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Heron of Alexandria (10--75) --- Historical Sketch

Historians generally refer to this impressive historical figure as Heron of Alexandria, for from his writings it seems clear that he taught mathematics and related topics at the Museum in Alexandria. It seems to be agreed that he lived from about 10 A.D. to about 75 A.D. He is often referred to as Hero rather than Heron, just as Platon is also known as Plato.

His lecture notes and textbooks that have survived reveal that he taught mathematics and physics in addition to mechanics, measurement, astronomy, surveying, plane and spatial geometry, number theory, approximation theory, and numerous other topics.

Heron is best known for Heron's formula (not a formula for the birds), also called Hero's formula, for computing the area $A$ of a triangle: If $a$, $b$, and $c$ are the lengths of the sides of a triangle, then the area of the triangle is MATHwhere $s$ is the semiperimeter of the triangle, that is,MATHYou may have used this formula in high school geometry. Heron proved it in Book I of his treatise Metrica, where he also developed formulas for areas of regular polygons of four sides, five sides, etc., through twelve sides, surfaces of cones, cylinders, prisms, pyramids, spheres, and so forth.

Heron approximated $\sqrt{p^{2}+q}$ by $p+\dfrac{q}{2p}$. Use this formula to find rational number approximations to each of the following: $\sqrt{11}$, $\sqrt{48}$, $\sqrt{77}$, $\sqrt{150}$. Choose $p$ so that $p^{2}$ is as close to the radicand as possible. Square your answers as a measure of how good your approximations are.


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Heron's formula

Historical sketches


Copyright © 2006 Darel Hardy, Fred Richman, Carol Walker, Robert Wisner. All rights reserved. Except upon the express prior permission in writing, from the authors, no part of this work may be reproduced, transcribed, stored electronically, or transmitted in any form by any method.

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