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<title>Mathematics Batch 07 - Historical & Educational - Programming Framework Analysis</title>
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<h1>Mathematics Batch 07 - Historical & Educational - Programming Framework Analysis</h1>
<p>This document presents historical and educational mathematics processes analyzed using the Programming Framework methodology. Each process is represented as a computational flowchart with standardized color coding: Red for triggers/inputs, Yellow for structures/objects, Green for processing/operations, Blue for intermediates/states, and Violet for products/outputs. Yellow nodes use black text for optimal readability, while all other colors use white text.</p>
<h2>1. Euclid's Geometry Process</h2>
<div class="figure">
<div class="mermaid">
graph TD
A1[Geometric Problem] --> B1[Euclidean Axioms]
C1[Postulates] --> D1[Common Notions]
E1[Geometric Construction] --> F1[Proof Method]
B1 --> G1[Point Line Plane]
D1 --> H1[Equality Axioms]
F1 --> I1[Logical Deduction]
G1 --> J1[Postulate One]
H1 --> K1[Transitive Property]
I1 --> L1[Theorem Statement]
J1 --> M1[Postulate Two]
K1 --> N1[Reflexive Property]
L1 --> O1[Hypothesis Analysis]
M1 --> P1[Postulate Three]
N1 --> O1
O1 --> Q1[Conclusion Deduction]
P1 --> R1[Postulate Four]
Q1 --> S1[Proof Construction]
R1 --> T1[Postulate Five]
S1 --> U1[Geometric Proof]
T1 --> V1[Parallel Postulate]
U1 --> W1[Euclidean Result]
V1 --> X1[Euclidean Geometry]
W1 --> Y1[Geometric Analysis]
X1 --> Z1[Euclidean Geometry Output]
Y1 --> AA1[Euclidean Geometry Analysis]
Z1 --> BB1[Euclidean Geometry Final Result]
AA1 --> CC1[Euclidean Geometry Analysis Complete]
style A1 fill:#ff6b6b,color:#fff
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style E1 fill:#ff6b6b,color:#fff
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style D1 fill:#ffd43b,color:#000
style F1 fill:#ffd43b,color:#000
style G1 fill:#ffd43b,color:#000
style H1 fill:#ffd43b,color:#000
style I1 fill:#ffd43b,color:#000
style J1 fill:#ffd43b,color:#000
style K1 fill:#ffd43b,color:#000
style L1 fill:#ffd43b,color:#000
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style N1 fill:#ffd43b,color:#000
style O1 fill:#ffd43b,color:#000
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style Q1 fill:#ffd43b,color:#000
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style Y1 fill:#ffd43b,color:#000
style Z1 fill:#ffd43b,color:#000
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style X1 fill:#51cf66,color:#fff
style Y1 fill:#51cf66,color:#fff
style Z1 fill:#51cf66,color:#fff
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</div>
<div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs
</div>
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Euclidean Methods
</div>
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Geometric Operations
</div>
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates
</div>
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products
</div>
</div>
<div class="figure-caption">
<strong>Figure 1.</strong> Euclid's Geometry Process. This historical mathematics process visualization demonstrates the axiomatic method of Euclidean geometry. The flowchart shows geometric problem inputs and postulates, Euclidean methods and axioms, geometric operations and proofs, intermediate results, and final Euclidean geometry outputs.
</div>
</div>
</div>
<h2>2. Aristotle's Syllogism Process</h2>
<div class="figure">
<div class="mermaid">
graph TD
A2[Logical Premises] --> B2[Syllogistic Form]
C2[Major Premise] --> D2[Minor Premise]
E2[Conclusion] --> F2[Logical Validity]
B2 --> G2[Major Term]
D2 --> H2[Middle Term]
F2 --> I2[Minor Term]
G2 --> J2[Universal Affirmative]
H2 --> K2[Universal Negative]
I2 --> L2[Particular Affirmative]
J2 --> M2[Particular Negative]
K2 --> N2[Barbara Syllogism]
L2 --> O2[Celarent Syllogism]
M2 --> P2[Darii Syllogism]
N2 --> O2
O2 --> Q2[Ferio Syllogism]
P2 --> R2[Syllogistic Figure]
Q2 --> S2[Logical Form Analysis]
R2 --> T2[Validity Check]
S2 --> U2[Syllogistic Result]
T2 --> V2[Logical Conclusion]
U2 --> W2[Aristotelian Logic]
V2 --> X2[Syllogistic Analysis]
W2 --> Y2[Logical Analysis]
X2 --> Z2[Aristotelian Logic Output]
Y2 --> AA2[Aristotelian Logic Analysis]
Z2 --> BB2[Aristotelian Logic Final Result]
AA2 --> CC2[Aristotelian Logic Analysis Complete]
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style CC2 fill:#b197fc,color:#fff
</div>
<div style="margin-top: 1rem; display: flex; flex-wrap: wrap; gap: 0.5rem; justify-content: center;">
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ff6b6b;"></span>Triggers & Inputs
</div>
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#ffd43b;"></span>Syllogistic Methods
</div>
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#51cf66;"></span>Logical Operations
</div>
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#74c0fc;"></span>Intermediates
</div>
<div style="display:inline-flex; align-items:center; gap:.5rem; padding:.25rem .5rem; border-radius: 999px; border: 1px solid rgba(0,0,0,.08); background:#fff;">
<span style="width: 12px; height: 12px; border-radius: 2px; border:1px solid rgba(0,0,0,.15); background:#b197fc;"></span>Products
</div>
</div>
<div class="figure-caption">
<strong>Figure 2.</strong> Aristotle's Syllogism Process. This historical logic process visualization demonstrates syllogistic reasoning and logical deduction. The flowchart shows logical premise inputs and syllogistic forms, syllogistic methods and figures, logical operations and validity checks, intermediate results, and final Aristotelian logic outputs.
</div>
</div>
</div>
<div class="navigation">
<h3>Navigation</h3>
<div class="nav-links">
<a href="mathematics_index.html" class="nav-link">← Back to Mathematics Index</a>
<a href="mathematics_batch_06.html" class="nav-link">← Previous: Discrete Mathematics</a>
<a href="index.html" class="nav-link">Programming Framework Home</a>
</div>
</div>
<div class="footer">
<p><strong>Generated using the Programming Framework methodology</strong></p>
<p>Each flowchart preserves maximum detail through optimized Mermaid configuration</p>
<div class="contact-info">
<p><strong>Gary Welz</strong></p>
<p>Retired Faculty Member</p>
<p>John Jay College, CUNY (Department of Mathematics and Computer Science)</p>
<p>Borough of Manhattan Community College, CUNY</p>
<p>CUNY Graduate Center (New Media Lab)</p>
<p>Email: gwelz@jjay.cuny.edu</p>
</div>
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