Diophanti redivivi pars prior [- posterior], in qua, non casu, ut putatum est, sed certissimâ methodo, & analysi subtiliore, innumera enodantur problemata, quae triangulum rectangulum spectant

BILLY, Jacques de

Published by J. Thiolly, Lyon, 1670

About this Item

DIOPHANTINE PROBLEMS SOLVED USING FERMAT'S METHODS. First edition, rare, of this early treatise on number theory, explicating and extending the indeterminate problems in the Arithmetica of Diophantus (fl. 3rd century AD). It represents an important testament to the early development of this branch of mathematics. Some of the methods used by Billy (1602-79) derived from Pierre de Fermat (1601-65), the inventor of modern number theory, with whom Billy corresponded starting in 1659. Billy's purpose in the present work was to develop some general methods of solution of such Diophantine problems (Fermat undoubtedly had such methods but did not publish them). "This mathematician, pronounced by M. Charles 'géomètre d'un grand merite', was highly esteemed by Fermat and Bachet de Meziriac. All his writings are rare. This work contains many of the discoveries on the theory of numbers made by Fermat, who was in frequent communication with Billy. It is a curious fact that Father Billy, without any mention of the name of Fermat, gives here, as his own, the resolution of some equations, which, in the Diophantus published with Fermat's annotations in the same year, he [Billy] acknowledges to have found in the letters of Fermat to himself" (Libri Catalogue, lot 1037). The first edition of the Greek text of the Arithmetica was first published by Claude-Gaspard Bachet de Méziriac, of whom Billy was a pupil, in 1621. One of the annotations in Fermat's own copy of this edition was a statement of his famous "last theorem". Fermat's son published a reprint of Bachet's edition in 1670, including his father's annotations. This edition also included a treatise by Billy, the Doctrinae Analyticae Inventum Novum, in which Billy gives an account of some of Fermat's methods of proof. The present work can be seen as an extension of the Doctrinae Analyticae, giving further applications of Fermat's methods. The problems treated in the first volume of the work typically ask whether there exist right-angled triangles whose sides are positive rational numbers (fractions) satisfying certain additional conditions. For example, if we require that each side of the triangle is the square of an integer (whole number) we have the question of whether there are positive whole numbers x,y, z such that x4 + y4 = z4. That no such numbers exist is one case of Fermat's last theorem (actually the first case to be proved). The second volume contains problems of a more arithmetic nature, such as (generalizations of) the problem of finding squares in arithmetic progression (for example, 1, 25, 49) - this is related to the first part, since if x2, y2, z2 is an arithmetic progression, then y is the hypotenuse of a right-angled triangle of which x is the difference and z the sum of the other two sides. Billy also considers the problem of determining when a given cubic or quartic with numerical coefficients can be equal to a square (Dickson, p. 569). RBH lists four copies since Honeyman, with only the Macclesfield copy being in comparable condition to ours. Almost all copies (including Macclesfield and Honeyman) lack some of the blank leaves A1 and T8 in part I and l7-8 in part II (almost all copies lack the blank A1) - all these blanks are present in our copy. OCLC lists Brown, Cincinnati, Harvard, and Huntington in US. Provenance: Old collectors stamp with the initials F.T. to front pastedown. "Jacques de Billy entered the Jesuit order and studied theology at the Colleges of the Order. He was ordained a Jesuit. The Jesuit Order had been created about sixty years before de Billy was born and, from the very beginning, education and scholarship became the principal work of the Order. By the time Billy entered the Order it contained around 15,000 men. "Billy taught mathematics and theology at Jesuit colleges all his life, in particular those colleges which were in the administrative region of Champagne, a region which covered the present-day northeastern French districts of Marne and p. Seller Inventory # 6209

Bibliographic Details

Title: Diophanti redivivi pars prior [- posterior],...

Publisher: J. Thiolly, Lyon

Publication Date: 1670

Edition: First edition.