Engineering Calculus Notes 30

Engineering Calculus Notes 30 - manuscripts the Caho Pei...

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18 CHAPTER 1. COORDINATES AND VECTORS that c 2 = a 2 + b 2 . In this problem, we outline two quite diferent prooFs oF this Fact. First Proof: Consider the pair oF ±gures in ²igure 1.13 . b a c b a c b a c b a c a b a b ²igure 1.13: Pythagoras’ Theorem by Dissection (a) Show that the white quadrilateral on the leFt is a square (that is, show that the angles at the corners are right angles). (b) Explain how the two ±gures prove Pythagoras’ theorem. A variant oF ²igure 1.13 was used by the twelFth-century Indian writer Bh¯askara (b. 1114) to prove Pythagoras’ Theorem. His prooF consisted oF a ±gure related to ²igure 1.13 (without the shading) together with the single word “Behold!”. According to Eves [ 13 , p. 158] and Maor [ 35 , p. 63], reasoning based on ²igure 1.13 appears in one oF the oldest Chinese mathematical
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Unformatted text preview: manuscripts, the Caho Pei Suang Chin , thought to date From the Han dynasty in the third century B.C. The Pythagorean Theorem appears as Proposition 47, Book I oF Euclid’s Elements with a diferent prooF (see below). In his translation oF the Elements , Heath has an extensive commentary on this theorem and its various prooFs [ 27 , vol. I, pp. 350-368]. In particular, he (as well as Eves) notes that the prooF above has been suggested as possibly the kind oF prooF that Pythagoras himselF might have produced. Eves concurs with this judgement, but Heath does not. Second Proof: The prooF above represents one tradition in prooFs oF the Pythagorean Theorem, which Maor [ 35 ] calls “dissection...
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