Contributions to the founding of the theory of transfinite by Georg Cantor

By Georg Cantor

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Thirdly, as will be seen presently, there is much, in Diophantus and in ViZtte's Zetetica, which in our view pertains to algebraic geometry. Moreover, modern developments have led to a better understanding of the analogies (already dimly perceived on some occasions by Leibniz and Euler) between functionfields and number-fields, showing that there is sometimes little difference between solving a problem in rational numbers and solving it in a field of rational functions. All these points of view acquire significance when one tries to classify the problems and methods which occur in Diophantus.

Supra, §IV), determine x so that Y =p qx is a multiple of m. Writing + we see that M is a multiple of m, since q is prime to m. Now the relation X' = N Y ~ mM shows that X' is a multiple of m', so that X is a multiple of m. Writing X = mp', Y = mq', we get the triple (p', q'; m'). T o make m' small, observe that we can choose x, within its congruence class modulo m, such that x < < x /mi; if then f l x were <0, we would have 2 f i < Imj; if therefore we assume Iml < 2 f i , this cannot happen, and we get + m + 0

Claude Gaspar BACHET, sieur de Mkziriac, was a country gentleman of independent means, with classical tastes, and no mathematician. Somehow he developed an interest for mathematical recreations and puzzles of the kind found in many epigrams of the Greek Anthology as well as in medieval and Renaissance mathematical texts, or nowadays in the puzzle columns of our newspapers and magazines. In 1612 Protohistory 33 he published in Lyon a collection of such puzzles under the title Probl2mes plaisants et dblectables qui sefont par les nombres.

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