Heller M, Woodin W.H. (eds.) Infinity. New research frontiers (CUP, 2011)(ISBN 1107003873)(O)(327s)_MAml_

three famous problems of antiquity 57Dante was not off the mark, because a proof of the impossibility of a purely geometricsolution to this problem was eventually found by Lindemann in 1882. This resultrepresents a triumph of modern mathematics, in view of its profound historical meaningand philosophical implications.These three problems of antiquity have been the hunting grounds of literally thousandsand thousands of amateur mathematicians, all in search of satisfying their egos.The reader will find an edifying account of the subject, up to the year 1872, in thebook A Budget of Paradoxes by Augustus De Morgan, astronomer and logician DeMorgan (1872). Notwithstanding the complete solution, in the negative, of all threeproblems, the flood of false solutions continues unabated. The truth has not yet reachedour politicians. In the book Mathematical Carnival by Martin Gardner, a selection ofhis Mathematical Games columns in Scientific American, we find the following:. . . On June 3, 1960, the Honorable Daniel K. Inouye, then a representative from Hawaii,later a senator and member of the Watergate investigation committee, read into theCongressional Record (Appendix, pages A4733-A4734) of the 86th Congress a longtribute to Maurice Kidjel, a Honolulu portrait artist who has not only trisected the anglebut also squared the circle and duplicated the cube. Kidjel and Kenneth W.K. Young havewritten a book about it called ‘The Two Hours that Shook the Mathematical World’, anda booklet, ‘Challenging and Solving the Three Impossibles’. Through a company calledthe Kidjel Ratio they sell this literature along with the Kidjel Ratio calipers with whichone can apply the system. In 1959 the two men lectured on their work in a number ofU.S. cities, and a San Francisco radio station, KPIX, produced a documentary about themcalled ‘The Riddle of the Ages’. According to Inouye, ‘The Kidjel solutions are beingtaught in hundreds of schools and colleges throughout Hawaii, the United States, andCanada.Indeed, a very instructive story.The purist view that permeates Euclid’s geometry can also be found in the arithmeticof the Elements. The following example deals with prime numbers, a favorite subject ofnumber theorists. In the ninth book of his Elements (IX.20) Euclid notes that “Given anarbitrary number of primes, there exists another prime different from them.” He givesa startlingly simple proof of this statement: 2 Consider the product of the given primesand add 1. Then we get a number, larger than 1, that divided by anyone of these primesyields a remainder 1, so it is not divisible by any of them. However, this number canbe factored as a product of prime numbers necessarily distinct from the given primes,concluding the proof.Nowadays, we refer to this as Euclid’s proof that the sequence of primes is infinite,but we must note that Euclid studiously avoids using the word “infinite” in the statementof the proposition.With Archimedes, things are quite different. Archimedes often used limiting processes.His method for calculating 2π, the length of the circumference of a circle ofradius 1, is exemplary. He started from the remark that this number is always largerthan the perimeter of a regular polygon inscribed in the circle and smaller than the2 See Busard (2001, p. 227). Euclid’s writing is always rather formal and lengthy, so we give his proof in modernterms.