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Download A Source Book in Mathematics, 1200-1800 by D. J. Struik PDF

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By D. J. Struik

These chosen mathematical writings conceal the years while the rules have been laid for the speculation of numbers, analytic geometry, and the calculus.

Originally released in 1986.

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Extra info for A Source Book in Mathematics, 1200-1800

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1 The properties of binomial coefficients were discussed by the Persian mathematician Jamshid Al-Kashi in his Key to arithmetic of c. 2 Both in China and in Persia the knowledge of these properties may be much older. This knowledge was shared by some of the Renaissance mathematicians, and we see Pascal's triangle on the title page of Peter Apian's German arithmetic of 1527. 3 Pascal wrote his treatise probably by the end of 1654. It can be found in the Oeuvres, ed. L. Brunschvicg and P. , and in other editions of Pascal's work.

Our notation is Euler's, except that we have written alb where Euler writes %. b 2 "Theorematum quorundam ad numeros primos spectantium demonstrate," Com- mentarii Academiae Scientiarum Petropolitanae 8 (1736, publ. 1741), 141-146, Opera omnia, set1. I, vol. 2, 35-37. The proof runs as follows. First it is proved by means of the binomial expansion of (1 + l ) " " 1 that p is divisible by p if p is an odd prime. Then by a similar expansion of (1 + a) it is shown that (1 + aY - (1 + a) - (a" - a) = 0 (mod p) if a is not a multiple of p.

2, 35-37. The proof runs as follows. First it is proved by means of the binomial expansion of (1 + l ) " " 1 that p is divisible by p if p is an odd prime. Then by a similar expansion of (1 + a) it is shown that (1 + aY - (1 + a) - (a" - a) = 0 (mod p) if a is not a multiple of p. Since 2" - 2 = 0, the proof follows by complete induction. 36 I I ARITHMETIC 9 EULER. 6, 7). Fermat's theorem that xn + yn = Zn cannot be solved for positive integers x, y, ζ, η, η > 2, attracted him and he gave proofs for the cases η = 3 and η = 4.

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