Download Beautiful Geometry by Eli Maor, Eugen Jost PDF

By Eli Maor, Eugen Jost

If you've ever concept that arithmetic and paintings don't combine, this gorgeous visible background of geometry will swap your brain. As a lot a piece of artwork as a publication approximately arithmetic, appealing Geometry provides greater than sixty beautiful colour plates illustrating quite a lot of geometric styles and theorems, followed via short bills of the interesting heritage and other people in the back of each one. With paintings via Swiss artist Eugen Jost and textual content via acclaimed math historian Eli Maor, this particular get together of geometry covers various topics, from straightedge-and-compass structures to fascinating configurations related to infinity. the result's a pleasant and informative illustrated travel in the course of the 2,500-year-old heritage of 1 of crucial and gorgeous branches of arithmetic.

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That is, the area of the square built on the hypotenuse (“the side subtending the right angle”) is equal to the combined area of the squares built on the other two sides. Pythagoras of Samos (ca. 580–ca. 500 BCE) may have been the first to prove the theorem that made his name immortal, but he was not the first to discover it: the Babylonians, and possibly the Chinese, knew it at least twelve hundred years before him, as is clear from several clay tablets discovered in Mesopotamia. Furthermore, if indeed he had a proof, it is lost to us.

The answer is yes, but in order to do so we must first prove a rather surprising result: if we multiply together the expressions A = (a + b)/2 and H = 2ab/(a + b), we get AH = [(a + b)/2] ⋅ [2ab/ (a + b)] = ab = G2, or G = AH : the geometric mean of a and b is also the geometric mean of A and H. This result is the key to the construction of H. For, if we rewrite the equation G2 = AH as a proportion, A/G = G/H, we see that A and H play the same role vis-à-vis G as did a and b in our construction of the geometric mean.

Note that the phrase the square on the tangent actually means the area of a square whose side equals the length of the tangent line. 33 Plate 11. 3 The proof of theorem 35 is quite simple. 3, P is a point inside the circle, and AB and CD are two chords passing through P. We have ∠APC = ∠BPD and ∠ACD = ∠DBA, the latter equality because an- gles ∠ACD and ∠DBA subtend the same arc, AD , on the circumference (Euclid III, 21). Thus, triangles PAC and PDB are similar, having two pairs (and, therefore, three) of equal angles.

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