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@@ -1046,9 +1046,9 @@ <h3 itemprop="name" style="margin:7px">Trigonometry</h3>
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},
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"rad(1.33333333)": {
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"sin": 0.9656,
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"cos": 0.26,
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"tan": 3.7,
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"sin": 0.9659,
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"cos": 0.2588,
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"tan": 3.732,
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"deg": 75.0
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},
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@@ -2663,19 +2663,168 @@ <h4>Archimedes and the Illusion of Limits</h4>
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But this identity is not derivable from Euclid.
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It is an axiom of the analytic system.
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<br><br>
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It holds for the Euclidean angles 90°, 60°, 45°, and 30° because those triangles are special.
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There is no geometric guarantee that it holds for arbitrary angles.</p>
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It holds for the Euclidean angles 90°, 60°, 45°, and 30° because those triangles are special.
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<br>
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<p>Thus, when Archimedes computed</p>
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There is no geometric guarantee that it holds for arbitrary angles.
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<br>
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Thus, when Archimedes computed sin(3.75°) he was not using Euclid.
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<br>
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He was using a numerical trigonometric ladder built on analytic assumptions that Euclid never supplied.
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<br><br>
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If the angle‑bisection formulas are slightly inaccurate for non‑special angles, then by the time Archimedes reached the 96‑gon, that error had compounded — even though the construction of the angles themselves was exact.</p>
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<br><br><br>
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<p>A deeper geometric issue is that the “circumscribed polygon” is not guaranteed to be an upper bound.
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<br><br>
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Even if we assume Archimedes computed sin(3.75°) exactly, a more fundamental geometric problem appears when we examine how polygonal slices compare to circular slices.
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<br><br>
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Consider a circular sector of angle 2x and radius r.</p>
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<br>
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<figure itemprop="image" class="imgbox" itemscope itemtype="http://schema.org/ImageObject">
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<img class="center-fit" src="polygonApproximation.png" alt="A slice of a circle with angle=2x and the corresponding isosceles triangle split in half">
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</figure>
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<br>
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<p>Now look at the isosceles triangle formed by joining the endpoints of the arc to the center.
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<br><br>
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Focus on half of this configuration: a right triangle with</p>
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<ul style="margin:6px">
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<li>sin(7.5°)</li>
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<li>sin(3.75°)</li>
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<li>one leg = r</li>
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<li>the other leg = half the polygon side</li>
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<li>hypotenuse = half the polygon diagonal</li>
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</ul>
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<p>he was not using Euclid.
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<br>
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He was using a numerical trigonometric ladder built on analytic assumptions that Euclid never supplied.
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<p>This right triangle always contains the corresponding half‑sector of the circle, so its area is larger.
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<br>
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But the key detail is the ratio of these areas.</p>
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<br>
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<math xmlns="http://www.w3.org/1998/Math/MathML">
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<mrow>
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<mi>Area of the half‑sector</mi>
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<mo>=</mo>
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<mfrac>
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<mrow>
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<mi>x</mi>
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<mo>×</mo>
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<msup>
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<mi>r</mi>
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<mn>2</mn>
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</msup>
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</mrow>
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<mn>2</mn>
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</mfrac>
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</mrow>
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</math>
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<br><br>
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<math xmlns="http://www.w3.org/1998/Math/MathML">
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<mrow>
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<mi>Area of the right triangle</mi>
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<mo>=</mo>
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<mfrac>
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<mrow>
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<mi>sin</mi>
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<mo>&#x2061;</mo>
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<mo>(</mo><mi>x</mi><mo>)</mo>
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</mrow>
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<mrow>
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<mn>2</mn>
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<mo>×</mo>
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<mi>cos</mi>
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<mo>&#x2061;</mo>
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<mo>(</mo><mi>x</mi><mo>)</mo>
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</mrow>
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</mfrac>
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<mo>×</mo>
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<msup>
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<mi>r</mi>
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<mn>2</mn>
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</msup>
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</mrow>
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</math>
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<br><br>
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<strong>The angle‑bisection formulas are slightly inaccurate for non‑special angles, and by the time Archimedes reached the 96‑gon, that error had compounded — even though the construction of the angles themselves was exact.</strong></p>
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<p>Since</p>
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<br>
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<math xmlns="http://www.w3.org/1998/Math/MathML">
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<mrow>
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<mi></mi>
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<mi>cos</mi>
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<mo>&#x2061;</mo>
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<mo>(</mo><mi>x</mi><mo>)</mo>
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<mo>&lt;</mo>
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<mn>1</mn>
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</mrow>
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</math>
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<br><br>
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<p>the inequality</p>
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<br>
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<math xmlns="http://www.w3.org/1998/Math/MathML">
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<mrow>
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<mfrac>
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<mrow>
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<mi>sin</mi>
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<mo>&#x2061;</mo>
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<mo>(</mo><mi>x</mi><mo>)</mo>
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</mrow>
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<mrow>
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<mn>2</mn>
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<mo>×</mo>
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<mi>cos</mi>
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<mo>&#x2061;</mo>
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<mo>(</mo><mi>x</mi><mo>)</mo>
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</mrow>
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</mfrac>
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<mo>&gt;</mo>
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<mi>x</mi>
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</mrow>
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</math>
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<br><br>
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<p>can hold even when</p>
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<br>
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<math xmlns="http://www.w3.org/1998/Math/MathML">
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<mrow>
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<mi>x</mi>
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<mo>&gt;</mo>
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<mi>sin</mi>
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<mo>&#x2061;</mo>
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<mo>(</mo><mi>x</mi><mo>)</mo>
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</mrow>
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</math>
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<br><br>
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<p>This means that a polygon can enclose more area than the circle even when its perimeter is smaller than the circle’s circumference.
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<br><br>
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As the number of sides increases, the polygon’s vertices flatten toward 180°.
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<br>
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The polygon begins to enclose area in a fundamentally different way than the circle.
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<br>
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The usual assumption — that a tangent circumscribed polygon must have a larger perimeter — is not guaranteed.
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<br><br>
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A circle has constant curvature. A straight line has zero curvature. Treating the two as interchangeable is a category error disguised as approximation.
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<br><br>
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Imagine two rigid plates with a straight lid wedged between them.
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<br>
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If we bend the lid into a curve, it can slip lower between the plates because the distance between its endpoints decreases — even if the curved lid is longer.
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<br>
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The polygon still encloses more area, but the arc sits lower.
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<br><br>
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These are the overlooked geometric facts.</p>
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<br><br>
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<p>Why this matters
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<br>
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<br>
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The classical argument assumes:</p>
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<ul style="margin:6px">
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<li>the inscribed polygon is always an under‑estimate</li>
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<li>the circumscribed polygon is always an over‑estimate</li>
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<li>and the circle lies cleanly between them in both area and perimeter</li>
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</ul>
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<br>
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<br>
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<p>But the geometry shows:</p>
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<ul style="margin:6px">
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<li>a tangent polygon can have smaller perimeter than the circle, yet still enclose larger area</li>
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<li>so “circumscribed” and “upper bound” are not equivalent concepts</li>
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<li>The geometric ordering required by the polygon‑approximation method is not structurally guaranteed.</li>
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</ul>
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<br>
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<p>Thus the isoperimetric limit argument that Archimedes’ method relies on is not as straightforward as it is usually presented.</p>
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</section>
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<br><br><br>
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<section itemprop="disambiguatingDescription" id="symbol">

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