By D. J. Struik
These chosen mathematical writings conceal the years while the rules have been laid for the idea of numbers, analytic geometry, and the calculus.
Originally released in 1986.
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Additional info for A Source Book in Mathematics, 1200-1800
2, 38-58, he gave a proof for the case η = 4, adding a number of related theorems. A proof for η = 4 had already been given by Fermat's corre spondent Bernard Frenicle (c. 1605-1675) in his Traite des triangles rectangles en nombres (Paris, 1676). Frenicle here used the so-called method of infinite descent, which Euler also employed. This method is as follows: suppose that a solution of the problem in question is possible in positive integers: then we show how to derive from it a solution in smaller positive integers, and so on.
Corollary 4. Suppose therefore ρ = α2 and 2q = 4¾/2, then j32 — q2 = Xi — 4«/4. Then it could not at all happen, that Xi — 4y4 were a square. Nor could 4xl — j/4 be a square; for then 16¾4 — 4y* would be a square, which reduces it to the former case, because 16¾4 is a biquadratic number. Corollary 5. From this it follows that also ab(a2 + b2) can never be a square. For the factors a, b, a2 + b2, all relative primes, would have to be squares, which is impossible. Corollary 6. In a similar way, there cannot exist relatively prime numbers a and b such as to make 2ab(a2 — b2) a square.
8 Only here does Napier begin to introduce angles into the construction of his tables. Napier proves Arts. 55-57 by geometric principles and the preceding theorems concerning logarithms. He then often speaks of the logarithms of the arcs, meaning logarithms of the corresponding sines. 20 I ARITHMETIC I 57. The sum of the logarithms of half radius and any given arc is equal to the sum of the logarithms of half the arc and the complement of the half arc. Whence the logarithm of the half arc may be found if the logarithms of the other three be given...