Update index.html

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@@ -2559,11 +2559,13 @@ <h3 itemprop="name" style="margin:7px">Calculate the Circumference of a Circle</
 <section itemprop="disambiguatingDescription" id="Archimedes">
 <h4>Archimedes and the Illusion of Limits</h4>
 <br><br>
-<p>Archimedes’ polygon method for estimating the pi is often celebrated as a triumph of geometric reasoning.  
+<p>Archimedes’ method marked a turning point: instead of calculating the properties of the circle directly, he introduced an analytic process that relied on polygonal limits.
+<br>
+His polygon method for estimating the pi is often celebrated as a triumph of geometric reasoning.  
 <br>
 But the pi, as obtained by that method, is not a Euclidean constant — it is an analytic approximation derived from limits.  
 <br>
-And the method itself quietly imports assumptions that Euclid never provided.
+And the method itself quietly imports assumptions that elementary geometry never provided.
 <br><br>
 Archimedes approximated the circumference of a circle using inscribed and circumscribed polygons.  
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@@ -2663,8 +2665,7 @@ <h4>Archimedes and the Illusion of Limits</h4>
 <br><br>
 for all x.
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-But this identity is not derivable from Euclid.  
-It is an axiom of the analytic system.
+But this identity is not derivable from Euclid. It is an axiom of the analytic system.
 <br><br>
 It holds for the Euclidean angles 90°, 60°, 45°, and 30° because those triangles are special.
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@@ -2674,11 +2675,11 @@ <h4>Archimedes and the Illusion of Limits</h4>
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 He was using a numerical trigonometric ladder built on analytic assumptions that Euclid never supplied.
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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>
-<br><br><br>
-<p>A deeper geometric issue is that the “circumscribed polygon” is not guaranteed to be an upper bound.
+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.
+<br><br>
+But even if we momentarily accept these analytic assumptions, a deeper geometric inconsistency emerges when we compare circumscribed polygons to circles.
 <br><br>
-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.
+A “circumscribed polygon” is not guaranteed to be an upper bound in every sense.
 <br><br>
 Consider a circular sector of angle 2x and radius r.</p>
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@@ -2827,7 +2828,9 @@ <h4>Archimedes and the Illusion of Limits</h4>
       <li>The geometric ordering required by the polygon‑approximation method is not structurally guaranteed.</li>
       </ul>
 <br>
-<p>Thus the isoperimetric limit argument that Archimedes’ method relies on is not as straightforward as it is usually presented.</p>
+<p>Thus the isoperimetric limit argument that Archimedes’ method relies on is not as straightforward as it is usually presented.
+<br><br>
+These structural issues in the polygon‑limit method set the stage for a second misconception: the symbolic fusion of an approximation with the geometric ratio it was meant to represent.</p>
 </section>
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 <section itemprop="disambiguatingDescription" id="symbol">
@@ -2974,7 +2977,9 @@ <h4>The true Ratio: 3.2</h4>
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 <strong>Since the true ratio is exactly 3.2, and that is a rational number, then we can—and should—write it as it is. Let the pi remain in the history books. Geometry deserves better.
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-That makes the arc value of 360° = 6.4radian, and trigonometric functions that rely on arc value have to be aligned to 3.2 respectively.</strong></p>
+That makes the arc value of 360° = 6.4radian, and trigonometric functions that rely on arc value have to be aligned to 3.2 respectively.</strong>
+<br><br>
+Once the historical assumptions are stripped away, the geometry itself becomes simple—and the technical construction follows naturally.</p>
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 </section>
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