect a finite right line by describing two circles.
2. every point equally distant from the points a, b is in the line cd.
from a given point (c) in a given right line (ab) to draw a right line perpendicular to the given line.
sol.-in ac take any point d, and make ce equal to cd iii. . upon de describe an equilateral triangle dfe i. . join cf. then cf shall be at right angles to ab.
dem.-the two triangles dcf, ecf have cd equal to ce (const.) and cf common; therefore the two sides cd, cf in one are respectively equal to the two sides ce, cf in the other, and the base df is equal to the base ef, being the sides of an equilateral
triangle (def. xxi.); therefore viii. the angle dce is equal to the angle ecf, and they are adjacent angles. therefore (def. xiii.) each of them is a right angle, and cf is perpendicular to ab at the point c.
1. the diagonals of a lozenge bisect each other perpendicularly.
2. prove prop. xi. without using prop. viii.
3. erect a line at right angles to a given line at one of its extremities without producing the line.
4. find a point in a given line that shall be equally distant from two given points.
5. find a point in a given line such that, if it be joined to two given points on opposite sides of the line, the angle formed by the joining lines shall be bisected by the given line.
6. find a point that shall be equidistant from three given points.
to draw a perpendicular to a given indefinite right line (ab) from a given point (c) without it.
sol.-take any point d on the other side of ab, and describe (post. iii.) a circle, with c as centre, and cd as radius, meeting ab in the points f and g. bisect fg in h x. . join ch (post. i.). ch shall be at right angles to ab.
dem.-join cf, cg. then the two triangles fhc, ghc have fh equal to gh (const.), and hc common; and the base cf equal to the base cg, being radii of the circle fdg (def. xxxii.). therefore the angle chf is equal to the angle chg viii.
, and, being adjacent angles, they are right angles (def. xiii.). therefore ch is perpendicular to ab.
1. prove that the circle cannot meet ab in more than two points.
2. if one angle of a triangle be equal to the sum of the other two, the triangle can be divided into the sum of two isosceles triangles, and the base is equal to twice the line from its middle point to the opposite angle.
the adjacent angles (abc, abd) which one right line (ab) standing on another (cd) makes with it are either both right angles, or their sum is equal to two right angles.
dem.-if ab is perpendicular to cd, as in fig. 1, the angles abc, abd are right angles. if not, draw be perpendicular to cd xi. . now the angle cba is equal to the sum of the two angles cbe, eba (def. xi.).
hence, adding the angle abd, the sum of the angles cba, abd is equal to the sum of the three angles cbe, eba, abd. in like manner, the sum of the angles cbe, ebd is equal to the sum of the three angles cbe, eba, abd.
and things which are equal to the same are equal to one another. therefore the sum of the angles cba, abd is equal to the sum of the angles cbe, ebd; but cbe, ebd are right angles; therefore the sum of the angles cba, abd is two right angles.
or thus: denote the angle eba by ; then evidently
cor. 1.-the sum of two supplemental angles is two right angles.
cor. 2.-two right lines cannot have a common segment.
cor. 3.-the bisector of any angle bisects the corresponding re-entrant angle.
cor. 4.-the bisectors of two supplemental angles are at right angles to each other.
cor. 5.-the angle eba is half the difference of the angles cba, abd.
if at a point (b) in a right line (ba) two other right lines (cb, bd) on opposite sides make the adjacent angles (cba, abd) together equal to two right angles, these two right lines form one continuous line.
dem.-if bd be not the continuation of cb, let be be its continuation.
now, since cbe is a right line, and ba stands on it, the sum of the angles cba, abe is two right angles (xiii.); and the sum of the angles cba, abd is two right angles (hyp.); therefore the sum of the angles cba, abe is equal to the sum of the angles cba,
abd. reject the angle cba, which is common, and we have the angle abe equal to the angle abd-that is, a part equal to the whole-which is absurd. hence bd must be in the same right line with cb.
if two right lines (ab, cd) intersect one another, the opposite angles are equal (cea deb, and bec aed).
dem.-because the line ae stands on cd, the sum of the angles cea, aed is two right angles xiii.
; and because the line ce stands on ab, the sum of the angles bec, cea is two right angles; therefore the sum of the angles cea, aed is equal to the sum of the angles bec, cea. reject the angle cea, which is common, and we have the angle aed equal to bec.
in like manner, the angle cea is equal to deb.
the foregoing proof may be briefly given, by saying that opposite angles are equal because they have a common supplement.
questions for examination on props. xiii., xiv., xv.
1. what problem is required in euclid's proof of prop. xiii.?
2. what theorem? ans. no theorem, only the axioms.
3. if two lines intersect, how many pairs of supplemental angles do they make?
4. what relation does prop. xiv. bear to prop. xiii.?
5. what three lines in prop. xiv. are concurrent?
6. what caution is required in the enunciation of prop. xiv.?
7. state the converse of prop. xv. prove it.
8. what is the subject of props. xiii., xiv., xv.? ans. angles at a point.
if any side (bc) of a triangle (abc) be produced, the exterior angle (acd) is greater than either of the interior non-adjacent angles.
dem.-bisect ac in e x. . join be (post. i.). produce it, and from the produced part cut off ef equal to be iii . join cf.
now because ec is equal to ea (const.), and ef is equal to eb, the triangles cef, aeb have the sides ce, ef in one equal to the sides ae, eb in the other; and the angle cef equal to aeb xv. . therefore iv.
the angle ecf is equal to eab; but the angle acd is greater than ecf; therefore the angle acd is greater than eab.
in like manner it may be shown, if the side ac be produced, that the exterior angle bcg is greater than the angle abc; but bcg is equal to acd xv. . hence acd is greater than abc.
therefore acd is greater than either of the interior non-adjacent angles a or b of the triangle abc.
cor. 1.-the sum of the three interior angles of the triangle bcf is equal to the sum of the three interior angles of the triangle abc.
cor. 2.-the area of bcf is equal to the area of abc.
cor. 3.-the lines ba and cf, if produced, cannot meet at any finite distance. for, if they met at any finite point x, the triangle cax would have an exterior angle bac equal to the interior angle acx.
any two angles (b, c) of a triangle (abc) are together less than two right angles.
dem.-produce bc to d; then the exterior angle acd is greater than abc xvi. : to each add the angle acb, and we have the sum of the angles acd, acb greater than the sum of the angles abc, acb; but the sum of the angles acd, acb is two right angles xiii. .
therefore the sum of the angles abc, acb is less than two right angles.
in like manner we may show that the sum of the angles a, b, or of the angles a, c, is less than two right angles.
cor. 1.-every triangle must have at least two acute angles.
cor. 2.-if two angles of a triangle be unequal, the lesser must be acute.
prove prop. xvii. without producing a side.
if in any triangle (abc) one side (ac) be greater than another (ab), the angle opposite to the greater side is grater than the angle opposite to the less.
dem.-from ac cut off ad equal to ab iii . join bd (post. i.). now since ab is equal to ad, the triangle abd is isosceles; therefore v. the angle adb is equal to abd; but the angle adb is greater than the angle acb xvi. ; therefore abd is greater than acb.
much more is the angle abc greater than the angle acb.
or thus: from a as centre, with the lesser side ab as radius, describe the circle bed, cutting bc in e. join ae. now since ab is equal to ae, the angle aeb is equal to abe; but aeb is greater than acb (xvi.); therefore abe is greater than acb.
1. if in the second method the circle cut the line cb produced through b, prove the proposition.
2. this proposition may be proved by producing the less side.
3. if two of the opposite sides of a quadrilateral be respectively the greatest and least, the angles adjacent to the least are greater than their opposite angles.
4. in any triangle, the perpendicular from the vertex opposite the side which is not less than either of the remaining sides falls within the triangle.
if one angle (b) of a triangle (abc) be greater than another angle (c), the side (ac) which it opposite to the greater angle is greater than the side (ab) which is opposite to the less.
dem.-if ac be not greater than ab, it must be either equal to it or less than it. let us examine each case:-
1. if ac were equal to ab, the triangle acb would be isosceles, and then the angle b would be equal to c v. ; but it is not by hypothesis; therefore ab is not equal to ac.
2. if ac were less than ab, the angle b would be less than the angle c xviii. ; but it is not by hypothesis; therefore ac is not less than ab; and since ac is neither equal to ab nor less than it, it must be greater.
1. prove this proposition by a direct demonstration.
2. a line from the vertex of an isosceles triangle to any point in the base is less than either of the equal sides, but greater if the point be in the base produced.
3. three equal lines could not be drawn from the same point to the same line.
4. the perpendicular is the least line which can be drawn from a given point to a given line; and of all others that may be drawn to it, that which is nearest to the perpendicular is less than any one more remote.
5. if in the fig., prop. xvi., ab be the greatest side of the abc, bf is the greatest side of the fbc, and the angle bfc is less than half the angle abc.
6. if abc be a having ab not greater than ac, a line ag, drawn from a to any point g in bc, is less than ac. for the angle acb xviii. is not greater than abc; but agc xvi. is greater than abc; therefore agc is greater than acg. hence ac is greater than ag.
the sum of any two sides (ba, ac) of a triangle (abc) is greater than the third.
dem.-produce ba to d (post. ii.), and make ad equal to ac iii. . join cd.
then because ad is equal to ac, the angle acd is equal to adc (v.); therefore the angle bcd is greater than the angle bdc; hence the side bd opposite to the greater angle is greater than bc opposite to the less xix. .
again, since ac is equal to ad, adding ba to both, we have the sum of the sides ba, ac equal to bd. therefore the sum of ba, ac is greater than bc.
or thus: bisect the angle bac by ae ix. then the angle bea is greater than eac; but eac eab (const.); therefore the angle bea is greater than eab. hence ab is greater than be xix. . in like manner ac is greater than ec.
therefore the sum of ba, ac is greater than bc.
1. in any triangle, the difference between any two sides is less than the third.
2. if any point within a triangle be joined to its angular points, the sum of the joining lines is greater than its semiperimeter.
3. if through the extremities of the base of a triangle, whose sides are unequal, lines be drawn to any point in the bisector of the vertical angle, their difference is less than the difference of the sides.
4. if the lines be drawn to any point in the bisector of the external vertical angle, their sum is greater than the sum of the sides.
5. any side of any polygon is less than the sum of the remaining sides.
6. the perimeter of any triangle is greater than that of any inscribed triangle, and less than that of any circumscribed triangle.
7. the perimeter of any polygon is greater than that of any inscribed, and less than that of any circumscribed, polygon of the same number of sides.
8. the perimeter of a quadrilateral is greater than the sum of its diagonals.
def.-a line drawn from any angle of a triangle to the middle point of the opposite side is called a median of the triangle.
9. the sum of the three medians of a triangle is less than its perimeter.
10. the sum of the diagonals of a quadrilateral is less than the sum of the lines which can be drawn to its angular points from any point except the intersection of the diagonals.
if two lines (bd, cd) be drawn to a point (d) within a triangle from the extremities of its base (bc), their sum is less than the sum of the remaining sides (ba, ca), but they contain a greater angle.
dem.-1. produce bd (post. ii.) to meet ac in e. then, in the triangle bae, the sum of the sides ba, ae is greater than the side be xx. : to each add ec, and we have the sum of ba, ac greater than the sum of be, ec.
again, the sum of the sides de, ec of the triangle dec is greater than dc: to each add bd, and we get the sum of be, ec greater than the sum of bd, dc; but it has been proved that the sum of ba, ac is greater than the sum of be, ec.
therefore much more is the sum of ba, ac greater than the sum of bd, dc.
2. the external angle bdc of the triangle dec is greater than the internal angle bec xvi. , and the angle bec, for a like reason, is greater than bac. therefore much more is bdc greater than bac.
part 2 may be proved without producing either of the sides bd, dc. thus: join ad and produce it to meet bc in f; then the angle bdf is greater than the angle baf xvi. , and fdc is greater than fac. therefore the whole angle bdc is greater than bac.
1. the sum of the lines drawn from any point within a triangle to its angular points is less than the perimeter. (compare ex. 2, last prop.)
2. if a convex polygonal line abcd lie within a convex polygonal line amnd terminating in the same extremities, the length of the former is less than that of the latter.
to construct a triangle whose three sides shall be respectively equal to three given lines (a, b, c), the sum of every two of which is greater than the third.
sol.-take any right line de, terminated at d, but unlimited towards e, and cut off iii. df equal to a, fg equal to b, and gh equal to c. with f as centre, and fd as radius, describe the circle kdl (post.
iii.); and with g as centre, and gh as radius, describe the circle khl, intersecting the former circle in k. join kf, kg. kfg is the triangle required.
dem.-since f is the centre of the circle kdl, fk is equal to fd; but fd is equal to a (const.); therefore (axiom i.) fk is equal to a.
in like manner gk is equal to c, and fg is equal to b (const.) hence the three sides of the triangle kfg are respectively equal to the three lines a, b, c.
1. what is the reason for stating in the enunciation that the sum of every two of the given lines must be greater than the third?
2. prove that when that condition is fulfilled the two circles must intersect.
3. under what conditions would the circles not intersect?
4. if the sum of two of the lines were equal to the third, would the circles meet? prove that they would not intersect.
at a given point (a) in a given right line (ab) to make an angle equal to a given rectilineal angle (def).
sol.-in the sides ed, ef of the given angle take any arbitrary points d and f. join df, and construct xxii.
the triangle bac, whose sides, taken in order, shall be equal to those of def-namely, ab equal to ed, ac equal to ef, and cb equal to fd; then the angle bac will viii. be equal to def. hence it is the required angle.
1. construct a triangle, being given two sides and the angle between them.
2. construct a triangle, being given two angles and the side between them.
3. construct a triangle, being given two sides and the angle opposite to one of them.
4. construct a triangle, being given the base, one of the angles at the base, and the sum or difference of the sides.
5. given two points, one of which is in a given line, it is required to find another point in the given line, such that the sum or difference of its distances from the former points may be given. show that two such points may be found in each case.
if two triangles (abc, def) have two sides (ab, ac) of one respectively equal to two sides (de, df) of the other, but the contained angle (bac) of one greater than the contained angle (edf) of the other, the base of that which has the greater angle is
dem.-of the two sides ab, ac, let ab be the one which is not the greater, and with it make the angle bag equal to edf xxiii. . then because ab is not greater than ac, ag is less than ac xix., exer. 6 . produce ag to h, and make ah equal to df or ac iii. .
in the triangles bah, edf, we have ab equal to de (hyp.), ah equal to df (const.), and the angle bah equal to the angle edf (const.); therefore the base iv. bh is equal to ef.
again, because ah is equal to ac (const.), the triangle ach is isosceles; therefore the angle ach is equal to ahc v.
; but ach is greater than bch; therefore ahc is greater than bch: much more is the angle bhc greater than bch, and the greater angle is subtended by the greater side xix. .
therefore bc is greater than bh; but bh has been proved to be equal to ef; therefore bc is greater than ef.
the concluding part of this proposition may be proved without joining ch, thus:-
or thus: bisect the angle cah by ao. join oh. now in the s cao, hao we have the sides ca, ao in one equal to the sides ah, ao in the other, and the contained angles equal; therefore the base oc is equal to the base oh iv.
: to each add bo, and we have bc equal to the sum of bo, oh; but the sum of bo, oh is greater than bh xx. . therefore bc is greater than bh, that is, greater than ef.
1. prove this proposition by making the angle abh to the left of ab.
2. prove that the angle bca is greater than efd.
if two triangles (abc, def) have two sides (ab, ac) of one respectively equal to two sides (de, df) of the other, but the base (bc) of one greater than the base (ef) of the other, the angle (a) contained by the sides of that which has the greater base is
greater them the angle (d) contained by the sides of the other.
dem.-if the angle a be not greater than d, it must be either equal to it or less than it. we shall examine each case:-
1. if a were equal to d, the triangles abc, def would have the two sides ab, ac of one respectively equal to the two sides de, df of the other, and the angle a contained by the two sides of one equal to the angle d contained by the two sides of the other.
hence iv. bc would be equal to ef; but bc is, by hypothesis, greater than ef; hence the angle a is not equal to the angle d.
2. if a were less than d, then d would be greater than a, and the triangles def, abc would have the two sides de, df of one respectively equal to the two sides ab, ac of the other, and the angle d contained by the two sides of one greater than the angle a
contained by the two sides of the other. hence xxiv. ef would be greater than bc; but ef (hyp.) is not greater than bc. therefore a is not less than d, and we have proved that it is not equal to it; therefore it must be greater.
or thus, directly: construct the triangle acg, whose three sides ag, gc, ca shall be respectively equal to the three sides de, ef, fd of the triangle def xxii. . join bg. then because bc is greater than ef, bc is greater than cg. hence xviii.
the angle bgc is greater than gbc; and make (xxiii.) the angle bgh equal to gbh, and join ah. then vi. bh is equal to gh. therefore the triangles abh, agh have the sides ab, ah of one equal to the sides ag, ah of the other, and the base bh equal to gh.
therefore viii. the angle bah is equal to gah. hence the angle bac is greater than cag, and therefore greater than edf.
demonstrate this proposition directly by cutting off from bc a part equal to ef.
if two triangles (abc, def) have two angles (b, c) of one equal respectively to two angles (e, f) of the other, and a side of one equal to a side similarly placed with respect to the equal angles of the other, the triangles are equal in every respect.
dem.-this proposition breaks up into two according as the sides given to be equal are the sides adjacent to the equal angles, namely bc and ef, or those opposite equal angles.
1. let the equal sides be bc and ef; then if de be not equal to ab, suppose ge to be equal to it.
join gf; then the triangles abc, gef have the sides ab, bc of one respectively equal to the sides ge, ef of the other, and the angle abc equal to the angle gef (hyp.); therefore iv.
the angle acb is equal to the angle gfe; but the angle acb is (hyp.) equal to dfe; hence gfe is equal to dfe-a part equal to the whole, which is absurd; therefore ab and de are not unequal, that is, they are equal.
consequently the triangles abc, def have the sides ab, bc of one respectively equal to the sides de, ef of the other; and the contained angles abc and def equal; therefore iv. ac is equal to df, and the angle bac is equal to the angle edf.
2. let the sides given to be equal be ab and de; it is required to prove that bc is equal to ef, and ac to df. if bc be not equal to ef, suppose bg to be equal to it. join ag.
then the triangles abg, def have the two sides ab, bg of one respectively equal to the two sides de, ef of the other, and the angle abg equal to the angle def; therefore iv. the angle agb is equal to dfe; but the angle acb is equal to dfe (hyp.).
hence (axiom i.) the angle agb is equal to acb, that is, the exterior angle of the triangle acg is equal to the interior and non-adjacent angle, which xvi. is impossible.
hence bc must be equal to ef, and the same as in 1, ac is equal to df, and the angle bac is equal to the angle edf.
this proposition, together with iv. and viii., includes all the cases of the congruence of two triangles. part i. may be proved immediately by superposition.
for it is evident if abc be applied to def, so that the point b shall coincide with e, and the line bc with ef, since bc is equal to ef, the point c shall coincide with f; and since the angles b, c are respectively equal to the angles e, f, the lines ba,
ca shall coincide with ed and fd. hence the triangles are congruent.
def.-if every point on a geometrical figure satisfies an assigned condition, that figure is called the locus of the point satisfying the condition. thus, for example, a circle is the locus of a point whose distance from the centre is equal to its radius.
1. the extremities of the base of an isosceles triangle are equally distant from any point in the perpendicular from the vertical angle on the base.
2. if the line which bisects the vertical angle of a triangle also bisects the base, the triangle is isosceles.
3. the locus of a point which is equally distant from two fixed lines is the pair of lines which bisect the angles made by the fixed lines.
4. in a given right line find a point such that the perpendiculars from it on two given lines may be equal. state also the number of solutions.
5. if two right-angled triangles have equal hypotenuses, and an acute angle of one equal to an acute angle of the other, they are congruent.
6. if two right-angled triangles have equal hypotenuses, and a side of one equal to a side of the other, they are congruent.
7. the bisectors of the three internal angles of a triangle are concurrent.
8. the bisectors of two external angles and the bisector of the third internal angle are concurrent.
9. through a given point draw a right line, such that perpendiculars on it from two given points on opposite sides may be equal to each other.
10. through a given point draw a right line intersecting two given lines, and forming an isosceles triangle with them.
def. i.-if two right lines in the same plane be such that, when produced indefinitely, they do not meet at any finite distance, they are said to be parallel.
def. ii.-a parallelogram is a quadrilateral, both pairs of whose opposite sides are parallel.
def. iii.-the right line joining either pair of opposite angles of a quadrilateral is called a diagonal.
def. iv.-if both pairs of opposite sides of a quadrilateral be produced to meet, the right line joining their points of intersection is called its third diagonal.
def. v.-a quadrilateral which has one pair of opposite sides parallel is called a trapezium.
def. vi.-if from the extremities of one right line perpendiculars be drawn to another, the intercept between their feet is called the projection of the first line on the second.
def. vii.-when a right line intersects two other right lines in two distinct points it makes with them eight angles, which have received special names in relation to one another.
thus, in the figure-1, 2; 7, 8 are called exterior angles; 3, 4; 5, 6, interior angles. again, 4; 6; 3, 5 are called alternate angles; lastly, 1, 5; 2, 6; 3, 8; 4, 7 are called corresponding angles.
if a right line (ef) intersecting two right lines (ab, cd) makes the alternate angles (aef, efd) equal to each other, these lines are parallel.
dem.-if ab and cd are not parallel they must meet, if produced, at some finite distance: if possible let them meet in g; then the figure egf is a triangle, and the angle aef is an exterior angle, and efd a non-adjacent interior angle. hence xvi.
aef is greater than efd; but it is also equal to it (hyp.), that is, both equal and greater, which is absurd. hence ab and cd are parallel.
or thus: bisect ef in o; turn the whole figure round o as a centre, so that ef shall fall on itself; then because oe of, the point e shall fall on f; and because the angle aef is equal to the angle efd, the line ea will occupy the place of fd, and the
line fd the place of ea; therefore the lines ab, cd interchange places, and the figure is symmetrical with respect to the point o.
hence, if ab, cd meet on one side of o, they must also meet on the other side; but two right lines cannot enclose a space (axiom x.); therefore they do not meet at either side. hence they are parallel.
if a right line (ef) intersecting two right lines (ab, cd) makes the exterior angle (egb) equal to its corresponding interior angle (ghd), or makes two interior angles (bgh, ghd) on the same side equal to two right angles, the two right lines are parallel.
dem.-1. since the lines ab, ef intersect, the angle agh is equal to egb xv. ; but egb is equal to ghd (hyp.); therefore agh is equal to ghd, and they are alternate angles. hence xxvii. ab is parallel to cd.
2. since agh and bgh are adjacent angles, their sum is equal to two right angles xiii.
; but the sum of bgh and ghd is two right angles (hyp.); therefore rejecting the angle bgh we have agh equal ghd, and they are alternate angles; therefore ab is parallel to cd xxvii. .
if a right line (ef) intersect two parallel right lines (ab, cd), it makes-1. the alternate angles (agh,ghd) equal to one another; 2. the exterior angle (egb) equal to the corresponding interior angle (ghd); 3.
the two interior angles (bgh, ghd) on the same side equal to two right angles.
dem.-if the angle agh be not equal to ghd, one must be greater than the other.
let agh be the greater; to each add bgh, and we have the sum of the angles agh, bgh greater than the sum of the angles bgh, ghd; but the sum of agh, bgh is two right angles; therefore the sum of bgh, ghd is less than two right angles, and therefore (axiom
xii.) the lines ab, cd, if produced, will meet at some finite distance: but since they are parallel (hyp.) they cannot meet at any finite distance. hence the angle agh is not unequal to ghd-that is, it is equal to it.
2. since the angle egb is equal to agh xv. , and ghd is equal to agh (1), egb is equal to ghd (axiom i.).
3. since agh is equal to ghd (1), add hgb to each, and we have the sum of the angles agh, hgb equal to the sum of the angles ghd, hgb; but the sum of the angles agh, hgb xiii.
is two right angles; therefore the sum of the angles bgh, ghd is two right angles.
1. demonstrate both parts of prop. xxviii. without using prop. xxvii.
2. the parts of all perpendiculars to two parallel lines intercepted between them are equal.
3. if acd, bcd be adjacent angles, any parallel to ab will meet the bisectors of these angles in points equally distant from where it meets cd.
4. if through the middle point o of any right line terminated by two parallel right lines any other secant be drawn, the intercept on this line made by the parallels is bisected in o.
5. two right lines passing through a point equidistant from two parallels intercept equal portions on the parallels.
6. the perimeter of the parallelogram, formed by drawing parallels to two sides of an equilateral triangle from any point in the third side, is equal to twice the side.
7. if the opposite sides of a hexagon be equal and parallel, its diagonals are concurrent.
8. if two intersecting right lines be respectively parallel to two others, the angle between the former is equal to the angle between the latter.
for if ab, ac be respectively parallel to de, df, and if ac, de meet in g, the angles a, d are each equal to g xxix. .
if two right lines (ab, cd) be parallel to the same right line (ef), they are parallel to one another.
dem.-draw any secant ghk. then since ab and ef are parallel, the angle agh is equal to ghf xxix. . in like manner the angle ghf is equal to hkd xxix. . therefore the angle agk is equal to the angle gkd (axiom i.). hence xxvii. ab is parallel to cd.
through a given point (c) to draw a right line parallel to a given right line.
sol.-take any point d in ab. join cd (post. i.), and make the angle dcf equal to the angle adc xxiii. . the line ce is parallel to ab xxvii. .
1. given the altitude of a triangle and the base angles, construct it.
2. from a given point draw to a given line a line making with it an angle equal to a given angle. show that there will be two solutions.
3. prove the following construction for trisecting a given line ab:-on ab describe an equilateral abc.
bisect the angles a, b by the lines ad, bd, meeting in d; through d draw parallels to ac, bc, meeting ab in e, f: e, f are the points of trisection of ab.
4. inscribe a square in a given equilateral triangle, having its base on a given side of the triangle.
5. draw a line parallel to the base of a triangle so that it may be-1. equal to the intercept it makes on one of the sides from the extremity of the base; 2. equal to the sum of the two intercepts on the sides from the extremities of the base; 3.
equal to their difference. show that there are two solutions in each case.
6. through two given points in two parallel lines draw two lines forming a lozenge with the given parallels.
7. between two lines given in position place a line of given length which shall be parallel to a given line. show that there are two solutions.
if any side (ab) of a triangle (abc) be produced (to d), the external angle (cbd) is equal to the sum of the two internal non-adjacent angles (a, c), and the sum of the three internal angles is equal to two right angles.
dem.-draw be parallel to ac xxxi. . now since bc intersects the parallels be, ac, the alternate angles ebc, acb are equal xxix. . again, since ab intersects the parallels be, ac, the angle ebd is equal to bac xxix.
; hence the whole angle cbd is equal to the sum of the two angles acb, bac: to each of these add the angle abc and we have the sum of cbd, abc equal to the sum of the three angles acb, bac, abc: but the sum of cbd, abc is two right angles xiii.
; hence the sum of the three angles acb, bac, abc is two right angles.
cor. 1.-if a right-angled triangle be isosceles, each base angle is half a right angle.
cor. 2.-if two triangles have two angles in one respectively equal to two angles in the other, their remaining angles are equal.
cor. 3.-since a quadrilateral can be divided into two triangles, the sum of its angles is equal to four right angles.
cor. 4.-if a figure of n sides be divided into triangles by drawing diagonals from any one of its angles there will be (n 2) triangles; hence the sum of its angles is equal 2(n 2) right angles.
cor. 5.-if all the sides of any convex polygon be produced, the sum of the external angles is equal to four right angles.
cor. 6.-each angle of an equilateral triangle is two-thirds of a right angle.
cor. 7.-if one angle of a triangle be equal to the sum of the other two, it is a right angle.
cor. 8.-every right-angled triangle can be divided into two isosceles triangles by a line drawn from the right angle to the hypotenuse.
2. any angle of a triangle is obtuse, right, or acute, according as the opposite side is greater than, equal to, or less than, twice the median drawn from that angle.
3. if the sides of a polygon of n sides be produced, the sum of the angles between each alternate pair is equal to 2(n 4) right angles.
4. if the line which bisects the external vertical angle be parallel to the base, the triangle is isosceles.
5. if two right-angled s abc, abd be on the same hypotenuse ab, and the vertices c and d be joined, the pair of angles subtended by any side of the quadrilateral thus formed are equal.
6. the three perpendiculars of a triangle are concurrent.
7. the bisectors of two adjacent angles of a parallelogram are at right angles.
8. the bisectors of the external angles of a quadrilateral form a circumscribed quadrilateral, the sum of whose opposite angles is equal to two right angles.
9. if the three sides of one triangle be respectively perpendicular to those of another triangle, the triangles are equiangular.
10. construct a right-angled triangle, being given the hypotenuse and the sum or difference of the sides.
11. the angles made with the base of an isosceles triangle by perpendiculars from its extremities on the equal sides are each equal to half the vertical angle.
12. the angle included between the internal bisector of one base angle of a triangle and the external bisector of the other base angle is equal to half the vertical angle.
13. in the construction of prop. xviii. prove that the angle dbc is equal to half the difference of the base angles.
14. if a, b, c denote the angles of a , prove that 12(a b), 12(b c), 12(c a) will be the angles of a formed by any side and the bisectors of the external angles between that side and the other sides produced.
the right lines (ac, bd) which join the adjacent extremities of two equal and parallel right lines (ab, cd) are equal and parallel.
dem.-join bc. now since ab is parallel to cd, and bc intersects them, the angle abc is equal to the alternate angle dcb xxix. .
again, since ab is equal to cd, and bc common, the triangles abc, dcb have the sides ab, bc in one respectively equal to the sides dc, cb in the other, and the angles abc, dcb contained by those sides equal; therefore iv.
the base ac is equal to the base bd, and the angle acb is equal to the angle cbd; but these are alternate angles; hence xxvii. ac is parallel to bd, and it has been proved equal to it. therefore ac is both equal and parallel to bd.
1. if two right lines ab, bc be respectively equal and parallel to two other right lines de, ef, the right line ac joining the extremities of the former pair is equal to the right line df joining the extremities of the latter.
2. right lines that are equal and parallel have equal projections on any other right line; and conversely, parallel right lines that have equal projections on another right line are equal.
3. equal right lines that have equal projections on another right line are parallel.
4. the right lines which join transversely the extremities of two equal and parallel right lines bisect each other.
the opposite sides (ab, cd; ac, bd) and the opposite angles (a, d; b, c) of a parallelogram are equal to one another, and either diagonal bisects the parallelogram.
dem.-join bc. since ab is parallel to cd, and bc intersects them, the angle abc is equal to the angle bcd xxix. .
again, since bc intersects the parallels ac, bd, the angle acb is equal to the angle cbd; hence the triangles abc, dcb have the two angles abc, acb in one respectively equal to the two angles bcd, cbd in the other, and the side bc common. therefore xxvi.
ab is equal to cd, and ac to bd; the angle bac to the angle bdc, and the triangle abc to the triangle bdc.
again, because the angle acb is equal to cbd, and dcb equal to abc, the whole angle acd is equal to the whole angle abd.
cor. 1.-if one angle of a parallelogram be a right angle, all its angles are right angles.
cor. 2.-if two adjacent sides of a parallelogram be equal, it is a lozenge.
cor. 3.-if both pairs of opposite sides of a quadrilateral be equal, it is a parallelogram.
cor. 4.-if both pairs of opposite angles of a quadrilateral be equal, it is a parallelogram.
cor. 5.-if the diagonals of a quadrilateral bisect each other, it is a parallelogram.
cor. 6.-if both diagonals of a quadrilateral bisect the quadrilateral, it is a parallelogram.
cor. 7.-if the adjacent sides of a parallelogram be equal, its diagonals bisect its angles.
cor. 8.-if the adjacent sides of a parallelogram be equal, its diagonals intersect at right angles.
cor. 9.-in a right-angled parallelogram the diagonals are equal.
cor. 10.-if the diagonals of a parallelogram be perpendicular to each other, it is a lozenge.
cor. 11.-if a diagonal of a parallelogram bisect the angles whose vertices it joins, the parallelogram is a lozenge.
1. the diagonals of a parallelogram bisect each other.
2. if the diagonals of a parallelogram be equal, all its angles are right angles.
3. divide a right line into any number of equal parts.
4. the right lines joining the adjacent extremities of two unequal parallel right lines will meet, if produced, on the side of the shorter parallel.
5. if two opposite sides of a quadrilateral be parallel but not equal, and the other pair equal but not parallel, its opposite angles are supplemental.
6. construct a triangle, being given the middle points of its three sides.
7. the area of a quadrilateral is equal to the area of a triangle, having two sides equal to its diagonals, and the contained angle equal to that between the diagonals.
parallelograms on the same base (bc) and between the same parallels are equal.
dem.-1. let the sides ad, df of the parallelograms ac, bf opposite to the common base bc terminate in the same point d, then xxxiv. each parallelogram is double of the triangle bcd. hence they are equal to one another.
2. let the sides ad, ef (figures (), ()) opposite to bc not terminate in the same point.
then because abcd is a parallelogram, ad is equal to bc xxxiv. ; and since bcef is a parallelogram, ef is equal to bc; therefore (see fig. ()) take away ed, and in fig. () add ed, and we have in each case ae equal to df, and ba is equal to cd xxxiv. .
hence the triangles bae, cdf have the two sides ba, ae in one respectively equal to the two sides cd, df in the other, and the angle bae xxix. equal to the angle cdf; hence iv.
the triangle bae is equal to the triangle cdf; and taking each of these triangles in succession from the quadrilateral bafc, there will remain the parallelogram bcfe equal to the parallelogram bcda.
or thus: the triangles abe, dcf have xxxiv. the sides ab, be in one respectively equal to the sides dc, cf in the other, and the angle abe equal to the angle dcf xxix., ex. 8 .
hence the triangle abe is equal to the triangle dcf; and, taking each away from the quadrilateral bafc, there will remain the parallelogram bcfe equal to the parallelogram bcda.
observation.-by the second method of proof the subdivision of the demonstration into cases is avoided. it is easy to see that either of the two parallelograms abcd, ebcf can be divided into parts and rearranged so as to make it congruent with the other.
this proposition affords the first instance in the elements in which equality which is not congruence occurs. this equality is expressed algebraically by the symbol , while congruence is denoted by , called also the symbol of identity.
figures that are congruent are said to be identically equal.
parallelograms (bd, fh) on equal bases (bc, fg) and between the same parallels are equal.
dem.-join be, ch. now since fh is a parallelogram, fg is equal to eh xxxiv. ; but bc is equal to fg (hyp.); therefore bc is equal to eh (axiom i.).
hence be, ch, which join their adjacent extremities, are equal and parallel; therefore bh is a parallelogram. again, since the parallelograms bd, bh are on the same base bc, and between the same parallels bc, ah, they are equal xxxv. .
in like manner, since the parallelograms hb, hf are on the same base eh, and between the same parallels eh, bg, they are equal. hence bd and fh are each equal to bh. therefore (axiom i.) bd is equal to fh.
exercise.-prove this proposition without joining be, ch.
triangles (abc, dbc) on the same base (bc) and between the same parallels (ad, bc) are equal.
dem.-produce ad both ways. draw be parallel to ac, and cf parallel to bd xxxi. then the figures aebc, dbcf are parallelograms; and since they are on the same base bc, and between the same parallels bc, ef they are equal xxxv. .
again, the triangle abc is half the parallelogram aebc xxxiv. , because the diagonal ab bisects it. in like manner the triangle dbc is half the parallelogram dbcf, because the diagonal dc bisects it, and halves of equal things are equal (axiom vii.).
therefore the triangle abc is equal to the triangle dbc.
1. if two equal triangles be on the same base, but on opposite sides, the right line joining their vertices is bisected by the base.
2. construct a triangle equal in area to a given quadrilateral figure.
3. construct a triangle equal in area to a given rectilineal figure.
4. construct a lozenge equal to a given parallelogram, and having a given side of the parallelogram for base.
5. given the base and the area of a triangle, find the locus of the vertex.
6. if through a point o, in the production of the diagonal ac of a parallelogram abcd, any right line be drawn cutting the sides ab, bc in the points e, f, and ed, fd be joined, the triangle efd is less than half the parallelogram.
two triangles on equal bases and between the same parallels are equal.
dem.-by a construction similar to the last, we see that the triangles are the halves of parallelograms, on equal bases, and between the same parallels. hence they are the halves of equal parallelograms xxxvi. . therefore they are equal to one another.
1. every median of a triangle bisects the triangle.
2. if two triangles have two sides of one respectively equal to two sides of the other, and the contained angles supplemental, their areas are equal.
3. if the base of a triangle be divided into any number of equal parts, right lines drawn from the vertex to the points of division will divide the whole triangle into as many equal parts.
4. right lines from any point in the diagonal of a parallelogram to the angular points through which the diagonal does not pass, and the diagonal, divide the parallelogram into four triangles which are equal, two by two.
5. if one diagonal of a quadrilateral bisects the other, it also bisects the quadrilateral, and conversely.
6. if two s abc, abd be on the same base ab, and between the same parallels, and if a parallel to ab meet the sides ac, bc in the point e, f; and the sides ad, bd in the point g, h; then ef gh.
7. if instead of triangles on the same base we have triangles on equal bases and between the same parallels, the intercepts made by the sides of the triangles on any parallel to the bases are equal.
8. if the middle points of any two sides of a triangle be joined, the triangle so formed with the two half sides is one-fourth of the whole.
9. the triangle whose vertices are the middle points of two sides, and any point in the base of another triangle, is one-fourth of that triangle.
10. bisect a given triangle by a right line drawn from a given point in one of the sides.
11. trisect a given triangle by three right lines drawn from a given point within it.
12. prove that any right line through the intersection of the diagonals of a parallelogram bisects the parallelogram.
13. the triangle formed by joining the middle point of one of the non-parallel sides of a trapezium to the extremities of the opposite side is equal to half the trapezium.
equal triangles (bac, bdc) on the same base (bc) and on the same side of it are between the same parallels.
dem.-join ad. then if ad be not parallel to bc, let ae be parallel to it, and let it cut bd in e. join ec. now since the triangles bec, bac are on the same base bc, and between the same parallels bc, ae, they are equal xxxvii.
; but the triangle bac is equal to the triangle bdc (hyp.). therefore (axiom i.) the triangle bec is equal to the triangle bdc-that is, a part equal to the whole which is absurd. hence ad must be parallel to bc.
equal triangles (abc, def) on equal bases (bc, ef) which form parts of the same right line, and on the same side of the line, are between the same parallels.
dem.-join ad. if ad be not parallel to bf, let ag be parallel to it. join gf. now since the triangles gef and abc are on equal bases bc, ef, and between the same parallels bf, ag, they are equal xxxviii.
; but the triangle def is equal to the triangle abc (hyp.). hence gef is equal to def (axiom i.)-that is, a part equal to the whole, which is absurd. therefore ad must be parallel to bf.
def.-the altitude of a triangle is the perpendicular from the vertex on the base.
1. triangles and parallelograms of equal bases and altitudes are respectively equal.
2. the right line joining the middle points of two sides of a triangle is parallel to the third; for the medians from the extremities of the base to these points will each bisect the original triangle.
hence the two triangles whose base is the third side and whose vertices are the points of bisection are equal.
3. the parallel to any side of a triangle through the middle point of another bisects the third.
4. the lines of connexion of the middle points of the sides of a triangle divide it into four congruent triangles.
5. the line of connexion of the middle points of two sides of a triangle is equal to half the third side.
6. the middle points of the four sides of a convex quadrilateral, taken in order, are the angular points of a parallelogram whose area is equal to half the area of the quadrilateral.
7. the sum of the two parallel sides of a trapezium is double the line joining the middle points of the two remaining sides.
8. the parallelogram formed by the line of connexion of the middle points of two sides of a triangle, and any pair of parallels drawn through the same points to meet the third side, is equal to half the triangle.
9. the right line joining the middle points of opposite sides of a quadrilateral, and the right line joining the middle points of its diagonals, are concurrent.
if a parallelogram (abcd) and a triangle (ebc) be on the same base (bc) and between the same parallels, the parallelogram is double of the triangle.
dem.-join ac. the parallelogram abcd is double of the triangle abc xxxiv. ; but the triangle abc is equal to the triangle ebc xxxvii. . therefore the parallelogram abcd is double of the triangle ebc.
cor. 1.-if a triangle and a parallelogram have equal altitudes, and if the base of the triangle be double of the base of the parallelogram, the areas are equal.
cor. 2.-the sum of the triangles whose bases are two opposite sides of a parallelogram, and which have any point between these sides as a common vertex, is equal to half the parallelogram.
to construct a parallelogram equal to a given triangle (abc), and having an angle equal to a given angle (d).
sol.-bisect ab in e. join ec. make the angle bef xxiii. equal to d. draw cg parallel to ab xxxi. , and bg parallel to ef. eg is a parallelogram fulfilling the required conditions.
dem.-because ae is equal to eb (const.), the triangle aec is equal to the triangle ebc xxxviii. , therefore the triangle abc is double of the triangle ebc; but the parallelogram eg is also double of the triangle ebc xli.
, because they are on the same base eb, and between the same parallels eb and cg. therefore the parallelogram eg is equal to the triangle abc, and it has (const.) the angle bef equal to d. hence eg is a parallelogram fulfilling the required conditions.
the parallels (ef, gh) through any point (k) in one of the diagonals (ac) of a parallelogram divide it into four parallelograms, of which the two (bk, kd) through which the diagonal does not pass, and which are called the complements of the other two, are
dem.-because the diagonal bisects the parallelograms ac, ak, kc, we have xxxiv. the triangle adc equal to the triangle abc, the triangle ahk equal to aek, and the triangle kfc equal to the triangle kgc.
hence, subtracting the sums of the two last equalities from the first, we get the parallelogram dk equal to the parallelogram kb.
cor. 1.-if through a point k within a parallelogram abcd lines drawn parallel to the sides make the parallelograms dk, kb equal, k is a point in the diagonal ac.
cor. 2.-the parallelogram bh is equal to af, and bf to hc.
cor. 2. supplies an easy demonstration of a fundamental proposition in statics.
1. if ef, gh be parallels to the adjacent sides of a parallelogram abcd, the diagonals eh, gf of two of the four picts into which they divide it and one of the diagonals of abcd are concurrent.
dem.-let eh, gf meet in m; through m draw mp, mj parallel to ab, bc. produce ad, gh, bc to meet mp, and ab, ef, dc to meet mj. now the complement of fj: to each add the pict fl, and we get the figure ofl pict cj. again, the complement ph hk xliii.
: to each add the pict oc, and we get the pict pc figure ofl. hence the pict pc cj. therefore they are about the same diagonal xliii., cor. 1 . hence ac produced will pass through m.
2. the middle points of the three diagonals ac, bd, ef of a quadrilateral abcd are collinear.
dem.-complete the pict aebg. draw dh, ci parallel to ag, bg. join ih, and produce; then ab, cd, ih are concurrent (ex. 1); therefore ih will pass through f. join ei, eh. now xi., ex. 2, 3 the middle points of ei, eh, ef are collinear, but xxxiv., ex.
1 the middle points of ei, eh are the middle points of ac, bd. hence the middle points of ac, bd, ef are collinear.
to a given, right line (ab) to apply a parallelogram which shall be equal to a given triangle (c), and have one of its angles equal to a given angle (d).
sol.-construct the parallelogram befg xlii. equal to the given triangle c, and having the angle b equal to the given angle d, and so that its side be shall be in the same right line with ab. through a draw ah parallel to bg xxxi.
, and produce fg to meet it in h. join hb. then because ha and fe are parallels, and hf intersects them, the sum of the angles ahf, hfe is two right angles xxix.
; therefore the sum of the angles bhf, hfe is less than two right angles; and therefore (axiom xii.) the lines hb, fe, if produced, will meet as at k. through k draw kl parallel to ab xxxi. , and produce ha and gb to meet it in the points l and m.
then am is a parallelogram fulfilling the required conditions.
dem.-the parallelogram am is equal to ge xliii. ; but ge is equal to the triangle c (const.); therefore am is equal to the triangle c. again, the angle abm is equal to ebg xv.
, and ebg is equal to d (const.); therefore the angle abm is equal to d; and am is constructed on the given line; therefore it is the parallelogram required.
to construct a parallelogram equal to a given rectilineal figure (abcd), and having an angle equal to a given rectilineal angle (x).
sol.-join bd. construct a parallelogram eg xlii. equal to the triangle abd, and having the angle e equal to the given angle x; and to the right line gh apply the parallelogram hi equal to the triangle bcd, and having the angle ghk equal to x xliv.
, and so on for additional triangles if there be any. then ei is a parallelogram fulfilling the required conditions.
dem.-because the angles ghk, feh are each equal to x (const.), they are equal to one another: to each add the angle ghe, and we have the sum of the angles ghk, ghe equal to the sum of the angles feh, ghe; but since hg is parallel to ef, and eh intersects
them, the sum of feh, ghe is two right angles xxix. . hence the sum of ghk, ghe is two right angles; therefore eh, hk are in the same right line xiv. .
again, because gh intersects the parallels fg, ek, the alternate angles fgh, ghk are equal xxix.
: to each add the angle hgi, and we have the sum of the angles fgh, hgi equal to the sum of the angles ghk, hgi; but since gi is parallel to hk, and gh intersects them, the sum of the angles ghk, hgi is equal to two right angles xxix. .
hence the sum of the angles fgh, hgi is two right angles; therefore fg and gi are in the same right line xiv. .
again, because eg and hi are parallelograms, ef and ki are each parallel to gh; hence xxx.
ef is parallel to ki, and the opposite sides ek and fi are parallel; therefore ei is a parallelogram; and because the parallelogram eg (const.) is equal to the triangle abd, and hi to the triangle bcd, the whole parallelogram ei is equal to the
rectilineal figure abcd, and it has the angle e equal to the given angle x. hence ei is a parallelogram fulfilling the required conditions.
it would simplify problems xliv., xlv., if they were stated as the constructing of rectangles, and in this special form they would be better understood by the student, since rectangles are the simplest areas to which others are referred.
1. construct a rectangle equal to the sum of two or any number of rectilineal figures.
2. construct a rectangle equal to the difference of two given figures.
on a given right line (ab) to describe a square.
sol.-erect ad at right angles to ab xi. , and make it equal to ab iii. . through d draw dc parallel to ab xxxi. , and through b draw bc parallel to ad; then ac is the square required.
dem.-because ac is a parallelogram, ab is equal to cd xxxiv. ; but ab is equal to ad (const.); therefore ad is equal to cd, and ad is equal to bc xxxiv. . hence the four sides are equal; therefore ac is a lozenge, and the angle a is a right angle.
1. the squares on equal lines are equal; and, conversely, the sides of equal squares are equal.
2. the parallelograms about the diagonal of a square are squares.
3. if on the four sides of a square, or on the sides produced, points be taken equidistant from the four angles, they will be the angular points of another square, and similarly for a regular pentagon, hexagon, c.
4. divide a given square into five equal parts; namely, four right-angled triangles, and a square.
in a right-angled triangle (abc) the square on the hypotenuse (ab) is equal to the sum of the squares on the other two sides (ac, bc).
dem.-on the sides ab, bc, ca describe squares xlvi. . draw cl parallel to ag. join cg, bk.
then because the angle acb is right (hyp.), and ach is right, being the angle of a square, the sum of the angles acb, ach is two right angles; therefore bc, ch are in the same right line xiv. . in like manner ac, cd are in the same right line.
again, because bag is the angle of a square it is a right angle: in like manner cak is a right angle. hence bag is equal to cak: to each add bac, and we get the angle cag equal to kab. again, since bg and ck are squares, ba is equal to ag, and ca to ak.
hence the two triangles cag, kab have the sides ca, ag in one respectively equal to the sides ka, ab in the other, and the contained angles cag, kab also equal. therefore iv.
the triangles are equal; but the parallelogram al is double of the triangle cag xli. , because they are on the same base ag, and between the same parallels ag and cl.
in like manner the parallelogram ah is double of the triangle kab, because they are on the same base ak, and between the same parallels ak and bh; and since doubles of equal things are equal (axiom vi.), the parallelogram al is equal to ah.
in like manner it can be proved that the parallelogram bl is equal to bd. hence the whole square af is equal to the sum of the two squares ah and bd.
or thus: let all the squares be made in reversed directions. join cg, bk, and through c draw ol parallel to ag. now, taking the bac from the right s bag, cak, the remaining s cag, bak are equal.
hence the s cag, bak have the side ca ak, and ag ab, and the cag bak; therefore iv. they are equal; and since xli. the picts al, ah are respectively the doubles of these triangles, they are equal.
in like manner the picts bl, bd are equal; hence the whole square af is equal to the sum of the two squares ah, bd.
this proof is shorter than the usual one, since it is not necessary to prove that ac, cd are in one right line. in a similar way the proposition may be proved by taking any of the eight figures formed by turning the squares in all possible directions.
another simplification of the proof would be got by considering that the point a is such that one of the s cag, bak can be turned round it in its own plane until it coincides with the other; and hence that they are congruent.
1. the square on ac is equal to the rectangle ab.ao, and the square on bc ab.bo.
4. find a line whose square shall be equal to the sum of two given squares.
5. given the base of a triangle and the difference of the squares of its sides, the locus of its vertex is a right line perpendicular to the base.
6. the transverse lines bk, cg are perpendicular to each other.
7. if eg be joined, its square is equal to ac2 4bc2.
8. the square described on the sum of the sides of a right-angled triangle exceeds the square on the hypotenuse by four times the area of the triangle (see fig., xlvi., ex. 3).
more generally, if the vertical angle of a triangle be equal to the angle of a regular polygon of n sides, then the regular polygon of n sides, described on a line equal to the sum of its sides, exceeds the area of the regular polygon of n sides described
on the base by n times the area of the triangle.
9. if ac and bk intersect in p, and through p a line be drawn parallel to bc, meeting ab in q; then cp is equal to pq.
10. each of the triangles agk and bef, formed by joining adjacent corners of the squares, is equal to the right-angled triangle abc.
11. find a line whose square shall be equal to the difference of the squares on two lines.
12. the square on the difference of the sides ac, cb is less than the square on the hypotenuse by four times the area of the triangle.
13. if ae be joined, the lines ae, bk, cl, are concurrent.
14. in an equilateral triangle, three times the square on any side is equal to four times the square on the perpendicular to it from the opposite vertex.
15. on be, a part of the side bc of a square abcd, is described the square befg, having its side bg in the continuation of ab; it is required to divide the figure agfecd into three parts which will form a square.
16. four times the sum of the squares on the medians which bisect the sides of a right-angled triangle is equal to five times the square on the hypotenuse.
17. if perpendiculars be let fall on the sides of a polygon from any point, dividing each side into two segments, the sum of the squares on one set of alternate segments is equal to the sum of the squares on the remaining set.
18. the sum of the squares on lines drawn from any point to one pair of opposite angles of a rectangle is equal to the sum of the squares on the lines from the same point to the remaining pair.
19. divide the hypotenuse of a right-angled triangle into two parts, such that the difference between their squares shall be equal to the square on one of the sides.
20. from the extremities of the base of a triangle perpendiculars are let fall on the opposite sides; prove that the sum of the rectangles contained by the sides and their lower segments is equal to the square on the base.
if the square on one side (ab) of a triangle be equal to the sum of the squares on the remaining sides (ac, cb), the angle (c) opposite to that side is a right angle.
dem.-erect cd at right angles to cb xi. , and make cd equal to ca iii. . join bd.
then because ac is equal to cd, the square on ac is equal to the square on cd: to each add the square on cb, and we have the sum of the squares on ac, cb equal to the sum of the squares on cd, cb; but the sum of the squares on ac, cb is equal to the
square on ab (hyp.), and the sum of the squares on cd, cb is equal to the square on bd xlvii. . therefore the square on ab is equal to the square on bd. hence ab is equal to bd xlvi., ex. 1 .
again, because ac is equal to cd (const.), and cb common to the two triangles acb, dcb, and the base ab equal to the base db, the angle acb is equal to the angle dcb; but the angle dcb is a right angle (const.). hence the angle acb is a right angle.
the foregoing proof forms an exception to euclid's demonstrations of converse propositions, for it is direct. the following is an indirect proof:-if cb be not at right angles to ac, let cd be perpendicular to it. make cd cb. join ad.
then, as before, it can be proved that ad is equal to ab, and cd is equal to cb (const.). this is contrary to prop. vii. hence the angle acb is a right angle.
2. what is geometric magnitude? ans. that which has extension in space.
3. name the primary concepts of geometry. ans. points, lines, surfaces, and solids.
4. how may lines be divided? ans. into straight and curved.
5. how is a straight line generated? ans. by the motion of a point which has the same direction throughout.
6. how is a curved line generated? ans. by the motion of a point which continually changes its direction.
7. how may surfaces be divided? ans. into planes and curved surfaces.
8. how may a plane surface be generated. ans. by the motion of a right line which crosses another right line, and moves along it without changing its direction.
10. why has a line neither breadth nor thickness?
13. what portion of plane geometry forms the subject of the "first six books of euclid's elements"? ans. the geometry of the point, line, and circle.
14. what is the subject-matter of book i.?
15. how many conditions are necessary to fix the position of a point in a plane? ans. two; for it must be the intersection of two lines, straight or curved.
17. in order to construct a line, how many conditions must be given? ans. two; as, for instance, two points through which it must pass; or one point through which it must pass and a line to which it must be parallel or perpendicular, c.
18. what problems on the drawing of lines occur in book i.? ans. ii., ix., xi., xii., xxiii., xxxi., in each of which, except problem 2, there are two conditions. the direction in problem 2 is indeterminate.
19. how many conditions are required in order to describe a circle? ans. three; as, for instance, the position of the centre (which depends on two conditions) and the length of the radius (compare post. iii.).
20. how is a proposition proved indirectly? ans. by proving that its contradictory is false.
21. what is meant by the obverse of a proposition?
22. what propositions in book i. are the obverse respectively of propositions iv., v., vi., xxvii.?
23. what proposition is an instance of the rule of identity?
25. what other name is applied to them? ans. they are said to be identically equal.
26. mention all the instances of equality which are not congruence that occur in book i.
27. what is the difference between the symbols denoting congruence and identity?
28. classify the properties of triangles and parallelograms proved in book i.
29. what proposition is the converse of prop. xxvi., part i.?
30. define adjacent, exterior, interior, alternate angles respectively.
31. what is meant by the projection of one line on another?
32. what are meant by the medians of a triangle?
33. what is meant by the third diagonal of a quadrilateral?
34. mention some propositions in book i. which are particular cases of more general ones that follow.
35. what is the sum of all the exterior angles of any rectilineal figure equal to?
36. how many conditions must be given in order to construct a triangle? ans. three; such as the three sides, or two sides and an angle, c.
1. any triangle is equal to the fourth part of that which is formed by drawing through each vertex a line parallel to its opposite side.
2. the three perpendiculars of the first triangle in question 1 are the perpendiculars at the middle points of the sides of the second triangle.
3. through a given point draw a line so that the portion intercepted by the legs of a given angle may be bisected in the point.
4. the three medians of a triangle are concurrent.
5. the medians of a triangle divide each other in the ratio of 2 : 1.
6. construct a triangle, being given two sides and the median of the third side.
7. in every triangle the sum of the medians is less than the perimeter, and greater than three-fourths of the perimeter.
8. construct a triangle, being given a side and the two medians of the remaining sides.
9. construct a triangle, being given the three medians.
10. the angle included between the perpendicular from the vertical angle of a triangle on the base, and the bisector of the vertical angle, is equal to half the difference of the base angles.
11. find in two parallels two points which shall be equidistant from a given point, and whose line of connexion shall be parallel to a given line.
12. construct a parallelogram, being given two diagonals and a side.
13. the smallest median of a triangle corresponds to the greatest side.
14. find in two parallels two points subtending a right angle at a given point and equally distant from it.
15. the sum of the distances of any point in the base of an isosceles triangle from the equal sides is equal to the distance of either extremity of the base from the opposite side.
16. the three perpendiculars at the middle points of the sides of a triangle are concurrent. hence prove that perpendiculars from the vertices on the opposite sides are concurrent see ex. 2 .
17. inscribe a lozenge in a triangle having for an angle one angle of the triangle.
18. inscribe a square in a triangle having its base on a side of the triangle.
19. find the locus of a point, the sum or the difference of whose distance from two fixed lines is equal to a given length.
20. the sum of the perpendiculars from any point in the interior of an equilateral triangle is equal to the perpendicular from any vertex on the opposite side.
21. the distance of the foot of the perpendicular from either extremity of the base of a triangle on the bisector of the vertical angle, from the middle point of the base, is equal to half the difference of the sides.
22. in the same case, if the bisector of the external vertical angle be taken, the distance will be equal to half the sum of the sides.
23. find a point in one of the sides of a triangle such that the sum of the intercepts made by the other sides, on parallels drawn from the same point to these sides, may be equal to a given length.
24. if two angles have their legs respectively parallel, their bisectors are either parallel or perpendicular.
25. if lines be drawn from the extremities of the base of a triangle to the feet of perpendiculars let fall from the same points on either bisector of the vertical angle, these lines meet on the other bisector of the vertical angle.
26. the perpendiculars of a triangle are the bisectors of the angles of the triangle whose vertices are the feet of these perpendiculars.
27. inscribe in a given triangle a parallelogram whose diagonals shall intersect in a given point.
28. construct a quadrilateral, the four sides being given in magnitude, and the middle points of two opposite sides being given in position.
29. the bases of two or more triangles having a common vertex are given, both in magnitude and position, and the sum of the areas is given; prove that the locus of the vertex is a right line.
30. if the sum of the perpendiculars let fall from a given point on the sides of a given rectilineal figure be given, the locus of the point is a right line.
31. abc is an isosceles triangle whose equal sides are ab, ac; bc is any secant cutting the equal sides in b, c, so that ab ac ab ac: prove that bc is greater than bc.
32. a, b are two given points, and p is a point in a given line l; prove that the difference of ap and pb is a maximum when l bisects the angle apb; and that their sum is a minimum if it bisects the supplement.
33. bisect a quadrilateral by a right line drawn from one of its angular points.
34. ad and bc are two parallel lines cut obliquely by ab, and perpendicularly by ac; and between these lines we draw bed, cutting ac in e, such that ed 2ab; prove that the angle dbc is one-third of abc.
35. if o be the point of concurrence of the bisectors of the angles of the triangle abc, and if ao produced meet bc in d, and from o, oe be drawn perpendicular to bc; prove that the angle bod is equal to the angle coe.
36. if the exterior angles of a triangle be bisected, the three external triangles formed on the sides of the original triangle are equiangular.
37. the angle made by the bisectors of two consecutive angles of a convex quadrilateral is equal to half the sum of the remaining angles; and the angle made by the bisectors of two opposite angles is equal to half the difference of the two other angles.
38. if in the construction of the figure, proposition xlvii., ef, kg be joined,
39. given the middle points of the sides of a convex polygon of an odd number of sides, construct the polygon.
40. trisect a quadrilateral by lines drawn from one of its angles.
41. given the base of a triangle in magnitude and position and the sum of the sides; prove that the perpendicular at either extremity of the base to the adjacent side, and the external bisector of the vertical angle, meet on a given line perpendicular to
42. the bisectors of the angles of a convex quadrilateral form a quadrilateral whose opposite angles are supplemental. if the first quadrilateral be a parallelogram, the second is a rectangle; if the first be a rectangle, the second is a square.
43. the middle points of the sides ab, bc, ca of a triangle are respectively d, e, f; dg is drawn parallel to bf to meet ef; prove that the sides of the triangle dcg are respectively equal to the three medians of the triangle abc.
44. find the path of a billiard ball started from a given point which, after being reflected from the four sides of the table, will pass through another given point.
45. if two lines bisecting two angles of a triangle and terminated by the opposite sides be equal, the triangle is isosceles.
46. state and prove the proposition corresponding to exercise 41, when the base and difference of the sides are given.
47. if a square be inscribed in a triangle, the rectangle under its side and the sum of the base and altitude is equal to twice the area of the triangle.
48. if ab, ac be equal sides of an isosceles triangle, and if bd be a perpendicular on ac; prove that bc2 2ac.cd.
49. the sum of the equilateral triangles described on the legs of a right-angled triangle is equal to the equilateral triangle described on the hypotenuse.
50. given the base of a triangle, the difference of the base angles, and the sum or difference of the sides; construct it.
51. given the base of a triangle, the median that bisects the base, and the area; construct it.
52. if the diagonals ac, bd of a quadrilateral abcd intersect in e, and be bisected in the points f, g, then
53. if squares be described on the sides of any triangle, the lines of connexion of the adjacent corners are respectively-(1) the doubles of the medians of the triangle; (2) perpendicular to them.
every proposition in the second book has either a square or a rectangle in its enunciation.
before commencing it the student should read the following preliminary explanations: by their assistance it will be seen that this book, which is usually considered difficult, will be rendered not only easy, but almost intuitively evident.
1. as the linear unit is that by which we express all linear measures, so the square unit is that to which all superficial measures are referred.
again, as there are different linear units in use, such as in this country, inches, feet, yards, miles, c., and in france, metres, and their multiples or sub-multiples, so different square units are employed.
2. a square unit is the square described on a line whose length is the linear unit. thus a square inch is the square described on a line whose length is an inch; a square foot is the square described on a line whose length is a foot, c.
3. if we take a linear foot, describe a square on it, divide two adjacent sides each into twelve equal parts, and draw parallels to the sides, we evidently divide the square foot into square inches; and as there will manifestly be 12 rectangular
parallelograms, each containing 12 square inches, the square foot contains 144 square inches.
in the same manner it can be shown that a square yard contains 9 square feet; and so in general the square described on any line contains n2 times the square described on the nth part of the line.
thus, as a simple case, the square on a line is four times the square on its half. on account of this property the second power of a quantity is called its square; and, conversely, the square on a line ab is expressed symbolically by ab2.
4. if a rectangular parallelogram be such that two adjacent sides contain respectively m and n linear units, by dividing one side into m and the other into n equal parts, and drawing parallels to the sides, the whole area is evidently divided into mn
square units. hence the area of the parallelogram is found by multiplying its length by its breadth, and this explains why we say (see def.
iv.) a rectangle is contained by any two adjacent sides; for if we multiply the length of one by the length of the other we have the area. thus, if ab, ad be two adjacent sides of a rectangle, the rectangle is expressed by ab.ad.
i. if a point c be taken in a line ab, the parts ac, cb are called segments, and c a point of division.
ii. if c be taken in the line ab produced, ac, cb are still called the segments of the line ab; but c is called a point of external division.
iii. a parallelogram whose angles are right angles is called a rectangle.
iv. a rectangle is said to be contained by any two adjacent sides. thus the rectangle abcd is said to be contained by ab, ad, or by ab, bc, c.
v. the rectangle contained by two separate lines such as ab and cd is the parallelogram formed by erecting a perpendicular to ab, at a, equal to cd, and drawing parallels: the area of the rectangle will be ab.cd.
vi. in any parallelogram the figure which is composed of either of the parallelograms about a diagonal and the two complements see i., xliii. is called a gnomon.
thus, if we take away either of the parallelograms ao, oc from the parallelogram ac, the remainder is called a gnomon.
if there be two lines (a, bc), one of which is divided into any number of parts (bd, de, ec), the rectangle contained by the two lines (a, bc), is equal to the sum of the rectangles contained by the undivided line (a) and the several parts of the divided
dem.-erect bf at right angles to bc i., xi. and make it equal to a. complete the parallelogram bk (def. v.). through d, e draw dg, eh parallel to bf. because the angles at b, d, e are right angles, each of the quadrilaterals bg, dh, ek is a rectangle.
again, since a is equal to bf (const.), the rectangle contained by a and bc is the rectangle contained by bf and bc (def. v.); but bk is the rectangle contained by bf and bc. hence the rectangle contained by a and bc is bk.
in like manner the rectangle contained by a and bd is bg. again, since a is equal to bf (const.), and bf is equal to dg i. xxxiv. , a is equal to dg. hence the rectangle contained by a and de is the figure dh (def. v.).
in like manner the rectangle contained by a and ec is the figure ek. hence we have the following identities:-
but bk is equal to the sum of bg, dh, ek (i., axiom ix.). therefore the rectangle contained by a and bc is equal to the sum of the rectangles contained by a and bd, a and de, a and ec.
if we denote the lines bd, de, ec by a, b, c, the proposition asserts that the rectangle contained by a, and a b c is equal to the sum of the rectangles contained by a and a, a and b, a and c, or, as it may be written, a(a b c) aa ab ac.
this corresponds to the distributive law in multiplication, and shows that rectangles in geometry, and products in arithmetic and algebra, are subject to the same rules.
illustration.-suppose a to be 6 inches; bd, 5 inches; de, 4 inches; ec, 3 inches; then bc will be 12 inches; and the rectangles will have the following values:-
now the sum of the three last rectangles, viz. 30, 24, 18, is 72. hence the rectangle a.bc a.bd a.de a.ec.
the second book is occupied with the relations between the segments of a line divided in various ways. all these can be proved in the most simple manner by algebraic multiplication.
we recommend the student to make himself acquainted with the proofs by this method as well as with those of euclid. he will thus better understand the meaning of each proposition.
cor. 1.-the rectangle contained by a line and the difference of two others is equal to the difference of the rectangles contained by the line and each of the others.
cor. 2.-the area of a triangle is equal to half the rectangle contained by its base and perpendicular.
dem.-from the vertex c let fall the perpendicular cd. draw ef parallel to ab, and ae, bf each parallel to cd. then af is the rectangle contained by ab and bf; but bf is equal to cd. hence af ab.cd; but i. xli.
the triangle abc is half the parallelogram af. therefore the triangle abc is 12ab.cd.
if a line (ab) be divided into any two parts (at c), the square on the whole line is equal to the sum of the rectangles contained by the whole and each of the segments (ac, cb).
dem.-on ab describe the square abdf i. xlvi. , and through c draw ce parallel to af i. xxxi. .
now, since ab is equal to af, the rectangle contained by ab and ac is equal to the rectangle contained by af and ac; but ae is the rectangle contained by af and ac. hence the rectangle contained by ab and ac is equal to ae.
in like manner the rectangle contained by ab and cb is equal to the figure cd. therefore the sum of the two rectangles ab.ac, ab.cb is equal to the square on ab.
hence, multiplying, we get ab2 ab.ac ab.cb.
this proposition is the particular case of i. when the divided and undivided lines are equal, hence it does not require a separate demonstration.
if a line (ab) be divided into two segments (at c), the rectangle contained by the whole line and either segment (cb) is equal to the square on that segment together with the rectangle contained by the segments.
dem.-on bc describe the square bcde i. xlvi. . through a draw af parallel to cd: produce ed to meet af in f.
now since cb is equal to cd, the rectangle contained by ac, cb is equal to the rectangle contained by ac, cd; but the rectangle contained by ac, cd is the figure ad.
hence the rectangle ac.cb is equal to the figure ad, and the square on cb is the figure ce. hence the rectangle ac.cb, together with the square on cb, is equal to the figure ae.
again, since cb is equal to be, the rectangle ab.cb is equal to the rectangle ab.be; but the rectangle ab.be is equal to the figure ae. hence the rectangle ab.cb is equal to the figure ae.
and since things which are equal to the same are equal to one another, the rectangle ac.cb, together with the square on cb, is equal to the rectangle ab.cb.
prop. iii. is the particular case of prop. i., when the undivided line is equal to a segment of the divided line.
if a line (ab) be divided into any two parts (at c), the square on the whole line is equal to the sum of the squares on the parts (ac, cb), together with twice their rectangle.
dem.-on ab describe a square abde. join eb; through c draw cf parallel to ae, intersecting be in g; and through g draw hi parallel to ab.
now since ae is equal to ab, the angle abe is equal to aeb i. v. ; but since be intersects the parallels ae, cf, the angle aeb is equal to cgb i. xxix. . hence the angle cbg is equal to cgb, and therefore i. vi.
cg is equal to cb; but cg is equal to bi and cb to gi. hence the figure cbig is a lozenge, and the angle cbi is right. hence (i., def. xxx.) it is a square. in like manner the figure efgh is a square.
again, since cb is equal to cg, the rectangle ac.cb is equal to the rectangle ac.cg; but ac.cg is the figure ag (def. iv.). therefore the rectangle ac.cb is equal to the figure ag. now the figures ag, gd are equal i. xliii.
, being the complements about the diagonal of the parallelogram ad. hence the parallelograms ag, gd are together equal to twice the rectangle ac.cb. again, the figure hf is the square on hg, and hg is equal to ac.
therefore hf is equal to the square on ac, and ci is the square on cb; but the whole figure ad, which is the square on ab, is the sum of the four figures hf, ci, ag, gd.
therefore the square on ab is equal to the sum of the squares on ac, cb, and twice the rectangle ac.cb.
or thus: on ab describe the square abde, and cut off ah, eg, df each equal to cb. join cf, fg, gh, hc. now the four s ach, cbf, fdg, geh are evidently equal; therefore their sum is equal to four times the ach; but the ach is half the rectangle ac.ah (i.
cor. 2)-that is, equal to half the rectangle ac.cb. therefore the sum of the four triangles is equal to 2ac.cb.
again, the figure cfgh is a square i. xlvi., cor. 3 , and equal to ac2 ah2 i. xlvii. -that is, equal to ac2 cb2. hence the whole figure abde ac2 cb2 2ac.cb.
cor. 1.-the parallelograms about the diagonal of a square are squares.
cor. 2.-the square on a line is equal to four times the square on its half.
this cor. may be proved by the first book thus: erect cd at right angles to ab, and make cd ac or cb. join ad, db.
but since the angle adb is right, ad2 db2 ab2;
cor. 3.-if a line be divided into any number of parts, the square on the whole is equal to the sum of the squares on all the parts, together with twice the sum of the rectangles contained by the several distinct pairs of parts.
1. prove proposition iv. by using propositions ii. and iii.
2. if from the vertical angle of a right-angled triangle a perpendicular be let fall on the hypotenuse, its square is equal to the rectangle contained by the segments of the hypotenuse.
3. from the hypotenuse of a right-angled triangle portions are cut off equal to the adjacent sides; prove that the square on the middle segment is equal to twice the rectangle contained by the extreme segments.
4. in any right-angled triangle the square on the sum of the hypotenuse and perpendicular, from the right angle on the hypotenuse, exceeds the square on the sum of the sides by the square on the perpendicular.
5. the square on the perimeter of a right-angled triangle is equal to twice the rectangle contained by the sum of the hypotenuse and one side, and the sum of the hypotenuse and the other side.
if a line (ab) be divided into two equal parts (at c), and also into two unequal parts (at d), the rectangle (ad.db) contained by the unequal parts, together with the square on the part (cd) between the points of section, is equal to the square on half
dem.-on cb describe the square cbef i. xlvi. . join bf. through d draw dg parallel to cf, meeting bf in h. through h draw km parallel to ab, and through a draw ak parallel to cl i. xxxi. .
the parallelogram cm is equal to de i. xliii., cor. 2 ; but al is equal to cm i. xxxvi.
, because they are on equal bases ac, cb, and between the same parallels; therefore al is equal to de: to each add ch, and we get the parallelogram ah equal to the gnomon cmg; but ah is equal to the rectangle ad.dh, and therefore equal to the rectangle
ad.db, since dh is equal to db iv., cor. 1 ; therefore the rectangle ad.db is equal to the gnomon cmg, and the square on cd is equal to the figure lg.
hence the rectangle ad.db, together with the square on cd, is equal to the whole figure cbef-that is, to the square on cb.
cor. 1.-the rectangle ad.db is the rectangle contained by the sum of the lines ac, cd and their difference; and we have proved it equal to the difference between the square on ac and the square on cd.
hence the difference of the squares on two lines is equal to the rectangle contained by their sum and their difference.
cor. 2.-the perimeter of the rectangle ah is equal to 2ab, and is therefore independent of the position of the point d on the line ab; and the area of the same rectangle is less than the square on half the line by the square on the segment between d and
the middle point of the line; therefore, when d is the middle point, the rectangle will have the maximum area. hence, of all rectangles having the same perimeter, the square has the greatest area.
1. divide a given line so that the rectangle contained by its parts may have a maximum area.
2. divide a given line so that the rectangle contained by its segments may be equal to a given square, not exceeding the square on half the given line.
3. the rectangle contained by the sum and the difference of two sides of a triangle is equal to the rectangle contained by the base and the difference of the segments of the base, made by the perpendicular from the vertex.
4. the difference of the sides of a triangle is less than the difference of the segments of the base, made by the perpendicular from the vertex.
5. the difference between the square on one of the equal sides of an isosceles triangle, and the square on any line drawn from the vertex to a point in the base, is equal to the rectangle contained by the segments of the base.
6. the square on either side of a right-angled triangle is equal to the rectangle contained by the sum and the difference of the hypotenuse and the other side.
if a line (ab) be bisected (at c), and divided externally in any point (d), the rectangle (ad.bd) contained by the segments made by the external point, together with the square on half the line, is equal to the square on the segment between the middle
point and the point of external division.
dem.-on cd describe the square cdfe i. xlvi. , and join de; through b draw bhg parallel to ce i. xxxi. , meeting de in h; through h draw klm parallel to ad; and through a draw ak parallel to cl.
then because ac is equal to cb, the rectangle al is equal to ch i. xxxvi. ; but the complements ch, hf are equal i. xliii. ; therefore al is equal to hf.
to each of these equals add cm and lg, and we get am and lg equal to the square cdfe; but am is equal to the rectangle ad.dm, and therefore equal to the rectangle ad.db, since db is equal to dm; also lg is equal to the square on cb, and cdfe is the square
on cd. hence the rectangle ad.db, together with the square on cb, is equal to the square on cd.
dem.-on cb describe the square cbef i. xlvi. . join bf. through d draw dg parallel to cf, meeting fb produced in h. through h draw km parallel to ab. through a draw ak parallel to cl i. xxxi. .
the parallelogram cm is equal to de i. xliii. ; but al is equal to cm i. xxxvi. , because they are on equal bases ac, cb, and between the same parallels; therefore al is equal to de.
to each add ch, and we get the parallelogram ah equal to the gnomon cmg; but ah is equal to the rectangle ad.dh, and therefore equal to the rectangle ad.db, since dh is equal to db iv., cor.
1 ; therefore the rectangle ad.db is equal to the gnomon cmg, and the square on cb is the figure ce. therefore the rectangle ad.db, together with the square on cb, is equal to the whole figure lhgf-that is, equal to the square on lh or to the square on cd.
1. show that proposition vi. is reduced to proposition v. by producing the line in the opposite direction.
2. divide a given line externally, so that the rectangle contained by its segments may be equal to the square on a given line.
3. given the difference of two lines and the rectangle contained by them; find the lines.
4. the rectangle contained by any two lines is equal to the square on half the sum, minus the square on half the difference.
5. given the sum or the difference of two lines and the difference of their squares; find the lines.
6. if from the vertex c of an isosceles triangle a line cd be drawn to any point in the base produced, prove that cd2 cb2 ad.db.
7. give a common enunciation which will include propositions v. and vi.
if a right line (ab) be divided into any two parts (at c), the sum of the squares on the whole line (ab) and either segment (cb) is equal to twice the rectangle (2ab.cb) contained by the whole line and that segment, together with the square on the other
dem.-on ab describe the square abde. join be. through c draw cg parallel to ae, intersecting be in f. through f draw hk parallel to ab.
now the square ad is equal to the three figures ak, fd, and gh: to each add the square ck, and we have the sum of the squares ad, ck equal to the sum of the three figures ak, cd, gh; but cd is equal to ak; therefore the sum of the squares ad, ck is equal
to twice the figure ak, together with the figure gh.
now ak is the rectangle ab.bk; but bk is equal to bc; therefore ak is equal to the rectangle ab.bc, and ad is the square on ab; ck the square on cb; and gh is the square on hf, and therefore equal to the square on ac.
hence the sum of the squares on ab and bc is equal to twice the rectangle ab.bc, together with the square on ac.
or thus: on ac describe the square acde. produce the sides cd, de, ea, and make each produced part equal to cb. join bf, fg, gh, hb. then the figure bfgh is a square i. xlvi., ex.
3 , and it is equal to the square on ac, together with the four equal triangles hab, bcf, fdg, geh. now i. xlvii.
, the figure bfgh is equal to the sum of the squares on ab, ah-that is, equal to the sum of the squares on ab, bc; and the sum of the four triangles is equal to twice the rectangle ab.bc, for each triangle is equal to half the rectangle ab.bc.
hence the sum of the squares on ab, bc is equal to twice the rectangle ab.bc, together with the square on ac.
by iv., square on sum sum of squares twice rectangle.
by vii., square on difference sum of squares-twice rectangle.
1. square on the sum, the sum of the squares, and the square on the difference of any two lines, are in arithmetical progression.
2. square on the sum square on the difference of any two lines twice the sum of the squares on the lines (props. ix. and x.).
3. the square on the sum the square on the difference of any two lines four times the rectangle under lines (prop. viii.).
if a line (ab) be divided into two parts (at c), the square on the sum of the whole line (ab) and either segment (bc) is equal to four times the rectangle contained by the whole line (ab) and that segment, together with the square on the other segment
dem.-produce ab to d. make bd equal to bc. on ad describe the square aefd i. xlvi. . join de. through c, b draw ch, bl parallel to ae i. xxxi. , and through k, i draw mn, po parallel to ad.
since co is the square on cd, and ck the square on cb, and cb is the half of cd, co is equal to four times ck iv., cor. 1 . again, since cg, gi are the sides of equal squares, they are equal i. xlvi., cor. 1 . hence the parallelogram ag is equal to mi i.
xxxvi. . in like manner il is equal to jf; but mi is equal to il i. xliii. . therefore the four figures ag, mi, il, jf are all equal; hence their sum is equal to four times ag; and the square co has been proved to be equal to four times ck.
hence the gnomon aoh is equal to four times the rectangle ak-that is, equal to four times the rectangle ab.bc, since bc is equal to bk.
again, the figure ph is the square on pi, and therefore equal to the square on ac. hence the whole figure af, that is, the square on ad, is equal to four times the rectangle ab.bc, together with the square on ac.
or thus: produce ba to d, and make ad bc. on db describe the square dbef. cut off bg, ei, fl each equal to bc. through a and i draw lines parallel to df, and through g and l, lines parallel to ab.
now it is evident that the four rectangles. ag, gi, il, la are all equal; but ag is the rectangle ab.bg or ab.bc. therefore the sum of the four rectangles is equal to 4ab.bc. again, the figure np is evidently equal to the square on ac.
hence the whole figure, which is the square on bd, or the square on the sum of ab and bc, is equal to 4ab.bc ac2.
since by v. or vi. the rectangle contained by any two lines is the square on half their sum the square on half their difference; therefore four times the rectangle contained by any two lines the square on their sum the square on their difference.
direct sequence of viii. from iv. and vii.
by iv., the square on the sum the sum of the squares twice the rectangle.
by vii., the square on the difference the sum of the squares twice the rectangle. therefore, by subtraction, the square on the sum the square on the difference four times the rectangle.
1. in the figure i. xlvii. if ef, gk be joined, prove ef2 co2 (ab bo)2.
3.1 1ex. 3 occurs in the solution of the problem of the inscription of a regular polygon of seventeen sides in a circle. see note c. given the difference of two lines r, and their rectangle 4r2; find the lines.
if a line (ab) be bisected (at c) and divided into two unequal parts (at d), the sum of the squares on the unequal parts (ad, db) is double the sum of the squares on half the line (ac), and on the segment (cd) between the points of section.
dem.-erect ce at right angles to ab, and make it equal to ac or cb. join ae, eb. draw df parallel to ce, and fg parallel to cd. join af.
because ac is equal to ce, and the angle ace is right, the angle cea is half a right angle. in like manner the angles ceb, cbe are half right angles; therefore the whole angle aef is right.
again, because gf is parallel to cb, and ce intersects them, the angle egf is equal to ecb; but ecb is right (const.); therefore egf is right; and gef has been proved to be half a right angle; therefore the angle gfe is half a right angle i. xxxii. .
therefore i. vi. ge is equal to gf. in like manner fd is equal to db.
again, since ac is equal to ce, ac2 is equal to ce2; but ae2 is equal to ac2 ce2 i. xlvii. . therefore ae2 is equal to 2ac2. in like manner ef2 is equal to 2gf2 or 2cd2. therefore ae2 ef2 is equal to 2ac2 2cd2; but ae2 ef2 is equal to af2 i. xlvii. .
again, since df is equal to db, df2 is equal to db2: to each add ad2, and we get ad2 df2 equal to ad2 db2; but ad2 df2 is equal to af2; therefore af2 is equal to ad2 db2; and we have proved af2 equal to 2ac2 2cd2. therefore ad2 db2 is equal to 2ac2 2cd2.
square and add, and we get ad2 db2 2ac2 2cd2.
1. the sum of the squares on the segments of a line of given length is a minimum when it is bisected.
2. divide a given line internally, so that the sum of the squares on the parts may be equal to a given square, and state the limitation to its possibility.
3. if a line ab be bisected in c and divided unequally in d,
4. twice the square on the line joining any point in the hypotenuse of a right-angled isosceles triangle to the vertex is equal to the sum of the squares on the segments of the hypotenuse.
5. if a line be divided into any number of parts, the continued product of all the parts is a maximum, and the sum of their squares is a minimum when all the parts are equal.
if a line (ab) be bisected (at c) and divided externally (at d), the sum of the squares on the segments (ad, db) made by the external point is equal to twice the square on half the line, and twice the square on the segment between the points of section.
dem.-erect ce at right angles to ab, and make it equal to ac or cb. join ae, eb. draw df parallel to ce, and produce eb. now since df is parallel to ec, the angle bdf is to bce i. xxix. , and i. xv.
the angle dbf is to ebc; but the sum of the angles bce, ebc is less than two right angles i. xvii. ; therefore the sum of the angles bdf, dbf is less than two right angles, and therefore i., axiom xii. the lines eb, df, if produced, will meet.
let them meet in f. through f draw fg parallel to ab, and produce ec to meet it in g. join af.
because ac is equal to ce, and the angle ace is right, the angle cea is half a right angle. in like manner the angles ceb, cbe are half right angles; therefore the whole angle aef is right.
again, because gf is parallel to cb, and ge intersects them, the angle egf is equal to ecb i. xxix. ; but ecb is right (const.); therefore egf is right, and gef has been proved to be half a right angle; therefore i. xxxii.
gfe is half a right angle, and therefore i. vi. ge is equal to gf. in like manner fd is equal to db.
again, since ac is equal to ce, ac2 is equal to ce2; but ae2 is equal to ac2 ce2 i. xlvii. ; therefore ae2 is equal to 2ac2. in like manner ef2 is equal to 2gf2 or 2cd2; therefore ae2 ef2 is equal to 2ac2 2cd2; but ae2 ef2 is equal to af2 i. xlvii. .
again, since df is equal to db, df2 is equal to db2: to each add ad2, and we get ad2 df2 equal to ad2 db2; but ad2 df2 is equal to af2; therefore af2 is equal to ad2 db2; and af2 has been proved equal to 2ac2 2cd2. therefore ad2 db2 is equal to 2ac2 2cd2.
square and add, and we get ad2 bd2 2cd2 2ac2.
the following enunciations include propositions ix. and x.:-
1. the square on the sum of any two lines plus the square on their difference equal twice the sum of their squares.
2. the sum of the squares on any two lines it equal to twice the square on half the sum plus twice the square on half the difference of the lines.
3. if a line be cut into two unequal parts, and also into two equal parts, the sum of the squares on the two unequal parts exceeds the sum of the squares on the two equal parts by the sum of the squares of the two differences between the equal and unequal
1. given the sum or the difference of any two lines, and the sum of their squares; find the lines.
2. the sum of the squares on two sides ac, cb of a triangle is equal to twice the square on half the base ab, and twice the square on the median which bisects ab.
3. if the base of a triangle be given both in magnitude and position, and the sum of the squares on the sides in magnitude, the locus of the vertex is a circle.
4. if in the abc a point d in the base bc be such that
prove that the middle point of ad is equally distant from b and c.
5. the sum of the squares on the sides of a parallelogram is equal to the sum of the squares on its diagonals.
to divide a given finite line (ab) into two segments (in h), so that the rectangle (ab.bh) contained by the whole line and one segment may be equal to the square on the other segment.
sol.-on ab describe the square abdc i. xlvi. . bisect ac in e. join be. produce ea to f, and make ef equal to eb. on af describe the square afgh. h is the point required.
dem.-produce gh to k. then because ca is bisected in e, and divided externally in f, the rectangle cf.af, together with the square on ea, is equal to the square on ef vi.
; but ef is equal to eb (const.); therefore the rectangle cf.af, together with ea2, is equal to eb2-that is i. xlvii. equal to ea2 ab2. rejecting ea2, which is common, we get the rectangle cf.af equal to ab2.
again, since af is equal to fg, being the sides of a square, the rectangle cf.af is equal to cf.fg-that is, to the figure cg; and ab2 is equal to the figure ad; therefore cg is equal to ad.
reject the part ak, which is common, and we get the figure fh equal to the figure hd; but hd is equal to the rectangle ab.bh, because bd is equal to ab, and fh is the square on ah. therefore the rectangle ab.bh is equal to the square on ah.
def.-a line divided as in this proposition is said to be divided in "extreme and mean ratio."
cor. 1.-the line cf is divided in "extreme and mean ratio" at a.
cor. 2.-if from the greater segment ca of cf we take a segment equal to af, it is evident that ca will be divided into parts respectively equal to ah, hb.
hence, if a line be divided in extreme and mean ratio, the greater segment will be cut in the same manner by taking on it a part equal to the less; and the less will be similarly divided by taking on it a part equal to the difference, and so on, c.
cor. 3.-let ab be divided in "extreme and mean ratio" in c, then it is evident (cor. 2) that ac is greater than cb. cut off cd cb; then (cor. 2) ac is cut in "extreme and mean ratio" at d, and cd is greater than ad.
next, cut off de equal to ad, and in the same manner we have de greater than ec, and so on. now since cd is greater than ad, it is evident that cd is not a common measure of ac and cb, and therefore not a common measure of ab and ac.
in like manner ad is not a common measure of ac and cd, and therefore not a common measure of ab and ac. hence, no matter how far we proceed we cannot arrive at any remainder which will be a common measure of ab and ac.
hence, the parts of a line divided in "extreme and mean ratio" are incommensurable.
1. cut a line externally in "extreme and mean ratio."
2. the difference between the squares on the segments of a line divided in "extreme and mean ratio" is equal to their rectangle.
3. in a right-angled triangle, if the square on one side be equal to the rectangle contained by the hypotenuse and the other side, the hypotenuse is cut in "extreme and mean ratio" by the perpendicular on it from the right angle.
4. if ab be cut in "extreme and mean ratio" at c, prove that
5. the three lines joining the pairs of points g, b; f, d; a, k, in the construction of proposition xi., are parallel.
6. if ch intersect be in o, ao is perpendicular to ch.
7. if ch be produced, it meets bf at right angles.
8. abc is a right-angled triangle having ab 2ac: if ah be made equal to the difference between bc and ac, ab is divided in "extreme and mean ratio" at h.
in an obtuse-angled triangle (abc), the square on the side (ab) subtending the obtuse angle exceeds the sum of the squares on the sides (bc, ca) containing the obtuse angle, by twice the rectangle contained by either of them (bc), and its continuation
(cd) to meet a perpendicular (ad) on it from the opposite angle.
dem.-because bd is divided into two parts in c, we have
hence, adding, since i. xlvii. bd2 ad2 ab2, and cd2 ad2 ca2, we get
therefore ab2 is greater than bc2 ca2 by 2bc.cd.
the foregoing proof differs from euclid's only in the use of symbols. i have found by experience that pupils more readily understand it than any other method.
or thus: by the first book: describe squares on the three sides. draw ae, bf, cg perpendicular to the sides of the squares. then it can be proved exactly as in the demonstration of i. xlvii. , that the rectangle bg is equal to be, ag to af, and ce to cf.
hence the sum of the two squares on ac, cb is less than the square on ab by twice the rectangle ce; that is, by twice the rectangle bc.cd.
cor. 1.-if perpendiculars from a and b to the opposite sides meet them in h and d, the rectangle ac.ch is equal to the rectangle bc.cd.
1. if the angle acb of a triangle be equal to twice the angle of an equilateral triangle, ab2 bc2 ca2 bc.ca.
2. abcd is a quadrilateral whose opposite angles b and d are right, and ad, bc produced meet in e; prove ae.de be.ce.
3. abc is a right-angled triangle, and bd is a perpendicular on the hypotenuse ac; prove ab.dc bd.bc.
4. if a line ab be divided in c so that ac2 2cb2; prove that ab2 bc2 2ab.ac.
5. if ab be the diameter of a semicircle, find a point c in ab such that, joining c to a fixed point d in the circumference, and erecting a perpendicular ce meeting the circumference in e, ce2 cd2 may be equal to a given square.
6. if the square of a line cd, drawn from the angle c of an equilateral triangle abc to a point d in the side ab produced, be equal to 2ab2; prove that ad is cut in "extreme and mean ratio" at b.
in any triangle (abc), the square on any side subtending an acute angle (c) is less than the sum of the squares on the sides containing that angle, by twice the rectangle (bc, cd) contained by either of them (bc) and the intercept (cd) between the acute
angle and the foot of the perpendicular on it from the opposite angle.
dem.-because bc is divided into two segments in d,
therefore ab2 is less than bc2 ac2 by 2bc.cd.
or thus: describe squares on the sides. draw ae, bf, cg perpendicular to the sides; then, as in the demonstration of i. xlvii. , the rectangle bg is equal to be; ag to af, and ce to cf.
hence the sum of the squares on ac, cb exceeds the square on ab by twice ce-that is, by 2bc.cd.
observation.-by comparing the proofs of the pairs of props. iv. and vii.; v. and vi.; ix. and x.; xii. and xiii., it will be seen that they are virtually identical.
in order to render this identity more apparent, we have made some slight alterations in the usual proofs. the pairs of propositions thus grouped are considered in modern geometry not as distinct, but each pair is regarded as one proposition.
1. if the angle c of the acb be equal to an angle of an equilateral , ab2 ac2 bc2 ac.bc.
2. the sum of the squares on the diagonals of a quadrilateral, together with four times the square on the line joining their middle points, is equal to the sum of the squares on its sides.
3. find a point c in a given line ab produced, so that ac2 bc2 2ac.bc.
to construct a square equal to a given rectilineal figure (x).
sol.-construct i. xlv. the rectangle ac equal to x.
then, if the adjacent sides ab, bc be equal, ac is a square, and the problem is solved; if not, produce ab to e, and make be equal to bc; bisect ae in f; with f as centre and fe as radius, describe the semicircle age; produce cb to meet it in g.
the square described on bg will be equal to x.
dem.-join fg. then because ae is divided equally in f and unequally in b, the rectangle ab.be, together with fb2 is equal to fe2 v. , that is, to fg2; but fg2 is equal to fb2 bg2 i. xlvii. . therefore the rectangle ab.be fb2 is equal to fb2 bg2.
reject fb2, which is common, and we have the rectangle ab.be bg2; but since be is equal to bc, the rectangle ab.be is equal to the figure ac. therefore bg2 is equal to the figure ac, and therefore equal to the given rectilineal figure (x).
cor.-the square on the perpendicular from any point in a semicircle on the diameter is equal to the rectangle contained by the segments of the diameter.
1. given the difference of the squares on two lines and their rectangle; find the lines.
2. divide a given line, so that the rectangle contained by another given line and one segment may be equal to the square on the other segment.
1. what is the subject-matter of book ii.? ans. theory of rectangles.
3. what is a square inch? a square foot? a square perch? a square mile? ans. the square described on a line whose length is an inch, a foot, a perch, c.
4. what is the difference between linear and superficial measurement? ans. linear measurement has but one dimension; superficial has two.
5. when is a line said to be divided internally? when externally?
6. how is the area of a rectangle found?
7. how is a line divided so that the rectangle contained by its segments may be a maximum?
8. how is the area of a parallelogram found?
9. what is the altitude of a parallelogram whose base is 65 metres and area 1430 square metres?
10. how is a line divided when the sum of the squares on its segments is a minimum?
11. the area of a rectangle is 108.60 square metres and its perimeter is 48.20 linear metres; find its dimensions.
12. what proposition in book ii. expresses the distributive law of multiplication?
13. on what proposition is the rule for extracting the square root founded?
14. compare i. xlvii. and ii. xii. and xiii.
15. if the sides of a triangle be expressed by x2 1, x2 1, and 2x linear units, respectively; prove that it is right-angled.
16. how would you construct a square whose area would be exactly an acre? give a solution by i. xlvii.
17. what is meant by incommensurable lines? give an example from book ii.
18. prove that a side and the diagonal of a square are incommensurable.
19. the diagonals of a lozenge are 16 and 30 metres respectively; find the length of a side.
20. the diagonal of a rectangle is 4.25 perches, and its area is 7.50 square perches; what are its dimensions?
21. the three sides of a triangle are 8, 11, 15; prove that it has an obtuse angle.
22. the sides of a triangle are 13, 14, 15; find the lengths of its medians; also the lengths of its perpendiculars, and prove that all its angles are acute.
23. if the sides of a triangle be expressed by m2 n2, m2 n2, and 2mn linear units, respectively; prove that it is right-angled.
24. if on each side of a square containing 5.29 square perches we measure from the corners respectively a distance of 1.5 linear perches; find the area of the square formed by joining the points thus found.
1. the squares on the diagonals of a quadrilateral are together double the sum of the squares on the lines joining the middle points of opposite sides.
2. if the medians of a triangle intersect in o, ab2 bc2 ca2 3(oa2 ob2 oc2).
3. through a given point o draw three lines oa, ob, oc of given lengths, such that their extremities may be collinear, and that ab bc.
4. if in any quadrilateral two opposite sides be bisected, the sum of the squares on the other two sides, together with the sum of the squares on the diagonals, is equal to the sum of the squares on the bisected sides, together with four times the square
on the line joining the points of bisection.
5. if squares be described on the sides of any triangle, the sum of the squares on the lines joining the adjacent corners is equal to three times the sum of the squares on the sides of the triangle.
6. divide a given line into two parts, so that the rectangle contained by the whole and one segment may be equal to any multiple of the square on the other segment.
7. if p be any point in the diameter ab of a semicircle, and cd any parallel chord, then
8. if a, b, c, d be four collinear points taken in order,
9. three times the sum of the squares on the sides of any pentagon exceeds the sum of the squares on its diagonals, by four times the sum of the squares on the lines joining the middle points of the diagonals.
10. in any triangle, three times the sum of the squares on the sides is equal to four times the sum of the squares on the medians.
11. if perpendiculars be drawn from the angular points of a square to any line, the sum of the squares on the perpendiculars from one pair of opposite angles exceeds twice the rectangle of the perpendiculars from the other pair by the area of the square.
12. if the base ab of a triangle be divided in d, so that mad nbd, then
13. if the point d be taken in ab produced, so that mad ndb, then
14. given the base of a triangle in magnitude and position, and the sum or the difference of m times the square on one side and n times the square on the other side, in magnitude, the locus of the vertex is a circle.
15. any rectangle is equal to half the rectangle contained by the diagonals of squares described on its adjacent sides.
16. if a, b, c. c., be any number of fixed points, and p a variable point, find the locus of p, if ap2 bp2 cp2 c., be given in magnitude.
17. if the area of a rectangle be given, its perimeter is a minimum when it is a square.
18. if a transversal cut in the points a, c, b three lines issuing from a point d, prove that
19. upon the segments ac, cb of a line ab equilateral triangles are described: prove that if d, d be the centres of circles described about these triangles, 6dd2 ab2 ac2 cb2.
20. if a, b, p denote the sides of a right-angled triangle about the right angle, and the perpendicular from the right angle on the hypotenuse, -12 a 12- b 12- p.
21. if, upon the greater segment ab of a line ac, divided in extreme and mean ratio, an equilateral triangle abd be described, and cd joined, cd2 2ab2.
22. if a variable line, whose extremities rest on the circumferences of two given concentric circles, subtend a right angle at any fixed point, the locus of its middle point is a circle.
i. equal circles are those whose radii are equal.
this is a theorem, and not a definition. for if two circles have equal radii, they are evidently congruent figures, and therefore equal. from this way of proving this theorem props. xxvi.-xxix. follow as immediate inferences.
ii. a chord of a circle is the line joining two points in its circumference.
if the chord be produced both ways, the whole line is called a secant, and each of the parts into which a secant divides the circumference is called an arc-the greater the major conjugate arc, and the lesser the minor conjugate arc.-newcomb.
iii. a right line is said to touch a circle when it meets the circle, and, being produced both ways, does not cut it; the line is called a tangent to the circle, and the point where it touches it the point of contact.
in modern geometry a curve is considered as made up of an infinite number of points, which are placed in order along the curve, and then the secant through two consecutive points is a tangent.
euclid's definition for a tangent is quite inadequate for any curve but the circle, and those derived from it by projection (the conic sections); and even for these the modern definition is better.
iv. circles are said to touch one another when they meet, but do not intersect. there are two species of contact:-
1. when each circle is external to the other.
the following is the modern definition of curve-contact:- when two curves have two, three, four, c., consecutive points common, they have contact of the first, second, third, c., orders.
v. a segment of a circle is a figure bounded by a chord and one of the arcs into which it divides the circumference.
vi. chords are said to be equally distant from the centre when the perpendiculars drawn to them from the centre are equal.
vii. the angle contained by two lines, drawn from any point in the circumference of a segment to the extremities of its chord, is called an angle in the segment.
viii. the angle of a segment is the angle contained between its chord and the tangent at either extremity.
a theorem is tacitly assumed in this definition, namely, that the angles which the chord makes with the tangent at its extremities are equal. we shall prove this further on.
ix. an angle in a segment is said to stand on its conjugate arc.
x. similar segments of circles are those that contain equal angles.
xi. a sector of a circle is formed of two radii and the arc included between them.
to a pair of radii may belong either of the two conjugate arcs into which their ends divide the circle.-newcomb.
xii. concentric circles are those that have the same centre.
xiii. points which lie on the circumference of a circle are said to be concyclic.
xiv. a cyclic quadrilateral is one which is inscribed in a circle.
xv. it will be proper to give here an explanation of the extended meaning of the word angle in modern geometry. this extension is necessary in trigonometry, in mechanics-in fact, in every application of geometry, and has been partly given in i. def. ix.
thus, if a line oa revolve about the point o, as in figures 1, 2, 3, 4, until it comes into the position ob, the amount of the rotation from oa to ob is called an angle. from the diagrams we see that in fig. 1 it is less than two right angles; in fig.
2 it is equal to two right angles; in fig. 3 greater than two right angles, but less than four; and in fig. 4 it is greater than four right angles. the arrow-heads denote the direction or sense, as it is technically termed, in which the line oa turns.
it is usual to call the direction indicated in the above figures positive, and the opposite negative. a line such as oa, which turns about a fixed point, is called a ray, and then we have the following definition:-
xvi. a ray which turns in the sense opposite to the hands of a watch describes a positive angle, and one which turns in the same direction as the hands, a negative angle.
to find the centre of a given circle (adb).
sol.-take any two points a, b in the circumference. join ab. bisect it in c. erect cd at right angles to ab. produce dc to meet the circle again in e. bisect de in f. then f is the centre.
dem.-if possible, let any other point g be the centre. join ga, gc, gb.
then in the triangles acg, bcg we have ac equal to cb (const.), cg common, and the base ga equal to gb, because they are drawn from g, which is, by hypothesis, the centre, to the circumference. hence i. viii.
the angle acg is equal to the adjacent angle bcg, and therefore i. def. xiii. each is a right angle; but the angle acd is right (const.); therefore acd is equal to acg-a part equal to the whole-which is absurd.
hence no point can be the centre which is not in the line de. therefore f, the middle point of de, must be the centre.
the foregoing proof may be abridged as follows:-
because ed bisects ab at right angles, every point equally distant from, the points a, b must lie in ed i. x. ex.
2 ; but the centre is equally distant from a and b; hence the centre must be in ed; and since it must be equally distant from e and d, it must be the middle point of de.
cor. 1.-the line which bisects any chord of a circle perpendicularly passes through the centre of the circle.
cor. 2.-the locus of the centres of the circles which pass through two fixed points is the line bisecting at right angles that connecting the two points.
cor. 3.-if a, b, c be three points in the circumference of a circle, the lines bisecting perpendicularly the chords ab, bc intersect in the centre.
if any two points (a, b) be taken in the circumference of a circle-1. the segment (ab) of the indefinite line through these points which lies between them falls within the circle. 2. the remaining parts of the line are without the circle.
dem.-1. let c be the centre. take any point d in ab. join ca,cd,cb. now the angle adc is i. xvi. greater than abc; but the angle abc is equal to cab i. v. , because the triangle cab is isosceles; therefore the angle adc is greater than cad.
hence ac is greater than cd i. xix. ; therefore cd is less than the radius of the circle, consequently the point d must be within the circle (note on i. def. xxxiii.).
in the same manner every other point between a and b lies within the circle.
2. take any point e in ab produced either way. join ce. then the angle abc is greater than aec i. xvi. ; therefore cab is greater than aec. hence ce is greater than ca, and the point e is without the circle.
we have added the second part of this proposition. the indirect proof given of the first part in several editions of euclid is very inelegant; it is one of those absurd things which give many students a dislike to geometry.
cor. 1.-three collinear points cannot be concyclic.
cor. 2.-a line cannot meet a circle in more than two points.
cor. 3.-the circumference of a circle is everywhere concave towards the centre.
if a line (ab) passing through the centre of a circle bisect a chord (cd), which does not pass through the centre, it cuts it at right angles. 2. if it cuts it at right angles, it bisects it.
dem.-1. let o be the centre of the circle. join oc, od. then the triangles ceo, deo have ce equal to ed (hyp.), eo common, and oc equal to od, because they are radii of the circle; hence i. viii.
the angle ceo is equal to deo, and they are adjacent angles. therefore i. def. xiii. each is a right angle. hence ab cuts cd at right angles.
2. the same construction being made: because oc is equal to od, the angle ocd is equal to odc i. v. , and ceo is equal to deo (hyp.), because each is right.
therefore the triangles ceo, deo have two angles in one respectively equal to two angles in the other, and the side eo common. hence i. xxvi. the side ce is equal to ed. therefore cd is bisected in e.
oc2 oe2 ec2 i. xlvii. , and od2 oe2 ed2;
observation.-the three theorems, namely, cor. 1., prop. i., and parts 1, 2, of prop. iii., are so related, that any one being proved directly, the other two follow by the rule of identity.
cor. 1.-the line which bisects perpendicularly one of two parallel chords of a circle bisects the other perpendicularly.
cor. 2.-the locus of the middle points of a system of parallel chords of a circle is the diameter of the circle perpendicular to them all.
cor. 3.-if a line intersect two concentric circles, its intercepts between the circles are equal.
cor. 4.-the line joining the centres of two intersecting circles bisects their common chord perpendicularly.
1. if a chord of a circle subtend a right angle at a given point, the locus of its middle point is a circle.
2. every circle passing through a given point, and having its centre on a given line, passes through another given point.
3. draw a chord in a given circle which shall subtend a right angle at a given point, and be parallel to a given line.
two chords of a circle (ab, cd) which are not both diameters cannot bisect each other, though either may bisect the other.
dem.-let o be the centre. let ab, cd intersect in e; then since ab, cd are not both diameters, join oe. if possible let ae be equal to eb, and ce equal to ed.
now, since oe passing through the centre bisects ab, which does not pass through the centre, it is at right angles to it; therefore the angle aeo is right. in like manner the angle ceo is right.
hence aeo is equal to ceo-that is, a part equal to the whole-which is absurd. therefore ab and cd do not bisect each other.
cor.-if two chords of a circle bisect each other, they are both diameters.
if two circles (abc, abd) cut one another in any point (a), they are not concentric.
dem.-if possible let them have a common centre at o. join oa, and draw any other line od, cutting the circles in c and d respectively. then because o is the centre of the circle abc, oa is equal to oc.
again, because o is the centre of the circle abd, oa is equal to od. hence oc is equal to od-a part equal to the whole-which is absurd. therefore the circles are not concentric.
1. if two non-concentric circles intersect in one point, they must intersect in another point. for, let o, o be the centres, a the point of intersection; from a let fall the ac on the line oo.
produce ac to b, making bc ca: then b is another point of intersection.
2. two circles cannot have three points in common without wholly coinciding.
if one circle (abc) touch another circle (ade) internally in any point (a), it is not concentric with it.
dem.-if possible let the circles be concentric, and let o be the centre of each. join oa, and draw any other line od, cutting the circles in the points b, d respectively.
then because o is the centre of each circle (hyp.), ob and od are each equal to oa; therefore ob is equal to od, which is impossible. hence the circles cannot have the same centre.
if from any point (p) within a circle, which is not the centre, lines (pa, pb, pc, c.), one of which passes through the centre, be drawn to the circumference, then-1. the greatest is the line (pa) which passes through the centre. 2.
the production (pe) of this in the opposite direction is the least. 3. of the others, that which is nearest to the line through the centre is greater than every one more remote. 4.
any two lines making equal angles with the diameter on opposite sides are equal. 5. more than two equal right lines cannot be drawn from the given point (p) to the circumference.
dem.-1. let o be the centre. join ob. now since o is the centre, oa is equal to ob: to each add op, and we have ap equal to the sum of ob, op; but the sum of ob, op is greater than pb i. xx. . therefore pa is greater than pb.
2. join od. then i. xx. the sum of op, pd is greater than od; but od is equal to oe i. def. xxx. . therefore the sum of op, pd is greater than oe. reject op, which is common, and we have pd greater than pe.
3. join oc; then two triangles pob, poc have the side ob equal to oc i. def. xxx. , and op common; but the angle pob is greater than poc; therefore i. xxiv. the base pb is greater than pc. in like manner pc is greater than pd.
4. make at the centre o the angle pof equal to pod. join pf. then the triangles pod, pof have the two sides op, od in one respectively equal to the sides op, of in the other, and the angle pod equal to the angle pof; hence pd is equal to pf i. iv.
, and the angle opd equal to the angle opf. therefore pd and pf make equal angles with the diameter.
5. a third line cannot be drawn from p equal to either of the equal lines pd, pf.
if possible let pg be equal to pd, then pg is equal to pf-that is, the line which is nearest to the one through the centre is equal to the one which is more remote, which is impossible. hence three equal lines cannot be drawn from p to the circumference.
cor. 1.-if two equal lines pd, pf be drawn from a point p to the circumference of a circle, the diameter through p bisects the angle dpf formed by these lines.
cor. 2.-if p be the common centre of circles whose radii are pa, pb, pc, c., then-1. the circle whose radius is the maximum line (pa) lies outside the circle ade, and touches it in a def. iv. . 2.
the circle whose radius is the minimum line (pe) lies inside the circle ade, and touches it in e. 3. a circle having any of the remaining lines (pd) as radius cuts ade in two points (d, f).
observation.-proposition vii. affords a good illustration of the following important definition (see sequel to euclid, p.
13):-if a geometrical magnitude varies its position continuously according to any law, and if it retains the same value throughout, it is said to be a constant, such as the radius of a circle revolving round the centre; but if it goes on increasing for
some time, and then begins to decrease, it is said to be a maximum at the end of the increase. thus, in the foregoing figure, pa, supposed to revolve round p and meet the circle, is a maximum.
again, if it decreases for some time, and then begins to increase, it is a minimum at the commencement of the increase. thus pe, supposed as before to revolve round p and meet the circle, is a minimum. proposition viii. will give other illustrations.
if from any point (p) outside a circle, lines (pa, pb, pc, c.) be drawn to the concave circumference, then-1. the maximum is that which passes through the centre. 2.
of the others, that which is nearer to the one through the centre is greater than the one more remote. again, if lines be drawn to the convex circumference-3. the minimum is that whose production passes through the centre. 4.
of the others, that which is nearer to the minimum is less than one more remote. 5.
from the given point (p) there can be drawn two equal lines to the concave or the convex circumference, both of which make equal angles with the line passing through the centre. 6.
more than two equal lines cannot be drawn from the given point (p) to either circumference.
dem.-1. let o be the centre. join ob. now since o is the centre, oa is equal to ob: to each add op, and we have ap equal to the sum of ob, op; but the sum of ob, op is greater than bp i. xx. . therefore ap is greater than bp.
2. join oc, od. the two triangles bop, cop have the side ob equal to oc, and op common, and the angle bop greater than cop; therefore the base bp is greater than cp i. xxiv. . in like manner cp is greater than dp, c.
3. join of. now in the triangle ofp the sum of the sides of, fp is greater than op i. xx. ; but of is equal to oe i. def. xxx. . reject them, and fp will remain greater than ep.
4. join og, oh. the two triangles gop, fop have two sides go, op in one respectively equal to two sides fo, op in the other; but the angle gop is greater than fop; therefore i. xxiv. the base gp is greater than fp. in like manner hp is greater than gp.
5. make the angle poi equal pof i. xxiii. . join ip. now the triangles iop, fop have two sides io, op in one respectively equal to two sides fo, op in the other, and the angle iop equal to fop (const.); therefore i. iv. ip is equal to fp.
6. a third line cannot be drawn from p equal to either of the lines ip, fp. for if possible let pk be equal to pf; then pk is equal to pi-that is, one which is nearer to the minimum equal to one more remote-which is impossible.
cor. 1.-if pi be produced to meet the circle again in l, pl is equal to pb.
cor. 2.-if two equal lines be drawn from p to either the convex or concave circumference, the diameter through p bisects the angle between them, and the parts of them intercepted by the circle are equal.
cor. 3.-if p be the common centre of circles whose radii are lines drawn from p to the circumference of hde, then-1. the circle whose radius is the minimum line (pe) has contact of the first kind with ade def. iv. . 2.
the circle whose radius is the maximum line (pa) has contact of the second kind. 3. a circle having any of the remaining lines (pf) as radius intersects hde in two points (f, i).
a point (p) within a circle (abc), from which more than two equal lines (pa, pb, pc, c.) can be drawn to the circumference, is the centre.
dem.-if p be not the centre, let o be the centre. join op, and produce it to meet the circle in d and e; then de is the diameter, and p is a point in it which is not the centre: therefore vii.
only two equal lines can be drawn from p to the circumference; but three equal lines are drawn (hyp.), which is absurd. hence p must be the centre.
or thus: since the lines ap, bp are equal, the line bisecting the angle apb vii. cor. 1 must pass through the centre: in like manner the line bisecting the angle bpc must pass through the centre.
hence the point of intersection of these bisectors, that is, the point p, must be the centre.
if two circles have more than two points common, they must coincide.
dem.-let x be one of the circles; and if possible let another circle y have three points, a, b, c, in common with x, without coinciding with it. find p, the centre of x. join pa, pb, pc.
then since p is the centre of x, the three lines pa, pb, pc are equal to one another.
again, since y is a circle and p a point, from which three equal lines pa, pb, pc can be drawn to its circumference, p must be the centre of y . hence x and y are concentric, which v. is impossible.
cor.-two circles not coinciding cannot have more than two points common. compare i., axiom x., that two right lines not coinciding cannot have more than one point common.
if one circle (cpd) touch another circle (apb) internally at any point p, the line joining the centres must pass through that point.
dem.-let o be the centre of apb. join op. i say the centre of the smaller circle is in the line op. if not, let it be in any other position such as e. join oe, ep, and produce oe through e to meet the circles in the points c, a.
now since e is a point in the diameter of the larger circle between the centre and a, ea is less than ep vii. 2 ; but ep is equal to ec (hyp.), being radii of the smaller circle.
hence ea is less than ec; which is impossible; consequently the centre of the smaller circle must be in the line op. let it be h; then we see that the line joining the centres passes through the point p.
or thus: since ep is a line drawn from a point within the circle apb to the circumference, but not forming part of the diameter through e, the circle whose centre is e and radius ep cuts vii., cor.
2 apb in p, but it touches it (hyp.) also in p, which is impossible. hence the centre of the smaller circle cpd must be in the line op.
if two circles (pcf, pde) have external contact at any point p, the line joining their centres must pass through that point.
dem.-let a be the centre of one of the circles. join ap, and produce it to meet the second circle again in e. i say the centre of the second circle is in the line pe. if not, let it be elsewhere, as at b.
join ab, intersecting the circles in c and d, and join bp. now since a is the centre of the circle pcf, ap is equal to ac; and since b is the centre of the circle pde, bp is equal to bd.
hence the sum of the lines ap, bp is equal to the sum of the lines ac, db; but ab is greater than the sum of ac and db; therefore ab is greater than the sum of ap, pb-that is, one side of a triangle greater than the sum of the other two-which i. xx.
is impossible. hence the centre of the second circle must be in the line pe. let it be g, and we see that the line through the centres passes through the point p.
or thus: since bp is a line drawn from a point without the circle pcf to its circumference, and when produced does not pass through the centre, the circle whose centre is b and radius bp must cut the circle pcf in p viii., cor.
3 ; but it touches it (hyp.) also in p, which is impossible. hence the centre of the second circle must be in the line pe.
observation.-propositions xi, xii., may both be included in one enunciation as follows:-"if two circles touch each other at any point, the centres and that point are collinear."
and this latter proposition is a limiting case of the theorem given in proposition iii., cor. 4, that "the line joining the centres of two intersecting circles bisects the common chord perpendicularly."
suppose the circle whose centre is o and one of the points of intersection a to remain fixed, while the second circle turns round that point in such a manner that the second point of intersection b becomes ultimately consecutive to a; then, since the line
oo always bisects ab, we see that when b ultimately becomes consecutive to a, the line oo passes through a.
in consequence of the motion, the common chord will become in the limit a tangent to each circle, as in the second diagram.-comberousse, geometrie plane, page 57.
cor. 1.-if two circles touch each other, their point of contact is the union of two points of intersection. hence a contact counts for two intersections.
cor. 2.-if two circles touch each other at any point, they cannot have any other common point. for, since two circles cannot have more than two points common x.
, and that the point of contact is equivalent to two common points, circles that touch cannot have any other point common.
the following is a formal proof of this proposition:-let o, o be the centres of the two circles, a the point of contact, and let o lie between o and a; take any other point b in the circumference of o. join ob; then vii.
ob is greater than oa; therefore the point c is outside the circumference of the smaller circle. hence b cannot be common to both circles. in like manner, they cannot have any other common point but a.
two circles cannot have double contact, that it, cannot touch each other in two points.
dem.-1. if possible let two circles touch each other at two points a and b. now since the two circles touch each other in a, the line joining their centres passes through a xi. . in like manner, it passes through b.
hence the centres and the points a, b are in one right line; therefore ab is a diameter of each circle. hence, if ab be bisected in e, e must be the centre of each circle-that is, the circles are concentric-which v. is impossible.
2. if two circles touched each other externally in two distinct points, then xii. the line joining the centres should pass through each point, which is impossible.
or thus: draw a line bisecting ab at right angles. then this line i., cor. 1 must pass through the centre of each circle, and therefore xi. xii. must pass through each point of contact, which is impossible. hence two circles cannot have double contact.
this proposition is an immediate inference from the theorem xii., cor. 1 , that a point of contact counts for two intersections, for then two contacts would be equivalent to four intersections; but there cannot be more than two intersections x. .
it also follows from prop. xii., cor. 2, that if two circles touch each other in a point a, they cannot have any other point common; hence they cannot touch again in b.
1. if a variable circle touch two fixed circles externally, the difference of the distances of its centre from the centres of the fixed circles is equal to the difference or the sum of their radii, according as the contacts are of the same or of opposite
2. if a variable circle be touched by one of two fixed circles internally, and touch the other fixed circle either externally or internally, the sum of the distances of its centre from the centres of the fixed circles is equal to the sum or the difference
of their radii, according as the contact with the second circle is of the first or second kind.
3. if through the point of contact of two touching circles any secant be drawn cutting the circles again in two points, the radii drawn to these points are parallel.
4. if two diameters of two touching circles be parallel, the lines from the point of contact to the extremities of one diameter pass through the extremities of the other.
in equal circles-1. equal chords (ab, cd) are equally distant from the centre. 2. chords which are equally distant from the centre are equal.
dem.-1. let o be the centre. draw the perpendiculars oe, of. join ao, co. then because ab is a chord in a circle, and oe is drawn from the centre cutting it at right angles, it bisects it iii. ; therefore ae is the half of ab.
in like manner, cf is the half of cd; but ab is equal to cd (hyp.). therefore ae is equal to cf i., axiom vii. . and because e is a right angle, ao2 is equal to ae2 eo2. in like manner, co2 is equal to cf2 fo2; but ao2 is equal to co2.
therefore ae2 eo2 is equal to cf2 fo2; and ae2 has been proved equal to cf2. hence eo2 is equal to fo2; therefore eo is equal to fo. hence ab, cd are (def. vi.) equally distant from the centre.
2. let eo be equal to fo, it is required to prove ab equal to cd. the same construction being made, we have, as before, ae2 eo2 equal to cf2 fo2; but eo2 is equal to fo2 (hyp.).
hence ae2 is equal to cf2, and ae is equal to cf; but ab is double of ae, and cd double of cf. therefore ab is equal to cd.
if a chord of given length slide round a fixed circle-1. the locus of its middle point is a circle; 2. the locus of any point fixed in the chord is a circle.
the diameter (ab) is the greatest chord in a circle; and of the others, the chord (cd) which is nearer to the centre is greater than (ef) one more remote, and the greater is nearer to the centre than the less.
dem.-1. join oc, od, oe, and draw the perpendiculars og, oh; then because o is the centre, oa is equal to oc i., def. xxxii. , and ob is equal to od. hence ab is equal to the sum of oc and od; but the sum of oc, od is greater than cd i. xx. .
2. because the chord cd is nearer to the centre than ef, og is less than oh; and since the triangles ogc, ohe are right-angled, we have oc2 og2 gc2, and oe2 oh2 he2; therefore og2 gc2 oh2 he2; but og2 is less than oh2; therefore gc2 is greater than he2,
and gc is greater than he, but cd and ef are the doubles of gc and he. hence cd is greater than ef.
3. let cd be greater than ef, it is required to prove that og is less than oh.
as before, we have og2 gc2 equal to oh2 he2; but cg2 is greater than eh2; therefore og2 is less than oh2. hence og is less than oh.
1. the shortest chord which can be drawn through a given point within a circle is the perpendicular to the diameter which passes through that point.
2. through a given point, within or without a given circle, draw a chord of length equal to that of a given chord.
3. through one of the points of intersection of two circles draw a secant-1. the sum of whose segments intercepted by the circles shall be a maximum; 2. which shall be of any length less than that of the maximum.
4. three circles touch each other externally at a, b, c; the chords ab, ac of two of them are produced to meet the third again in the points d and e; prove that de is a diameter of the third circle, and parallel to the line joining the centres of the
1. the perpendicular (bi) to the diameter (ab) of a circle at its extremity (b) touches the circle at that point. 2. any other line (bh) through the same point cuts the circle.
dem.-1. take any point i, and join it to the centre c. then because the angle cbi is a right angle, ci2 is equal to cb2 bi2 i. xlvii. ; therefore ci2 is greater than cb2. hence ci is greater than cb, and the point i note on i., def. xxxii.
is without the circle. in like manner, every other point in bi, except b, is without the circle. hence, since bi meets the circle at b, but does not cut it, it must touch it.
2. to prove that bh, which is not perpendicular to ab, cuts the circle. draw cg perpendicular to hb. now bc2 is equal to cg2 gb2. therefore bc2 is greater than cg2, and bc is greater than cg. hence note on i., def. xxxii.
the point g must be within the circle, and consequently the line bg produced must meet the circle again, and must therefore cut it.
this proposition may be proved as follows:
at every point on a circle the tangent is perpendicular to the radius.
let p and q be two consecutive points on the circumference. join cp, cq, pq; produce pq both ways. now since p and q are consecutive points, pq is a tangent (def. iii.).
again, the sum of the three angles of the triangle cpq is equal to two right angles; but the angle c is infinitely small, and the others are equal. hence each of them is a right angle. therefore the tangent is perpendicular to the diameter.
or thus: a tangent is a limiting position of a secant, namely, when the secant moves out until the two points of intersection with the circle become consecutive; but the line through the centre which bisects the part of the secant within the circle iii.
is perpendicular to it. hence, in the limit the tangent is perpendicular to the line from the centre to the point of contact.
or again: the angle cpr is always equal to cqs; hence, when p and q come together each is a right angle, and the tangent is perpendicular to the radius.
1. if two circles be concentric, all chords of the greater which touch the lesser are equal.
2. draw a parallel to a given line to touch a given circle.
3. draw a perpendicular to a given line to touch a given circle.
4. describe a circle having its centre at a given point-1. and touching a given line; 2. and touching a given circle. how many solutions of this case?
5. describe a circle of given radius that shall touch two given lines. how many solutions?
6. find the locus of the centres of a system of circles touching two given lines.
7. describe a circle of given radius that shall touch a given circle and a given line, or that shall touch two given circles.
from a given point (p) without a given circle (bcd) to draw a tangent to the circle.
sol.-let o (fig. 1) be the centre of the given circle. join op, cutting the circumference in c. with o as centre, and op as radius, describe the circle ape. erect ca at right angles to op. join oa, intersecting the circle bcd in b.
join bp; it will be the tangent required.
dem.-since o is the centre of the two circles, we have oa equal to op, and oc equal to ob. hence the two triangles aoc, pob have the sides oa, oc in one respectively equal to the sides op, ob in the other, and the contained angle common to both. hence i.
iv. the angle oca is equal to obp; but oca is a right angle (const.); therefore obp is a right angle, and xvi. pb touches the circle at b.
cor.-if ac (fig. 2) be produced to e, oe joined, cutting the circle bcd in d, and the line dp drawn, dp will be another tangent from p.
1. the two tangents pb, pd (fig. 2) are equal to one another, because the square of each is equal to the square of op minus the square of the radius.
2. if two circles be concentric, all tangents to the inner from points on the outer are equal.
3. if a quadrilateral be circumscribed to a circle, the sum of one pair of opposite sides is equal to the sum of the other pair.
4. if a parallelogram be circumscribed to a circle it must be a lozenge, and its diagonals intersect in the centre.
5. if bd be joined, intersecting op in f, op is perpendicular to bd.
6. the locus of the intersection of two equal tangents to two circles is a right line (called the radical axis of the two circles).
7. find a point such that tangents from it to three given circles shall be equal. (this point is called the radical centre of the three circles.)
8. the rectangle of.op is equal to the square of the radius.
def. two points, such as f and p, the rectangle of whose distances of, op from the centre is equal to the square of the radius, are called inverse points with respect to the circle.
9. the intercept made on a variable tangent by two fixed tangents subtends a constant angle at the centre.
10. draw a common tangent to two circles. hence, show how to draw a line cutting two circles, so that the intercepted chords shall be of given lengths.
if a line (cd) touch a circle, the line (oc) from the centre to the point of contact is perpendicular to it.
dem.-if not, suppose another line og drawn from the centre to be perpendicular to cd. let og cut the circle in f. then because the angle ogc is right (hyp.) the angle ocg i. xvii. must be acute. therefore i. xix.
oc is greater than og; but oc is equal to of i. def. xxxii. ; therefore of is greater than og-that is, a part greater than the whole, which is impossible. hence oc must be perpendicular to cd.
or thus: since the perpendicular must be the shortest line from o to cd, and oc is evidently the shortest line; therefore oc must be perpendicular to cd.
if a line (ab) be a tangent to a circle, the line (ac) drawn at right angles to it from the point of contact passes through the centre.
if the centre be not in ac, let o be the centre. join ao. then because ab touches the circle, and oa is drawn from the centre to the point of contact, oa is at right angles to ab xviii.
; therefore the angle oab is right, and the angle cab is right (hyp.); therefore oab is equal to cab-a part equal to the whole, which is impossible. hence the centre must be in the line ac.
cor.-if a number of circles touch the same line at the same point, the locus of their centres is the perpendicular to the line at the point.
observation.-propositions xvi., xviii., xix., are so related that any two can be inferred from the third by the "rule of identity." hence it would, in strict logic, be sufficient to prove any one of the three, and the others would follow.
again, these three theorems are limiting cases of proposition i., cor. 1., and parts 1, 2, of proposition iii., namely, when the points in which the chord cuts the circle become consecutive.
the angle (aob) at the centre (o) of a circle is double the angle (acb) at the circumference standing on the same arc.
dem.-join co, and produce it to e. then because oa is equal to oc, the angle aco is equal to oac; but the angle aoe is equal to the sum of the two angles oac, aco. hence the angle aoe is double the angle aco.
in like manner the angle eob is double the angle ocb. hence (by adding in figs. (), (), and subtracting in ()), the angle aob is double of the angle acb.
cor.-if aob be a straight line, acb will be a right angle-that is, the angle in a semicircle is a right angle (compare xxxi.).
the angles (acb, adb) in the same segment of a circle are equal.
dem.-let o be the centre. join oa, ob. then the angle aob is double of the angle acb xx. , and also double of the angle adb. therefore the angle acb is equal to the angle adb.
the following is the proof of the second part-that is, when the arc ab is not greater than a semicircle, without using angles greater than two right angles:-
let o be the centre. join co, and produce it to meet the circle again in e. join de. now since o is the centre, the segment ace is greater than a semicircle; hence, by the first case, fig. (), the angle ace is equal to ade.
in like manner the angle ecb is equal to edb. hence the whole angle acb is equal to the whole angle adb.
cor. 1.-if two triangles acb, adb on the same base ab, and on the same side of it, have equal vertical angles, the four points a, c, d, b are concyclic.
cor. 2.-if a, b be two fixed points, and if c varies its position in such a way that the angle acb retains the same value throughout, the locus of c is a circle.
in other words-given the base of a triangle and the vertical angle, the locus of the vertex is a circle.
1. given the base of a triangle and the vertical angle, find the locus-
(1) of the intersection of its perpendiculars;
(2) of the intersection of the internal bisectors of its base angles;
(3) of the intersection of the external bisectors of the base angles;
(4) of the intersection of the external bisector of one base angle and the internal bisector of the other.
2. if the sum of the squares of two lines be given, their sum is a maximum when the lines are equal.
3. of all triangles having the same base and vertical angle, the sum of the sides of an isosceles triangle is a maximum.
4. of all triangles inscribed in a circle, the equilateral triangle has the maximum perimeter.
5. of all concyclic figures having a given number of sides, the area is a maximum when the sides are equal.
the sum of the opposite angles of a quadrilateral (abcd) inscribed in a circle is two right angles.
dem.-join ac, bd. the angle abd is equal to acd, being in the same segment abcd xxi. ; and the angle dbc is equal to dac, because they are in the same segment dabc. hence the whole angle abc is equal to the sum of the two angles acd, dac.
to each add the angle cda, and we have the sum of the two angles abc, cda equal to the sum of the three angles acd, dac, cda of the triangle acd; but the sum of the three angles of a triangle is equal to two right angles i. xxxii. .
therefore the sum of abc, cda is two right angles.
or thus: let o be the centre of the circle. join oa, oc (see fig. 2). now the angle aoc is double of cda xx. , and the angle coa is double of abc. hence the sum of the angles i. def.
ix., note aoc, coa is double of the sum of the angles cda, abc; but the sum of two angles aoc, coa is four right angles. therefore the sum of the angles cda, abc is two right angles.
or again: let o be the centre (fig. 2). join oa, ob oc, od. then the four triangles aob, boc, cod, doa are each isosceles.
hence the angle oab is equal to the angle oba, and the angle oad equal to the angle oda; therefore the angle bad is equal to the sum of the angles oba, oda. in like manner the angle bcd is equal to the sum of the angles obc, odc.
hence the sum of the two angles bad, bcd is equal to the sum of the two angles abc, adc, and hence each sum is two right angles.
cor.-if a parallelogram be inscribed in a circle it is a rectangle.
1. if the opposite angles of a quadrilateral be supplemental, it is cyclic.
2. if a figure of six sides be inscribed in a circle, the sum of any three alternate angles is four right angles.
3. a line which makes equal angles with one pair of opposite sides of a cyclic quadrilateral, makes equal angles with the remaining pair and with the diagonals.
4. if two opposite sides of a cyclic quadrilateral be produced to meet, and a perpendicular be let fall on the bisector of the angle between them from the point of intersection of the diagonals, this perpendicular will bisect the angle between the
5. if two pairs of opposite sides of a cyclic hexagon be respectively parallel to each other, the remaining pair of sides are also parallel.
6. if two circles intersect in the points a, b, and any two lines acd, bfe, be drawn through a and b, cutting one of the circles in the points c, e, and the other in the points d, f, the line ce is parallel to df.
7. if equilateral triangles be described on the sides of any triangle, the lines joining the vertices of the original triangle to the opposite vertices of the equilateral triangles are concurrent.
8. in the same case prove that the centres of the circles described about the equilateral triangles form another equilateral triangle.
9. if a quadrilateral be described about a circle, the angles at the centre subtended by the opposite sides are supplemental.
10. the perpendiculars of a triangle are concurrent.
11. if a variable tangent meets two parallel tangents it subtends a right angle at the centre.
12. the feet of the perpendiculars let fall on the sides of a triangle from any point in the circumference of the circumscribed circle are collinear (simson).
def.-the line of collinearity is called simson's line.
13. if a hexagon be circumscribed about a circle, the sum of the angles subtended at the centre by any three alternate sides is equal to two right angles.
two similar segments of circles which do not coincide cannot be constructed on the same chord (ab), and on the same side of that chord.
dem.-if possible, let acb, adb, be two similar segments constructed on the same side of ab. take any point d in the inner one. join ad, and produce it to meet the outer one in c. join bc, bd.
then since the segments are similar, the angle adb is equal to acb (def. x.), which is impossible i. xvi. . hence two similar segments not coinciding cannot be described on the same chord and on the same side of it.
similar segments of circles (aeb, cfd) on equal chords (ab, cd) are equal to one another.
dem.-since the lines are equal, if ab be applied to cd, so that the point a will coincide with c, and the line ab with cd, the point b shall coincide with d; and because the segments are similar, they must coincide xxiii. . hence they are equal.
this demonstration may be stated as follows:-since the chords are equal, they are congruent; and therefore the segments, being similar, must be congruent.
an arc (abc) of a circle being given, it is required to describe the whole circle.
sol.-take any three points a, b, c in the arc. join ab, bc. bisect ab in d, and bc in e. erect df, ef at right angles to ab, bc; then f, the point of intersection, will be the centre of the circle.
dem.-because df bisects the chord ab and is perpendicular to it, it passes through the centre i., cor. 1 . in like manner ef passes through the centre.
hence the point f must be the centre; and the circle described from f as centre, with fa as radius, will be the circle required.
the four propositions xxvi.-xxix. are so like in their enunciations that students frequently substitute one for another. the following scheme will assist in remembering them:-
in proposition xxvi. are given angles , to prove arcs ,
so that proposition xxvii. is the converse of xxvi., and xxix. of xxviii.
in equal circles (acb, dfe), equal angles at the centres (aob, dhe) or at the circumferences (acb, dfe) stand upon equal arcs.
dem.-1. suppose the angles at the centres to be given equal. now because the circles are equal their radii are equal (def. i.).
therefore the two triangles aob, dhe have the sides ao, ob in one respectively equal to the sides dh, he in the other, and the angle aob equal to dhe (hyp.). therefore i. iv. the base ab is equal to de.
again, since the angles acb, dfe are xx. the halves of the equal angles aob, dhe, they are equal i. axiom vii. . therefore (def. x.) the segments acb, dfe are similar, and their chords ab, de have been proved equal; therefore xxiv. the segments are equal.
and taking these equals from the whole circles, which are equal (hyp.), the remaining segments agb, dke are equal. hence the arcs agb, dke are equal.
2. the demonstration of this case is included in the foregoing.
cor. 1.-if the opposite angles of a cyclic quadrilateral be equal, one of its diagonals must be a diameter of the circumscribed circle.
cor. 2.-parallel chords in a circle intercept equal arcs.
cor. 3.-if two chords intersect at any point within a circle, the sum of the opposite arcs which they intercept is equal to the arc which parallel chords intersecting on the circumference intercept. 2.
if they intersect without the circle, the difference of the arcs they intercept is equal to the arc which parallel chords intersecting on the circumference intercept.
cor. 4.-if two chords intersect at right angles, the sum of the opposite arcs which they intercept on the circle is a semicircle.
in equal circles (acb, dfe), angles at the centres (aob, dhe), or at the circumferences (acb, dfe), which stand on equal arcs (ab, de), are equal.
dem.-if possible let one of them, such as aob, be greater than the other, dhe; and suppose a part such as aol to be equal to dhe. then since the circles are equal, and the angles aol, dhe at the centres are equal (hyp.), the arc al is equal to de xxvi.
; but ab is equal to de (hyp.). hence al is equal to ab-that is, a part equal to the whole, which is absurd. therefore the angle aob is equal to dhe.
2. the angles at the circumference, being the halves of the central angles, are therefore equal.
in equal circles (acb, dfe), equal chords (ab, de) divide the circumferences into arcs, which are equal each to each-that is, the lesser to the lesser, and the greater to the greater.
dem.-if the equal chords be diameters, the proposition is evident. if not, let o, h be the centres. join ao, ob, dh, he; then because the circles are equal their radii are equal (def. i.).
hence the two triangles aob, dhe have the sides ao, ob in one respectively equal to the sides dh, he in the other, and the base ab is equal to de (hyp.). therefore i. viii. the angle aob is equal to dhe. hence the arc agb is equal to dke xxvi.
; and since the whole circumference agbc is equal to the whole circumference dkef, the remaining arc acb is equal to the remaining arc dfe.
1. the line joining the feet of perpendiculars from any point in the circumference of a circle, on two diameters given in position, is given in magnitude.
2. if a line of given length slide between two lines given in position, the locus of the intersection of perpendiculars to the given lines at its extremities is a circle. (this is the converse of 1.)
in equal circles (acb, dfe), equal arcs (agb, dck) are subtended by equal chords.
dem.-let o, h be the centres (see last fig.). join ao, ob, dh, he; then because the circles are equal, the angles aob, dhe at the centres, which stand on the equal arcs agb, dke, are equal xxvii. .
again, because the triangles aob, dhe have the two sides ao, ob in one respectively equal to the two sides dh, he in the other, and the angle aob equal to the angle dhe, the base ab of one is equal to the base de of the other.
observation.-since the two circles in the four last propositions are equal, they are congruent figures, and the truth of the propositions is evident by superposition.
sol.-draw the chord ab; bisect it in d; erect dc at right angles to ab, meeting the arc in c; then the arc is bisected in c.
dem.-join ac, bc. then the triangles adc, bdc have the side ad equal to db (const.), and dc common to both, and the angle adc equal to the angle bdc, each being right. hence the base ac is equal to the base bc. therefore xxviii.
the arc ac is equal to the arc bc. hence the arc ab is bisected in c.
1. abcd is a semicircle whose diameter is ad; the chord bc produced meets ad produced in e: prove that if ce is equal to the radius, the arc ab is equal to three times cd.
2. the internal and the external bisectors of the vertical angle of a triangle inscribed in a circle meet the circumference again in points equidistant from the extremities of the base.
3. if from a, one of the points of intersection of two given circles, two chords acd, acd be drawn, cutting the circles in the points c, d; c, d, the triangles bcd, bcd, formed by joining these to the second point b of intersection of the circles, are
4. if the vertical angle acb of a triangle inscribed in a circle be bisected by a line cd, which meets the circle again in d, and from d perpendiculars de, df be drawn to the sides, one of which must be produced: prove that ea is equal to bf, and hence
show that ce is equal to half the sum of ac, bc.
in a circle-(1). the angle in a semicircle is a right angle. (2). the angle in a segment greater than a semicircle is an acute angle. (3). the angle in a segment less than a semicircle is an obtuse angle.
dem.-(1). let ab be the diameter, c any point in the semicircle. join ac, cb. the angle acb is a right angle.
for let o be the centre. join oc, and produce ac to f. then because ao is equal to oc, the angle aco is equal to the angle oac. in like manner, the angle ocb is equal to cbo. hence the angle acb is equal to the sum of the two angles bac, cba; but i.
xxxii. the angle fcb is equal to the sum of the two interior angles bac, cba of the triangle abc. hence the angle acb is equal to its adjacent angle fcb, and therefore it is a right angle i. def. xiii. .
(2). let the arc ace be greater than a semicircle. join ce. then the angle ace is evidently less than acb; but acb is right; therefore ace is acute.
(3). let the arc acd be less than a semicircle; then evidently, from (1), the angle acd is obtuse.
cor. 1.-if a parallelogram be inscribed in a circle, its diagonals intersect at the centre of the circle.
cor. 2.-find the centre of a circle by means of a carpenter's square.
cor. 3.-from a point outside a circle draw two tangents to the circle.
if a line (ef) be a tangent to a circle, and from the point of contact (a) a chord (ac) be drawn cutting the circle, the angles made by this line with the tangent are respectively equal to the angles in the alternate segments of the circle.
dem.-(1). if the chord passes through the centre, the proposition is evident, for the angles are right angles; but if not, from the point of contact a draw ab at right angles to the tangent. join bc.
then because ef is a tangent to the circle, and ab is drawn from the point of contact perpendicular to ef, ab passes through this centre xix. . therefore the angle acb is right xxxi. .
hence the sum of the two remaining angles abc, cab is one right angle; but the angle baf is right (const.); therefore the sum of the angles abc, bac is equal to baf. reject bac, which is common, and we get the angle abc equal to the angle fac.
(2). take any point d in the arc ac. it is required to prove that the angle cae is equal to cda.
since the quadrilateral abcd is cyclic, the sum of the opposite angles abc, cda is two right angles xxii. , and therefore equal to the sum of the angles fac, cae; but the angles abc, fac are equal (1). reject them, and we get the angle cda equal to cae.
or thus: take any point g in the semicircle agb. join ag, gb, gc. then the angle agb fab, each being right, and cgb cab xxi. . therefore the remaining angle agc fac. again, join bd, cd. the angle bda bae, each being right, and cdb cab xxi. .
or by the method of limits, see townsend's modern geometry, vol. i., page 14.
the angle bac is equal to bdc xxi. . now let the point b move until it becomes consecutive to a; then ab will be a tangent, and bd will coincide with ad, and the angle bdc with adc.
hence, if ax be a tangent at a, ac any chord, the angle which the tangent makes with the chord is equal to the angle in the alternate segment.
1. if two circles touch, any line drawn through the point of contact will cut off similar segments.
2. if two circles touch, and any two lines be drawn through the point of contact, cutting both circles again, the chord connecting their points of intersection with one circle is parallel to the chord connecting their points of intersection with the other
3. acb is an arc of a circle, ce a tangent at c, meeting the chord ab produced in e, and ad a perpendicular to ab in d: prove, if de be bisected in c, that the arc ac 2cb.
4. if two circles touch at a point a, and abc be a chord through a, meeting the circles in b and c: prove that the tangents at b and c are parallel to each other, and that when one circle is within the other, the tangent at b meets the outer circle in two
5. if two circles touch externally, their common tangent at either side subtends a right angle at the point of contact, and its square is equal to the rectangle contained by their diameters.
on a given right line (ab) to describe a segment of a circle which shall contain an angle equal to a given rectilineal angle (x).
sol.-if x be a right angle, describe a semicircle on the given line, and the thing required is done; for the angle in a semicircle is a right angle.
if not, make with the given line ab the angle bae equal to x. erect ac at right angles to ae, and bc at right angles to ab. on ac as diameter describe a circle: it will be the circle required.
dem.-the circle on ac as diameter passes through b, since the angle abc is right xxxi. and touches ae, since the angle cae is right xvi. . therefore the angle bae xxxii.
is equal to the angle in the alternate segment; but the angle bae is equal to the angle x (const.). therefore the angle x is equal to the angle in the segment described on ab.
1. construct a triangle, being given base, vertical angle, and any of the following data:-1. perpendicular. 2. the sum or difference of the sides. 3. sum or difference of the squares of the sides. 4. side of the inscribed square on the base. 5.
2. if lines be drawn from a fixed point to all the points of the circumference of a given circle, the locus of all their points of bisection is a circle.
3. given the base and vertical angle of a triangle, find the locus of the middle point of the line joining the vertices of equilateral triangles described on the sides.
4. in the same case, find the loci of the angular points of a square described on one of the sides.
to cut off from a given circle (abc) a segment which shall contain an angle equal to a given angle (x).
sol.-take any point a in the circumference. draw the tangent ad, and make the angle dac equal to the given angle x. ac will cut off the required segment.
dem.-take any point b in the alternate segment. join ba, bc. then the angle dac is equal to abc xxxii. ; but dac is equal to x (const.). therefore the angle abc is equal to x.
if two chords (ab, cd) of a circle intersect in a point (e) within the circle, the rectangles (ae.eb, ce.ed) contained by the segments are equal.
dem.-1. if the point of intersection be the centre, each rectangle is equal to the square of the radius. hence they are equal.
2. let one of the chords ab pass through the centre o, and cut the other chord cd, which does not pass through the centre, at right angles. join oc.
now because ab passes through the centre, and cuts the other chord cd, which does not pass through the centre at right angles, it bisects it iii. .
again, because ab is divided equally in o and unequally in e, the rectangle ae.eb, together with oe2, is equal to ob2-that is, to oc2 ii. v. ; but oc2 is equal to oe2 ec2 i. xlvii. therefore
reject oe2, which is common, and we have ae.eb ec2; but ce2 is equal to the rectangle ce.ed, since ce is equal to ed. therefore the rectangle ae.eb is equal to the rectangle ce.ed.
3. let ab pass through the centre, and cut cd, which does not pass through the centre obliquely. let o be the centre. draw of perpendicular to cd i. xi. . join oc, od.
then, since cd is cut at right angles by of, which passes through the centre, it is bisected in f iii. , and divided unequally in e. hence
hence, adding, since fe2 of2 oe2 i. xlvii. , and fd2 of2 od2, we get
again, since ab is bisected in o and divided unequally in e,
4. let neither chord pass through the centre. through the point e, where they intersect, draw the diameter fg. then by 3, the rectangle fe.eg is equal to the rectangle ae.eb, and also to the rectangle ce.ed.
hence the rectangle ae.eb is equal to the rectangle ce.ed.
cor. 1.-if a chord of a circle be divided in any point within the circle, the rectangle contained by its segments is equal to the difference between the square of the radius and the square of the line drawn from the centre to the point of section.
cor. 2.-if the rectangle contained by the segments of one of two intersecting lines be equal to the rectangle contained by the segments of the other, the four extremities are concyclic.
cor. 3.-if two triangles be equiangular, the rectangle contained by the non-corresponding sides about any two equal angles are equal.
let abo, dco be the equiangular triangles, and let them be placed so that the equal angles at o may be vertically opposite, and that the non-corresponding sides ao, co may be in one line; then the non-corresponding sides bo, od shall be in one line.
now, since the angle abd is equal to acd, the points a, b, c, d are concyclic xxi., cor. 1 . hence the rectangle ao.oc is equal to the rectangle bo.od xxxv. .
1. in any triangle, the rectangle contained by two sides is equal to the rectangle contained by the perpendicular on the third side and the diameter of the circumscribed circle.
def.-the supplement of an arc is the difference between it and a semicircle.
2. the rectangle contained by the chord of an arc and the chord of its supplement is equal to the rectangle contained by the radius and the chord of twice the supplement.
3. if the base of a triangle be given, and the sum of the sides, the rectangle contained by the perpendiculars from the extremities of the base on the external bisector of the vertical angle is given.
4. if the base and the difference of the sides be given, the rectangle contained by the perpendiculars from the extremities of the base on the internal bisector is given.
5. through one of the points of intersection of two circles draw a secant, so that the rectangle contained by the intercepted chords may be given, or a maximum.
6. if the sum of two arcs, ac, cb of a circle be less than a semicircle, the rectangle ac.cb contained by their chords is equal to the rectangle contained by the radius, and the excess of the chord of the supplement of their difference above the chord of
dem.-draw de, the diameter which is perpendicular to ab, and draw the chords cf, bg parallel to de. now it is evident that the difference between the arcs ac, cb is equal to 2cd, and therefore cd ef.
hence the arc cbf is the supplement of the difference, and cf is the chord of that supplement. again, since the angle abg is right, the arc abg is a semicircle.
hence bg is the supplement of the sum of the arcs ac, cb; therefore the line bg is the chord of the supplement of the sum. now (ex.
1), the rectangle ac.cb is equal to the rectangle contained by the diameter and ci, and therefore equal to the rectangle contained by the radius and 2ci; but the difference between cf and bg is evidently equal to 2ci.
hence the rectangle ac.cb is equal to the rectangle contained by the radius and the difference between the chords cf, bg.
7. if we join af, bf we find, as before, the rectangle af.fb equal to the rectangle contained by the radius and 2fi-that is, equal to the rectangle contained by the radius and the sum of cf and bg.
hence-if the sum of two arcs of a circle be greater than a semicircle, the rectangle contained by their chords is equal to the rectangle contained by the radius, and the sum of the chords of the supplements of their sum and their difference.
8. through a given point draw a transversal cutting two lines given in position, so that the rectangle contained by the segments intercepted between it and the line may be given.
if from any point (p) without a circle two lines be drawn to it, one of which (pt) is a tangent, and the other (pa) a secant, the rectangle (ap, bp) contained by the segments of the secant is equal to the square of the tangent.
dem.-1. let pa pass through the centre o. join ot. then because ab is bisected in o and divided externally in p, the rectangle ap.bp ob2 is equal to op2 ii. vi. .
but since pt is a tangent, and ot drawn from the centre to the point of contact, the angle otp is right xviii. . hence ot2 pt2 is equal to op2.
2. if ab does not pass through the centre o, let fall the perpendicular oc on ab. join ot, ob, op. then because oc, a line through the centre, cuts ab, which does not pass through the centre at right angles, it bisects it iii. .
hence, since ab is bisected in c and divided externally in p, the rectangle
hence, adding, since cb2 oc2 ob2 i. xlvii. , and cp2 oc2 op2, we get
and rejecting the equals ob2 and ot2, we have the rectangle
the two propositions xxxv., xxxvi., may be included in one enunciation, as follows:-the rectangle ap.bp contained by the segments of any chord of a given circle passing through a fixed point p, either within or without the circle, is constant.
for let o be the centre: join oa, ob, op. then oab is an isosceles triangle, and op is a line drawn from its vertex to a point p in the base, or base produced.
then the rectangle ap.bp is equal to the difference of the squares of ob and op, and is therefore constant.
cor. 1.-if two lines ab, cd produced meet in p, and if the rectangle ap.bp cp.dp, the points a, b, c, d are concyclic (compare xxxv., cor. 2).
cor. 2.-tangents to two circles from any point in their common chord are equal (compare xvii., ex. 6).
cor. 3.-the common chords of any three intersecting circles are concurrent (compare xvii., ex. 7).
if from the vertex a of a abc, ad be drawn, meeting cb produced in d, and making the angle bad acb, prove db.dc da2.
if the rectangle (ap.bp) contained by the segments of a secant, drawn from any point (p) without a circle, be equal to the square of a line (pt) drawn from the same point to meet the circle, the line which meets the circle is a tangent.
dem.-from p draw pq touching the circle xvii. . let o be the centre. join op, oq, ot. now the rectangle ap.bp is equal to the square on pt (hyp.), and equal to the square on pq xxxvi. . hence pt2 is equal to pq2, and therefore pt is equal to pq.
again, the triangles otp, oqp have the side ot equal oq, tp equal qp, and the base op common; hence i. viii. the angle otp is equal to oqp; but oqp is a right angle, since pq is a tangent xviii. ; hence otp is right, and therefore xvi. pt is a tangent.
1. describe a circle passing through two given points, and fulfilling either of the following conditions: 1, touching a given line; 2, touching a given circle.
2. describe a circle through a given point, and touching two given lines; or touching a given file and a given circle.
3. describe a circle passing through a given point, having its centre on a given line and touching a given circle.
4. describe a circle through two given points, and intercepting a given arc on a given circle.
5. a, b, c, d are four collinear points, and ef is a common tangent to the circles described upon ab, cd as diameters: prove that the triangles aeb, cfd are equiangular.
6. the diameter of the circle inscribed in a right-angled triangle is equal to half the sum of the diameters of the circles touching the hypotenuse, the perpendicular from the right angle of the hypotenuse, and the circle described about the right-angled
1. what is the subject-matter of book iii.?
3. what is the difference between a chord and a secant?
5. what is the difference between a segment of a circle and a sector?
6. what is meant by an angle in a segment?
7. if an arc of a circle be one-sixth of the whole circumference, what is the magnitude of the angle in it?
9. what is meant by an angle standing on a segment?
12. how many intersections can a line and a circle have?
13. what does the line become when the points of intersection become consecutive?
14. how many points of intersection can two circles have?
15. what is the reason that if two circles touch they cannot have any other common point?
16. give one enunciation that will include propositions xi., xii. of book iii.
17. what proposition is this a limiting case of?
18. explain the extended meaning of the word angle.
20. state the relations between propositions xvi., xviii., xix.
21. what propositions are these limiting cases of?
22. how many common tangents can two circles have?
23. what is the magnitude of the rectangle of the segments of a chord drawn through a point 3.65 metres distant from the centre of a circle whose radius is 4.25 metres?
24. the radii of two circles are 4.25 and 1.75 feet respectively, and the distance between their centres 6.5 feet; find the lengths of their direct and their transverse common tangents.
25. if a point be h feet outside the circumference of a circle whose diameter is 7920 miles, prove that the length of the tangent drawn from it to the circle is --- 3h 2 miles.
26. two parallel chords of a circle are 12 perches and 16 perches respectively, and their distance asunder is 2 perches; find the length of the diameter.
27. what is the locus of the centres of all circles touching a given circle in a given point?
28. what is the condition that must be fulfilled that four points may be concyclic?
29. if the angle in a segment of a circle be a right angle and a-half, what part of the whole circumference is it?
30. mention the converse propositions of book iii. which are proved directly.
31. what is the locus of the middle points of equal chords in a circle?
32. the radii of two circles are 6 and 8, and the distance between their centres 10; find the length of their common chord.
33. if a figure of any even number of sides be inscribed in a circle, prove that the sum of one set of alternate angles is equal to the sum of the remaining angles.
1. if two chords of a circle intersect at right angles, the sum of the squares on their segments is equal to the square on the diameter.
2. if a chord of a given circle subtend a right angle at a fixed point, the rectangle of the perpendiculars on it from the fixed point and from the centre of the given circle is constant.
also the sum of the squares of perpendiculars on it from two other fixed points (which may be found) is constant.
3. if through either of the points of intersection of two equal circles any line be drawn meeting them again in two points, these points are equally distant from the other intersection of the circles.
4. draw a tangent to a given circle so that the triangle formed by it and two fixed tangents to the circle shall be-1, a maximum; 2, a minimum.
5. if through the points of intersection a, b of two circles any two lines acd, bef be drawn parallel to each other, and meeting the circles again in c, d, e, f; then cd ef.
6. in every triangle the bisector of the greatest angle is the least of the three bisectors of the angles.
7. the circles whose diameters are the four sides of any cyclic quadrilateral intersect again in four concyclic points.
8. the four angular points of a cyclic quadrilateral determine four triangles whose orthocentres (the intersections of their perpendiculars) form an equal quadrilateral.
9. if through one of the points of intersection of two circles we draw two common chords, the lines joining the extremities of these chords make a given angle with each other.
10. the square on the perpendicular from any point in the circumference of a circle, on the chord of contact of two tangents, is equal to the rectangle of the perpendiculars from the same point on the tangents.
11. find a point in the circumference of a given circle, the sum of the squares on whose distances from two given points may be a maximum or a minimum.
12. four circles are described on the sides of a quadrilateral as diameters. the common chord of any two on adjacent sides is parallel to the common chord of the remaining two.
13. the rectangle contained by the perpendiculars from any point in a circle, on the diagonals of an inscribed quadrilateral, is equal to the rectangle contained by the perpendiculars from the same point on either pair of opposite sides.
14. the rectangle contained by the sides of a triangle is greater than the square on the internal bisector of the vertical angle, by the rectangle contained by the segments of the base.
15. if through a, one of the points of intersection of two circles, we draw any line abc, cutting the circles again in b and c, the tangents at b and c intersect at a given angle.
16. if a chord of a given circle pass through a given point, the locus of the intersection of tangents at its extremities is a right line.
17. the rectangle contained by the distances of the point where the internal bisector of the vertical angle meets the base, and the point where the perpendicular from the vertex meets it from the middle point of the base, is equal to the square on half
18. state and prove the proposition analogous to 17 for the external bisector of the vertical angle.
19. the square on the external diagonal of a cyclic quadrilateral is equal to the sum of the squares on the tangents from its extremities to the circumscribed circle.
20. if a variable circle touch a given circle and a given line, the chord of contact passes through a given point.
21. if a, b, c be three points in the circumference of a circle, and d, e the middle points of the arcs ab, ac; then if the line de intersect the chords ab, ac in the points f, g, af is equal to ag.
22. given two circles, o, o; then if any secant cut o in the points b, c, and o in the points b, c, and another secant cuts them in the points d, e; d, e respectively; the four chords bd, ce, bd, ce form a cyclic quadrilateral.
23. if a cyclic quadrilateral be such that a circle can be inscribed in it, the lines joining the points of contact are perpendicular to each other.
24. if through the point of intersection of the diagonals of a cyclic quadrilateral the minimum chord be drawn, that point will bisect the part of the chord between the opposite sides of the quadrilateral.
25. given the base of a triangle, the vertical angle, and either the internal or the external bisector at the vertical angle; construct it.
26. if through the middle point a of a given arc bac we draw any chord ad, cutting bc in e, the rectangle ad.ae is constant.
27. the four circles circumscribing the four triangles formed by any four lines pass through a common point.
28. if x, y , z be any three points on the three sides of a triangle abc, the three circles about the triangles y az, zbx, xcy pass through a common point.
29. if the position of the common point in the last question be given, the three angles of the triangle xy z are given, and conversely.
30. place a given triangle so that its three sides shall pass through three given points.
31. place a given triangle so that its three vertices shall lie on three given lines.
32. construct the greatest triangle equiangular to a given one whose sides shall pass through three given points.
33. construct the least triangle equiangular to a given one whose vertices shall lie on three given lines.
34. construct the greatest triangle equiangular to a given one whose sides shall touch three given circles.
35. if two sides of a given triangle pass through fixed points, the third touches a fixed circle.
36. if two sides of a given triangle touch fixed circles, the third touches a fixed circle.
37. construct an equilateral triangle having its vertex at a given point, and the extremities of its base on a given circle.
38. construct an equilateral triangle having its vertex at a given point, and the extremities of its base on two given circles.
39. place a given triangle so that its three sides shall touch three given circles.
40. circumscribe a square about a given quadrilateral.
41. inscribe a square in a given quadrilateral.
42. describe circles-(1) orthogonal (cutting at right angles) to a given circle and passing through two given points; (2) orthogonal to two others, and passing through a given point; (3) orthogonal to three others.
43. if from the extremities of a diameter ab of a semicircle two chords ad, be be drawn, meeting in c, ac.ad bc.be ab2.
44. if abcd be a cyclic quadrilateral, and if we describe any circle passing through the points a and b, another through b and c, a third through c and d, and a fourth through d and a; these circles intersect successively in four other points e, f, g, h,
45. if abc be an equilateral triangle, what is the locus of the point m, if ma mb mc?
46. in a triangle, given the sum or the difference of two sides and the angle formed by these sides both in magnitude and position, the locus of the centre of the circumscribed circle is a right line.
47. describe a circle-(1) through two given points which shall bisect the circumference of a given circle; (2) through one given point which shall bisect the circumference of two given circles.
48. find the locus of the centre of a circle which bisects the circumferences of two given circles.
49. describe a circle which shall bisect the circumferences of three given circles.
50. ab is a diameter of a circle; ac, ad are two chords meeting the tangent at b in the points e, f respectively: prove that the points c, d, e, f are concyclic.
51. cd is a perpendicular from any point c in a semicircle on the diameter ab; efg is a circle touching db in e, cd in f, and the semicircle in g; prove-(1) that the points a, f, g are collinear; (2) that ac ae.
52. being given an obtuse-angled triangle, draw from the obtuse angle to the opposite side a line whose square shall be equal to the rectangle contained by the segments into which it divides the opposite side.
53. o is a point outside a circle whose centre is e; two perpendicular lines passing through o intercept chords ab, cd on the circle; then ab2 cd2 4oe2 8r2.
54. the sum of the squares on the sides of a triangle is equal to twice the sum of the rectangles contained by each perpendicular and the portion of it comprised between the corresponding vertex and the orthocentre; also equal to 12r2 minus the sum of the
squares of the distances of the orthocentre from the vertices.
55. if two circles touch in c, and if d be any point outside the circles at which their radii through c subtend equal angles, if de, df be tangent from d, de.df dc2.
inscription and circumscription of triangles and of regular polygons in and about circles
i. if two rectilineal figures be so related that the angular points of one lie on the sides of the other-1, the former is said to be inscribed in the latter; 2, the latter is said to be described about the former.
ii. a rectilineal figure is said to be inscribed in a circle when its angular points are on the circumference. reciprocally, a rectilineal figure is said to be circumscribed to a circle when each side touches the circle.
iii. a circle is said to be inscribed in a rectilineal figure when it touches each side of the figure. reciprocally, a circle is said to be circumscribed to a rectilineal figure when it passes through each angular point of the figure.
iv. a rectilineal figure which is both equilateral and equiangular is said to be regular.
observation.-the following summary of the contents of the fourth book will assist the student in remembering it:-
1. it contains sixteen propositions, of which four relate to triangles, four to squares, four to pentagons, and four miscellaneous propositions.
2. of the four propositions occupied with triangles-
() one is to inscribe a triangle in a circle.
() its reciprocal, to describe a triangle about a circle.
() its reciprocal, to describe a circle about a triangle.
3. if we substitute in (), (), (), () squares for triangles, and pentagons for triangles, we have the problems for squares and pentagons respectively.
4. every proposition in the fourth book is a problem.
in a given circle (abc) to place a chord equal to a given line (d) not greater than the diameter.
sol.-draw any diameter ac of the circle; then, if ac be equal to d, the thing required is done; if not, from ac cut off the part ae equal to d i. iii.
; and with a as centre and ae as radius, describe the circle ebf, cutting the circle abc in the points b, f. join ab. then ab is the chord required.
dem.-because a is the centre of the circle ebf, ab is equal to ae i. def. xxxii. ; but ae is equal to d (const.); therefore ab is equal to d.
in a given circle (abc) to inscribe a triangle equiangular to a given triangle (def).
sol.-take any point a in the circumference, and at it draw the tangent gh; then make the angle hac equal to e, and gab equal to f i. xxiii. join bc. abc is a triangle fulfilling the required conditions.
dem.-the angle e is equal to hac (const.), and hac is equal to the angle abc in the alternate segment iii. xxxii. . hence the angle e is equal to abc. in like manner the angle f is equal to acb. therefore i. xxxii. the remaining angle d is equal to bac.
hence the triangle abc inscribed in the given circle is equiangular to def.
about a given circle (abc) to describe a triangle equiangular to a given triangle (def).
sol.-produce any side de of the given triangle both ways to g and h, and from the centre o of the circle draw any radius oa; make the angle aob equal to gef i. xxiii. , and the angle aoc equal to hdf.
at the points a, b, c draw the tangents lm, mn, nl to the given circle. lmn is a triangle fulfilling the required conditions.
dem.-because am touches the circle at a, the angle oam is right. in like manner, the angle mbo is right; but the sum of the four angles of the quadrilateral oamb is equal to four right angles.
therefore the sum of the two remaining angles aob, amb is two right angles; and i. xiii. the sum of the two angles gef, fed is two right angles. therefore the sum of aob, amb is equal to the sum of gef, fed; but aob is equal to gef (const.).
hence amb is equal to fed. in like manner, alc is equal to edf; therefore i. xxxii. the remaining angle bnc is equal to dfe. hence the triangle lmn is equiangular to def.
to inscribe a circle in a given triangle (abc).
sol.-bisect any two angles a, b of the given triangle by the lines ao, bo; then o, their point of intersection, is the centre of the required circle.
dem.-from o let fall the perpendiculars od, oe, of on the sides of the triangle. now, in the triangles oae, oaf the angle oae is equal to oaf (const.), and the angle aeo equal to afo, because each is right, and the side oa common. hence i. xxvi.
the side oe is equal to of. in like manner od is equal to of; therefore the three lines od, oe, of are all equal.
and the circle described with o as centre and od as radius will pass through the points e, f; and since the angles d, e, f are right, it will iii. xvi.
touch the three sides of the triangle abc; and therefore the circle def is inscribed in the triangle abc.
1. if the points o, c be joined, the angle c is bisected. hence "the bisectors of the angles of a triangle are concurrent" (compare i. xxvi., ex. 7).
2. if the sides bc, ca, ab of the triangle abc be denoted by a, b, c, and half their sum by s, the distances of the vertices a, b, c of the triangle from the points of contact of the inscribed circle are respectively s a, s b, s c.
3. if the external angles of the triangle abc be bisected as in the annexed diagram, the three angular points o, o, o, of the triangle formed by the three bisectors will be the centres of three circles, each touching one side externally, and the other two
produced. these three circles are called the escribed circles of the triangle abc.
4. the distances of the vertices a, b, c from the points of contact of the escribed circle which touches ab externally are s b, s a, s.
5. the centre of the inscribed circle, the centre of each escribed circle, and two of the angular points of the triangle, are concyclic. also any two of the escribed centres are concyclic with the corresponding two of the angular points of the triangle.
6. of the four points o, o, o, o, any one is the orthocentre of the triangle formed by the remaining three.
7. the three triangles bco, cao, abo are equiangular.
8. the rectangle co.co ab; ao.ao bc; bo.bo ca.
9. since the whole triangle abc is made up of the three triangles aob, boc, coa, we see that the rectangle contained by the sum of the three sides, and the radius of the inscribed circle, is equal to twice the area of the triangle.
hence, if r denote the radius of the inscribed circle, rs area of the triangle.
10. if r denote the radius of the escribed circle which touches the side a externally, it may be shown in like manner that r(s a) area of the triangle.
14. if the triangle abc be right-angled, having the angle c right,
15. given the base of a triangle, the vertical angle, and the radius of the inscribed, or any of the escribed circles: construct it.
to describe a circle about a given triangle (abc).
sol.-bisect any two sides bc, ac in the points d, e. erect do, eo at right angles to bc, ca; then o, the point of intersection of the perpendiculars, is the centre of the required circle.
dem.-join oa, ob, oc. the triangles bdo, cdo have the side bd equal cd (const.), and do common, and the angle bdo equal to the angle cdo, because each is right. hence i. iv. bo is equal to oc. in like manner ao is equal to oc.
therefore the three lines ao, bo, co are equal, and the circle described with o as centre, and oa as radius, will pass through the points a, b, c, and be described about the triangle abc.
cor. 1.-since the perpendicular from o on ab bisects it iii. iii. , we see that the perpendiculars at the middle points of the sides of a triangle are concurrent.
def.-the circle abc is called the circumcircle, its radius the circumradius, and its centre the circumcentre of the triangle.
1. the three perpendiculars of a triangle (abc) are concurrent.
dem.-describe a circle about the triangle. let fall the perpendicular cf. produce cf to meet the circle in g. make fo fg. join ag, ao. produce ao to meet bc in d.
then the triangles gfa, ofa have the sides gf, fa in one equal to the sides of, fa in the other, and the contained angles equal. hence i. iv. the angle gaf equal oaf; but gaf gcb iii. xxi.
; hence oaf ocd, and foa doc; hence ofa odc; but ofa is right, hence odc is right. in like manner, if bo be joined to meet ac in e, be will be perpendicular to ac. hence the three perpendiculars pass through o, and are concurrent.
this proposition may be proved simply as follows:-
draw parallels to the sides of the original triangle abc through its vertices, forming a new triangle abc described about abc; then the three perpendiculars at the middle points of the sides of abc are concurrent v. cor.
1 , and these are evidently the perpendiculars from the vertices on the opposite sides of the triangle abc (compare ex. 16, book i.).
def.-the point o is called the orthocentre of the triangle abc.
2. the three rectangles oa.op, ob.oq, oc.or are equal.
def.-the circle round o as centre, the square of whose radius is equal oa.op ob.oq oc.or, is called the polar circle of the triangle abc.
observation.-if the orthocentre of the triangle abc be within the triangle, the rectangles oa.op, ob.oq, oc.or are negative, because the lines oa.op, c., are measured in opposite directions, and have contrary signs; hence the polar circle is imaginary;
but it is real when the point o is without the triangle-that is, when the triangle has an obtuse angle.
3. if the perpendiculars of a triangle be produced to meet the circumscribed circle, the intercepts between the orthocentre and the circle are bisected by the sides of the triangle.
4. the point of bisection (i) of the line (op) joining the orthocentre (o) to the circumference (p) of any triangle is equally distant from the feet of the perpendiculars, from the middle points of the sides, and from the middle points of the distances of
dem.-draw the perpendicular ph; then, since of, ph are perpendiculars on ab, and op is bisected in i, it is easy to see that ih if. again, since op, og are bisected in i, f; if 1 2pg-that is, if 1 2 the radius.
hence the distance of i from the foot of each perpendicular, and from the middle point of each side, is 1 2 the radius. in like manner, if oc be bisected in k, then ik 1 2 the radius.
hence we have the following theorem:-the nine points made up of the feet of the perpendiculars, the middle points of the sides, and the middle points of the lines from the vertices to the orthocentre, are concyclic.
def.-the circle through these nine points is called the "nine points circle" of the triangle.
5. the circumcircle of a triangle is the "nine points circle" of each of the four triangles formed by joining the centres of the inscribed and escribed circles.
6. the distances between the vertices of a triangle and its orthocentre are respectively the doubles of the perpendiculars from the circumcentre on the sides.
7. the radius of the "nine points circle" of a triangle is equal to half its circumradius.
in a given circle (abcd) to inscribe a square.
sol.-draw any two diameters ac, bd at right angles to each other. join ab, bc, cd, da. abcd is a square.
dem.-let o be the centre. then the four angles at o, being right angles, are equal. hence the arcs on which they stand are equal iii. xxvi. , and hence the four chords are equal iii. xxix. . therefore the figure abcd is equilateral.
again, because ac is a diameter, the angle abc is right iii. xxxi. . in like manner the remaining angles are right. hence abcd is a square.
about a given circle (abcd) to describe a square.
sol.-through the centre o draw any two diameters at right angles to each other, and draw at the points a, b, c, d the lines he, ef, fg, gh touching the circle. efgh is a square.
dem.-because ae touches the circle at a, the angle eao is right iii. xviii. , and therefore equal to boc, which is right (const.). hence ae is parallel to ob.
in like manner eb is parallel to ao; and since ao is equal to ob, the figure aobe is a lozenge, and the angle aob is right; hence aobe is a square. in like manner each of the figures bc, cd, da is a square. hence the whole figure is a square.
cor.-the circumscribed square is double of the inscribed square.
in a given square (abcd) to inscribe a circle.
sol.-bisect (see last diagram) two adjacent sides eh, ef in the points a, b, and through a, b draw the lines ac, bd, respectively parallel to ef, eh; then o, the point of intersection of these parallels, is the centre of the required circle.
dem.-because aobe is a parallelogram, its opposite sides are equal; therefore ao is equal to eb; but eb is half the side of the given square; therefore ao is equal to half the side of the given square; and so in like manner is each of the lines ob, oc,
od; therefore the four lines oa, ob, oc, od are all equal; and since they are perpendicular to the sides of the given square, the circle described with o as centre, and oa as radius, will be inscribed in the square.
about a given square (abcd) to describe a circle.
sol.-draw the diagonals ac, bd intersecting in o (see diagram to proposition vi.). o is the centre of the required circle.
dem.-since abc is an isosceles triangle, and the angle b is right, each of the other angles is half a right angle; therefore bao is half a right angle. in like manner abo is half a right angle; hence the angle bao equal abo; therefore i. vi.
ao is equal to ob. in like manner ob is equal to oc, and oc to od. hence the circle described, with o as centre and oa as radius, will pass through the points b, c, d, and be described about the square.
to construct an isosceles triangle having each base angle double the vertical angle.
sol.-take any line ab. divide it in c, so that the rectangle ab.bc shall be equal to ac2 ii. xi. . with a as centre, and ab as radius, describe the circle bde, and in it place the chord bd equal to ac i. . join ad.
adb is a triangle fulfilling the required conditions.
dem.-join cd. about the triangle acd describe the circle cde v. . then, because the rectangle ab.bc is equal to ac2 (const.), and that ac is equal to bd (const.); therefore the rectangle ab.bc is equal to bd2. hence iii. xxxvii. bd touches the circle acd.
hence the angle bdc is equal to the angle a in the alternate segment iii. xxxii. .
to each add cda, and we have the angle bda equal to the sum of the angles cda and a; but the exterior angle bcd of the triangle acd is equal to the sum of the angles cda and a.
hence the angle bda is equal to bcd; but since ab is equal to ad, the angle bda is equal to abd; therefore the angle cbd is equal to bcd. hence i. vi. bd is equal to cd; but bd is equal to ac (const.); therefore ac is equal to cd, and therefore i. v.
the angle cda is equal to a; but bda has been proved to be equal to the sum of cda and a. hence bda is double of a. hence each of the base angles of the triangle abd is double of the vertical angle.
1. prove that acd is an isosceles triangle whose vertical angle is equal to three times each of the base angles.
2. prove that bd is the side of a regular decagon inscribed in the circle bde.
3. if db, de, ef be consecutive sides of a regular decagon inscribed in a circle, prove bf bd radius of circle.
4. if e be the second point of intersection of the circle acd with bde, de is equal to db; and if ae, be, ce, de be joined, each of the triangles ace, ade is congruent with abd.
5. ac is the side of a regular pentagon inscribed in the circle acd, and eb the side of a regular pentagon inscribed in the circle bde.
6. since ace is an isosceles triangle, eb2 ea2 ab.bc-that is bd2; therefore eb2 bd2 ea2-that is, the square of the side of a pentagon inscribed in a circle exceeds the square of the side of the decagon inscribed in the same circle by the square of the
to inscribe a regular pentagon in a given circle (abcde).
sol.-construct an isosceles triangle x. , having each base angle double the vertical angle, and inscribe in the given circle a triangle abd equiangular to it. bisect the angles dab, abd by the lines ac, be.
join ea, ed, dc, cb; then the figure abcde is a regular pentagon.
dem.-because each of the base angles bad, abd is double of the angle adb, and the lines ac, be bisect them, the five angles bac, cad, adb, dbe, eba are all equal; therefore the arcs on which they stand are equal; and therefore the five chords, ab, bc, cd,
de, ea are equal. hence the figure abcde is equilateral.
again, because the arcs ab, de are equal, adding the arc bcd to both, the arc abcd is equal to the arc bcde, and therefore iii. xxvii. the angles aed, bae, which stand on them, are equal.
in the same manner it can be proved that all the angles are equal; therefore the figure abcde is equiangular. hence it is a regular pentagon.
1. the figure formed by the five diagonals of a regular pentagon is another regular pentagon.
2. if the alternate sides of a regular pentagon be produced to meet, the five points of meeting form another regular pentagon.
3. every two consecutive diagonals of a regular pentagon divide each other in extreme and mean ratio.
4. being given a side of a regular pentagon, construct it.
5. divide a right angle into five equal parts.
to describe a regular pentagon about a given circle (abcde).
sol.-let the five points a, b, c, d, e on the circle be the vertices of any inscribed regular pentagon: at these points draw tangents fg, gh, hi, ij, jf: the figure fghij is a circumscribed regular pentagon.
dem.-let o be the centre of the circle. join oe, oa, ob. now, because the angles a, e of the quadrilateral aoef are right angles iii. xviii. , the sum of the two remaining angles aoe, afe is two right angles.
in like manner the sum of the angles aob, agb is two right angles; therefore the sum of aoe, afe is equal to the sum of aob, agb; but the angles aoe, aob are equal, because they stand on equal arcs ae, ab iii. xxvii. . hence the angle afe is equal to agb.
in like manner the remaining angles of the figure fghij are equal. therefore it is equiangular.
again, join of, og. now the triangles eof, aof have the sides af, fe equal iii. xvii., ex. 1 , and fo common, and the base ao equal to the base eo. hence the angle afo is equal to efo i. viii. . therefore the angle afo is half the angle afe.
in like manner ago is half the angle agb; but afe has been proved equal to agb; hence afo is equal to ago, and fao is equal to gao, each being right, and ao common to the two triangles fao, gao; hence i. xxvi.
the side af is equal to ag; therefore gf is double af. in like manner jf is double ef; but af is equal to ef; hence gf is equal to jf.
in like manner the remaining sides are equal; therefore the figure fghij is equilateral, and it has been proved equiangular. hence it is a regular pentagon.
this proposition is a particular case of the following general theorem, of which the proof is the same as the foregoing:-
"if tangents be drawn to a circle, at the angular points of an inscribed polygon of any number of sides, they will form a regular polygon of the same number of sides circumscribed to the circle."
to inscribe a circle in a regular pentagon (abcde).
sol.-bisect two adjacent angles a, b by the lines ao, bo; then o, the point of intersection of the bisectors, is the centre of the required circle.
dem.-join co, and let fall perpendiculars from o on the five sides of the pentagon. now the triangles abo, cbo have the side ab equal to bc (hyp.), and bo common, and the angle abo equal to cbo (const.). hence the angle bao is equal to bco i. iv.
; but bao is half bae (const.). therefore bco is half bcd, and therefore co bisects the angle bcd. in like manner it may be proved that do bisects the angle d, and eo the angle e.
again, the triangles bof, bog have the angle f equal to g, each being right; and obf equal to obg, because ob bisects the angle abc (const.), and ob common; hence i. xxvi. of is equal to og.
in like manner all the perpendiculars from o on the sides of the pentagon are equal; hence the circle whose centre is o, and radius of, will touch all the sides of the pentagon, and will therefore be inscribed in it.
in the same manner a circle may be inscribed in any regular polygon.
to describe a circle about a regular pentagon (abcde).
sol.-bisect two adjacent angles a, b by the lines ao, bo. then o, the point of intersection of the bisectors, is the centre of the required circle.
dem.-join oc, od, oe. then the triangles abo, cbo have the side ab equal to bc (hyp.), bo common, and the angle abo equal to cbo (const.). hence the angle bao is equal to bco i. iv.
; but the angle bae is equal to bcd (hyp.); and since bao is half bae (const.), bco is half bcd. hence co bisects the angle bcd. in like manner it may be proved that do bisects cde, and eo the angle dea.
again, because the angle eab is equal to abc, their halves are equal. hence oab is equal to oba; therefore i. vi. oa is equal to ob. in like manner the lines oc, od, oe are equal to one another and to oa.
therefore the circle described with o as centre, and oa as radius, will pass through the points b, c, d, e, and be described about the pentagon.
in the same manner a circle may be described about any regular polygon.
propositions xiii., xiv. are particular cases of the following theorem:-
"a regular polygon of any number of sides has one circle inscribed in it, and another described about it, and both circles are concentric."
in a given circle (abcdef) to inscribe a regular hexagon.
sol.-take any point a in the circumference, and join it to o, the centre of the given circle; then with a as centre, and ao as radius, describe the circle obf, intersecting the given circle in the points b, f.
join ob, of, and produce ao, bo, fo to meet the given circle again in the points d, e, c. join ab, bc, cd, de, ef, fa; abcdef is the required hexagon.
dem.-each of the triangles aob, aof is equilateral (see dem., i. i.). hence the angles aob, aof are each one-third of two right angles; therefore eof is one-third of two right angles. again, the angles boc, cod, doe are i. xv.
respectively equal to the angles eof, foa, aob. therefore the six angles at the centre are equal, because each is one-third of two right angles. therefore the six chords are equal iii. xxix. . hence the hexagon is equilateral.
again, since the arc af is equal to ed, to each add the arc abcd; then the whole arc fabcd is equal to abcde; therefore the angles def, efa which stand on these arcs are equal iii. xxvii. .
in the same manner it may be shown that the other angles of the hexagon are equal. hence it is equiangular, and is therefore a regular hexagon inscribed in the circle.
cor. 1.-the side of a regular hexagon inscribed in a circle is equal to the radius.
cor. 2.-if three alternate angles of a hexagon be joined, they form an inscribed equilateral triangle.
1. the area of a regular hexagon inscribed in a circle is equal to twice the area of an equilateral triangle inscribed in the circle; and the square of the side of the triangle is three times the square of the side of the hexagon.
2. if the diameter of a circle be produced to c until the produced part is equal to the radius, the two tangents from c and their chord of contact form an equilateral triangle.
3. the area of a regular hexagon inscribed in a circle is half the area of an equilateral triangle, and three-fourths of the area of a regular hexagon circumscribed to the circle.
to inscribe a regular polygon of fifteen sides in a given circle.
sol.-inscribe a regular pentagon abcde in the circle xi. , and also an equilateral triangle agh ii. . join cg. cg is a side of the required polygon.
dem.-since abcde is a regular pentagon, the arc abc is 2 5ths of the circumference; and since agh is an equilateral triangle, the arc abg is 1 3rd of the circumference.
hence the arc gc, which is the difference between these two arcs, is equal to 2 5ths 1 3rd, or -1 15th of the entire circumference; and therefore, if chords equal to gc i.
be placed round the circle, we shall have a regular polygon of fifteen sides, or quindecagon, inscribed in it.
scholium.-until the year 1801 no regular polygon could be described by constructions employing the line and circle only, except those discussed in this book, and those obtained from them by the continued bisection of the arcs of which their sides are the
chords; but in that year the celebrated gauss proved that if 2n 1 be a prime number, regular polygons of 2n 1 sides are inscriptable by elementary geometry.
for the case n 4, which is the only figure of this class except the pentagon for which a construction has been given, see note at the end of this work.
1. what is the subject-matter of book iv.?
2. when is one rectilineal figure said to be inscribed in another?
4. when is a circle said to be inscribed in a rectilineal figure?
6. what is meant by reciprocal propositions? ans. in reciprocal propositions, to every line in one there corresponds a point in the other; and, conversely, to every point in one there corresponds a line in the other.
7. give instances of reciprocal propositions in book iv.
9. what figures can be inscribed in, and circumscribed about, a circle by means of book iv.?
10. what regular polygons has gauss proved to be inscriptable by the line and circle?
12. how many circles can be described to touch three lines forming a triangle?
20. name the special nine points through which it passes.
21. what three regular figures can be used in filling up the space round a point? ans. equilateral triangles, squares, and hexagons.
22. if the sides of a triangle be 13, 14, 15, what are the values of the radii of its inscribed and escribed circles?
23. what is the radius of the circumscribed circle?
24. what is the radius of its nine-points circle?
25. what is the distance between the centres of its inscribed and circumscribed circles?
26. if r be the radius of a circle, what is the area of its inscribed equilateral triangle?-of its inscribed square?-its inscribed pentagon?-its inscribed hexagon?-its inscribed octagon?-its inscribed decagon?
27. with the same hypothesis, find the sides of the same regular figures.
1. if a circumscribed polygon be regular, the corresponding inscribed polygon is also regular, and conversely.
2. if a circumscribed triangle be isosceles, the corresponding inscribed triangle is isosceles, and conversely.
3. if the two isosceles triangles in ex. 2 have equal vertical angles, they are both equilateral.
4. divide an angle of an equilateral triangle into five equal parts.
5. inscribe a circle in a sector of a given circle.
6. the line de is parallel to the base bc of the triangle abc: prove that the circles described about the triangles abc, ade touch at a.
7. the diagonals of a cyclic quadrilateral intersect in e: prove that the tangent at e to the circle about the triangle abe is parallel to cd.
8. inscribe a regular octagon in a given square.
9. a line of given length slides between two given lines: find the locus of the intersection of perpendiculars from its extremities to the given lines.
10. if the perpendicular to any side of a triangle at its middle point meet the internal and external bisectors of the opposite angle in the points d and e; prove that d, e are points on the circumscribed circle.
11. through a given point p draw a chord of a circle so that the intercept ef may subtend a given angle x.
12. in a given circle inscribe a triangle having two sides passing through two given points, and the third parallel to a given line.
13. given four points, no three of which are collinear; describe a circle which shall be equidistant from them.
14. in a given circle inscribe a triangle whose three sides shall pass through three given points.
the radius of the inscribed circle, the vertical angle, and the perpendicular from the vertical angle on the base.
the base, the sum or difference of the other sides, and the radius of the inscribed circle, or of one of the escribed circles.
16. if f be the middle point of the base of a triangle, de the diameter of the circumscribed circle which passes through f, and l the point where a parallel to the base through the vertex meets de: prove dl.fe is equal to the square of half the sum, and
df.le equal to the square of half the difference of the two remaining sides.
17. if from any point within a regular polygon of n sides perpendiculars be let fall on the sides, their sum is equal to n times the radius of the inscribed circle.
18. the sum of the perpendiculars let fall from the angular points of a regular polygon of n sides on any line is equal to n times the perpendicular from the centre of the polygon on the same line.
19. if r denotes the radius of the circle circumscribed about a triangle abc, r, r, r, r the radii of its inscribed and escribed circles, , , the perpendiculars from its circumcentre on the sides; , , the segments of these perpendiculars between the sides
and circumference of the circumscribed circle, we have the relations-
the relation (3) supposes that the circumcentre is inside the triangle.
20. through a point d, taken on the side bc of a triangle abc, is drawn a transversal edf, and circles described about the triangles dbf, ecd. the locus of their second point of intersection is a circle.
21. in every quadrilateral circumscribed about a circle, the middle points of its diagonals and the centre of the circle are collinear.
22. find on a given line a point p, the sum or difference of whose distances from two given points may be given.
23. find a point such that, if perpendiculars be let fall from it on four given lines, their feet may be collinear.
24. the line joining the orthocentre of a triangle to any point p, in the circumference of its circumscribed circle, is bisected by the line of collinearity of perpendiculars from p on the sides of the triangle.
25. the orthocentres of the four triangles formed by any four lines are collinear.
26. if a semicircle and its diameter be touched by any circle, either internally or externally, twice the rectangle contained by the radius of the semicircle, and the radius of the tangential circle, is equal to the rectangle contained by the segments of
any secant to the semicircle, through the point of contact of the diameter and touching circle.
27. if , be the radii of two circles, touching each other at the centre of the inscribed circle of a triangle, and each touching the circumscribed circle, prove
and state and prove corresponding theorems for the escribed circles.
28. if from any point in the circumference of the circle, circumscribed about a regular polygon of n sides, lines be drawn to its angular points, the sum of their squares is equal to 2n times the square of the radius.
29. in the same case, if the lines be drawn from any point in the circumference of the inscribed circle, prove that the sum of their squares is equal to n times the sum of the squares of the radii of the inscribed and the circumscribed circles.
30. state the corresponding theorem for the sum of the squares of the lines drawn from any point in the circumference of any concentric circle.
31. if from any point in the circumference of any concentric circle perpendiculars be let fall on all the sides of any regular polygon, the sum of their squares is constant.
32. for the inscribed circle, the constant is equal to 3n 2 times the square of the radius.
33. for the circumscribed circle, the constant is equal to n times the square of the radius of the inscribed circle, together with 1 2n times the square of the radius of the circumscribed circle.
34. if the circumference of a circle whose radius is r be divided into seventeen equal parts, and ao be the diameter drawn from one of the points of division (a), and if 1, 2......8 denote the chords from o to the points of division, a1, a2......a8 on one
dem.-let the supplemental chords corresponding to 1, 2, c., be denoted by r1, r2, c.; then iii. xxxv. ex. 2 , we have
and it may be proved in the same manner that
35. if from the middle point of the line joining any two of four concyclic points a perpendicular be let fall on the line joining the remaining two, the six perpendiculars thus obtained are concurrent.
36. the greater the number of sides of a regular polygon circumscribed about a given circle, the less will be its perimeter.
37. the area of any regular polygon of more than four sides circumscribed about a circle is less than the square of the diameter.
38. four concyclic points taken three by three determine four triangles, the centres of whose nine-points circles are concyclic.
39. if two sides of a triangle be given in position, and if their included angle be equal to an angle of an equilateral triangle, the locus of the centre of its nine-points circle is a right line.
40. if, in the hypothesis and notation of ex. 34, , denote any two suffixes whose sum is less than 8, and of which is the greater,
for instance, 14 r(3 5) iii. xxxv., ex. 7 .
in the same case, if the suffixes be greater than 8,
for instance, 82 r(6 7) iii. xxxv., ex. 6 .
41. two lines are given in position: draw a transversal through a given point, forming with the given lines a triangle of given perimeter.
42. given the vertical angle and perimeter of a triangle, construct it with either of the following data: 1. the bisector of the vertical angle; 2. the perpendicular from the vertical angle on the base; 3. the radius of the inscribed circle.
43. in a given circle inscribe a triangle so that two sides may pass through two given points, and that the third side may be a maximum or a minimum.
44. if s be the semiperimeter of a triangle, r, r, r, the radii of its escribed circles,
45. the feet of the perpendiculars from the extremities of the base on either bisector of the vertical angle, the middle point of the base, and the foot of the perpendicular from the vertical angle on the base, are concyclic.
46. given the base of a triangle and the vertical angle; find the locus of the centre of the circle passing through the centres of the escribed circles.
47. the perpendiculars from the centres of the escribed circles of a triangle on the corresponding sides are concurrent.
48. if ab be the diameter of a circle, and pq any chord cutting ab in o, and if the lines ap, aq intersect the perpendicular to ab at o, in d and e respectively, the points a, b, d, e are concyclic.
49. if the sides of a triangle be in arithmetical progression, and if r, r be the radii of the circumscribed and inscribed circles; then 6rr is equal to the rectangle contained by the greatest and least sides.
50. inscribe in a given circle a triangle having its three sides parallel to three given lines.
51. if the sides ab, bc, c., of a regular pentagon be bisected in the points a, b, c, d, e, and if the two pairs of alternate sides, bc, ae; ab, de, meet in the points a, e, respectively, prove
52. in a circle, prove that an equilateral inscribed polygon is regular, and also an equilateral circumscribed polygon, if the number of sides be odd.
53. prove also that an equiangular circumscribed polygon is regular, and an equiangular inscribed polygon, if the number of sides be odd.
54. the sum of the perpendiculars drawn to the sides of an equiangular polygon from any point inside the figure is constant.
55. express the sides of a triangle in terms of the radii of its escribed circles.
introduction.-every proposition in the theories of ratio and proportion is true for all descriptions of magnitude. hence it follows that the proper treatment is the algebraic. it is, at all events, the easiest and the most satisfactory.
euclid's proofs of the propositions, in the theory of proportion, possess at present none but a historical interest, as no student reads them now.
but although his demonstrations are abandoned, his propositions are quoted by every writer, and his nomenclature is universally adopted.
for these reasons it appears to us that the best method is to state euclid's definitions, explain them, or prove them when necessary, for some are theorems under the guise of definitions, and then supply simple algebraic proofs of his propositions.
i. a less magnitude is said to be a part or submultiple of a greater magnitude, when the less measures the greater-that is, when the less is contained a certain number of times exactly in the greater.
ii. a greater magnitude is said to be a multiple of a less when the greater is measured by the less-that is, when the greater contains the less a certain number of times exactly.
iii. ratio is the mutual relation of two magnitudes of the same kind with respect to quantity.
iv. magnitudes are said to have a ratio to one another when the less can be multiplied so as to exceed the greater.
these definitions require explanation, especially def. iii., which has the fault of conveying no precise meaning-being, in fact, unintelligible.
the following annotations will make them explicit:-
1. if an integer be divided into any number of equal parts, one, or the sum of any number of these parts, is called a fraction.
thus, if the line ab represent the integer, and if it be divided into four equal parts in the points c, d, e, then ac is 14; ad, 24; ae, 34.
thus, a fraction is denoted by two numbers parted by a horizontal line; the lower, called the denominator, denotes the number of equal parts into which the integer is divided; and the upper, called the numerator, denotes the number of these equal parts
which are taken. hence it follows, that if the numerator be less than the denominator, the fraction is less than unity.
if the numerator be equal to the denominator, the fraction is equal to unity; and if greater than the denominator, it is greater than unity.
it is evident that a fraction is an abstract quantity-that is, that its value is independent of the nature of the integer which is divided.
2. if we divide each of the equal parts ac, cd, de, eb into two equal parts, the whole, ab, will be divided into eight equal parts; and we see that ac 2 8; ad 4 8; ae 6 8; ab 8 8; now, we saw in 1, that ae 3 4 of the integer, and we have just shown that
it is equal to 6 8. hence 3 4 6 8; but 6 8 would be got from 3 4 by multiplying its terms (numerator and denominator) by 2. hence we infer generally that multiplying the terms of any fraction by 2 does not alter its value.
in like manner it may be shown that multiplying the terms of a fraction by any whole number does not alter its value. hence it follows conversely, that dividing the terms of a fraction by a whole number does not alter the value.
hence we have the following important and fundamental theorem:-two transformations can be made on any fraction without changing its value; namely, its terms can be either multiplied or divided by any whole number, and in either case the value of the new
fraction is equal to the value of the original one.
3. if we take any number, such as 3, and multiply it by any whole number, the product is called a multiple of 3.
thus 6, 9, 12, 15, c., are multiples of 3; but 10, 13, 17, c., are not, because the multiplication of 3 by any whole number will not produce them.
conversely, 3 is a submultiple, or measure, or part of 6, 9, 12, 15, c., because it is contained in each of these without a remainder; but not of 10, 13, 17, c., because in each case it leaves a remainder.
4. if we consider two magnitudes of the same kind, such as two lines ab, cd, and if we suppose that ab is equal to 3 4 of cd, it is evident, if ab be divided into 3 equal parts, and cd into 4 equal parts, that one of the parts into which ab is divided is
equal to one of the parts into which cd is divided. and as there are 3 parts in ab, and 4 in cd, we express this relation by saying that ab has to cd the ratio of 3 to 4; and we denote it thus, 3 : 4.
hence the ratio 3 : 4 expresses the same idea as the fraction 34. in fact, both are different ways of expressing and writing the same thing. when written 3 : 4 it is called a ratio, and when 3 4 a fraction.
in the same manner it can be shown that every ratio whose terms are commensurable can be converted into a fraction; and, conversely, every fraction can be turned into a ratio.
from this explanation we see that the ratio of any two commensurable magnitudes is the same as the ratio of the numerical quantities which denote these magnitudes.
thus, the ratio of two commensurable lines is the ratio of the numbers which express their lengths, measured with the same unit. and this may be extended to the case where the lines are incommensurable.
thus, if a be the side and b the diagonal of a square, the ratio of a : b is
when two quantities are incommensurable, such as the diagonal and the side of a square, although their ratio is not equal to that of any two commensurable numbers, yet a series of pairs of fractions can be found whose difference is continually
diminishing, and which ultimately becomes indefinitely small; such that the ratio of the incommensurable quantities is greater than one, and less than the other fraction of each pair. these fractions are called convergents.
by their means we can approximate as nearly as we please to the exact value of the ratio. in the case of the diagonal and the side of a square, the following are the pairs of convergents:-
14, 15; 141, 142; 1414, 1415; c., 10 10 100 100 1000 1000
and the ratio is intermediate to each pair. it is evident we may continue the series as far as we please.
now if we denote the first of any of the foregoing pairs of fractions by m- n, the second will be m 1 --n--; and in general, in the case of two incommensurable quantities, two fractions m -n and m 1 -n--- can always be found, where n can be made as large
as we please, one of which is less and the other greater than the true value of the ratio. for let a and b be the incommensurable quantities; then, evidently, we cannot find two multiples na, mb, such that na mb.
in this case, take any multiple of a, such as na, then this quantity must lie between some two consecutive multiples of b, such as mb, and (m 1)b; therefore -na- mb is greater than unity, and ---na-- (m 1)b less than unity.
hence a b lies between m- n and m- -1 n.
now, since the difference between m- n and m- -1 n namely, 1 n becomes small as n increases, we see that the difference between the ratio of two incommensurable quantities and that of two commensurable numbers m and n can be made as small as we please.
hence, ultimately, the ratio of incommensurable quantities may be regarded as the limit of the ratio of commensurable quantities.
5. the two terms of a ratio are called the antecedent and the consequent. these correspond to the numerator and the denominator of a fraction.
hence we have the following definition:-"a ratio is the fraction got by making the antecedent the numerator and the consequent the denominator."
6. the reciprocal of a ratio is the ratio obtained by interchanging the antecedent and the consequent. thus, 4 : 3 is the reciprocal of the ratio 3 : 4. hence we have the following theorem:-"the product of a ratio and its reciprocal is unity."
7. if we multiply any two numbers, as 5 and 7, by any number such as 4, the products 20, 28 are called equimultiples of 5 and 7. in like manner, 10 and 15 are equimultiples of 2 and 3, and 18 and 30 of 3 and 5, c.
v. the first of four magnitudes has to the second the same ratio which the third has to the fourth, when any equimultiples whatsoever of the first and third being taken, and any equimultiples whatsoever of the second and fourth, if, according as the
multiple of the first is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth.
vi. magnitudes which have the same ratio are called proportionals. when four magnitudes are proportionals, it is usually expressed by saying, "the first is to the second as the third is to the fourth."
viii. analogy or proportion is the similitude of ratios.
we have given the foregoing definitions in the order of euclid, as given by simson, lardner, and others;2 2except that viii. is put before vii., because it relates, as v. and vi., to the equality of ratios, whereas vii. is a test of their inequality.
but it is evidently an inverted order; for vi. viii. are definitions of proportion, and v. is only a test of proportion, and is not a definition but a theorem, and one which, instead of being taken for granted, requires proof.
the following explanations will give the student clear conceptions of their meaning:-
1. if we take two ratios, such as 6 : 9 and 10 : 15, which are each equal to the same thing (in this example each is equal to 2 3), they are equal to one another (i. axiom i.). then we may write it thus-
this would be the most intelligible way, but it is not the usual one, which is as follows:-6 : 9 :: 10 : 15. in this form it is called a proportion. hence a proportion consists of two ratios which are asserted by it to be equal.
its four terms consist of two antecedents and two consequents. the 1st and 3rd terms are the antecedents, and the 2nd and 4th the consequents. also the first and last terms are called the extremes, and the two middle terms the means.
2. since a proportion consists of two equal ratios, and each ratio can be written as a fraction, whenever we have a proportion such as
we can write it in the form of two equal fractions. thus:
conversely, an equation between two fractions can be put into a proportion. by means of these simple principles all the various properties of proportion can be proved in the most direct and easy manner.
3. if we take the proportion a : b :: c : d, and multiply the first and third terms, each by m, and second and fourth, each by n, we get the four multiples, ma, nb, mc, nd; and we want to prove that if ma is greater than nb, mc is greater than nd; if
now, it is evident that if ma- nb is greater than unity, mc- nd is greater than unity; but if ma- nb is greater than unity, ma is greater than nb; and if mc- nd is greater than unity, mc is greater than nd.
in like manner, if ma be equal to nb, mc is equal to nd; and if less, less.
the foregoing is an easy proof of the converse of the theorem which is contained in euclid's celebrated fifth definition.
next, to prove euclid's theorem-that if, according as the multiple of the first of four magnitudes is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the
fourth; the ratio of the first to the second is equal to the ratio of the third to the fourth.
dem.-let, a, b, c, d be the four magnitudes. first suppose that a and b are commensurable, then it is evident that we can take multiples na, mb, such that na mb. hence, by hypothesis, nc md. thus,
next, suppose a and b are incommensurable. then, as in a recent note, we can find two numbers m and n, such that na mb- is greater than unity, but na (m -1)b less than unity. hence a b lies between m n- and m -1- n.
now, since by hypothesis, when na- mb is greater than unity, -nc md is greater than unity; and when na (m- -1)b- is less than unity, nc (m- -1)d- is less than unity. hence, since a b lies between m n- and m -1- n, c d lies between the same quantities.
therefore the difference between a b and c d is less than 1 n; and since n may be as large as we please, the difference is nothing; therefore
vii. when of the multiples of four magnitudes (taken as in def.
v.) the multiple of the first is greater than that of the second, but the multiple of the third not greater than that of the fourth, the first has to the second a greater ratio than the third has to the fourth.
this, instead of being a definition, is a theorem. we have altered the last clause from that given in simson's euclid, which runs thus:-"the first is said to have to the second a greater ratio than the third has to the fourth."
this is misleading, as it implies that it is, by convention, that the first ratio is greater than the second, whereas, in fact, such is not the case; for it follows from the hypothesis that the first ratio is greater than the second; and if it did not, it
could not be made so by definition. we have made a similar change in the enunciation of the fifth definition.
let a, b, c, d be the four magnitudes, and m and n the multiples taken, it is required to prove, that if ma be greater than nb, but mc not greater than nd, that the ratio a : b is greater than the ratio c : d.
dem.-since ma is greater than nb, but mc not greater than nd, it is evident that
that is, the ratio a : b is greater than the ratio c : d.
ix. proportion consists of three terms at least.
this has the same fault as some of the others-it is not a definition, but an inference. it occurs when the means in a proportion are equal, so that, in fact, there are four terms. as an illustration, let us take the numbers 4, 6, 9.
here the ratio of 4 : 6 is 23, and the ratio of 6 : 9 is 23, so that 4, 6, 9 are continued proportionals; but, in reality, there are four terms, for the full proportion is 4 : 6 :: 6 : 9.
x. when three magnitudes are continual proportionals, the first is said to have to the third the duplicate ratio of that which it has to the second.
xi. when four magnitudes are continual proportionals, the first is said to have to the fourth the triplicate ratio of that which it has to the second.
xii. when there is any number of magnitudes of the same kind greater than two, the first is said to have to the last the ratio compounded of the ratios of the first to the second, of the second to the third, of the third to the fourth, c.
we have placed these definitions in a group; but their order is inverted, and, as we shall see, def. xii. is a theorem, and x. and xi. are only inferences from it.
1. if we have two ratios, such as 5 : 7 and 3 : 4, and if we convert each ratio into a fraction, and multiply these fractions together, we get a result which is called the ratio compounded of the two ratios; viz. in this example it is 1258, or 15 : 28.
it is evident we get the same result if we multiply the two antecedents together for a new antecedent, and the two consequents for a new consequent.
hence we have the following definition:-"the ratio compounded of any number of ratios it the ratio of the product of all the antecedents to the product of all the consequents."
2. to prove the theorem contained in def. xii.
let the magnitudes be a, b, c, d. then the ratio of
hence the ratio compounded of the ratio of 1st : 2nd, of 2nd : 3rd, of 3rd : 4th
3. if three magnitudes be proportional, the ratio of the 1st : 3rd is equal to the square of the ratio of the 1st : 2nd.
for the ratio of the 1st : 3rd is compounded of the ratio of the 1st : 2nd, and of the ratio of the 2nd : 3rd; and since these ratios are equal, the ratio compounded of them will be equal to the square of one of them.
or thus: let the proportionals be a, b, c, that is, let a : b :: b : c; hence we have
or a : c :: a2 : b2-that is, 1st : 3rd :: square of 1st : square of 2nd. now, the ratio of 1st : 3rd is, by def. x., the duplicate ratio of 1st : 2nd.
hence the duplicate ratio of two magnitudes means the square of their ratio, or, what is the same thing, the ratio of their squares (see book vi. xx.).
4. if four magnitudes be continual proportionals, the ratio of 1st : 4th is equal to the cube of the ratio of 1st : 2nd. this may be proved exactly like 3.
hence we see that what euclid calls triplicate ratio of two magnitudes is the ratio of their cubes, or the cube of their ratio.
we also see that there is no necessity to introduce extraneous magnitudes for the purpose of defining duplicate and triplicate ratios, as euclid does. in fact, the definitions by squares and cubes are more explicit.
xiii. in proportionals, the antecedent terms are called homologous to one another; as also the consequents to one another.
if one proportion be given, from it an indefinite number of other proportions can be inferred, and a great part of the theory of proportion consists in proving the truth of these derived proportions.
geometers make use of certain technical terms to denote the most important of these processes. we shall indicate these terms by including them in parentheses in connexion with the propositions to which they refer.
they are useful as indicating, by one word, the whole enunciation of a theorem.
every proposition in the fifth book is a theorem.
if any number of magnitudes of the same kind (a, b, c, c.), be equimultiples of as many others (a, b, c, c.), then the sum of the first magnitudes (a b c, c.) shall be the same multiple of the sum of the second which any magnitude of the first system is
of the corresponding magnitude of the second system.
dem.-let m denote the multiple which the magnitudes of the first system are of those of the second system.
if two magnitudes of the same kind (a, b) be the same multiples of another (c) which two corresponding magnitudes (a, b) are of another (c), then the sum of the two first is the same multiple of their submultiple which the sum of their corresponding
dem.-let m and n be the multiples which a and b are of c.
hence a b is the same multiple of c that a b is of c.
this proposition is evidently true for any number of multiples.
if two magnitudes (a, b) be equimultiples of two others (a, b); then any equimultiples of the first magnitudes (a, b) will be also equimultiples of the second magnitudes (a, b).
dem.-let m denote the multiples which a, b are of a, b; then we have
hence, multiplying each equation by n, we get
hence, na, nb are equimultiples of a, b.
if four magnitudes be proportional, and if any equimultiples of the first and third be taken, and any other equimultiples of the second and fourth; then the multiple of the first : the multiple of the second :: the multiple of the third : the multiple of
let a : b :: c : d; then ma : nb :: mc : nd.
hence, multiplying each fraction by m-n, we get
if two magnitudes of the same kind (a, b) be the same multiples of another (c) which two corresponding magnitudes (a, b) are of another (c), then the difference of the two first is the same multiple of their submultiple (c), which the difference of their
corresponding magnitudes is of their submultiple (c) (compare proposition ii.).
dem.-let m and n be the multiples which a and b are of c.
therefore (ab) (mn)c, and (ab) (mn)c. hence ab is the same multiple of c that a b is of c.
cor.-if a b c, a b c; for if a b c, m n 1.
if a magnitude (a) be the same multiple of another (b), which a magnitude (a) taken from the first is of a magnitude (b) taken from the second, the remainder is the same multiple of the remainder that the whole is of the whole (compare proposition i.).
dem.-let m denote the multiples which the magnitudes a, a are of b, b; then we have
if two ratios be equal, then according as the antecedent of the first ratio is greater than, equal to, or less than its consequent, the antecedent of the second ratio is greater than, equal to, or less than its consequent.
and if a be greater than b, a b- is greater than unity; therefore c d is greater than unity, and c is greater than d.
in like manner, if a be equal to b, c is equal to d, and if less, less.
if two ratios are equal their reciprocals are equal (invertendo).
let a : b :: c : d, then b : a :: d : c.
if the first of four magnitudes be the same multiple of the second which the third is of the fourth, the first is to the second as the third is to the fourth.
in like manner, c d m; therefore a- b -c d.
if the first be to the second as to the third is to the fourth, and if the first be a multiple or submultiple of the second, the third is the same multiple or submultiple of the fourth.
1. let a : b :: c : d, and let a be a multiple of b, then c is the same multiple of d.
but a- b -c d; therefore c d m, and c md.
1. equal magnitudes have equal ratios to the same magnitude.
2. the same magnitude has equal ratios to equal magnitudes.
let a and b be equal magnitudes, and c any other magnitude.
dem.-since a b, dividing each by c, we have
again, since a b, dividing c by each, we have
observation.-2 follows at once from 1 by proposition b.
1. of two unequal magnitudes, the greater has a greater ratio to any third magnitude than the less has; 2. any third magnitude has a greater ratio to the less of two unequal magnitudes than it has to the greater.
1. let a be greater than b, and let c be any other magnitude of the same kind, then the ratio a : c is greater than the ratio b : c.
dem.-since a is greater than b, dividing each by c,
therefore the ratio a : c is greater than the ratio b : c.
2. to prove that the ratio c : b is greater than the ratio c : a.
dem.-since b is less than a, the quotient which is the result of dividing any magnitude by b is greater than the quotient which is got by dividing the same magnitude by a;
hence the ratio c : b is greater than the ratio c : a.
magnitudes which have equal ratios to the same magnitude are equal to one another; 2. magnitudes to which the same magnitude has equal ratios are equal to one another.
hence, multiplying each by c, we get a b.
of two unequal magnitudes, that which has the greater ratio to any third is the greater of the two; and that to which any third has the greater ratio is the less of the two.
1. if the ratio a : c be greater than the ratio b : c, to prove a greater than b.
dem.-since the ratio a : c is greater than the ratio b : c,
hence, multiplying each by c, we get a greater than b.
2. if the ratio c : b is greater than the ratio c : a, to prove b is less than a.
dem.-since the ratio c : b is greater than the ratio c : a,
ratios that are equal to the same ratio are equal to one another.
let a : b :: e : f, and c : d :: e : f, to prove a : b :: c : d.
if any number of ratios be equal to one another, any one of these equal ratios is equal to the ratio of the sum of all the antecedents to the sum of all the consequents.
let the ratios a : b, c : d, e : f, be all equal to one another; it is required to prove that any of these ratios is equal to the ratio a c e : b d f.
since these fractions are all equal, let their common value be r; then we have
cor.-with the same hypotheses, if l, m, n be any three multipliers, a : b :: la mc ne : lb md nf.
if two ratios are equal, and if one of them be greater than any third ratio, then the other is also greater than that third ratio.
if a : b :: c : d, but the ratio of c : d greater than the ratio of e : f; then the ratio of a : b is greater than the ratio of e : f.
dem.-since the ratio of c : d is greater than the ratio of e : f,
or the ratio of a : b is greater than the ratio of e : f.
if two ratios be equal, then, according as the antecedent of the first ratio is greater than, equal to, or less than the antecedent of the second, the consequent of the first is greater than, equal to, or less than the consequent of the second.
let a : b :: c : d; then if a be greater than c, b is greater than d; if equal, equal; if less, less.
hence, proposition a , if a be greater than c, b is greater than d; if equal, equal; and if less, less.
magnitudes have the same ratio which all equimultiples of them have.
let a, b be two magnitudes, then the ratio a : b is equal to the ratio ma : mb.
dem.-the ratio a : b a- b, and the ratio of ma : mb ma- mb; but since the value of a fraction is not altered by multiplying its numerator and denominator by the same number,
if four magnitudes of the same kind be proportionals they are also proportionals by alternation (alternando).
let a : b :: c : d, then a : c :: b : d.
if four magnitudes be proportional, the difference between the first and second : the second :: the difference between the third and fourth : the fourth (dividendo).
let a : b :: c : d : then a b : b :: c d : d;
if four magnitudes be proportionals, the sum of the first and second : the second :: the sum of the third and fourth : the fourth (componendo).
let a : b :: c : d; then a b : b :: c d : d.
if a whole magnitude be to another whole at a magnitude taken from the first it to a magnitude taken from the second, the first remainder : the second remainder :: the first whole : the second whole.
let a : b :: c : d, c and d being less than a and b;
if four magnitudes be proportional, the first : its excess above the second :: the third : its excess above the fourth (convertendo).
let a : b :: c : d; then a : a b :: c : c d.
if there be two sets of three magnitudes, which taken two by two in direct order have equal ratios, then if the first of either set be greater than the third, the first of the other set is greater than the third; if equal, equal; and if less, less.
let a, b, c; a, b, c be the two sets of magnitudes, and let the ratio a : b a : b, and b : c b : c; then, if a be greater than, equal to, or less than c, a will be greater than, equal to, or less than c.
therefore if a be greater than c, a is greater than c; if equal, equal; and if less, less.
if there be two sets of three magnitudes, which taken two by two in transverse order have equal ratios; then, if the first of either set be greater than the third, the first of the other set is greater than the third; if equal, equal; and if less, less.
let a, b, c; a, b, c be the two sets of magnitudes, and let the ratio a : b b : c, and b : c a : b. then, if a be greater than, equal to, or less than c, a will be greater than, equal to, or less than c.
therefore, if a be greater than c, a is greater than c; if equal, equal; if less, less.
if there be two sets of magnitudes, which, taken two by two in direct order, have equal ratios, then the first : the last of the first set :: the first : the last of the second set ("ex aequali," or "ex aequo").
let a, b, c; a, b, c be the two sets of magnitudes, and if a : b :: a : b, and b : c :: b : c, then a : c :: a : c.
and similarly for any number of magnitudes in each set.
cor. 1.-if the ratio b : c be equal to the ratio a : b, then a, b, c will be in continued proportion, and so will a, b, c. hence def. xii. annotation 3 ,
or if four magnitudes be proportional, their squares are proportional.
cor. 2.-if four magnitudes be proportional, their cubes are proportional.
if there be two sets of magnitudes, which, taken two by two in transverse order, have equal ratios; then the first : the last of the first set :: the first : the last of the second set ("ex aequo perturbato").
let a, b, c; a, b, c be the two sets of magnitudes, and let the ratio a : b b : c, and b : c a : b; then a : c :: a : c.
and similarly for any number of magnitudes in each set.
this proposition and the preceding one may be included in one enunciation, thus: "ratios compounded of equal ratios are equal."
if two magnitudes of the same kind (a, b) have to a third magnitude (c) ratios equal to those which two other magnitudes (a, b) have to a third (c), then the sum (a b) of the first two has the same ratio to their third (c) which the sum (a b) of the other
if four magnitudes of the same kind be proportionals, the sum of the greatest and least is greater than the sum of the other two.
let a : b :: c : d; then, if a be the greatest, d will be the least xiv. and a . it is required to prove that a d is greater than b c.
therefore a c is greater than b d xiv. .
1. what is the subject-matter of this book?
2. when is one magnitude said to be a multiple of another?
5. what is the ratio of two commensurable magnitudes?
6. what is meant by the ratio of incommensurable magnitudes?
7. give an illustration of the ratio of incommensurables.
8. what are the terms of a ratio called?
9. what is a ratio of greater inequality?
10. what is a ratio of lesser inequality?
11. what is the product of two ratios called? ans. the ratio compounded of these ratios.
13. what is euclid's definition of duplicate ratio?
16. what is proportion? ans. equality of ratios.
17. give euclid's definition of proportion.
19. what are the latin terms in use to denote some of the propositions of book v.?
20. when is a line divided harmonically?
21. when a line is divided harmonically, what are corresponding pairs of points called? ans. harmonic conjugates.
23. give one enunciation that will include propositions xxii., xxiii. of book v.
def. i.-a ratio whose antecedent is greater than its consequent is called a ratio of greater inequality; and a ratio whose antecedent is less than its consequent, a ratio of lesser inequality.
def. ii.-a right line is said to be cut harmonically when it is divided internally and externally in any ratios that are equal in magnitude.
1. a ratio of greater inequality is increased by diminishing its terms by the same quantity, and diminished by increasing its terms by the same quantity.
2. a ratio of lesser inequality is diminished by diminishing its terms by the same quantity, and increased by increasing its terms by the same quantity.
3. if four magnitudes be proportionals, the sum of the first and second is to their difference as the sum of the third and fourth is to their difference (componendo et dividendo).
4. if two sets of four magnitudes be proportionals, and if we multiply corresponding terms together, the products are proportionals.
5. if two sets of four magnitudes be proportionals, and if we divide corresponding terms, the quotients are proportionals.
6. if four magnitudes be proportionals, their squares, cubes, c., are proportionals.
7. it two proportions have three terms of one respectively equal to three corresponding terms of the other, the remaining term of the first is equal to the remaining term of the second.
8. if three magnitudes be continual proportionals, the first is to the third as the square of the difference between the first and second is to the square of the difference between the second and third.
9. if a line ab, cut harmonically in c and d, be bisected in o; prove oc, ob, od are continual proportionals.
10. in the same case, if o be the middle point of cd; prove oo2 ob2 od2.
11. and ab(ac ad) 2ac.ad, or 1 ac- 1 ad- 2 ab-.
12. and cd(ad bd) 2ad.bd, or -1- bd -1- ad -2- cd.
i. similar rectilineal figures are those whose several angles are equal, each to each, and whose sides about the equal angles are proportional.
similar figures agree in shape; if they agree also in size, they are congruent.
1. when the shape of a figure is given, it is said to be given in species. thus a triangle whose angles are given is given in species. hence similar figures are of the same species.
2. when the size of a figure is given, it is said to be given in magnitude; for instance, a square whose side is of given length.
3. when the place which a figure occupies is known, it is said to be given in position.
ii. a right line is said to be cut at a point in extreme and mean ratio when the whole line is to the greater segment as the greater segment is to the less.
iii. if three quantities of the same kind be in continued proportion, the middle term is called a mean proportional between the other two.
magnitudes in continued proportion are also said to be in geometrical progression.
iv. if four quantities of the same kind be in continued proportion, the two middle terms are called two mean proportionals between the other two.
v. the altitude of any figure is the length of the perpendicular from its highest point to its base.
vi. two corresponding angles of two figures have the sides about them reciprocally proportional, when a side of the first is to a side of the second as the remaining side of the second is to the remaining side of the first.
this is evidently equivalent to saying that a side of the first is to a side of the second in the reciprocal ratio of the remaining side of the first to the remaining side of the second.
triangles (abc, acd) and parallelograms (ec, cf) which have the same altitude are to one another as their bases (bc, cd).
dem.-produce bd both ways, and cut off any number of parts bg, gh, c., each equal to cb, and any number dk, kl, each equal to cd. join ag, ah, ak, al.
now, since the several bases cb, bg, gh are all equal, the triangles acb, abg, agh are also all equal i. xxxviii. . therefore the triangle ach is the same multiple of acb that the base ch is of the base cb.
in like manner, the triangle acl is the same multiple of acd that the base cl is of the base cd; and it is evident that i. xxxviii. if the base hc be greater than cl, the triangle hac is greater than cal; if equal, equal; and if less, less.
now we have four magnitudes: the base bc is the first, the base cd the second, the triangle abc the third, and the triangle acd the fourth.
we have taken equimultiples of the first and third, namely, the base ch, and the triangle ach; also equimultiples of the second and fourth, namely, the base cl, and the triangle acl; and we have proved that according as the multiple of the first is
greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth. hence v. def. v. the base bc : cd :: the triangle abc : acd.
2. the parallelogram ec is double of the triangle abc i. xxxiv. , and the parallelogram cf is double of the triangle acd. hence v. xv. ec : cf :: the triangle abc : acd; but abc : acd :: bc : cd (part i.). therefore v. xi. ec : cf :: the base bc : cd.
or thus: let a, a denote the areas of the triangles abc, acd, respectively, and p their common altitude; then ii. i., cor. 1 ,
in extending this proof to parallelograms we have only to use p instead of 12p.
if a line (de) be parallel to a side (bc) of a triangle (abc), it divides the remaining sides, measured from the opposite angle (a), proportionally; and, conversely, if two sides of a triangle, measured from an angle, be cut proportionally, the line
joining the points of section is parallel to the third side.
1. it is required to prove that ad : db :: ae : ec.
dem.-join be, cd. the triangles bde, ced are on the same base de, and between the same parallels bc, de. hence i. xxxvii. they are equal, and therefore v. vii. the triangle ade : bde :: ade : cde;
2. if ad : db :: ae : ec, it is required to prove that de is parallel to bc.
then ad : db :: the triangle ade : bde i. .
and ae : ec :: the triangle ade : cde i. ;
therefore v. ix. the triangle bde is equal to cde, and they are on the same base de, and on the same side of it; hence they are between the same parallels i. xxxix. . therefore de is parallel to bc.
observation.-the line de may cut the sides ab, ac produced through b, c, or through the angle a; but evidently a separate figure for each of these cases is unnecessary.
if two lines be cut by three or more parallels, the intercepts on one are proportional to the corresponding intercepts on the other.
if a line (ad) bisect any angle (a) of a triangle (abc), it divides the opposite side (bc) into segments proportional to the adjacent sides.
conversely, if the segments (bd, dc) into which a line (ad) drawn from any angle (a) of a triangle divides the opposite side be proportional to the adjacent sides, that line bisects the angle (a).
dem.-1. through c draw ce parallel to ad, to meet ba produced in e. because ba meets the parallels ad, ec, the angle bad i. xxix.
is equal to aec; and because ac meets the parallels ad, ec, the angle dac is equal to ace; but the angle bad is equal to dac (hyp.); therefore the angle ace is equal to aec; therefore ae is equal to ac i. vi. .
again, because ad is parallel to ec, one of the sides of the triangle bec, bd : dc :: ba : ae ii. ; but ae has been proved equal to ac. therefore bd : dc :: ba : ac.
2. if bd : dc :: ba : ac, the angle bac is bisected.
because ad is parallel to ec, ba : ae :: bd : dc ii. ; but bd : dc :: ba : ac (hyp.). therefore v. xi. ba : ae :: ba : ac, and hence v. ix. ae is equal to ac; therefore the angle aec is equal to ace; but aec is equal to bad i. xxix.
, and ace to dac; hence bad is equal to dac, and the line ad bisects the angle bac.
1. if the line ad bisect the external vertical angle cae, ba : ac :: bd : dc, and conversely.
dem.-cut off ae ac. join ed. then the triangles acd, aed are evidently congruent; therefore the angle edb is bisected; hence iii. ba : ae :: bd : de; or ba : ac :: bd : dc.
2. exercise 1 has been proved by quoting proposition iii. prove it independently, and prove iii. as an inference from it.
3. the internal and the external bisectors of the vertical angle of a triangle divide the base harmonically (see definition, p. 191).
4. any line intersecting the legs of any angle is cut harmonically by the internal and external bisectors of the angle.
5. any line intersecting the legs of a right angle is cut harmonically by any two lines through its vertex which make equal angles with either of its sides.
6. if the base of a triangle be given in magnitude and position, and the ratio of the sides, the locus of the vertex is a circle which divides the base harmonically in the ratio of the sides.
7. if a, b, c denote the sides of a triangle abc, and d, d the points where the internal and external bisectors of a meet bc; prove
8. in the same case, if e, e, f, f be points similarly determined on the sides ca, ab, respectively; prove
the sides about the equal angles of equiangular triangles (bac, cde) are proportional, and those which are opposite to the equal angles are homologous.
dem.-let the sides bc, ce, which are opposite to the equal angles a and d, be conceived to be placed so as to form one continuous line, the triangles being on the same side, and so that the equal angles bca, ced may not have a common vertex.
now, the sum of the angles abc, bca is less than two right angles; but bca is equal to bed (hyp.). therefore the sum of the angles abe, bed is less than two right angles; hence i., axiom xii. the lines ab, ed will meet if produced. let them meet in f.
again, because the angle bca is equal to bef, the line ca i. xxviii. is parallel to ef. in like manner, bf is parallel to cd; therefore the figure acdf is a parallelogram; hence ac is equal to df, and cd is equal to af.
now, because ac is parallel to fe, ba : af :: bc : ce ii. ; but af is equal to cd, therefore ba : cd :: bc : ce; hence v. xvi. ; ab : bc :: dc : ce.
again, because cd is parallel to bf, bc : ce :: fd : de; but fd is equal to ac, therefore bc : ce :: ac : de; hence v. xvi. bc : ac :: ce : de. therefore we have proved that ab : bc :: dc : ce, and that bc : ca :: ce : ed.
hence (ex aequali) ab : ac :: dc : de. therefore the sides about the equal angles are proportional.
this proposition may also be proved very simply by superposition. thus (see fig., prop.
ii.): let the two triangles be abc, ade; let the second triangle ade be conceived to be placed on abc, so that its two sides ad, ae may fall on the sides ab, ac; then, since the angle ade is equal to abc, the side de is parallel to bc. hence ii.
ad : db :: ae : ec; hence ad : ab :: ae : ac, and v. xvi. ad : ae :: ab : ac. therefore the sides about the equal angles bac, dae are proportional, and similarly for the others.
it can be proved by this proposition that two lines which meet at infinity are parallel.
for, let i denote the point at infinity through which the two given lines pass, and draw any two parallels intersecting them in the points a, b; a, b; then the triangles aib, aib are equiangular; therefore ai : ab :: ai : ab; but the first term of the
proportion is equal to the third; therefore v. xiv. the second term ab is equal to the fourth ab, and, being parallel to it, the lines aa, bb i. xxxiii. are parallel.
1. if two circles intercept equal chords ab, ab on any secant, the tangents at, at to the circles at the points of intersection are to one another as the radii of the circles.
2. if two circles intercept on any secant chords that have a given ratio, the tangents to the circles at the points of intersection have a given ratio, namely, the ratio compounded of the direct ratio of the radii and the inverse ratio of the chords.
3. being given a circle and a line, prove that a point may be found, such that the rectangle of the perpendiculars let fall on the line from the points of intersection of the circle with any chord through the point shall be given.
4. ab is the diameter of a semicircle adb; cd a perpendicular to ab; draw through a a chord af of the semicircle meeting cd in e, so that the ratio ce : ef may be given.
if two triangles (abc, def) have their sides proportional (ba : ac :: ed : df; ac : cb :: df : fe) they are equiangular, and those angles are equal which are subtended by the homologous sides.
dem.-at the points d, e make the angles edg, deg equal to the angles a, b of the triangle abc. then i. xxxii. the triangles abc, deg are equiangular.
therefore dg is equal to df. in like manner it may be proved that eg is equal to ef. hence the triangles edf, edg have the sides ed, df in one equal to the sides ed, dg in the other, and the base ef equal to the base eg. hence i. viii.
they are equiangular; but the triangle deg is equiangular to abc. therefore the triangle def is equiangular to abc.
observation.-in vi. def. i. two conditions are laid down as necessary for the similitude of rectilineal figures. 1. the equality of angles; 2. the proportionality of sides. now, from propositions iv.
and v., we see that if two triangles possess either condition, they also possess the other. triangles are unique in this respect. in all other rectilineal figures one of the conditions may exist without the other.
thus, two quadrilaterals may have their sides proportional without having equal angles, or vice versa.
if two triangles (abc, def) have one angle (a) in one equal to one angle (d) in the other, and the sides about these angles proportional (ba : ac :: ed : df), the triangles are equiangular, and have those angles equal which are opposite to the homologous
dem.-make the same construction as in the last proposition; then the triangles abc, deg are equiangular.
therefore dg is equal to df. again, because the angle edg is equal to bac (const.), and bac equal to edf (hyp.), the angle edg is equal to edf; and it has been proved that dg is equal to df, and de is common; hence the triangles edg and edf are
equiangular; but edg is equiangular to bac. therefore edf is equiangular to bac.
it is easy to see, as in the case of proposition iv., that an immediate proof of this proposition can also be got from proposition ii. .
cor. 1.-if the ratio of two sides of a triangle be given, and the angle between them, the triangle is given in species.
if two triangles (abc, def) have one angle (a) one equal to one angle (d) in the other, the sides about two other angles (b, e) proportional (ab : bc :: de : ef), and the remaining angles (c, f) of the same species (i. e.
either both acute or both not acute), the triangles are similar.
dem.-if the angles b and e are not equal, one must be greater than the other. suppose abc to be the greater, and that the part abg is equal to def, then the triangles abg, def have two angles in one equal to two angles in the other, and are i. xxxii.
therefore bg is equal to bc. hence the angles bcg, bgc must be each acute i. xvii.
; therefore agb must be obtuse; hence dfe, which is equal to it, is obtuse; and it has been proved that acb is acute; therefore the angles acb, dfe are of different species; but (hyp.) they are of the same species, which is absurd.
hence the angles b and e are not unequal, that is, they are equal. therefore the triangles are equiangular.
cor. 1.-if two triangles abc, def have two sides in one proportional to two sides in the other, ab : bc :: de : ef, and the angles a, d opposite one pair of homologous sides equal, the angles c, f opposite the other are either equal or supplemental.
this proposition is nearly identical with vii.
cor. 2.-if either of the angles c, f be right, the other must be right.
the triangles (acd, bcd) into which a right-angled triangle (acb) is divided, by the perpendicular (cd) from the right angle (c) on the hypotenuse, are similar to the whole and to one another.
dem.-since the two triangles adc, acb have the angle a common, and the angles adc, acb equal, each being right, they are i. xxxii. equiangular; hence iv. they are similar. in like manner it may be proved that bdc is similar to abc.
hence adc, cdb are each similar to acd, and therefore they are similar to one another.
cor. 1.-the perpendicular cd is a mean proportional between the segments ad, db of the hypotenuse.
for, since the triangles adc, cdb are equiangular, we have ad : dc :: dc : db; hence dc is a mean proportional between ad, db (def. iii.).
cor. 2.-bc is a mean proportional between ab, bd; and ac between ab, ad.
cor. 3.-the segments ad, db are in the duplicate of ac : cb, or in other words, ad : db :: ac2 : cb2,
cor. 4.-ba : ad in the duplicate ratios of ba : ac; and ab : bd in the duplicate ratio of ab : bc.
from a given right line (ab) to cut off any part required (i.e. to cut off any required submultiple)
sol.-let it be required, for instance, to cut off the fourth part. draw af, making any angle with ab, and in af take any point c, and cut off (i. iii.) the parts cd, de, ef each equal to ac. join bf, and draw cg parallel to bf. ag is the fourth part of ab.
dem.-since cg is parallel to the side bf of the triangle abf, ac : af :: ag : ab ii. ; but ac is the fourth part of af (const.). hence ag is the fourth part of ab v., d. . in the same manner, any other required submultiple may be cut off.
proposition x., book i., is a particular case of this proposition.
to divide a given undivided line (ab) similarly to a given divided line (cd).
sol.-draw ag, making any angle with ab, and cut off the parts ah, hi, ig respectively equal to the parts ce, ef, fd of the given divided line cd. join bg, and draw hk, il, each parallel to bg. ab will be divided similarly to cd.
dem.-through h draw hn parallel to ab, cutting il in m. now in the triangle ali, hk is parallel to il. hence ii. ak : kl :: ah : hi, that is :: ce : ef (const.). again, in the triangle hng, mi is parallel to ng. therefore ii. hm : mn :: hi : ig; but i.
xxxiv. hm is equal to kl, mn is equal to lb, hi is equal to ef, and ig is equal to fd (const.). therefore kl : lb :: ef : fd. hence the line ab is divided similarly to the line cd.
1. to divide a given line ab internally or externally in the ratio of two given lines, m, n.
sol.-through a and b draw any two parallels ac and bd in opposite directions. cut off ac m, and bd n, and join cd; the joining line will divide ab internally at e in the ratio of m : n.
2. if bd be drawn in the same direction with ac, as denoted by the dotted line, then cd will cut ab externally at e in the ratio of m : n.
cor.-the two points e, e divide ab harmonically.
this problem is manifestly equivalent to the following:-given the sum or difference of two lines and their ratio, to find the lines.
3. any line ae, through the middle point b of the base dd of a triangle dcd, is cut harmonically by the sides of the triangle and a parallel to the base through the vertex.
4. given the sum of the squares on two lines and their ratio; find the lines.
5. given the difference of the squares on two lines and their ratio; find the lines.
6. given the base and ratio of the sides of a triangle; construct it when any of the following data is given:-1, the area; 2, the difference on the squares of the sides; 3, the sum of the squares on the sides; 4, the vertical angle; 5, the difference of
to find a third proportional to two given lines (x, y ).
sol.-draw any two lines ac, ae making an angle. cut off ab equal x, bc equal y , and ad equal y . join bd, and draw ce parallel to bd, then de is the third proportional required.
dem.-in the triangle cae, bd is parallel to ce; therefore ab : bc :: ad : de ii. ; but ab is equal to x, and bc, ad each equal to y . therefore x : y :: y : de. hence de is a third proportional to x and y .
another solution can be inferred from proposition viii. for if ad, dc in that proposition be respectively equal to x and y , then db will be the third proportional.
or again, if in the diagram, proposition viii., ad x, and ac y , ab will be the third proportional. hence may be inferred a method of continuing the proportion to any number of terms.
1. if ao be a triangle, having the side a greater than ao; then if we cut off ab ao, draw bb parallel to ao, cut off bc bb, c., the series of lines ab, bc, cd, c., are in continual proportion.
2. ab bc : ab :: ab : a. this is evident by drawing through b a parallel to a.
to find a fourth proportional to three given lines (x, y, z).
sol.-draw any two lines ac, ae, making an angle; then cut off ab equal x, bc equal y , ad equal z. join bd, and draw ce parallel to bd. de will be the fourth proportional required.
dem.-since bd is parallel to ce, we have ii. ab : bc :: ad : de; therefore x : y :: z : de. hence de is a fourth proportional to x, y , z.
or thus: take two lines ad, bc intersecting in o. make oa x, ob y , oc z, and describe a circle through the points a, b, c iv. v. cutting ad in d. od will be the fourth proportional required.
the demonstration is evident from the similarity of the triangles aob and cod.
to find a mean proportional between two given lines. (x, y ).
sol.-take on any line ac parts ab, bc respectively equal to x, y . on ac describe a semicircle adc. erect bd at right angles to ac, meeting the semicircle in d. bd will be the mean proportional required.
dem.-join ad, dc. since adc is a semicircle, the angle adc is right iii. xxxi. . hence, since adc is a right-angled triangle, and bd a perpendicular from the right angle on the hypotenuse, bd is a mean proportional viii. cor.
1 between ab, bc; that is, bd is a mean proportional between x and y .
1. another solution may be inferred from proposition viii., cor. 2.
2. if through any point within a circle the chord be drawn, which is bisected in that point, its half is a mean proportional between the segments of any other chord passing through the same point.
3. the tangent to a circle from any external point is a mean proportional between the segments of any secant passing through the same point.
4. if through the middle point c of any arc of a circle any secant be drawn cutting the chord of the arc in d, and the circle again in e, the chord of half the arc is a mean proportional between cd and ce.
5. if a circle be described touching another circle internally and two parallel chords, the perpendicular from the centre of the former on the diameter of the latter, which bisects the chords, is a mean proportional between the two extremes of the three
segments into which the diameter is divided by the chords.
6. if a circle be described touching a semicircle and its diameter, the diameter of the circle is a harmonic mean between the segments into which the diameter of the semicircle is divided at the point of contact.
7. state and prove the proposition corresponding to ex. 5, for external contact of the circles.
1. equiangular parallelograms (ab, cd) which are equal in area have the sides about the equal angles reciprocally proportional-ac : ce :: gc : cb.
2. equiangular parallelograms which have the sides about the equal angles reciprocally proportional are equal in area.
dem.-let ac, ce be so placed as to form one right line, and that the equal angles acb, ecg may be vertically opposite.
now, since the angle acb is equal to ecg, to each add bce, and we have the sum of the angles acb, bce equal to the sum of the angles ecg, bce; but the sum of acb, bce is i. xiii. two right angles. therefore the sum of ecg, bce is two right angles.
hence i. xiv. bc, cg form one right line. complete the parallelogram be.
again, since the parallelograms ab, cd are equal (hyp.),
that is, the sides about the equal angles are reciprocally proportional.
2. let ac : ce :: gc : cb, to prove the parallelograms ab, cd are equal.
dem.-let the same construction be made, we have
or thus: join he, be, hd, bd. the pict hc twice the hbe, and the pict cd twice the bde. therefore the hbe bde, and i. xxxix. hd is parallel to be. hence
hb :bf ::de :ef ; that is, ac :ce ::gc :cb.
2. may be proved by reversing this demonstration.
another demonstration of this proposition may be got by producing the lines ha and dg to meet in i. then i. xliii. the points i, c, f are collinear, and the proposition is evident.
1. two triangles equal in area (acb, dce), which have one angle (c) in one equal to one angle (c) in the other, have the sides about these angles reciprocally proportional.
2. two triangles, which have one angle in one equal to one angle in the other, and the sides about these angles reciprocally proportional, are equal in area.
dem.-1. let the equal angles be so placed as to be vertically opposite, and that ac, cd may form one right line; then it may be demonstrated, as in the last proposition, that bc, ce form one right line. join bd.
now since the triangles acb, dce are equal,
that is, the sides about the equal angles are reciprocally proportional.
2. if ac : cd :: ec : cb, to prove the triangle acb equal to dce.
dem.-let the same construction be made, then we have
hence the triangle acb dce v. ix. -that is, the triangles are equal.
this proposition might have been appended as a cor. to the preceding, since the triangles are the halves of equiangular parallelograms, or it may be proved by joining ae, and showing that it is parallel to bd.
1. if four right lines (ab, cd, e, f) be proportional, the rectangle (ab.f) contained by the extremes is equal to the rectangle (cd.e) contained by the means.
2. if the rectangle contained by the extremes of four right lines be equal to the rectangle contained by the means, the four lines are proportional.
dem.-1. erect ah, ci at right angles to ab and cd, and equal to f and e respectively, and complete the rectangles. then because ab : cd :: e : f (hyp.), and that e is equal to ci, and f to ah (const.), we have ab : cd :: ci : ah.
hence the parallelograms ag, ck are equiangular, and have the sides about their equal angles reciprocally proportional. therefore they are xiv. equal; but since ah is equal to f, ag is equal to the rectangle ab.f.
in like manner, ck is equal to the rectangle cd.e. hence ab.f cd.e; that is, the rectangle contained by the extremes is equal to the rectangle contained by the means.
2. if ab.f cd.e, to prove ab : cd :: e : f.
the same construction being made, because ab.f cd.e, and that f is equal to ah, and e to ci, we have the parallelogram ag ck; and since these parallelograms are equiangular, the sides about their equal angles are reciprocally proportional. therefore
ab : cd :: ci : ah; that is, ab : cd :: e : f.
or thus: place the four lines in a concurrent position so that the extremes may form one continuous line, and the means another. let the four lines so placed be ao, bo, od, oc. join ab, cd.
then because ao : ob :: od : oc, and the angle aob doc, the triangles aob, cod are equiangular. hence the four points a, b, c, d are concyclic. therefore iii. xxxv. ao.oc bo.od.
1. if three right lines (a, b, c) be proportional, the rectangle (a.c) contained by the extremes is equal to the square (b2) of the mean.
2. if the rectangle contained by the extremes of three right lines be equal to the square of the mean, the three lines are proportional.
dem.-1. assume a line d b; then because a : b :: b : c, we have a : b :: d : c. therefore xvi. ac bd; but bd b2. therefore ac b2; that is, the rectangle contained by the extremes is equal to the square of the mean.
2. the same construction being made, since ac b2, we have a.c b.d; therefore a : b :: d : c; but d b. hence a : b :: b : c; that is, the three lines are proportionals.
this proposition may be inferred as a cor. to the last, which is one of the fundamental propositions in mathematics.
1. if a line cd bisect the vertical angle c of any triangle acb, its square added to the rectangle ad.db contained by the segments of the base is equal to the rectangle contained by the sides.
dem.-describe a circle about the triangle, and produce cd to meet it in e; then it is easy to see that the triangles acd, ecb are equiangular. hence iv. ac : cd :: ce : cb; therefore ac.cb ce.cd cd2 cd.de cd2 ad.db iii. xxxv. .
2. if the line cd bisect the external vertical angle of any triangle acb, its square subtracted from the rectangle ad.db is equal to ac.cb.
3. the rectangle contained by the diameter of the circumscribed circle, and the radius of the inscribed circle of any triangle, is equal to the rectangle contained by the segments of any chord of the circumscribed circle passing through the centre of the
dem.-let o be the centre of the inscribed circle. join ob (see foregoing fig.); let fall the perpendicular og, draw the diameter ef of the circumscribed circle. now the angle abe ecb iii. xxvii. , and abo obc; therefore ebo sum of ocb, obc eob.
hence eb eo. again, the triangles ebf, ogc are equiangular, because efb, ecb are equal, and ebf, ogc are each right. therefore, ef : eb :: oc : og; therefore ef.og eb.oc eo.oc.
4. ex. 3 may be extended to each of the escribed circles of the triangle acb.
5. the rectangle contained by two sides of a triangle is equal to the rectangle contained by the perpendicular and the diameter of the circumscribed circle. for, let ce be the diameter. join ae.
then the triangles ace, dcb are equiangular; hence ac : ce :: cd : cb; therefore ac.cb cd.ce.
6. if a circle passing through one of the angles a of a parallelogram abcd intersect the two sides ab, ad again in the points e, g and the diagonal ac again in f; then ab.ae ad.ag ac.af.
dem.-join ef, fg, and make the angle abh afe. then the triangles abh, afe are equiangular. therefore ab : ah :: af : ae. hence ab.ae af.ah.
again, it is easy to see that the triangles bch, fag are equiangular; therefore bc : ch :: af : ag; hence bc.ag af.ch, or ad.ag af.ch; but we have proved ab.ae af.ah. hence ad.ag ab.ae af.ac.
7. if de, df be parallels to the sides of a triangle abc from any point d in the base, then ab.ae ac.af ad2 bd.dc. this is an easy deduction from 6.
8. if through a point o within a triangle abc parallels ef, gh, ik to the sides be drawn, the sum of the rectangles of their segments is equal to the rectangle contained by the segments of any chord of the circumscribing circle passing through o.
9. the rectangle contained by the side of an inscribed square standing on the base of a triangle, and the sum of the base and altitude, is equal to twice the area of the triangle.
10. the rectangle contained by the side of an escribed square standing on the base of a triangle, and the difference between the base and altitude, is equal to twice the area of the triangle.
11. if from any point p in the circumference of a circle a perpendicular be drawn to any chord, its square is equal to the rectangle contained by the perpendiculars from the extremities of the chord on the tangent at p.
12. if o be the point of intersection of the diagonals of a cyclic quadrilateral abcd, the four rectangles ab.bc, bd.cd, cd.da, da.ab, are proportional to the four lines bo, co, do, ao.
13. the sum of the rectangles of the opposite sides of a cyclic quadrilateral abcd is equal to the rectangle contained by its diagonals.
dem.-make the angle dao cab; then the triangles dao, cab are equiangular; therefore ad : do :: ac : cb; therefore ad.bc ac.do. again, the triangles dac, oab are equiangular, and cd : ac :: bo : ab; therefore ac.cd ac.bo.
hence ad.bc ab.cd ac.bd.3 3this proposition is known as ptolemy's theorem.
14. if the quadrilateral abcd is not cyclic, prove that the three rectangles ab.cd, bc.ad, ac.bd are proportional to the three sides of a triangle which has an angle equal to the sum of a pair of opposite angles of the quadrilateral.
15. prove by using theorem 11 that if perpendiculars be let fall on the sides and diagonals of a cyclic quadrilateral, from any point in the circumference of the circumscribed circle, the rectangle contained by the perpendiculars on the diagonals is equal
to the rectangle contained by the perpendiculars on either pair of opposite sides.
16. if ab be the diameter of a semicircle, and pa, pb chords from any point p in the circumference, and if a perpendicular to ab from any point c meet pa, pb in d and e, and the semicircle in f, cf is a mean proportional between cd and ce.
on a given right line (ab) to construct a rectilineal figure similar to a given one (cdefg), and similarly placed as regards any side (cd) of the latter.
def.-similar figures are said to be similarly described upon given right lines, when these lines are homologous sides of the figures.
sol.-join ce, cf, and construct a triangle abh on ab equiangular to cde, and similarly placed as regards cd; that is, make, the angle abh equal to cde, and bah equal to dce.
in like manner construct the triangle hai equiangular to ecf, and similarly placed, and lastly, the triangle iaj equiangular and similarly placed with fcg. then abhij is the figure required.
dem.-from the construction it is evident that the figures are equiangular, and it is only required to prove that the sides about the equal angles are proportional. now because the triangle abh is equiangular to cde, ab : bh :: cd : de iv.
; hence the sides about the equal angles b and d are proportional.
again, from the same triangles we have bh : ha :: de : ec, and from the triangles iha, fec; ha : hi :: ec : ef; therefore (ex quali) bh : hi :: de : ef; that is, the sides about the equal angles bhi, def are proportional, and so in like manner are the
sides about the other equal angles. hence (def. i.) the figures are similar.
observation.-in the foregoing construction, the line ab is homologous to cd, and it is evident that we may take ab to be homologous to any other side of the given figure cdefg.
again, in each case, it the figure abhij be turned round the line ab until it falls on the other side, it will still be similar to the figure cdefg.
hence on a given line ab there can be constructed two figures each similar to a given figure cdefg, and having the given line ab homologous to any given side cd of the given figure.
the first of the figures thus constructed is said to be directly similar, and the second inversely similar to the given figure. these technical terms are due to hamilton: see "elements of quaternions," page 112.
cor. 1.-twice as many polygons may be constructed on ab similar to a given polygon cdefg as that figure has sides.
cor. 2.-if the figure abhij be applied to cdefg so that the point a will coincide with c, and that the line ab may be placed along cd, then the points h, i, j will be respectively on the lines ce, cf, cg; also the sides bh, hi, ij of the one polygon will
be respectively parallel to their homologous sides de, ef, fg of the other.
cor. 3.-if lines drawn from any point o in the plane of a figure to all its angular points be divided in the same ratio, the lines joining the points of division will form a new figure similar to, and having every side parallel to, the homologous side of
similar triangles (abc, def) have their areas to one another in the duplicate ratio of their homologous sides.
dem.-take bg a third proportional to bc, ef xi. . join ag. then because the triangles abc, def are similar, ab : bc :: de : ef; hence (alternately) ab : de :: bc : ef; but bc : ef :: ef : bg (const.); therefore v. xi.
ab : de :: ef : bg; hence the sides of the triangles abg, def about the equal angles b, e are reciprocally proportional; therefore the triangles are equal.
again, since the lines bc, ef, bg are continual proportionals, bc : bg in the duplicate ratio of bc : ef v. def. x. ; but bc : bg :: triangle abc : abg.
therefore abc : abg in the duplicate ratio of bc : ef; but it has been proved that the triangle abg is equal to def. therefore the triangle abc is to the triangle def in the duplicate ratio of bc : ef.
this is the first proposition in euclid in which the technical term "duplicate ratio" occurs. my experience with pupils is, that they find it very difficult to understand either euclid's proof or his definition.
on this account i submit the following alternative proof, which, however, makes use of a new definition of the duplicate ratio of two lines, viz. the ratio of the squares (see annotations on v. def. x.) described on these lines.
on ab and de describe squares, and through c and f draw lines parallel to ab and de, and complete the rectangles ai, dn.
now, the triangles jac, odf are evidently equiangular.
again, ja : ag :: pictai : square ah i. ,
1. if one of two similar triangles has its sides 50 per cent. longer than the homologous sides of the other; what is the ratio of their areas?
2. when the inscribed and circumscribed regular polygons of any common number of sides to a circle have more than four sides, the difference of their areas is less than the square of the side of the inscribed polygon.
similar polygons may be divided (1) into the same number of similar triangles; (2) the corresponding triangles have the same ratio to one another which the polygons have; (3) the polygons are to each other in the duplicate ratio of their homologous sides.
dem.-let abhij, cdefg be the polygons, and let the sides ab, cd be homologous. join ah, ai, ce, cf.
1. the triangles into which the polygons are divided are similar. for, since the polygons are similar, they are equiangular, and have the sides about their equal angles proportional def. i.
; hence the angle b is equal to d, and ab : bh :: cd : de; therefore vi. the triangle abh is equiangular to cde; hence the angle bha is equal to dec; but bhi is equal to def (hyp.); therefore the angle ahi is equal to cef.
again, because the polygons are similar, ih : hb :: fe : ed; and since the triangles abh, cde are similar, hb : ha :: ed : ec; hence (ex aequali) ih : ha :: fe : ec, and the angle iha has been proved to be equal to the angle fec; therefore the triangles
iha, fec are equiangular. in the same manner it can be proved that the remaining triangles are equiangular.
2. since the triangle abh is similar to cde, we have xix. .
abh : cde in the duplicate ratio of ah : ce.
ahi : cef in the duplicate ratio of ah : ce;
in these equal ratios, the triangles abh, ahi, aij are the antecedents, and the triangles cde, cef, cfg the consequents, and v. xii.
any one of these equal ratios is equal to the ratio of the sum of all the antecedents to the sum of all the consequents; therefore the triangle abh : the triangle cde :: the polygon abhij : the polygon cdefg.
3. the triangle abh : cde in the duplicate ratio of ab : cd xix. . hence (2) the polygon abhij : the polygon cdefg in the duplicate ratio of ab : cd.
cor. 1.-the perimeters of similar polygons are to one another in the ratio of their homologous sides.
cor. 2.-as squares are similar polygons, therefore the duplicate ratio of two lines is equal to the ratio of the squares described on them (compare annotations, v. def. x.).
cor. 3.-similar portions of similar figures bear the same ratio to each other as the wholes of the figures.
cor. 4.-similar portions of the perimeters of similar figures are to each other in the ratio of the whole perimeters.
def. i.-homologous points in the planes of two similar figures are such, that lines drawn from them to the angular points of the two figures are proportional to the homologous sides of the two figures.
1. if two figures be similar, to each point in the plane of one there will be a corresponding point in the plane of the other.
dem.-let abcd, abcd be the two figures, p a point in the plane of abcd.
join ap, bp, and construct a triangle apb on ab, similar to apb; then it is easy to see that lines from p to the angular points of abcd are proportional to the lines from p to the angular points of abcd.
2. if two figures be directly similar, and in the same plane, there is in the plane a special point which, regarded as belonging to either figure, is its own homologous point with respect to the other.
for, let ab, ab be two homologous sides of the figures, c their point of intersection. through the two triads of points a, a, c; b, b, c describe two circles intersecting again in the point o: o will be the point required.
for it is evident that the triangles oab, oab are similar and that either may be turned round the point o, so that the two bases, ab, ab, will be parallel.
def. ii.-the point o is called the centre of similitude of the figures. it is also called their double point.
3. two regular polygons of n sides each have n centres of similitude.
4. if any number of similar triangles have their corresponding vertices lying on three given lines, they have a common centre of similitude.
5. if two figures be directly similar, and have a pair of homologous sides parallel, every pair of homologous sides will be parallel.
def. iii.-two figures, such as those in 5, are said to be homothetic.
6. if two figures be homothetic, the lines joining corresponding angular points are concurrent, and the point of concurrence is the centre of similitude of the figures.
7. if two polygons be directly similar, either may be turned round their centre of similitude until they become homothetic, and this may be done in two different ways.
dem.-let o, o be their centres; let the angle aob be indefinitely small, so that the arc ab may be regarded as a right line; make the angle aob equal to aob; then the triangles aob, aob are similar.
again, make the angle boc indefinitely small, and make boc equal to it; the triangles boc, boc are similar. proceeding in this way, we see that the circles can be divided into the same number of similar elementary triangles.
9. sectors of circles having equal central angles are similar figures.
10. as any two points of two circles may be regarded as homologous, two circles have in consequence an infinite number of centres of similitude; their locus is the circle, whose diameter is the line joining the two points for which the two circles are
11. the areas of circles are to one another as the squares of their diameters. for they are to one another as the similar elementary triangles into which they are divided, and these are as the squares of the radii.
12. the circumferences of circles are as their diameters (cor. 1).
13. the circumference of sectors having equal central angles are proportional to their radii. hence if a, a denote the arcs of two sectors, which subtend equal angles at the centres, and if r, r be their radii, a r a r.
14. the area of a circle is equal to half the rectangle contained by the circumference and the radius. this is evident by dividing the circle into elementary triangles, as in ex. 8.
15. the area of a sector of a circle is equal to half the rectangle contained by the arc of the sector and the radius of the circle.
rectilineal figures (a, b), which are similar to the same figure (c), are similar to one another.
dem.-since the figures a and c are similar, they are equiangular, and have the sides about their equal angles proportional. in like manner b and c are equiangular, and have the sides about their equal angles proportional.
hence a and b are equiangular, and have the sides about their equal angles proportional. therefore they are similar.
cor.-two similar rectilineal figures which are homothetic to a third are homothetic to one another.
if three similar rectilineal figures be homothetic, two by two, their three centres of similitudes are collinear.
if four lines (ab, cd, ef, gh) be proportional, and any pair of similar rectilineal figures (abk, cdl) be similarly described on the first and second, and also any pair (ei, gj) on the third and fourth, these figures are proportional.
conversely, if any rectilineal figure described on the first of four right lines: the similar and similarly described figure described on the second :: any rectilineal figure on the third : the similar and similarly described figure on the fourth, the
ab2 : cd2 :: ef2 : gh2 v. xxii., cor. 1 ;
if abk : cdl :: ei : gj, ab : cd :: ef : gh.
3 dem. 2.-abk : cdl :: ab2 : cd2 xx. , and ei : gj :: ef2 : gh2 xx. ; therefore ab2 : cd2 :: ef2 : gh2. hence ab : cd :: ef : gh.
the enunciation of this proposition is wrongly stated in simson's euclid, and in those that copy it. as given in those works, the four figures should be similar.
equiangular parallelograms (ad, cg) are to each other as the rectangles contained by their sides about a pair of equal angles.
dem.-let the two sides ab, bc about the equal angles abd, cbg, be placed so as to form one right line; then it is evident, as in prop. xiv., that gb, bd form one right line. complete the parallelogram bf.
now, denoting the parallelograms ab, bf, cg by x, y , z, respectively, we have-
observation.-since ab.bd : bc.bg is compounded of the two ratios ab : bc and bd : bg v. def.
of compound ratio , the enunciation is the same as if we said, "in the ratio compounded of the ratios of the sides," which is euclid's; but it is more easily understood as we have put it.
1. triangles which have one angle of one equal or supplemental to one angle of the other, are to one another in the ratio of the rectangles of the sides about those angles.
2. two quadrilaterals whose diagonals intersect at equal angles are to one another in the ratio of the rectangles of the diagonals.
in any parallelogram (ac), every two parallelograms (af, fc) which are about a diagonal are similar to the whole and to one another.
dem.-since the parallelograms ac, af have a common angle, they are equiangular i. xxxiv. , and all that is required to be proved is, that the sides about the equal angles are proportional.
now, since the lines ef, bc are parallel, the triangles aef, abc are equiangular; therefore iv.
ae : ef :: ab : bc, and the other sides of the parallelograms are equal to ae, ef; ab, bc: hence the sides about the equal angles are proportional; therefore the parallelograms af, ac are similar. in the same manner the parallelograms af, fc are similar.
cor.-the parallelograms af, fc, ac are, two by two, homothetic.
to describe a rectilineal figure equal to a given one (a), and similar to another given one (bcd).
sol.-on any side bc of the figure bcd describe the rectangle be equal to bcd i. xlv. , and on ce describe the rectangle ef equal to a. between bc, cf find a mean proportional gh, and on it describe the figure ghi similar to bcd xviii.
, so that bc and gh may be homologous sides. ghi is the figure required.
dem.-the three lines bc, gh, cf are in continued proportion; therefore bc : cf in the duplicate ratio of bc : gh v. def. x. ; and since the figures bcd, ghi are similar, bcd : ghi in the duplicate ratio of bc : gh xx.
; also bc : cf :: rectangle be : rectangle ef. hence rectangle be : ef :: figure bcd : ghi; but the rectangle be is equal to the figure bcd; therefore the rectangle ef is equal to the figure ghi; but ef is equal to a (const.).
therefore the figure ghi is equal to a, and it is similar to bcd. hence it is the figure required.
or thus: describe the squares efjk, lmno equal to the figures bcd and a respectively ii. xiv. ; then find gh a fourth proportional to ef, lm, and bc xii. . on gh describe the rectilineal figure ghi similar to the figure bcd xviii.
, so that bc and gh may be homologous sides. ghi is the figure required.
dem.-because ef : lm :: bc : gh (const.), the figure efjk : lmno :: bcd : ghi xxii. ; but efjk is equal to bcd (const.); therefore lmno is equal to ghi; but lmno is equal to a (const.). therefore ghi is equal to a, and it is similar to bcd.
if two similar and similarly situated parallelograms (aefg, abcd) have a common angle, they are about the same diagonal.
dem.-draw the diagonals (see fig., prop. xxiv.) af, ac. then because the parallelograms aefg, abcd are similar figures, they can be divided into the same number of similar triangles xx. .
hence the triangle fag is similar to cad, and therefore the angle fag is equal to the angle cad. hence the line ac must pass through the point f, and therefore the parallelograms are about the same diagonal.
observation.-proposition xxvi., being the converse of xxiv., has evidently been misplaced. the following would be a simpler enunciation:-"if two homothetic parallelograms have a common angle, they are about the same diagonal."
to inscribe in a given triangle (abc) the maximum parallelogram having a common angle (b) with the triangle.
sol.-bisect the side ac opposite to the angle b, at p : through p draw pe, pf parallel to the other sides of the triangle. bp is the parallelogram required.
dem.-take any other point d in ac : draw dg, dh parallel to the sides, and ck parallel to ab; produce ep, gd to meet ck in k and j, and produce hd to meet pk in i.
now, since ac is bisected in p, ek is also bisected in p; hence i. xxxvi. the parallelogram eo is equal to ok; therefore eo is greater than dk; but dk is equal to fd i. xliii. ; hence eo is greater than fd.
to each add bo, and we have the parallelogram bp greater than bd. hence bp is the maximum parallelogram which can be inscribed in the given triangle.
cor. 1.-the maximum parallelogram exceeds any other parallelogram about the same angle in the triangle, by the area of the similar parallelogram whose diagonal is the line between the middle point p of the opposite side and the point d, which is the
corner of the other inscribed parallelogram.
cor. 2.-the parallelograms inscribed in a triangle, and having one angle common with it, are proportional to the rectangles contained by the segments of the sides of the triangle, made by the opposite corners of the parallelograms.
cor. 3.-the parallelogram ac : gh :: ac2 : ad.dc.
to inscribe in a given triangle (abc) a parallelogram equal to a given rectilineal figure (x) not greater than the maximum inscribed parallelogram, and having an angle (b) common with the triangle.
sol.-bisect the side ac opposite to b, at p. draw pf, pe parallel to the sides ab, bc; then xxvii. bp is the maximum parallelogram that can be inscribed in the triangle abc; and if x be equal to it, the problem is solved.
if not, produce ep, and draw cj parallel to pf; then describe the parallelogram klmn xxv.
equal to the difference between the figure pjcf and x, and similar to pjcf, and so that the sides pj and kl will be homologous; then cut off pi equal to kl; draw ih parallel to ab, cutting ac in d, and draw dg parallel to bc.
dem.-since the parallelograms pc, pd are about the same diagonal, they are similar xxiv. ; but pc is similar to kpt (const.); therefore pd is similar to kn, and (const.) their homologous sides, pi and kl, are equal; hence xx. pd is equal to kn.
now, pd is the difference between ef and gh xxvii. cor. 1 , and kn is (const.) the difference between pc and x; therefore the difference between pc and x is equal to the difference between ef and gh; but ef is equal to pc. hence gh is equal to x.
to escribe to a given triangle (abc) a parallelogram equal to a given rectilineal figure (x), and having an angle common with an external angle (b) of the triangle.
sol.-the construction is the same as the last, except that, instead of making the parallelogram kn equal to the excess of the parallelogram pc over the rectilineal figure x, we make it equal to their sum; and then make pi equal to kl; draw ih parallel to
ab, and the rest of the construction as before.
dem.-now it can be proved, as in ii. vi., that the parallelogram bd is equal to the gnomon ohj; that is, equal to the difference between the parallelograms pd and pc, or the difference (const.) between kn and pc; that is (const.), equal to x, and bd is
escribed to the triangle abc, and has an angle common with the external angle b. hence the thing required is done.
observation.-the enunciations of the three foregoing propositions have been altered, in order to express them in modern technical language. some writers recommend the student to omit them-we think differently.
in the form we have given them they are freed from their usual repulsive appearance. the constructions and demonstrations are euclid's, but slightly modified.
to divide a given line (ab) in "extreme and mean ratio."
sol.-divide ab in c, so that the rectangle ab.bc may be equal to the square on ac ii. xi. then c is the point required.
dem.-because the rectangle ab.bc is equal to the square on ac,ab : ac :: ac : bc xvii. . hence ab is cut in extreme and mean ratio in c def. ii. .
1. if the three sides of a right-angled triangle be in continued proportion, the hypotenuse is divided in extreme and mean ratio by the perpendicular from the right angle on the hypotenuse.
2. in the same case the greater segment of the hypotenuse is equal to the least side of the triangle.
3. the square on the diameter of the circle described about the triangle formed by the points f, h, d (see fig. ii. xi.), is equal to six times the square on the line fd.
if any similar rectilineal figure be similarly described on the three sides of a right-angled triangle (abc), the figure on the hypotenuse is equal to the sum of those described on the two other sides.
dem.-draw the perpendicular cd i. xii. . then because abc is a right-angled triangle, and cd is drawn from the right angle perpendicular to the hypotenuse; bd : ad in the duplicate ratio of ba : ac viii. cor. 4 .
again, because the figures described on ba, ac are similar, they are in the duplicate ratio of ba : ac xx. . hence v. xi. ba : ad :: figure described on ba : figure described on ac.
in like manner, ab : bd :: figure described on ab : figure described on bc. hence v. xxiv. ab : sum of ad and bd :: figure described on the line ab : sum of the figures described on the lines ac, bc; but ab is equal to the sum of ad and bd. therefore v.
a. the figure described on the line ab is equal to the sum of the similar figures described on the lines ac and bc.
or thus: let us denote the sides by a, b, c, and the figures by , , ; then because the figures are similar, we have xx.
but a2 b2 c2 i. xlvii. . therefore ; that is, the sum of the figures on the sides is equal to the figure on the hypotenuse.
if semicircles be described on supplemental chords of a semicircle, the sum of the areas of the two crescents thus formed is equal to the area of the triangle whose sides are the supplemental chords and the diameter.
if two triangles (abc, cde) which have two sides of one proportional to two sides of the other (ab : bc :: cd : de), and the contained angles (b, d) equal, be joined at an angle (c), so as to have their homologous sides parallel, the remaining sides are
dem.-because the triangles abc, cde have the angles b and d equal, and the sides about these angles proportional, viz., ab : bc :: cd : de, they are equiangular vi. ; therefore the angle bac is equal to dce.
to each add acd, and we have the sum of the angles bac, acd equal to the sum of dce and acd; but the sum of bac, acd is i. xxix. two right angles; therefore the sum of dce and acd is two right angles. hence i. xiv. ac, ce are in the same right line.
in equal circles, angles (boc, epf) at the centres or (bac, edf) at the circumferences have the same ratio to one another as the arcs (bc, ef) on which they stand, and so also have the sectors (boc, epf).
dem.-1. take any number of arcs cg, gh in the first circle, each equal to bc. join og, oh, and in the second circle take any number of arcs fi, ij, each equal to ef. join ip, jp.
then because the arcs bc, cg, gh are all equal, the angles boc, cog, goh, are all equal iii. xxvii. . therefore the arc bh and the angle boh are equimultiples of the arc bc and the angle boc.
in like manner it may be proved that the arc ej and the angle epj are equimultiples of the arc ef and the angle epf.
again, since the circles are equal, it is evident that the angle boh is greater than, equal to, or less than the angle epj, according as the arc bh is greater than, equal to, or less than the arc ej.
now we have four magnitudes, namely, the arc bc, the arc ef, the angle boc, and the angle epf; and we have taken equimultiples of the first and third, namely, the arc bh, the angle boh, and other equimultiples of the second and fourth, namely, the arc ej
and the angle epj, and we have proved that, according as the multiple of the first is greater than, equal to, or less than the multiple of the second, the multiple of the third is greater than, equal to, or less than the multiple of the fourth. hence v.
again, since the angle bac is half the angle boc iii. xx. , and edf is half the angle epf,
2. the sector boc : sector epf :: bc : ef.
dem.-the same construction being made, since the arc bc is equal to cg, the angle boc is equal to cog. hence the sectors boc, cog are congruent (see observation, proposition xxix., book iii.); therefore they are equal.
in like manner the sectors cog, goh are equal. hence there are as many equal sectors as there are equal arcs; therefore the arc bh and the sector boh are equimultiples of the arc bc and the sector boc.
in the same manner it may be proved that the arc ej and the sector epj are equimultiples of the arc ef and the sector epf; and it is evident, by superposition, that if the arc bh is greater than, equal to, or less than the arc ej, the sector boh is
greater than, equal to, or less than the sector epj. hence v. def. v. the arc bc : ef :: sector boc : sector epf.
the second part may be proved as follows:-
sector boc 1 2 rectangle contained by the arc bc, and the radius of the circle abc xx. ex. 14 and sector epf 1 2 rectangle contained by the arc ef and the radius of the circle edf; and since the circles are equal, their radii are equal.
hence, sector boc : sector epf :: arc bc : arc ef.
1. what is the subject-matter of book vi.? ans. application of the theory of proportion.
2. what are similar rectilineal figures?
4. how many conditions are necessary to define similar triangles?
5. how many to define similar rectilineal figures of more than three sides?
6. when is a figure said to be given in species?
9. what is a mean proportional between two lines?
11. what is the altitude of a rectilineal figure?
12. if two triangles have equal altitudes, how do their areas vary?
13. how do these areas vary if they have equal bases but unequal altitudes?
14. if both bases and altitudes differ, how do the areas vary?
15. when are two lines divided proportionally?
16. if in two lines divided proportionally a pair of homologous points coincide with their point of intersection, what property holds for the lines joining the other pairs of homologous points?
18. if two triangles have equal areas, prove that their perpendiculars are reciprocally proportional to the bases.
19. what is meant by figures inversely similar?
20. if two figures be inversely similar, how can they be changed into figures directly similar?
21. give an example of two triangles inversely similar. ans. if two lines passing through any point o outside a circle intersect it in pairs of points a, a; b, b, respectively, the triangles oab, oab, are inversely similar.
22. what point is it round which a figure can be turned so as to bring its sides into positions of parallelism with the sides of a similar rectilineal figure. ans. the centre of similitude of the two figures.
23. how many figures similar to a given rectilineal figure of sides can be described on a given line?
24. how many centres of similitude can two regular polygons of n sides each have? ans. n centres, which lie on a circle.
26. how do the areas of similar rectilineal figures vary?
27. what proposition is xix. a special case of?
29. how many centres of similitude have two circles?
1. if in a fixed triangle we draw a variable parallel to the base, the locus of the points of intersection of the diagonals of the trapezium thus cut off from the triangle is the median that bisects the base.
2. find the locus of the point which divides in a given ratio the several lines drawn from a given point to the circumference of a given circle.
3. two lines ab, xy , are given in position: ab is divided in c in the ratio m : n, and parallels aa, bb, cc, are drawn in any direction meeting xy in the points a, b, c; prove
4. three concurrent lines from the vertices of a triangle abc meet the opposite sides in a, b, c; prove
5. if a transversal meet the sides of a triangle abc in the points a, b, c; prove
6. if on a variable line ac, drawn from a fixed point a to any point b in the circumference of a given circle, a point c be taken such that the rectangle ab.ac is constant, the locus of c is a circle.
7. if d be the middle point of the base bc of a triangle abc, e the foot of the perpendicular, l the point where the bisector of the angle a meets bc, h the point of contact of the inscribed circle with bc; prove de.hl he.hd.
8. in the same case, if k be the point of contact with bc of the escribed circle, which touches the other sides produced, lh.bk bd.le.
9. if r, r, r, r, r be the radii of the circumscribed, the inscribed, and the escribed circles of a plane triangle, d, d, d, d the distances of the centre of the circumscribed circle from the centres of the others, then r2 d2 2rr d2 2rr, c.
11. if p, p, p denote the perpendiculars of a triangle, then
12. in a given triangle inscribe another of given form, and having one of its angles at a given point in one of the sides of the original triangle.
13. if a triangle of given form move so that its three sides pass through three fixed points, the locus of any point in its plane is a circle.
14. the angle a and the area of a triangle abc are given in magnitude: if the point a be fixed in position, and the point b move along a fixed line or circle, the locus of the point c is a circle.
15. one of the vertices of a triangle of given form remains fixed; the locus of another is a right line or circle; find the locus of the third.
16. find the area of a triangle-(1) in terms of its medians; (2) in terms of its perpendiculars.
17. if two circles touch externally, their common tangent is a mean proportional between their diameters.
18. if there be given three parallel lines, and two fixed points a, b; then if the lines of connexion of a and b to any variable point in one of the parallels intersect the other parallels in the points c and d, e and f, respectively, cf and de pass each
19. if a system of circles pass through two fixed points, any two secants passing through one of the points are cut proportionally by the circles.
20. find a point o in the plane of a triangle abc, such that the diameters of the three circles, about the triangles oab, obc, oca, may be in the ratios of three given lines.
21. abcd is a cyclic quadrilateral: the lines ab, ad, and the point c, are given in position; find the locus of the point which divides bd in a given ratio.
22. ca, cb are two tangents to a circle; be is perpendicular to ad, the diameter through a; prove that cd bisects be.
23. if three lines from the vertices of a triangle abc to any interior point o meet the opposite sides in the points a, b, c; prove
24. if three concurrent lines oa, ob, oc be cut by two transversals in the two systems of points a, b, c; a, b, c, respectively: prove
-ab-.oc-- -bc-.oa- ca-cob--. a b oc b c oa ca ob
25. the line joining the middle points of the diagonals of a quadrilateral circumscribed to a circle-
divides each pair of opposite sides into inversely proportional segments;
is divided by each pair of opposite lines into segments which, measured from the centre, are proportional to the sides;
is divided by both pairs of opposite sides into segments which, measured from either diagonal, have the same ratio to each other.
26. if cd, cd be the internal and external bisectors of the angle c of the triangle acb, the three rectangles ad.db, ac.cb, ad.bd are proportional to the squares of ad, ac, ad; and are-(1) in arithmetical progression if the difference of the base angles
be equal to a right angle; (2) in geometrical progression if one base angle be right; (3) in harmonical progression if the sum of the base angles be equal to a right angle.
27. if a variable circle touch two fixed circles, the chord of contact passes through a fixed point on the line connecting the centres of the fixed circles.
dem.-let o, o be the centres of the two fixed circles; o the centre of the variable circle; a, b the points of contact. let ab and oo meet in c, and cut the fixed circles again in the points a, b respectively. join ao, ao, bo. then ao, bo meet in o iii.
xi. . now, because the triangles oaa, oab are isosceles, the angle oba oab oaa. hence oa is parallel to ob; therefore oc : oc :: oa : ob; that is, in a given ratio. hence c is a given point.
28. if dd be the common tangent to the two circles, dd2 ab.ab.
29. if r denote the radius of o and , , the radii of o, o, dd2 : ab2 :: (r)(r) : r2, the choice of sign depending on the nature of the contacts. this follows from 28.
30. if four circles be tangential to a fifth, and if we denote by 12 the common tangent to the first and second, c., then
31. the inscribed and escribed circles of any triangle are all touched by its nine-points circle.
32. the four triangles which are determined by four points, taken three by three, are such that their nine-points circles have one common point.
33. if a, b, c, d denote the four sides, and d, d the diagonals of a quadrilateral; prove that the sides of the triangle, formed by joining the feet of the perpendiculars from any of its angular points on the sides of the triangle formed by the three
remaining points, are proportional to the three rectangles ac, bd, dd.
34. prove the converse of ptolemy's theorem (see xvii., ex. 13).
35. describe a circle which shall-(1) pass through a given point, and touch two given circles; (2) touch three given circles.
36. if a variable circle touch two fixed circles, the tangent to it from their centre of similitude, through which the chord of contact passes (27), is of constant length.
37. if the lines ad, bd (see fig., ex. 27) be produced, they meet in a point on the circumference of o, and the line op is perpendicular to dd.
38. if a, b be two fixed points on two lines given in position, and a, b two variable points, such that the ratio aa : bb is constant, the locus of the point dividing ab in a given ratio is a right line.
39. if a line ef divide proportionally two opposite sides of a quadrilateral, and a line gh the other sides, each of these is divided by the other in the same ratio as the sides which determine them.
40. in a given circle inscribe a triangle, such that the triangle whose angular points are the feet of the perpendiculars from the extremities of the base on the bisector of the vertical angle, and the foot of the perpendicular from the vertical angle on
41. in a circle, the point of intersection of the diagonals of any inscribed quadrilateral coincides with the point of intersection of the diagonals of the circumscribed quadrilateral, whose sides touch the circle at the angular points of the inscribed
42. through two given points describe a circle whose common chord with another given circle may be parallel to a given line, or pass through a given point.
43. being given the centre of a circle, describe it so as to cut the legs of a given angle along a chord parallel to a given line.
44. if concurrent lines drawn from the angles of a polygon of an odd number of sides divide the opposite sides each into two segments, the product of one set of alternate segments is equal to the product of the other set.
45. if a triangle be described about a circle, the lines from the points of contact of its sides with the circle to the opposite angular points are concurrent.
46. if a triangle be inscribed in a circle, the tangents to the circle at its three angular points meet the three opposite sides at three collinear points.
47. the external bisectors of the angles of a triangle meet the opposite sides in three collinear points.
48. describe a circle touching a given line at a given point, and cutting a given circle at a given angle.
def.-the centre of mean position of any number of points a, b, c, d, c., is a point which may be found as follows:-bisect the line joining any two points a, b, in g. join g to a third point c; divide gc in h, so that gh 13gc.
join h to a fourth point d, and divide hd in k, so that hk 14hd, and so on. the last point found will be the centre of mean position of the given points.
49. the centre of mean position of the angular points of a regular polygon is the centre of figure of the polygon.
50. the sum of the perpendiculars let fall from any system of points a, b, c, d, c., whose number is n on any line l, is equal to n times the perpendicular from the centre of mean position on l.
51. the sum of the squares of lines drawn from any system of points a, b, c, d, c., to any point p, exceeds the sum of the squares of lines from the same points to their centre of mean position, o, by nop2.
52. if a point be taken within a triangle, so as to be the centre of mean position of the feet of the perpendiculars drawn from it to the sides of the triangle, the sum of the squares of the perpendiculars is a minimum.
53. construct a quadrilateral, being given two opposite angles, the diagonals, and the angle between the diagonals.
54. a circle rolls inside another of double its diameter; find the locus of a fixed point in its circumference.
55. two points, c, d, in the circumference of a given circle are on the same side of a given diameter; find a point p in the circumference at the other side of the given diameter, ab, such that pc, pd may cut ab at equal distances from the centre.
56. if the sides of any polygon be cut by a transversal, the product of one set of alternate segments is equal to the product of the remaining set.
57. a transversal being drawn cutting the sides of a triangle, the lines from the angles of the triangle to the middle points of the segments of the transversal intercepted by those angles meet the opposite sides in collinear points.
58. if lines be drawn from any point p to the angles of a triangle, the perpendiculars at p to these lines meet the opposite sides of the triangle in three collinear points.
59. divide a given semicircle into two parts by a perpendicular to the diameter, so that the radii of the circles inscribed in them may have a given ratio.
60. from a point within a triangle perpendiculars are let fall on the sides; find the locus of the point, when the sum of the squares of the lines joining the feet of the perpendiculars is given.
61. if a circle make given intercepts on two fixed lines, the rectangle contained by the perpendiculars from its centre on the bisectors of the angle formed by the lines is given.
62. if the base and the difference of the base angles of a triangle be given, the rectangle contained by the perpendiculars from the vertex on two lines through the middle point of the base, parallel to the internal and external bisectors of the vertical
63. the rectangle contained by the perpendiculars from the extremities of the base of a triangle, on the internal bisector of the vertical angle, is equal to the rectangle contained by the external bisector and the perpendicular from the middle of the
64. state and prove the corresponding theorem for perpendiculars on the external bisector.
65. if r, r denote the radii of the circles inscribed in the triangles into which a right-angled triangle is divided by the perpendicular from the right angle on the hypotenuse; then, if c be the hypotenuse, and s the semiperimeter, r2 r2 (s c)2.
66. if a, b, c, d be four collinear points, find a point o in the same line with them such that oa.od ob.oc.
67. the four sides of a cyclic quadrilateral are given; construct it.
68. being given two circles, find the locus of a point such that tangents from it to the circles may have a given ratio.
69. if four points a, b, c, d be collinear, find the locus of the point p at which ab and cd subtend equal angles.
70. if a circle touch internally two sides, ca, cb, of a triangle and its circumscribed circle, the distance from c to the point of contact on either side is a fourth proportional to the semiperimeter, and ca, cb.
71. state and prove the corresponding theorem for a circle touching the circumscribed circle externally and two sides produced.
72. pascal's theorem.-if the opposite sides of an irregular hexagon abcdef inscribed in a circle be produced till they meet, the three points of intersection g, h, i are collinear.
dem.-join ad. describe a circle about the triangle adi, cutting the lines af, cd produced, if necessary, in k and l. join ik, kl, li. now, the angles klg, fcg are each iii. xxi. equal to the angle gad. hence they are equal. therefore kl is parallel to cf.
similarly, li is parallel to ch, and ki to fh; hence the triangles kli, fch are homothetic. hence the lines joining corresponding vertices are concurrent. therefore the points i, h, g are collinear.
73. if two sides of a triangle circumscribed to a given circle be given in position, but the third side variable, the circle described about the triangle touches a fixed circle.
74. if two sides of a triangle be given in position, and if the area be given in magnitude, two points can be found, at each of which the base subtends a constant angle.
75. if a, b, c, d denote the sides of a cyclic quadrilateral, and s its semiperimeter, prove its area (s-a)(s-b)(s c)(s- d).
76. if three concurrent lines from the angles of a triangle abc meet the opposite side in the points a, b, c, and the points a, b, c be joined, forming a second triangle abc,
77. in the same case the diameter of the circle circumscribed about the triangle abc ab.bc.ca divided by the area of abc.
78. if a quadrilateral be inscribed in one circle, and circumscribed to another, the square of its area is equal to the product of its four sides.
79. if on the sides ab, ac of a triangle abc we take two points d, e, and on their line of connexion f, such that
80. if through the middle points of each of the two diagonals of a quadrilateral we draw a parallel to the other, the lines drawn from their points of intersection to the middle points of the sides divide the quadrilateral into four equal parts.
81. ce, df are perpendiculars to the diameter of a semicircle, and two circles are described touching ce, de, and the semicircle, one internally and the other externally; the rectangle contained by the perpendiculars from their centres on ab is equal to
82. if lines be drawn from any point in the circumference of a circle to the angular points of any inscribed regular polygon of an odd number of sides, the sums of the alternate lines are equal.
83. if at the extremities of a chord drawn through a given point within a given circle tangents be drawn, the sum of the reciprocals of the perpendiculars from the point upon the tangents is constant.
84. if a cyclic quadrilateral be such that three of its sides pass through three fixed collinear points, the fourth side passes through a fourth fixed point, collinear with the three given ones.
85. if all the sides of a polygon be parallel to given lines, and if the loci of all the angles but one be right lines, the locus of the remaining angle is also a right line.
86. if the vertical angle and the bisector of the vertical angle be given, the sum of the reciprocals of the containing sides is constant.
87. if p, p denote the areas of two regular polygons of any common number of sides, inscribed and circumscribed to a circle, and , the areas of the corresponding polygons of double the number of sides; prove is a geometric mean between p and p, and a
88. the difference of the areas of the triangles formed by joining the centres of the circles described about the equilateral triangles constructed-(1) outwards; (2) inwards-on the sides of any triangle, is equal to the area of that triangle.
89. in the same case, the sum of the squares of the sides of the two new triangles is equal to the sum of the squares of the sides of the original triangle.
90. if r, r denote the radii of the circumscribed and inscribed circles to a regular polygon of any number of sides, r, r, corresponding radii to a regular polygon of the same area, and double the number of sides; prove
91. if the altitude of a triangle be equal to its base, the sum of the distances of the orthocentre from the base and from the middle point of the base is equal to half the base.
92. in any triangle, the radius of the circumscribed circle is to the radius of the circle which is the locus of the vertex, when the base and the ratio of the sides are given, as the difference of the squares of the sides is to four times the area.
93. given the area of a parallelogram, one of its angles, and the difference between its diagonals; construct the parallelogram.
94. if a variable circle touch two equal circles, one internally and the other externally, and perpendiculars be let fall from its centre on the transverse tangents to these circles, the rectangle of the intercepts between the feet of these perpendiculars
and the intersection of the tangents is constant.
95. given the base of a triangle, the vertical angle, and the point in the base whose distance from the vertex is equal half the sum of the sides; construct the triangle.
96. if the middle point of the base bc of an isosceles triangle abc be the centre of a circle touching the equal sides, prove that any variable tangent to the circle will cut the sides in points d, e, such that the rectangle bd.ce will be constant.
97. inscribe in a given circle a trapezium, the sum of whose opposite parallel sides is given, and whose area is given.
98. inscribe in a given circle a polygon all whose sides pass through given points.
99. if two circles x, y be so related that a triangle may be inscribed in x and circumscribed about y , an infinite number of such triangles can be constructed.
100. in the same case, the circle inscribed in the triangle formed by joining the points of contact on y touches a given circle.
101. and the circle described about the triangle formed by drawing tangents to x, at the angular points of the inscribed triangle, touches a given circle.
102. find a point, the sum of whose distances from three given points may be a minimum.
103. a line drawn through the intersection of two tangents to a circle is divided harmonically by the circle and the chord of contact.
104. to construct a quadrilateral similar to a given one whose four sides shall pass through four given points.
105. to construct a quadrilateral, similar to a given one, whose four vertices shall lie on four given lines.
106. given the base of a triangle, the difference of the base angles, and the rectangle of the sides; construct the triangle.
107. abcd is a square, the side cd is bisected in e, and the line ef drawn, making the angle aef eab; prove that ef divides the side bc in the ratio of 2 : 1.
108. if any chord be drawn through a fixed point on a diameter of a circle, and its extremities joined to either end of the diameter, the joining lines cut off, on the tangent at the other end, portions whose rectangle is constant.
109. if two circles touch, and through their point of contact two secants be drawn at right angles to each other, cutting the circles respectively in the points a, a; b, b; then aa2 bb2 is constant.
110. if two secants at right angles to each other, passing through one of the points of intersection of two circles, cut the circles again, and the line through their centres in the two systems of points a, b, c; a, b, c respectively, then ab : bc :: ab :
111. two circles described to touch an ordinate of a semicircle, the semicircle itself, and the semicircles on the segments of the diameter, are equal to one another.
112. if a chord of a given circle subtend a right angle at a given point, the locus of the intersection of the tangents at its extremities is a circle.
113. the rectangle contained by the segments of the base of a triangle, made by the point of contact of the inscribed circle, is equal to the rectangle contained by the perpendiculars from the extremities of the base on the bisector of the vertical angle.
114. if o be the centre of the inscribed circle of the triangle prove
115. state and prove the corresponding theorems for the centres of the escribed circles.
116. four points a, b, c, d are collinear; find a point p at which the segments ab, bc, cd subtend equal angles.
117. the product of the bisectors of the three angles of a triangle whose sides are a, b, c, is
118. in the same case the product of the alternate segments of the sides made by the bisectors of the angles is
119. if three of the six points in which a circle meets the sides of any triangle be such, that the lines joining them to the opposite vertices are concurrent, the same property is true of the three remaining points.
120. if a triangle abc be inscribed in another abc, prove
is equal twice the triangle abc multiplied by the diameter of the circle abc.
121. construct a polygon of an odd number of sides, being given that the sides taken in order are divided in given ratios by fixed points.
122. if the external diagonal of a quadrilateral inscribed in a given circle be a chord of another given circle, the locus of its middle point is a circle.
123. if a chord of one circle be a tangent to another, the line connecting the middle point of each arc which it cuts off on the first, to its point of contact with the second, passes through a given point.
124. from a point p in the plane of a given polygon perpendiculars are let fall on its sides; if the area of the polygon formed by joining the feet of the perpendiculars be given, the locus of p is a circle.
theory of planes, coplanar lines, and solid angles
i. when two or more lines are in one plane they are said to be coplanar.
ii. the angle which one plane makes with another is called a dihedral angle.
iii. a solid angle is that which is made by more than two plane angles, in different planes, meeting in a point.
iv. the point is called the vertex of the solid angle.
v. if a solid angle be composed of three plane angles it is called a trihedral angle; if of four, a tetrahedral angle; and if of more than four, a polyhedral angle.
one part (ab) of a right line cannot be in a plane (x), and another part (bc) not in it.
dem.-since ab is in the plane x, it can be produced in it bk. i. post. ii. ; let it be produced to d. then, if bc be not in x, let any other plane passing through ad be turned round ad until it passes through the point c.
now, because the points b, c are in this second plane, the line bc i., def. vi. is in it. therefore the two right lines abc, abd lying in one plane have a common segment ab, which is impossible. therefore, c.
two right lines (ab, cd) which intersect one another in any point (e) are coplanar, and so also are any three right lines (ec, cb, be) which form a triangle.
dem.-let any plane pass through eb, and be turned round it until it passes through c. then because the points e, c are in this plane, the right line ec is in it i., def. vi. . for the same reason the line bc is in it.
therefore the lines ec, cb, be are coplanar; but ab and cd are two of these lines. hence ab and cd are coplanar.
if two planes (ab, bc) cut one another, their common section (bd) is a right line.
dem.-if not from b to d, draw in the plane ab the right line bed, and in the plane bc the right line bfd. then the right lines bed, bfd enclose a space, which i., axiom x. is impossible.
therefore the common section bd of the two planes must be a right line.
if a right line (ef) be perpendicular to each of two intersecting lines (ab, cd), it will be perpendicular to any line gh, which is both coplanar and concurrent with them.
dem.-through any point g in gh draw a line bc intersecting ab, cd, and so as to be bisected in g; and join any point f in ef to b, g, c. then, because ef is perpendicular to the lines eb, ec, we have
hence the angle gef is right, and ef is perpendicular to eg.
def. vi.-a line such as ef, which is perpendicular to a system of concurrent and coplanar lines, is said to be perpendicular to the plane of these lines, and is called a normal to it.
cor. 1.-the normal is the least line that may be drawn from a given point to a given plane; and of all others that may be drawn to it, the lines of any system making equal angles with the normal are equal to each other.
cor. 2.-a perpendicular to each of two intersecting lines is normal to their plane.
if three concurrent lines (bc, bd, be) have a common perpendicular (ab), they are coplanar.
dem.-for if possible let bc be not coplanar with bd, be, and let the plane of ab, bc intersect the plane of bd, be in the line bf. then xi. iii. bf is a right line; and, since it is coplanar with bd, be, which are each perpendicular to ab, it is xi. iv.
perpendicular to ab. therefore the angle abf is right; and the angle abc is right (hyp.). hence abc is equal to abf, which is impossible i., axiom ix. . therefore the lines bc, bd, be are coplanar.
if two right lines (ab, cd) be normals to the same plane (x), they shall be parallel to one another.
dem.-let ab, cd meet the plane x at the points b, d. join bd, and in the plane x draw de at right angles to bd; take any point e in de. join be, ae, ad. then because ab is normal to x, the angle abe is right.
therefore ae2 ab2 be2 ab2 bd2 de2; because the angle bde is right. but ab2 bd2 ad2, because the angle abd is right. hence ae2 ad2 de2. therefore the angle ade is right. i. xlviii . and since cd is normal to the plane x, de is perpendicular to cd.
hence de is a common perpendicular to the three concurrent lines cd, ad, bd. therefore these lines are coplanar xi. v. . but ab is coplanar with ad, bd xi. ii. .
therefore the lines ad, bd, cd are coplanar; and since the angles abd, bdc are right, the line ab is parallel to cd i. xxviii. .
def. vii.-if from every point in a given line normals be drawn to a given plane, the locus of their feet is called the projection of the given line on the plane.
1. the projection of any line on a plane is a right line.
2. the projection on either of two intersecting planes of a normal to the other plane is perpendicular to the line of intersection of the planes.
two parallel lines (ab, cd) and any line (ef) intersecting them are coplanar.
dem.-if possible let the intersecting line be out of the plane, as egf. and in the plane, of the parallels draw i. post. ii. the right line ehf. then we have two right lines egf, ehf, enclosing a space, which i. axiom x. is impossible.
hence the two parallel right lines and the transversal are coplanar.
or thus: since the points e, f are in the plane of the parallels, the line joining these points is in that plane i. def. vi .
if one (ab) of two parallel right lines (ab, cd), be normal to a plane (x), the other line (cd) shall be normal to the same plane.
dem.-let ab, cd meet the plane x in the points b, d. join bd. then the lines ab, bd, cd are coplanar. now in the plane x, to which ab is normal, draw de at right angles to bd. take any point e in de, and join be, ae, ad.
then because ab is normal to the plane x, it is perpendicular to the line be in that plane xi. def. vi. . hence the angle abe is right; therefore ae2 ab2 be2 ab2 bd2 de2 (because bde is right (const.)) ad2 de2 (because abd is right (hyp.)).
therefore the angle ade is right. hence de is at right angles both to ad and bd. therefore xi. iv. de is perpendicular to cd, which is coplanar and concurrent with ad and bd.
again, since ab and cd are parallel, the sum of the angles abd, bdc is two right angles i. xxix. ; but abd is right (hyp.); therefore bdc is right. hence cd is perpendicular to the two lines db, de, and therefore xi. iv.
it is normal to their plane, that is, it is normal to x.
two right lines (ab, cd) which are each parallel to a third line (ef) are parallel to one another.
dem.-if the three lines be coplanar, the proposition is evidently the same as i. xxx. if they are not coplanar, from any point g in ef draw in the planes of ef, ab; ef, cd, respectively, the lines gh, gk each perpendicular to ef i. xi. .
then because ef is perpendicular to each of the lines gh, gk, it is normal to their plane xi. iv. . and because ab is parallel to ef (hyp.), and ef is normal to the plane ghk, ab is normal to the plane ghk xi. viii. .
in like manner cd is normal to the plane hgk. hence, since ab and cd are normals to the same plane, they are parallel to one another.
if two intersecting right lines (ab, bc) be respectively parallel to two other intersecting right lines (de, ef), the angle (abc) between the former is equal to the angle (def) between the latter.
dem.-if both pairs of lines be coplanar, the proposition is the same as i. xxix., ex. 2. if not, take any points a, c in the lines ab, bc, and cut off ed ba, and ef bc i. iii. . join ad, be, cf, ac, df.
then because ab is equal and parallel to de, ad is equal and parallel to be i. xxxiii . in like manner cf is equal and parallel to be. hence xi. ix. ad is equal and parallel to cf. hence i. xxxiii. ac is equal to df.
therefore the triangles abc, def, have the three sides of one respectively equal to the three sides of the other. hence i. viii. the angle abc is equal to def.
def. viii.-two planes which meet are perpendicular to each other, when the right lines drawn in one of them perpendicular to their common section are normals to the other.
def. ix.-when two planes which meet are not perpendicular to each other, their inclination is the acute angle contained by two right lines drawn from any point of their common section at right angles to it-one in one plane, and the other in the other.
observation.-these definitions tacitly assume the result of props. iii. and x. of this book. on this account we have departed from the usual custom of placing them at the beginning of the book. we have altered the place of definition vi.
to draw a normal to a given plane (bh) from a given point (a) not in it.
sol.-in the given plane bh draw any line bc, and from a draw ad perpendicular to bc i. xii. ; then if ad be perpendicular to the plane, the thing required is done. if not, from d draw de in the plane bh at right angles to bc i. xi. , and from a draw af i.
xii. perpendicular to de. af is normal to the plane bh.
dem.-draw gh parallel to bc. then because bc is perpendicular both to ed and da, it is normal to the plane of ed, da xi. iv. ; and since gh is parallel to bc, it is normal to the same plane xi. viii. . hence af is perpendicular to gh xi. def. vi.
, and af is perpendicular to de (const.). therefore af is normal to the plane of gh and ed-that is, to the plane bh.
to draw a normal to a given plane from a given point (a) in the plane.
sol.-from any point b not in the plane draw xi. xi. bc normal to it. if this line pass through a it is the normal required. if not, from a draw ad parallel to bc i. xxxi. .
then because ad and bc are parallel, and bc is normal to the plane, ad is also normal to it xi. viii. , and it is drawn from the given point. hence it is the required normal.
from the same point (a) there can be but one normal drawn to a given plane (x).
dem.-1. let a be in the given plane, and if possible let ab, ac be both normals to it, on the same side. now let the plane of ba, ac cut the given plane x in the line de. then because ba is a normal, the angle bae is right. in like manner cae is right.
2. if the point be above the plane, there can be but one normal; for, if there could be two, they would be parallel xi. vi. to one another, which is absurd. therefore from the same point there can be drawn but one normal to a given plane.
planes (cd, ef) which have a common normal (ab) are parallel to each other.
dem.-if the planes be not parallel, they will meet when produced. let them meet, their common section being the line gh, in which take any point k. join ak, bk.
then because ab is normal to the plane cd, it is perpendicular to the line ak, which it meets in that plane xi. def. vi. . therefore the angle bak is right. in like manner the angle abk is right.
hence the plane triangle abk has two right angles, which is impossible. therefore the planes cd, ef cannot meet-that is, they are parallel.
1. the angle between two planes is equal to the angle between two intersecting normals to these planes.
2. if a line be parallel to each of two planes, the sections which any plane passing through it makes with them are parallel.
3. if a line be parallel to each of two intersecting planes, it is parallel to their intersection.
4. if two right lines be parallel, they are parallel to the common section of any two planes passing through them.
5. if the intersections of several planes be parallel, the normals drawn to them from any point are coplanar.
two planes (ac, df) are parallel, if two intersecting lines (ab, bc) on one of them be respectively parallel to two intersecting lines (de, ef) on the other.
dem.-from b draw bg perpendicular to the plane df xi. xi. , and let it meet that plane in g. through g draw gh parallel to ed, and gk to ef. now, since gh is parallel to ed (const.), and ab to ed (hyp.), ab is parallel to gh xi. ix. .
hence the sum of the angles abg, bgh is two right angles i. xxix ; but bgh is right (const.); therefore abg is right. in like manner cbg is right. hence bg is normal to the plane ac xi. def. vi. , and it is normal to df (const.).
hence the planes ac, df have a common normal bg; therefore they are parallel to one another.
if two parallel planes (ab, cd) be cut by a third plane (ef, hg), their common sections (ef, gh) with it are parallel.
dem.-if the lines ef, gh are not parallel, they must meet at some finite distance. let them meet in k. now since k is a point in the line ef, and ef is in the plane ab, k is in the plane ab. in like manner k is a point in the plane cd.
hence the planes ab, cd meet in k, which is impossible, since they are parallel. therefore the lines ef, gh must be parallel.
1. parallel planes intercept equal segments on parallel lines.
2. parallel lines intersecting the same plane make equal angles with it.
3. a right line intersecting parallel planes makes equal angles with them.
if two parallel lines (ab, cd) be cut by three parallel planes (gh, kl, mn) in two triads of points (a, e, b; c, f, d), their segments between those points are proportional.
dem.-join ac, bd, ad. let ad meet the plane kl in x. join ex, xf. then because the parallel planes kl, mn are cut by the plane abd in the lines ex, bd, these lines are parallel xi. xvi. . hence
if a right line (ab) be normal to a plane (ci), any plane (de) passing through it shall be perpendicular to that plane.
dem.-let ce be the common section of the planes de, ci. from any point f in ce draw fg in the plane de parallel to ab i. xxxi. . then because ab and fg are parallel, but ab is normal, to the plane ci; hence fg is normal to it xi. viii. .
now since fg is parallel to ab, the angles abf, bfg are equal to two right angles i. xxix. ; but abf is right (hyp.); therefore bfg is right-that is, fg is perpendicular to ce.
hence every line in the plane de, drawn perpendicular to the common section of the planes de, ci, is normal to the plane ci. therefore xi. def. viii. the planes de, ci are perpendicular to each other.
if two intersecting planes (ab, bc) be each perpendicular to a third plane (adc), their common section (bd) shall be normal to that plane.
dem.-if not, draw from d in the plane ab the line de perpendicular to ad, the common section of the planes ab, adc; and in the plane bc draw bf perpendicular to the common section dc of the planes bc, adc.
then because the plane ab is perpendicular to adc, the line de in ab is normal to the plane adc xi. def. viii. . in like manner df is normal to it. therefore from the point d there are two distinct normals to the plane adc, which xi. xiii. is impossible.
hence bd must be normal to the plane adc.
1. if three planes have a common line of intersection, the normals drawn to these planes from any point of that line are coplanar.
2. if two intersecting planes be respectively perpendicular to two intersecting lines, the line of intersection of the former is normal to the plane of the latter.
3. in the last case, show that the dihedral angle between the planes is equal to the rectilineal angle between the normals.
the sum of any two plane angles (bad, dac) of a trihedral angle (a) is greater than the third (bac).
dem.-if the third angle bac be less than or equal to either of the other angles the proposition is evident. if not, suppose it greater: take any point d in ad, and at the point a in the plane bac make the angle bae equal bad i. xxiii.
, and cut off ae equal ad. through e draw bc, cutting ab, ac in the points b, c. join db, dc.
then the triangles bad, bae have the two sides ba, ad in one equal respectively to the two sides ba, ae in the other, and the angle bad equal to bae; therefore the third side bd is equal to be.
but the sum of the sides bd, dc is greater than bc; hence dc is greater than ec. again, because the triangles dac, eac have the sides da, ac respectively equal to the sides ea, ac in the other, but the base dc greater than ec; therefore i. xxv.
the angle dac is greater than eac, but the angle dab is equal to bae (const.). hence the sum of the angles bad, dac is greater than the angle bac.
the sum of all the plane angles (bac, cad, c.) forming any solid angle (a) is less than four right angles.
dem.-suppose for simplicity that the solid angle a is contained by five plane angles bac, cad, dae, eaf, fab; and let the planes of these angles be cut by another plane in the lines bc, cd, de, ef, fb; then we have xi. xx. ,
hence, adding, we get the sum of the base angles of the five triangles bac, cad, c., greater than the sum of the interior angles of the pentagon bcdef -that is, greater than six right angles.
but the sum of the base angles of the same triangles, together with the sum of the plane angles bac, cad, c., forming the solid angle a, is equal to twice as many right angles as there are triangles bac, cad, c.-that is, equal to ten right angles.
hence the sum of the angles forming the solid angle is less than four right angles.
observation.-this prop. may not hold if the polygonal base bcdef contain re-entrant angles.
1. any face angle of a trihedral angle is less than the sum, but greater than the difference, of the supplements of the other two face angles.
2. a solid angle cannot be formed of equal plane angles which are equal to the angles of a regular polygon of n sides, except in the case of n 3, 4, or 5.
3. through one of two non-coplanar lines draw a plane parallel to the other.
4. draw a common perpendicular to two non-coplanar lines, and show that it is the shortest distance between them.
5. if two of the plane angles of a tetrahedral angle be equal, the planes of these angles are equally inclined to the plane of the third angle, and conversely.
if two of the planes of a trihedral angle be equally inclined to the third plane, the angles contained in those planes are equal.
6. the three lines of intersection of three planes are either parallel or concurrent.
7. if a trihedral angle o be formed by three right angles, and a, b, c be points along the edges, the orthocentre of the triangle abc is the foot of the normal from o on the plane abc.
8. if through the vertex o of a trihedral angle o-abc any line od be drawn interior to the angle, the sum of the rectilineal angles doa, dob, doc is less than the sum, but greater than half the sum, of the face angles of the trihedral.
9. if on the edges of a trihedral angle o-abc three equal lines oa, ob, oc be taken, each of these is greater than the radius of the circle described about the triangle abc.
10. given the three angles of a trihedral angle, find, by a plane construction, the angles between the containing planes.
11. if any plane p cut the four sides of a gauche quadrilateral (a quadrilateral whose angular points are not coplanar) abcd in four points, a, b, c, d, then the product of the four ratios
is plus unity, and conversely, if the product
12. if in the last exercise the intersecting plane be parallel to any two sides of the quadrilateral, it cuts the two remaining sides proportionally.
def. x.-if at the vertex o of a trihedral angle o-abc we draw normals oa, ob, oc to the faces obc, oca, oab, respectively, in such a manner that oa will be on the same side of the plane obc as oa, c., the trihedral angle o-abc is called the supplementary
13. if o-abc be the supplementary of o-abc, prove that o-abc is the supplementary of o-abc.
14. if two trihedral angles be supplementary, each dihedral angle of one is the supplement of the corresponding face angle of the other.
15. through a given point draw a right line which will meet two non-coplanar lines.
16. draw a right line parallel to a given line, which will meet two non-coplanar lines.
17. being given an angle aob, the locus of all the points p of space, such that the sum of the projections of the line op on oa and ob may be constant, is a plane.
prism, pyramid, cylinder, sphere, and cone
i. a polyhedron is a solid figure contained by plane figures: if it be contained by four plane figures it is called a tetrahedron; by six, a hexahedron; by eight, an octahedron; by twelve, a dodecahedron; and if by twenty, an icosahedron.
ii. if the plane faces of a polyhedron be equal and similar rectilineal figures, it is called a regular polyhedron.
iii. a pyramid is a polyhedron of which all the faces but one meet in a point. this point is called the vertex; and the opposite face, the base.
iv. a prism is a polyhedron having a pair of parallel faces which are equal and similar rectilineal figures, and are called its ends. the others, called its side faces, are parallelograms.
v. a prism whose ends are perpendicular to its sides is called a right prism; any other is called an oblique prism.
vi. the altitude of a pyramid is the length of the perpendicular drawn from its vertex to its base; and the altitude of a prism is the perpendicular distance between its ends.
vii. a parallelopiped is a prism whose bases are parallelograms. a parallelopiped is evidently a hexahedron.
viii. a cube is a rectangular parallelopiped, all whose sides are squares.
ix. a cylinder is a solid figure formed by the revolution of a rectangle about one of its sides, which remains fixed, and which is called its axis. the circles which terminate a cylinder are called its bases or ends.
x. a cone is the solid figure described by the revolution of a right-angled triangle about one of the legs, which remains fixed, and which is called the axis. the other leg describes the base, which is a circle.
xi. a sphere is the solid described by the revolution of a semicircle about a diameter, which remains fixed. the centre of the sphere is the centre of the generating semicircle.
any line passing through the centre of a sphere and terminated both ways by the surface is called a diameter.
right prisms (abcde-fghij, abcde-fghij) which have bases (abcde, abcde) that are equal and similar, and which have equal altitudes, are equal.
dem.-apply the bases to each other; then, since they are equal and similar figures, they will coincide-that is, the point a with a, b with b, c. and since af is equal to af, and each is normal to its respective base, the: point f will coincide with f.
in the same manner the points g, h, i, j will coincide respectively with the points g, h, i, j. hence the prisms are equal in every respect.
cor. 1.-right prisms which have equal bases (ef, ef) and equal altitudes are equal in volume.
dem.-since the bases are equal, but not similar, we can suppose one of them, ef, divided into parts a, b, c, and re-arranged so as to make them coincide with the other i.
xxxv., note ; and since the prism on ef can be subdivided in the same manner by planes perpendicular to the base, the proposition is evident.
cor. 2.-the volumes of right prisms (x, y ) having equal bases are proportional to their altitudes.
for, if the altitudes be in the ratio of m : n, x can be divided into m prisms of equal altitudes by planes parallel to the base; then these m prisms will be all equal. in like manner, y can be divided into n equal prisms. hence x : y :: m : n.
cor. 3.-in right prisms of equal altitudes the volumes are to one another as the areas of their bases.
this may be proved by dividing the bases into parts so that the subdivisions will be equal, and then the volumes proportional to the number of subdivisions in their respective bases, that is, to their areas.
cor. 4.-the volume of a rectangular parallelopiped is measured by the continued product of its three dimensions.
parallelopipeds (abcd-efgh, abcd-mnop), having a common base (abcd) and equal altitudes, are equal.
1. let the edges mn, ef be in one right line; then gh, op must be in one right line.
now ef mn, because each equal ab; therefore me nf; therefore the prisms aem-dhp, and bfn-cgo, have their triangular bases aem, bfn identically equal, and they have equal altitudes; hence they are equal; and supposing them taken away from the entire solid,
the remaining parallelopipeds abcd-efgh, abcd-mnop are equal.
2. let the edges ef, mn be in different lines; then produce on, pm to meet the lines ef and gh produced in the points j, k, l, i. then by 1 the parallelopipeds abcd-efgh, abcd-mnop are each equal to the parallelopiped abcd-ijkl.
cor.-the volume of any parallelopiped is equal to the product of its base and altitude.
a diagonal plane of a parallelopiped divides it into two prisms of equal volume.
1. when the parallelopiped is rectangular the proposition is evident.
2. when it is any parallelopiped, abcd-efgh, the diagonal plane bisects it.
dem.-through the vertices a, e let planes be drawn perpendicular to the edges and cutting them in the points i, j, k; l, m, n, respectively. then i. xxxiv. we have il bf, because each is equal to ae. hence ib lf. in like manner jc mg.
hence the pyramid a-ijcb agrees in everything but position with e-lmgf; hence it is equal to it in volume. to each add the solid abc-lme, and we have the prism aij-elm equal to the prism abc-efg.
in like manner ajk-emn acd-egh; but (1) the prism aij-elm ajk-emn. hence abc-efg acd-egh. therefore the diagonal plane bisects the parallelopiped.
cor. 1.-the volume of a triangular prism is equal to the product of its base and altitude; because it is half of a parallelopiped, which has a double base and equal altitude.
cor. 2.-the volume of any prism is equal to the product of its base and altitude; because it can be divided into triangular prisms.
if a pyramid (o-abcde) be cut by any plane (abcde) parallel to the base, the section is similar to the base.
dem.-because the plane aob cuts the parallel planes abcde, abcde, the sections ab, ab are parallel xi. xvi. in like manner bc, bc are parallel. hence the angle abc abc xi. x. .
in like manner the remaining angles of the polygon abcde are equal to the corresponding angles of abcde. again, because ab is parallel to ab, the triangles abo, abo are equiangular.
therefore the polygons abcde, abcde are equiangular, and have the sides about their equal angles proportional. hence they are similar.
cor. 1.-the edges and the altitude of the pyramid are similarly divided by the parallel plane.
cor. 2.-the areas of parallel sections are in the duplicate ratio of the distances of their planes from the vertex.
cor. 3.-in any two pyramids, sections parallel to their bases, which divide their altitudes in the same ratio, are proportional to their bases.
pyramids (p-abcd, p-abc), having equal altitudes (po, po) and bases (abcd, abc) of equal areas, have equal volumes.
dem.-if they be not equal in volume, let abc be the base of the greater; and let ox be the altitude of a prism, with an equal base, and whose volume is equal to their difference; then let the equal altitudes po, po be divided into such a number of equal
parts, by planes parallel to the bases of the pyramids, that each part shall be less than ox. then iv. cor. 3 the sections made by these planes will be equal each to each.
now let prisms be constructed on these sections as bases and with the equal parts of the altitudes of the pyramids as altitudes, and let the prisms in p-abcd be constructed below the sections, and in p-abc, above.
then it is evident that the sum of the prisms in p-abcd is less than that pyramid, and the sum of those on the sections of p-abc greater than p-abc.
therefore the difference between the pyramids is less than the difference between the sums of the prisms, that is, less than the lower prism in the pyramid p-abc; but the altitude of this prism is less than ox (const.).
hence the difference between the pyramids is less than the prism whose base is equal to one of the equal bases, and whose altitude is equal to ox, and the difference is equal to this prism (hyp.), which is impossible.
therefore the volumes of the pyramids are equal.
cor. 1.-the volume of a triangular pyramid e-abc is one third the volume of the prism abc-def, having the same base and altitude.
for, draw the plane eaf, then the pyramids e-afc, e-afd are equal, having equal bases afc, afd, and a common altitude; and the pyramids e-abc, f-abc are equal, having a common base and equal altitudes.
hence the pyramid e-abc is one of three equal pyramids into which the prism is divided. therefore it is one third of the prism.
cor. 2.-the volume of every pyramid is one-third of the volume of a prism having an equal base and altitude.
because it may be divided into triangular pyramids by planes through the vertex and the diagonals of the base.
the volume of a cylinder is equal to the product of the area of its base by its altitude
dem.-let o be the centre of its circular base; and take the angle aob indefinitely small, so that the arc ab may be regarded as a right line.
then planes perpendicular to the base, and cutting it in the lines oa, ob, will be faces of a triangular prism, whose base will be the triangle aob, and whose altitude will be the altitude of the cylinder.
the volume of this prism will be equal to the area of the triangle aob by the height of the cylinder.
hence, dividing the circle into elementary triangles, the cylinder will be equal to the sum of all the prisms, and therefore its volume will be equal to the area of the base multiplied by the altitude.
cor. 1.-if r be the radius, and h the height of the cylinder,
cor. 2.-if abcd be a rectangle; x a line in its plane parallel to the side ab; o the middle point of the rectangle; the volume of the solid described by the revolution of abcd round x is equal to the area of abcd multiplied by the circumference of the
dem.-produce the lines ad, bc to meet x in the points e, f. then when the rectangle revolves round x, the rectangles abfe, dcfe will describe cylinders whose bases will be circles having ae, de as radii, and whose common altitude will be ab.
hence the difference between the volumes of these cylinders will be equal to the differences between the areas of the bases multiplied by ab, that is (ae2 de2).ab. therefore the volume described by abcd
hence volume described by the rectangle abcd
rectangle abcd multiplied by the circumference of the circle described by its middle point o.
observation.-this cor. is a simple case of guldinus's celebrated theorem.
by its assistance we give in the two following corollaries original methods of finding the volumes of the cone and sphere, and it may be applied with equal facility to the solution of several other problems which are usually done by the integral calculus.
cor. 3.-the volume of a cone is one-third the volume of a cylinder having the same base and altitude.
dem.-let abcd be a rectangle whose diagonal is ac. the triangle abc will describe a cone, and the rectangle a cylinder by revolving round ab.
take two points e, f infinitely near each other in ac, and form two rectangles, eh, ek, by drawing lines parallel to ad, ab.
now if o, o be the middle points of these rectangles, it is evident that, when the whole figure revolves round ab, the circumference of the circle described by o will ultimately be twice the circumference of the circle described by o; and since the
parallelogram ek is equal to eh i. xliii. , the solid described by ek (cor. 1) will be equal to twice the solid described by eh.
hence, if ac be divided into an indefinite number of equal parts, and rectangles corresponding to eh, ek be inscribed in the triangles abc, adc, the sum of the solids described by the rectangles in the triangle adc is equal to twice the sum of the solids
described by the rectangles in the triangle abc-that is, the difference between the cylinder and cone is equal to twice the cone. hence the cylinder is equal to three times the cone.
or thus: we may regard the cone and the cylinder as limiting cases of a pyramid and prism having the same base and altitude; and since (v. cor.
2) the volume of a pyramid is one-third of the volume of a prism, having the same base and altitude, the volume of the cone is one-third of the volume of the cylinder.
cor. 4.-if r be the radius of the base of a cone, and h its height,
cor. 5.-the volume of a sphere is two-thirds of the volume of a circumscribed cylinder.
dem.-let ab be the diameter of the semicircle which describes the sphere; abcd the rectangle which describes the cylinder. take two points e, f indefinitely near each other in the semicircle.
join ef, which will be a tangent, and produce it to meet the diameter pq perpendicular to ab in n. let r be the centre.
join re; draw eg, fh, nl parallel to ab; and ei, fk parallel to pq; and produce to meet ln in m and l; and let o, o be the middle points of the rectangles eh, ek.
now the rectangle ng.gr pg.gq, because each is equal to ge2. hence ng : gp :: gq : gr, or me : ie :: rp rg : rg. now, denoting the radii of the circles described by the points o, o by , respectively, we have ultimately gr and 1 2(rp rg).
hence me : ie :: 2 : ; but me : ie :: rectangle el : rectangle ek :: i. xliii. eh : ek;
hence the solid described by eh equal twice the solid described by ek. therefore we infer, as in the last cor., that the whole volume of the sphere is equal to twice the difference between the cylinder and sphere.
therefore the sphere is two-thirds of the cylinder.
the surface of a sphere is equal to the convex surface of the circumscribed cylinder.
dem.-let ab be the diameter of the semicircle which describes the sphere. take two points, e, f, indefinitely near each other in the semicircle. join ef, and produce to meet the tangent cd parallel to ab in n. draw ei, fk parallel to pq.
produce ei to meet ab in g. let o be the centre. join oe.
now we have fe : ki :: en : in vi. ii. ;
because the triangles eni and oeg are similar.
hence ef : ik :: ig : eg; and ig : eg :: circumference of circle described by the point i : circumference of circle described by the point e.
hence the rectangle contained by ef, and circumference of circle described by e is equal to the rectangle contained by ik, and circumference of circle described by i-that is, the portion of the spherical surface described by ef is equal to the portion of
the cylindrical surface described by ik. hence it is evident, if planes be drawn perpendicular to the diameter ab-that the portions of cylindrical and spherical surface between any two of them are equal.
hence the whole spherical surface is equal to the cylindrical surface described by cd.
or thus: conceive the whole surface of the sphere divided into an indefinitely great number of equal parts, then it is evident that each of these may be regarded as the base of a pyramid having the centre of the sphere as a common vertex.
therefore the volume of the sphere is equal to the whole area of the surface multiplied by one-third of the radius. hence if s denote the surface, we have
that is, the area of the surface of a sphere is equal four times the area of one of its great circles.
1. the convex surface of a cone is equal to half the rectangle contained by the circumference of the base and the slant height.
2. the convex surface of a right cylinder is equal to the rectangle contained by the circumference of the base and the altitude.
3. if p be a point in the base abc of a triangular pyramid o-abc, and if parallels to the edges oa, ob, oc, through p, meet the faces in the points a, b, c, the sum of the ratios
4. the volume of the frustum of a cone, made by a plane parallel to the base, is equal to the sum of the three cones whose bases are the two ends of the frustum, and the circle whose diameter is a mean proportional between the end diameters, and whose
common altitude is equal to one-third of the altitude of the frustum.
5. if a point p be joined to the angular points a, b, c, d of a tetrahedron, and the joining lines, produced if necessary, meet the opposite faces in a, b, c, d, the sum of the ratios
6. the surface of a sphere is equal to the rectangle by its diameter, and the circumference of a great circle.
7. the surface of a sphere is two thirds of the whole surface of its circumscribed cylinder.
8. if the four diagonals of a quadrangular prism be concurrent, it is a parallelopiped.
9. if the slant height of a right cone be equal to the diameter of its base, its total surface is to the surface of the inscribed sphere as 9 : 4.
10. the middle points of two pairs of opposite edges of a triangular pyramid are coplanar, and form a parallelogram.
11. if the four perpendiculars from the vertices on the opposite faces of a pyramid abcd be concurrent, then
12. every section of a sphere by a plane is a circle.
13. the locus of the centres of parallel sections is a diameter of the sphere.
14. if any number of lines in space pass through a fixed point, the feet of the perpendiculars on them from another fixed point are homospheric.
15. extend the property of ex. 4 to the pyramid.
16. the volume of the ring described by a circle which revolves round a line in its plane is equal to the area of the circle, multiplied by the circumference of the circle described by its centre.
17. any plane bisecting two opposite edges of a tetrahedron bisects its volume.
18. planes which bisect the dihedral angles of a tetrahedron meet in a point.
19. planes which bisect perpendicularly the edges of a tetrahedron meet in a point.
20. the volumes of two triangular pyramids, having a common solid angle, are proportional to the rectangles contained by the edges terminating in that angle.
21. a plane bisecting a dihedral angle of a tetrahedron divides the opposite edge into portions proportional to the areas containing that edge.
22. the volume of a sphere: the volume of the circumscribed cube as : 6.
23. if h be the height, and the radius of a segment of a sphere, its volume is h 6(h2 32).
24. if h be the perpendicular distance between two parallel planes, which cut a sphere in sections whose radii are 1, 2, the volume of the frustum is h 6- h2 3(12 22) .
25. if be the distance of a point p from the centre of a sphere whose radius is r, the sum of the squares of the six segments at three rectangular chords passing through p is 6r2 22.
26. the volume of a sphere : the volume of an inscribed cube as : 2.
27. if o-abc be a tetrahedron whose angles aob, boc, coa are right, the square of the area of the triangle abc is equal to the sum of the squares of the three other triangular faces.
28. in the same case, if p be the perpendicular from o on the face abc,
29. if h be the height of an ronaut, and r the radius of the earth, the extent of surface visible 2r2h r- -h.
30. if the four sides of a gauche quadrilateral touch a sphere, the points of contact are concyclic.
in every plane there is one special line called the line at infinity. the point where any other line in the plane cuts the line at infinity is called the point at infinity in that line. all other points in the line are called finite points.
two lines in the plane which meet the line at infinity in the same point are said to have the same direction, and two lines which meet it in different points to have different directions.
two lines which have the same direction cannot meet in any finite point i. axiom x. , and are parallel. two lines which have different directions must intersect in some finite point, since, if produced, they meet the line at infinity in different points.
this is a fundamental conception in geometry, it is self-evident, and may be assumed as an axiom (see observations on the axioms, book i.).
hence we may infer the following general proposition:-"any two lines in the same plane must meet in some point in that plane; that is-(1) at infinity, when the lines have the same direction; (2) in some finite point, when they have different
directions."-see poncelet, proprietes projectives, page 52.
legendre's and hamilton's proofs of euclid, i. xxxii.
the discovery of the proposition that "the sum of the three angles of a triangle is equal to two right angles" is attributed to pythagoras. until modern times no proof of it, independent of the theory of parallels, was known.
we shall give here two demonstrations, each independent of that theory. these are due to two of the greatest mathematicians of modern times-one, the founder of the theory of elliptic functions; the other, the discoverer of the calculus of quaternions.
legendre's proof.-let abc be a triangle, of which the side ac is the greatest. bisect bc in d. join ad. then ad is less than ac i. xix. ex. 5 . now, construct a new triangle abc, having the side ac 2ad, and ab ac.
again, bisect bc in d, and form another triangle abc, having ac 2ad, and ab ac, c. (1) the sum of the angles of the triangle abc the sum of the angles of abc i. xvi. cor. 1 the sum of the angles of abc the sum of the angles of abc, c.
(2) the angle bac is less than half bac; the angle bac is less than half bac, and so on; hence the angle b(n)a(n) will ultimately become infinitely small.
(3) the sum of the base angles of any triangle of the series is equal to the angle of the preceding triangle (see dem. i. xvi.). hence, if the annexed diagram represent the triangle ab(n 1)c(n 1), the sum of the base angles a and c(n 1) is
equal to the angle b(n)c(n); and when n is indefinitely large, this angle is an infinitesimal; hence the point b(n 1) will ultimately be in the line ac, and the angle ab(n 1)c(n 1) will become a straight angle i. def. x.
, that is, it is equal to two right angles; but the sum of the angles of ab(n 1)c(n 1) is equal to the sum of the angles abc. hence the sum of the three angles of abc is equal to two right angles.
hamilton's quaternion proof.-let abc be the triangle. produce ba to d, and make ad equal to ac. produce cb to e, and make be equal to bd; finally, produce ac to f, and make cf equal to ce. denote the exterior angles thus formed by a, b, c.
now let the leg ac of the angle a be turned round the point a through the angle a; then the point c will coincide with d.
again, let the leg bd of the angle b be turned round the point b through the angle b, until bd coincides with be; then the point d will coincide with e.
lastly, let ce be turned round c, through the angle c, until ce coincides with cf, and the point e with f.
now, it is evident that by these rotations the point c has been brought successively into the positions d, e, f; hence, by a motion of mere translation along the line fc, the line ca can be brought into its former position.
therefore it follows, since rotation is independent of translation, that the amount of the three rotations is equal to one complete revolution round the point a; therefore a b c four right angles; but
observation.-the foregoing demonstration is the most elementary that was ever given of this celebrated proposition. i have reduced it to its simplest form, and without making any use of the language of quaternions.
the same method of proof will establish the more general proposition, that the sum of the external angles of any convex rectilineal figure is equal to four right angles.
mr. abbott, f.t.c.d., has informed me that this demonstration was first given by playfair in 1826, so that hamilton was anticipated.
it has been objected to on the ground that, applied verbatim to a spherical triangle, it would lead to the conclusion that the sum of the angles is two right angles, which being wrong, proves that the method is not valid.
a slight consideration will show that the cases are different.
in the proof given in the text there are three motions of rotation, in each of which a point describes an arc of a circle, followed by a motion of translation, in which the same point describes a right line, and returns to its original position.
on the surface of a sphere we should have, corresponding to these, three motions of rotation, in each of which the point would describe an arc of a circle, followed by a motion of rotation about the centre of the sphere, in which the point should describe
an arc of a great circle to return to its original position. hence, the proof for a plane triangle cannot be applied to a spherical triangle.
to inscribe a regular polygon of seventeen sides in a circle.
analysis.-let a be one of the angular points, ao the diameter, a1, a2, ... a8 the vertices at one side of ao. produce oa3 to m, and oa2 to p, making a3m oa5, and a2p oa8. again, cut off a6n oa7, and a1q oa4. lastly, cut off or on, and os oq.
r(3 8) r(1 6) r(2 7) r(2 5) iv. ex. 40 .
and performing the multiplication and substituting, we get
hence, the rectangle and the difference of the lines mr and ps being given, each is given; hence mr is given; but mr om on; therefore om on is given; and we have proved that the rectangle om.on r2; therefore om and on are each given.
in like manner, op and oq are each given.
hence, since oq and on are each given, 6 and 7 are each given; therefore we can draw these chords, and we have the arc a6a7 between their extremities given; that is, the seventeenth part of the circumference of a circle. hence the problem is solved.
the foregoing analysis is due to ampere: see catalan, theoremes et problemes de geometrie elementaire. we have abridged and simplified ampere's solution.
to find two mean proportionals between two given lines.
the problem to find two mean proportionals is one of the most celebrated in geometry on account of the importance which the ancients attached to it. it cannot be solved by the line and circle, but is very easy by conic sections.
the following is a mechanical construction by the ruler and compass.
sol.-let the extremes ab, bc be placed at right angles to each other; complete the rectangle abcd, and describe a circle about it.
produce da, dc, and let a graduated ruler be made to revolve round the point b, and so adjusted that be shall be equal to gf; then af, ce are two mean proportionals between ab, bc.
dem.-since be is equal to gf, the rectangle be.ge bf.gf. therefore de.ce df.af; hence de : df :: af : ce; and by similar triangles, ab : af :: de : df, and ce : cb :: de : df. hence ab : af :: af : ce; and af : ce :: ce : cb.
therefore ab, af, ce, cb are continual proportions. hence vi. def. iv. af, ce are two mean proportionals between ab and bc.
the foregoing elegant construction is due to the ancient geometer philo of byzantium. if we join dg it will be perpendicular to ef.
the line ef is called philo's line; it possesses the remarkable property of being the minimum line through the point b between the fixed lines de, df.
newton's construction.-let ab and l be the two given lines of which ab is the greater. bisect ab in c. with a as centre and ac as radius, describe a circle, and in it place the chord cd equal to the second line l.
join bd, and draw by trial through a a line meeting bd, cd produced in the points e, f, so that the intercept ef will be equal to the radius of the circle. de and fa are the mean proportionals required.
dem.-join ad. since the line bf cuts the sides of the ace, we have
therefore ab.cd de.fa, or cd- de fa- ab.
again, since the acd is isosceles, we have
therefore de2 fa.ab, and we have ab.cd de.fa.
hence ab, de, fa, cd are in continued proportion.
i am indebted to professor galbraith for the following proof of the minimum property of philo's line.
it is due to the late professor mac cullagh:-let ac, cb be two given lines, e a fixed point, cd a perpendicular on ab; it is required to prove, if ae is equal to db, that ab is a minimum.
dem.-through e draw em parallel to bc; make en em; produce ab until ep ab. through the points n, p draw nt, rp each parallel to ac, and through p draw pq parallel to bc.
it is easy to see from the figure that the parallelogram qr is equal to the parallelogram mf, and is therefore given. through p draw st perpendicular to ep. now, since ae db, bp is equal to db; therefore ps cd.
again, since op ad, pt is equal to cd; therefore ps pt. hence qr is the maximum parallelogram in the triangle sv t.
again, if any other line ab be drawn through e, and produced to p, so that ep ap, the point p must fall outside st, because the parallelogram qr, corresponding to qr, will be equal to mf, and therefore equal to qr.
hence the line ep is greater than ep, or ab is greater than ab. hence ab is a minimum.
the following mechanical method of trisecting an angle occurred to me several years ago. apart from the interest belonging to the problem, it is valuable to the student as a geometrical exercise:-
sol.-erect cd perpendicular to ca; bisect the angle bcd by cg, and make the angle eci equal half a right angle; it is evident that ci will fall between cb and ca.
then, if we use a jointed ruler-that is two equal rulers connected by a pivot-and make cb equal to the length of one of these rulers, and, with c as centre and cb as radius, describe the circle bam, cutting ci in i: at i draw the tangent ig, cutting cg in
then, since icg is half a right angle, and cig is right, igc is half a right angle; therefore ic is equal to ig; but ic equal cb; therefore ig cb-equal length of one of the two equal rulers.
hence, if the rulers be opened out at right angles, and placed so that the pivot will be at i, and one extremity at c, the other extremity at g; it is evident that the point b will be between the two rulers; then, while the extremity at c remains fixed,
let the other be made to traverse the line gf, until the edge of the second ruler passes through b: it is plain that the pivot moves along the circumference of the circle.
let ch, hf, be the positions of the rulers when this happens; draw the line ch; the angle ach is one-third of acb.
dem.-produce bc to m. join hm. erect bo at right angles to bm. then, because ch hf, the angle hcf hfc, and the angle dce ecb (const.). hence the angle hcd hbc i. xxxii.
, and the right angles acd, cbo are equal; therefore the angle ach is equal to hbo; that is iii. xxxii. , equal to hmb, or to half the angle hcb. hence ach is one-third of acb.
modern mathematicians denote the ratio of the circumference of a circle to its diameter by the symbol . hence, if r denote the radius, the circumference will be 2r; and, since the area of a circle vi. xx. ex.
15 is equal to half the rectangle contained by the circumference and the radius, the area will be r2. hence, if the area be known, the value of will be known; and, conversely, if the value of be known, the area is known.
on this account the determination of the value of is called "the problem of the quadrature of the circle," and is one of the most celebrated in mathematics.
it is now known that the value of is incommensurable; that is, that it cannot be expressed as the ratio of any two whole numbers, and therefore that it can be found only approximately; but the approximation can be carried as far as we please, just as in
extracting the square root we may proceed to as many decimal places as may be required. the simplest approximate value of was found by archimedes, namely, 22 : 7.
this value is tolerably exact, and is the one used in ordinary calculations, except where great accuracy is required. the next to this in ascending order, viz. 355 : 113, found by vieta, is correct to six places of decimals.
it differs very little from the ratio 3.1416 : 1, given in our elementary books.
several expeditious methods, depending on the higher mathematics, are known for calculating the value of . the following is an outline of a very simple elementary method for determining this important constant.
it depends on a theorem which is at once inferred from vi., ex.
87, namely "if a, a denote the reciprocals of the areas of any two polygons of the same number of sides inscribed and circumscribed to a circle; a, a the corresponding quantities for polygons of twice the number; a is the geometric mean between a and a,
and a the arithmetic mean between a and a." hence, if a and a be known, we can, by the processes of finding arithmetic and geometric means, find a and a. in like manner, from a, a we can find a, a related to a, a; as a, a are to a, a.
therefore, proceeding in this manner until we arrive at values a(n), a(n) that will agree in as many decimal places as there are in the degree of accuracy we wish to attain; and since the area of a circle is intermediate between the reciprocals of a(n)
and a(n), the area of the circle can be found to any required degree of approximation.
if for simplicity we take the radius of the circle to be unity, and commence with the inscribed and circumscribed squares, we have
a .5, a .25. a .3535533, a .3017766. a .3264853, a .3141315.
these numbers are found thus: a is the geometric mean between a and a; that is, between .5 and .25, and a is the arithmetic mean between a and a, or between .3535533 and .25.
again, a is the geometric mean between a and a; and a the arithmetic mean between a and a. proceeding in this manner, we find a(13) .3183099; a(13) .3183099.
hence the area of a circle radius unity, correct to seven decimal places, is equal to the reciprocal of .3183099; that is, equal to 3.1415926; or the value of correct to seven places of decimals is 3.1415926.
the number is of such fundamental importance in geometry, that mathematicians have devoted great attention to its calculation. mr. shanks, an english computer, carried the calculation to 707 places of decimals.
the following are the first 36 figures of his result:-
3.141,592,653,589,793,238,462,643,383,279,502,884.
the result is here carried far beyond all the requirements of mathematics.
ten decimals are sufficient to give the circumference of the earth to the fraction of an inch, and thirty decimals would give the circumference of the whole visible universe to a quantity imperceptible with the most powerful microscope.
in the foregoing treatise we have given the elementary geometry of the point, the line, and the circle, and figures formed by combinations of these.
but it is important to the student to remark, that points and lines, instead of being distinct from, are limiting cases of, circles; and points and planes limiting cases of spheres. thus, a circle whose radius diminishes to zero becomes a point.
if, on the contrary, the circle be continually enlarged, it may have its curvature so much diminished, that any portion of its circumference may be made to differ in as small a degree as we please from a right line, and become one when the radius becomes
infinite. this happens when the centre, but not the circumference, goes to infinity.
third edition, revised and enlarged-3 6, cloth.
first six books of the elements of euclid.
fellow of the royal university of ireland; vice-president, royal irish academy; c. c.
dublin: hodges, figgis, co. london: longmans, green, co.
"we have noticed ('nature,' vol. xxiv., p. 52; vol. xxvi., p. 219) two previous editions of this book, and are glad to find that our favourable opinion of it has been so convincingly indorsed by teachers and students in general.
the novelty of this edition is a supplement of additional propositions and exercises.
this contains an elegant mode of obtaining the circle tangential to three given circles by the methods of false positions, constructions for a quadrilateral, and a full account-for the first time in a text-book-of the brocard, triplicate ratio, and (what
the author proposes to call) the cosine circles. dr. casey has collected together very many properties of these circles, and, as usual with him, has added several beautiful results of his own.
he has done excellent service in introducing the circles to the notice of english students....we only need say we hope that this edition may meet with as much acceptance as its predecessors, it deserves greater acceptance."
the mathematical magazine," erie, pennsylvania.
"dr. casey, an eminent professor of the higher mathematics and mathematical physics in the catholic university of ireland, has just brought out a second edition of his unique 'sequel to the first six books of euclid,' in which he has contrived to arrange
and to pack more geometrical gems than have appeared in any single text-book since the days of the self-taught thomas simpson.
'the principles of modern geometry contained in the work are, in the present state of science, indispensable in pure and applied mathematics, and in mathematical physics; and it is important that the student should become early acquainted with them.'
"eleven of the sixteen sections into which the work is divided exhibit most excellent specimens of geometrical reasoning and research. these will be found to furnish very neat models for systematic methods of study.
the other five sections contain 261 choice problems for solution.
here the earnest student will find all that he needs to bring himself abreast with the amazing developments that are being made almost daily in the vast regions of pure and applied geometry. on pp.
152 and 153 there is an elegant solution of the celebrated malfatti's problem.
"as our space is limited, we earnestly advise every lover of the 'bright seraphic truth' and every friend of the 'mathematical magazine' to procure this invaluable book without delay."
"this book contains a large number of elementary geometrical propositions not given in euclid, which are required by every student of mathematics.
here are such propositions as that the three bisectors of the sides of a triangle are concurrent, needed in determining the position of the centre of gravity of a triangle; propositions in the circle needed in practical geometry and mechanics; properties
of the centres of similitudes, and the theories of inversion and reciprocations so useful in certain electrical questions. the proofs are always neat, and in many cases exceedingly elegant."
"we have certainly seen nowhere so good an introduction to modern geometry, or so copious a collection of those elementary propositions not given by euclid, but which are absolutely indispensable for every student who intends to proceed to the study of
the higher mathematics. the style and general get up of the book are, in every way, worthy of the 'dublin university press series,' to which it belongs."
"this book is a well-devised and useful work. it consists of propositions supplementary to those of the first six books of euclid, and a series of carefully arranged exercises which follow each section.
more than half the book is devoted to the sixth book of euclid, the chapters on the 'theory of inversion' and on the 'poles and polars' being especially good.
its method skilfully combines the methods of old and modern geometry; and a student well acquainted with its subject-matter would be fairly equipped with the geometrical knowledge he would require for the study of any branch of physical science."
"professor casey's aim has been to collect within reasonable compass all those propositions of modern geometry to which reference is often made, but which are as yet embodied nowhere....we can unreservedly give the highest praise to the matter of the
book. in most cases the proofs are extraordinarily neat....the notes to the sixth book are the most satisfactory.
feuerbach's theorem (the nine-points circle touches inscribed and escribed circles) is favoured with two or three proofs, all of which are elegant. dr.
hart's extension of it is extremely well proved....we shall have given sufficient commendation to the book when we say, that the proofs of these (malfatti's problem, and miquel's theorem), and equally complex problems, which we used to shudder to attack,
even by the powerful weapons of analysis, are easily and triumphantly accomplished by pure geometry.
"after showing what great results this book has accomplished in the minimum of space, it is almost superfluous to say more. our author is almost alone in the field, and for the present need scarcely fear rivals."
"dr. casey is an accomplished geometer, and this little book is worthy of his reputation. it is well adapted for use in the higher forms of our schools. it is a good introduction to the larger works of chasles, salmon, and townsend.
it contains both a text and numerous examples."
"dr. casey's 'sequel to euclid' will be found a most valuable work to any student who has thoroughly mastered euclid, and imbibed a real taste for geometrical reasoning....the higher methods of pure geometrical demonstration, which form by far the larger
and more important portion, are admirable; the propositions are for the most part extremely well given, and will amply repay a careful perusal to advanced students."
frequent applications having been made to dr.
casey requesting him to publish a "key" containing the solutions of the exercises in his "elements of euclid," but his professorial and other duties scarcely leaving him any time to devote to it, i undertook, under his direction, the task of preparing
one. every solution was examined and approved of by him before writing it for publication, so that the work may be regarded as virtually his.
the exercises are a joint selection made by him and the late lamented professor townsend, s.f.t.c.d., and form one of the finest collections ever published.
with copious annotations and numerous exercises.
fellow of the royal university of ireland; vice-president, royal irish academy; c. c.
dublin: hodges, figgis, co. london: longmans, green, co.
the following are a few of the opinions received by dr. casey on this work:-
"teachers no longer need be at a loss when asked which of the numerous 'euclids' they recommend to learners. dr. casey's will, we presume, supersede all others."-the dublin evening mail.
"dr. casey's work is one of the best and most complete treatises on elementary geometry we have seen. the annotations on the several propositions are specially valuable to students."-the northern whig.
"his long and successful experience as a teacher has eminently qualified dr. casey for the task which he has undertaken....we can unhesitatingly say that this is the best edition of euclid that has been yet offered to the public."-the freeman's journal.
"i have no doubt whatever of the general adoption of your work through all the schools of ireland immediately, and of england also before very long."
from george francis fitzgerald, esq., f.t.c.d.
"your work on euclid seems admirable, and is a great improvement in most ways on its predecessors.
it is a great thing to call the attention of students to the innumerable variations in statement and simple deductions from propositions....i should have preferred some modification of euclid to a reproduction, but i suppose people cannot be got to agree
from h. j. cooke, esq., the academy, banbridge.
"in the clearness, neatness, and variety of demonstrations, it is far superior to any text-book yet published, whilst the exercises are all that could be desired."
from james a. poole, m.a., 29, harcourt-street, dublin.
"this work proves that irish scholars can produce class-books which even the head masters of english schools will feel it a duty to introduce into their establishments."
from professor leebody, magee college, londonderry.
"so far as i have had time to examine it, it seems to me a very valuable addition to our text-books of elementary geometry, and a most suitable introduction to the 'sequel to euclid,' which i have found an admirable book for class teaching."
from mrs. bryant, f.c.p., principal of the north london collegiate school for girls.
"i am heartily glad to welcome this work as a substitute for the much less elegant text-books in vogue here. i have begun to use it already with some of my classes, and find that the arrangement of exercises after each proposition works admirably."
from the rev. j. e. reffe, french college, blackrock.
"i am sure you will soon be obliged to prepare a second edition. i have ordered fifty copies more of the euclid (this makes 250 copies for the french college). they all like the book here."
"the edition of the first six books of euclid by dr. john casey is a particularly useful and able work....the illustrative exercises and problems are exceedingly numerous, and have been selected with great care. dr.
casey has done an undoubted service to teachers in preparing an edition of euclid adapted to the development of the geometry of the present day."
"there is a simplicity and neatness of style in the solution of the problems which will be of great assistance to the students in mastering them....at the end of each proposition there is an examination paper upon it, with deductions and other
propositions, by means of which the student is at once enabled to test himself whether he has fully grasped the principles involved....dr.
casey brings at once the student face to face with the difficulties to be encountered, and trains him, stage by stage, to solve them."
"the preface states that this book 'is intended to supply a want much felt by teachers at the present day-the production of a work which, while giving the unrivalled original in all its integrity, would also contain the modern conceptions and developments
of the portion of geometry over which the elements extend.'
"the book is all, and more than all, it professes to be....the propositions suggested are such as will be found to have most important applications, and the methods of proof are both simple and elegant.
we know no book which, within so moderate a compass, puts the student in possession of such valuable results.
"the exercises left for solution are such as will repay patient study, and those whose solution are given in the book itself will suggest the methods by which the others are to be demonstrated.
we recommend everyone who wants good exercises in geometry to get the book, and study it for themselves."
"the editor has been very happy in some of the changes he has made.
the combination of the general and particular enunciations of each proposition into one is good; and the shortening of the proofs, by omitting the repetitions so common in euclid, is another improvement.
the use of the contra-positive of a proved theorem is introduced with advantage, in place of the reductio ad absurdum; while the alternative (or, in some cases, substituted) proofs are numerous, many of them being not only elegant but eminently
suggestive. the notes at the end of the book are of great interest, and much of the matter is not easily accessible. the collection of exercises, 'of which there are nearly eight hundred,' is another feature which will commend the book to teachers.
to sum up, we think that this work ought to be read by every teacher of geometry; and we make bold to say that no one can study it without gaining valuable information, and still more valuable suggestions."
from the journal of education, sept. 1, 1883.
"in the text of the propositions, the author has adhered, in all but a few instances, to the substance of euclid's demonstrations, without, however, giving way to a slavish following of his occasional verbiage and redundance.
the use of letters in brackets in the enunciations eludes the necessity of giving a second or particular enunciation, and can do no harm.
hints of other proofs are often given in small type at the end of a proposition, and, where necessary, short explanations. the definitions are also carefully annotated. the theory of proportion, book v., is given in an algebraical form.
this book has always appeared to us an exquisitely subtle example of greek mathematical logic, but the subject can be made infinitely simpler and shorter by a little algebra, and naturally the more difficult method has yielded place to the less.
it is not studied in schools, it is not asked for even in the cambridge tripos; a few years ago, it still survived in one of the college examinations at st. john's, but whether the reforming spirit which is dominant there has left it, we do not know.
the book contains a very large body of riders and independent geometrical problems.
the simpler of these are given in immediate connexion with the propositions to which they naturally attach; the more difficult are given in collections at the end of each book.
some of these are solved in the book, and these include many well-known theorems, properties of orthocentre, of nine-point circle, c. in every way this edition of euclid is deserving of commendation.
we would also express a hope that everyone who uses this book will afterwards read the same author's 'sequel to euclid,' where he will find an excellent account of more modern geometry."
a key to the exercises in the elements of euclid.
typographical errors corrected in project gutenberg edition
p. ??. "def. viii.-when a right line intersects ..." in original, amended to "def. vii" in sequence.
p. ??. 12 "bisects the parallellogram" in original, amended to match every other occurrence as "parallelogram".
p. ??. "ach is half the rectangle ac.ah (i. cor. 1)" in original. the reference is to prop. i. of the current book and misnumbered, it should be (i. cor. 2).
p. ??. "the parallelogram cm is equal to de i. xliii., cor. 3 ;" in original, amended to "cor. 2" following ms. correction: there is no cor. 3.
p. ??. "on cb describe the square cbef i. xlvi. ." in original, clearly meant to read i. xlvi. .
p. ??. "the remainiug parts of the line" in original, obvious error amended to "remaining".
p. ??. "that which is nearest to the line throuyh the centre" in original, obvious error amended to "through".
p. ??. "then this line i., cor. 1 " in original. the reference is to prop. i. of the current book, so it should be i., cor. 1 .
p. ??. "oa is equal to oc i., def. xxii. " in original. the reference should be i., def. xxxii. .
p. ??. "the four points a, c, d, b are concylic" in original, evidently intended is "concyclic".
p. ??. "through tho point e" in original, obvious error amended to "the".
p. ??., p. ??. "the points a, b, c, d are concylic" in original, as p. ??.
p. ??. "(ex. 2.) ...or touchlng a given file and a given circle." in original, obvious error amended to "touching".
p. ??. "21. what is the locus of the middle points ..." in original, amended to "31." in sequence.
p. ??. in (21) "the if then line de intersect the chords ..." in original, garbled phrase amended to "then if the".
p. ??. in (44) "these circle sintersect" in original, misplaced space amended to "these circles intersect".
p. ??. "4. the point of bisection (1) of the line (op)" in original, from the diagram and following discussion this should be (i).
p. ??. prop. ix. "about a given circle (abcd) to describe a circle." in original, clearly this is nonsense and must mean "about a given square".
p. ??. "then the traingles abo, cbo" in original, obvious error amended to "triangles".
p. ??. in (52) "and also en equilateral circumscribed polygon" in original, wrong letter amended to "an".
p. ??. heading "prop. xxv.-problem." in original, the preamble to this book says that every proposition in it is a theorem and this one seems to be no exception, so amended.
p. ??. reference " i. " is to proposition i. of current book, amended to " i. " (4 times).
p. ?? sqq. reference " ii. " is to proposition ii. of current book, amended to " ii. " (4 times).
p. ??. prop v. header "subtended bg the homologous sides" in original, obvious error amended to "by".
p. ??. reference " xvi. " corrected to " xvi. ".
p. ??. "from the construction is is evident ..." in original, obvious error amended to "it is".
p. ??. "20. find a poiat o" in original, obvious error amended to "point".
p. ??. "the lines gh, gk each perpendiclar to ef" in original, obvious error amended to "perpendicular".
p. ??. "o" when associated with a lower case letter was wrongly printed as o, which is not defined. these have been corrected (3 times).
p. ??. reference " vi., cor. 6 " corrected to " vi., cor. 6 ".
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