Janos Bolyai appendix: The theory of space by Janos Bolyai

By Janos Bolyai

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For a recent comprehensive treatment of π, see L. Berggren, J. Borwein, and P. Borwein, π: A Source Book, Springer, 1997. 6 7 2. “. . ). The Babylonians obtained the first value by stating that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle was “equal” to 57/60 + 36/602 . 1). The area of the octagon, 63 units, was rounded up to 82 , and the approximate value of π turned out to be 82 /(9/2)2 28 /34 . A well-known recorded approximation of π was made by Archimedes (c.

Cn ∈ Z, has a rational root x a/b, a, b ∈ Z. As usual, we may assume that a and b are relatively prime. Substituting, we have c0 + c1 (a/b) + · · · + cn (a/b)n 0. Multiplying through by bn−1 , we obtain c0 bn−1 + c1 abn−2 + · · · + cn−1 an−1 + cn an /b 0. This says that cn an /b must be an integer, or equivalently, b divides cn an . Since a and b are relatively prime, we conclude that b divides ¬ indicates indirect argument; that is, we assume that the statement is false and get (eventually) a contradiction (indicated by another ¬).

The origins of this equation go back to ancient times. This is easily understood if we consider the special case P(x) x3 + c, c ∈ N, and realize that integer solutions of this equation are nothing but the possible ways to write c as the difference of a square and a cube: y2 − x3 c. This special case is called the Bachet equation. The Bachet curve corresponding to the Bachet equation is nonsingular unless c 0, in which case it reduces to a cusp. In 1917, Thue proved that for any c, there are only finitely many integral solutions to this equation.

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