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

Lyon: J. Thiolly, 1670.

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 parts of Ardennes, Meuse, Haute-Marne, Aube, Yonne, Seine-et-Marne, and Aisne. From 1629 to 1630 he taught mathematics at the Jesuit College at Pont à Mousson, during this time he was still studying theology.

“From 1631 to 1633 Billy taught mathematics at the Jesuit college at Rheims. He became a close friend of Bachet. After this Billy taught in Grenoble and then was rector of a number of Jesuit Colleges in Chalons, Langres and in Sens. From 1665 to 1668 he was professor of mathematics at the College of Dijon” (MacTutor). There, “one of his students was Jacques Ozanam, whom he taught privately because there was no chair of mathematics at the college, and in whom he instilled a profound love for calculus. Finally, a professorship having been created in mathematics, he taught his favourite subject from 1665 to 1668” (DSB).

Ozanam composed a 1200-page manuscript on Diophantine problems, entitled The Six Books of Diophantus’ Arithmetic, which was only rediscovered in the twentieth century. This manuscript establishes the correctness of the methods used by Billy, to be described below.

In 1670 the Jesuit Father Jacques de Billy published his work Diophanti redivivi where he treats a very large number of diophantine problems [algebraic equations the solutions of which are required to be integers (whole numbers) or rational numbers (fractions)]. In the first of the two volumes which constitute the work, the author treats many problems on rational right triangles.Willingness to give general methods for certain classes of problems is obvious.The originality of the methods used provides elements new to Diophantine analysis.The treatment of “double equations” of the second degree and the "triple equations" of the first degree, necessary for the resolution of the aforementioned problems, is an important part of this new contribution.

In several places in the work, father Billy refers to Jacques Ozanam, whom he considers very competent on diophantine problems (see pp. 8-9 & 260-261 of Volume 1; pp. 101-103 of Volume 2) .The first quote concerns the following problem: find a right triangle [whose sides] x, y, z are [rational numbers] such that x 4xy, y 4xy, and z 4xy are all squares;of this Billy writes (pp. 8-9 of Volume 1): ‘This problem, proposed to all mathematicians of France and sent to England, Holland, Germany and other parts of Europe, was not solved by anyone except Ozanam, a man of admirable skill and Professor of mathematics in Lyon.’

“Volume 2 ends with an ‘Epilogue, or Specious Analysis for the Solution of Problems,’ in which Billy reconsiders 6 problems of his work, treating them this time using algebraic notation (the ‘specious algebra’ developed by F. Viète) in order to give these problems a general solution.We know that the preview given in this epilogue, is the main goal that Ozanam set itself in his monumental autograph manuscript, the title of which is: ‘The six books of arithmetic of Diophantus, extended and reduced to Specious Algebra by Mr. Ozanam, Professor of Mathematics.’

“In the first part of de Billy's work we see the appearance a class of problems with two general methods of solution. One of these methods (the second) uses a process whose validity is demonstrated by Ozanam in a part of his manuscript which constitutes a small treatise entitled ‘Treatise of simple, double and triple equalities for the solution of number problems’” (Cassinet, pp. 15-17, our translation).

The problems treated by Billy are of the following form. Find a right-angled triangle with positive rational number sides x, y, z (z being the hypotenuse) such that

Ax + kxy,

By + lxy,

Cz + Dy + Ex + mxy

are all squares, A, B, C, D, E, k, l, m being given rational numbers – note that the area of the triangle is xy/2. For example, Problem I in Chapter 1 is: ‘Find a right-angled triangle in which, if double the area is subtracted from each of the sides, the results are all squares.’ Thus, x – xy, y – xy, and z – xy should all be squares (of rational numbers).

Billy’s method of solving this type of problem is ingenious (and a little complicated). He supposes that X, Y, Z are the sides of an auxiliary right-angled triangle, and puts

x = X/(AX + rY), y = Y/(AX + rY), z = Z/(AX + rY),

where r = – k/A; obviously x, y, z are the sides of a right-angled triangle. He then shows, by straightforward algebra, that if B = – n2A/k and l = n2A2/k (n being any positive integer), then Ax + kxy and By + lxy are squares.

To make Cz + Dy + Ex + mxy a square, Billy recalls (from Euclid) that the equation

X2 + Y2 = Z2

can be ‘solved’: X = s2t2, Y = 2st, Z = s2 + t2. He puts s = s0 + u, t = t0, where u is an unknown rational number. Then Cz + Dy + Ex + mxy becomes a polynomial P(u) of degree 4 – we would like P(u) to be a square. Billy looks for a quadratic polynomial Q(u) such that P(u) – Q(u)2 is a sum of two monomials (e.g., au4 + bu3). Then it is easy to choose a value of u such that P(u) – Q(u)2 = 0, i.e., such that P(u) is a square.

Cassinet illustrates the method with Problem 12 (pp. 27-29 in Vol. I), which requires that

x – xy/2, y/2 – xy/2, and z – xy/2

are all squares. Then, A = 1, k = –1/2, B = 1/2, l = –1/2, r = 1/2. Billy first tries s = u + 3, t = 2, which gives P(u) = u4 + 12u3 + 54u2 +124u +113. Taking Q(u) = u2 + 6u + 9 leads to u = –2, and finally (X,Y,Z) = (–3,4,5). This is not acceptable as X,Y,Z must all be positive.

However, taking s = u – 5 and t = 2 leads to P(u) = u4 – 20u3 + 150u2 – 48u +529. Noting that 529 = 232, Billy takes Q(u) = 23 – 242u/23 + 10393u2/12167, which leads to the solution

x = 13745944625329/44896517387689,

y – 62301145524720/44896517387689,

z = 63799558990129/44896517387689.

Later (after 1676), Billy composed a manuscript entitled Novarum quaestionum libri tres, which should be considered as a kind of appendix of applications to Diophantus redivivus. This manuscript was never published.

Brunet I, 946: ‘Recherché et rare’; De Backer/Sommervogel I c. 1479 nr. 13; Sauvenier-Goffin (Liege) II p. 66; not in Poggendorf or Graesse. Cassinet, ‘Problèmes Diophantiens en France à la fin du 17ème siècle: J. de Billy, J. Ozanam, J. Prestet (1670–1689),’ Cahiers du séminaire d’histoire des mathématiques de Toulouse 9 (1987), pp. 13–41. Dickson, History of the Theory of Numbers, Vol. II, 1920.

Two parts in one volume, 8vo (166 x 102 mm), pp. [viii], [1-2], 3-302, [2]; [1-2], 3-140, [4], including separate title to each part, part I with blank leaves A1 and T8, part II with final two blanks I7-8, woodcut printer's device, woodcut initials, head- and tail-pieces (the three unnumbered preliminary leaves of the ‘Lectori benevolo’ misbound between pp. 16 and 17 of part I, occasional light browning and spotting, worm track to lower corner of some leaves of part I not affecting text). Contemporary French vellum, spine lettered in ink, sprinkled edges (spine and joints with worm holes, vellum soiled, three gatherings working loose). A very good copy, entirely untouched.

Item #6209

Price: $12,500.00

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