## Introduction en l’art analytic, ou Nouuelle algebre de François Viete. Oeuure dans lequel sont veus les plus miraculeux effects des sciences mathematiques, pour l'inuention & solution, tant des problemes, que theoremes, proposez en icelles. Traduit en nostre langue & commenté & illustré d’exemples. Paris: J. Jacquin, 1629. [Bound with:] VIÈTE, François; VAULEZARD, Jean-Louis (translator). Les cinq livres des Zetetiques de Francois Viette. Mis en francois, commentez et augmentez des exemples du poristique, & exegetique, parties restantes de l'analitique. Soit que l'exegetique, soit traitté en nombres ou en lignes. Paris: J. Jacquin, 1630. [Bound with:] VAULEZARD, Jean-Louis. Examen de la Traduction faicte par Anthoine Vasset, des cinq Livres des Zetetiques de M. Viette. Paris: n.p. [J. Jacquin?], 1631.

Paris: J. Jacquin, 1629;1630.

First edition, very rare, and in an unrecorded 1629 issue (usually dated 1630), of the first vernacular translation and exposition of Viète’s *In artem analyticum isagoge *(Tours, 1591), the earliest work on symbolic algebra, here bound with a first edition of Vaulezard’s translation of Viète’s *Zeteticorum libri quinque *(Tours, 1593), which gives examples of the application of his ‘analytic art’ to problems from Diophantus’ *Arithmetica*. The third work is a scathing criticism by Vaulezard of a later translation of the *Isagoge*, published in 1630, by Anthoine Vasset (generally believed to be a pseudonym for Claude Hardy); in a lengthy introduction to his translation, *L’alg**è**bre nouvelle de Mr. Vi**è**te* (Paris: Rocolet, 1630), Vasset criticized Vaulezard’s translation, to which Vaulezard responded in his *Examen* (thus, Vasset’s translation definitely post-dates Vaulezard’s). The greatest French mathematician of the sixteenth century, François Viète (1540-1603) was “the first extensively to use letters of the alphabet to represent numerical quantities” (*Hutchinson**’**s DSB*, p. 690), and “the first mathematician of his age to think occasionally as mathematicians habitually think today” (Bell, p. 99). Viète used letters “both for known … and for unknown quantities” and “this innovation, considered one of the most significant advances in the history of mathematics, prepared the way for the development of algebra” (DSB); it earned him the sobriquet “the father of algebra” (*ibid*.). *Zetetica *is a Greek word meaning “those things to be sought out”, and *zetetics* is the process of transforming a problem into an equation. In his preface to *Les cinq livres des Zetetiques*, Vaulezard tells us that in addition to the *Zetetica* he has added the second and third part of the *Isagoge*, i.e., the sections dealing with *poristics* (proving theorems through equations) and *exegetics* (solving equations), and that he has printed Viète's words in italic and his own commentary in Roman so that the difference may easily be seen. The first edition of *In artem analyticem isagoge* is among the rarest of the important works in the history of mathematics, and is hardly ever seen on the market. A copy was offered a few years ago by a prominent New York dealer for $450,000, and another copy sold at a German auction at about the same time for €200,000 (plus premium), exemplifying Haskell Norman’s dictum that rare books come in twos; we know of no other copy on the market since a copy offered by Sotheran in the 1920s (later in the Turner Collection at Keele University and now in private hands). The present works are almost as rare on the market, and are in fact even rarer than the *Isagoge* in institutional collections. ABPC/RBH records only one other copy (Macclesfield) of *Les cinq livres *(Sotheby’s, November 25, 2005, lot 2049, £3,120 = $5,506) and no other copies of the other two works. OCLC lists, in the US, Brown, Harvard, Michigan, & NYPL for *Introduction en l’art analytic* and *Les cinq livres*; and Brown & Harvard only for *Examen*. There is no copy of any of the three offered works on COPAC, which lists five copies of the *Isagoge*.

*In artem analyticum isagoge* is “a text in which Viète proposed nothing short of a complete refashioning of algebra as it was then understood … Rather than viewing algebra merely as the search for solutions of particular equations, he understood it as the analysis of an actual *theory* of equations” (Katz & Parshall, pp. 236-237). “The most important of Viète‘s many works on algebra … [It] introduced the use of letters both for known quantities, which were denoted by the consonants *B, C, D*, and so on, and for unknown quantities, which were denoted by the vowels. Furthermore, in using *A* to denote the unknown quantity *x*, Viète sometimes employed *A* quadratus, *A *cubus ... to represent *x*^{2}*, x*^{3}, ... This innovation, considered one of the most significant advances in the history of mathematics, prepared the way for the development of algebra” (DSB). “If this seems reminiscent in principle of our modern notation of *x*, *y*, and *z* for unknowns and *a*, *b*, *c*, etc. for indeterminate magnitudes, a convention which we owe to René Descartes in the seventeenth century, it is important to recognize that Viète’s symbols or ‘species’, unlike ours, carried explicit geometrical meaning. They had dimension, and only expressions of the same dimension were commensurate … To Viète’s way of thinking, then, the addition of a one-dimensional unknown *A* to a one-dimensional indeterminate *B* was denoted simply by *A + B* (we would write *x + b*), but in two dimensions, the sum appeared as *A* square + *B* plane (or our *x*^{2} + *b*) and in three dimensions as *A* cube + *B* solid (or our *x*^{3} + *b*)” (Katz & Parshall, pp. 238-239).

Perhaps the most important part of the work is chapter 4, in which “he presents a mode of calculation carried out completely in terms of ‘species’ of numbers and calls it *logistice speciosa* – in contrast with calculation using determinate numbers, which is *logistice numerosa*. Of significance for the formation of the concepts of modern mathematics, Viète devotes the *logistice speciosa* to pure algebra, understood as the most comprehensive possible analytic art, applicable indifferently to numbers and to geometric magnitudes” (DSB).

Viete’s ‘Analytic Art’ comprises three stages. At the first stage, *zetetics*, a problem, whether of arithmetic or geometry, is translated into Viète’s newly created symbol system or *logistice speciosa* in the form of an equation. At the second stage, *poristics*, equations are transformed according to rules into canonical forms; and finally at the third, *exegetics*, a solution to the problem is found on the basis of the derived equation. As Viète himself emphasizes, at this third stage the analyst turns either geometer, ‘by executing a true construction,’ or arithmetician, ‘solving numerically whatever powers, whether pure or affected, are exhibited.’

“In 1593 Viète published *Zeteticorum libri quinque* (pp. 42-81), which he very probably had completed in 1591. In it he offered a sample of *logistice speciosa* and contrasted it directly with Diophantus’ *Arithmetica*, which, in his opinion, remained too much within the limits of the *logistice numerosa* … The *Zetetics* is composed of five books, the first of which contains ten problems that seek to determine quantities of which the sum, difference, or ratio is known. The problems of the second book give the sum or difference of the squares or cubes of the unknown quantities, their product, and the ratio of this product to the sum or the difference of their squares. In the third book the unknown quantities are proportional, and one is required to find them if the sum or the difference of the extremes or means is given. This book contains the application of these problems to right triangles. The fourth book gives the solutions of second-and third-degree indeterminate problems, such as to divide a number, which is the sum of two squares, into two other squares. The fifth book contains problems of the same kind, but generally concerning three numbers” (*ibid*.).

Our copy of Vaulezard’s *Introduction en l’art analytic* is an unrecorded 1629 issue – all bibliographies describe the work as having been published in 1630, this is the date on the BNF copy digitized on Gallica, and no copy dated 1629 is recorded on OCLC or KVK. Vaulezard dedicated *Les cinq livres des Zetetiques *to Jean Beaugrand, who studied under Viète and had close ties, both personally and intellectually, to Fermat and probably introduced him to Viète’s analytic art. Apart from their inclusion in Viète’s*Opera Mathematica* (1646), neither of Viète’s works were published again in any language until the twentieth century.

Probably from the ancient French province of Perche, Jean-Louis Vaulezard, who may have been a pupil of Viète, is known for his works on perspective. In *Perspective cilindrique et conique, concave et convexe ou traité des apparences vueus par le moyen des miroirs* (1630), he discussed cylindrical mirror anamorphoses, a drawing that looks distorted unless it is viewed by reflection in a cylindrical mirror. In 1631, he followed up this work with *Abrégé ou racourcy de la perspective par l'imitation*, in which his aim was to present simplified perspective constructions. One of his means was to provide a sector with a special scale for perspective, which is used to determine the positions of the images of transversals that have a given distance to the ground line.

Remembered principally for his mathematical work, Viète (1540-1603) was a lawyer by trade, serving as a privy counsellor to Henry III and Henry IV, for whom he worked primarily as a code-breaker during the French Wars of Religion.

Macclesfield 2049 (*Les cinq livres*). Bell, *The Development of Mathematics*, 1945. Katz & Parshall, *Taming the Unknown*, 2014.

Three works in one vol., 8vo (170 x 107 mm). Introduction: pp. [1-8], 9-79; Les cinq livres: pp. [8], 9-66, 69-217, [3], engraved figures in text as far as p. 132, thereafter woodcut figures (collation exactly as the BNF copy on Gallica but without the blank leaf I2); Examen: pp. [2], 2, 8 (some gatherings browned, a few small water stains in the margin, contemporary ex-libris inscription to title of Introduction). Contemporary limp vellum with embossed coat of arms to front and rear boards. Preserved in a custom half-morocco clamshell box. An excellent untouched copy of a very rare book.

Item #3128

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Price:
$28,500.00
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