Update index.html

Author: Gmac4247

index.html

@@ -524,6 +524,7 @@ <h2 style="font-size:100%">This system doesn't require advanced calculus or imag
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 <h3 style="margin:7px">Key Points</h3>
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@@ -570,6 +571,7 @@ <h4 style="margin:7px">Algebraic Manipulation</h4>
 <p style="margin:12px"><strong>Simplifying equations to ensure consistency and precision.
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 <p style="margin:12px">Which is equivalent to 1 = 1 .
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-This is an identity; not tautology.
+Note: This is an identity; not tautology.
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 The area of both the square and the sum of the quadrants equals 16 right triangles with legs of a quarter, and a half of the square's sides, and its hypotenuse equal to the radius of the circle. 
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@@ -3099,9 +3099,7 @@ <h3 style="margin:7px">The circumference of a circle can be derived from its are
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 The length of the circumference approaches 6.4 × radius as its thickness approaces 0.
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-It can be verified physically by wrapping a piece of paper around a round object, or placing the paper in a cylindrical tube, making sure that its ends touch without overlapping, and measuring both the diameter and the length of the straightened piece of paper.</p>
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@@ -3262,7 +3260,7 @@ <h4 style="margin:12px">∫ Calculus: Summary, Not Source</h4>
 <strong>But this is not a magical formula—it’s a symbolic summary of prior assumptions.</strong>
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-There are at least a dozen different calculus methods out there, but each and every one of those are solved through basic operations. Each notation should correspond to a real, logical property of the circle. Yet upon inspection, inconsistencies emerge. The formula doesn’t derive the circumference from first principles; it assumes it. 
+There are at least a dozen different calculus methods in use, but each and every one of those are solved through basic operations. Each notation should correspond to a real, logical property of the circle. Yet upon inspection, inconsistencies emerge. The formula doesn’t derive the circumference from first principles; it assumes it. 
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 The classical polygon-based approach to approximate a circle’s circumference relies on inscribed and circumscribed polygons, calculated using trigonometric functions aligned to π. But this alignment is problematic if π itself is the quantity under investigation.