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L'Algèbre ... départie an deus Liures. Lyon: Jean de Tournes, 1554. First edition, very rare, of "the first printed book on algebra in French and the richest among vernacular books on algebra" (Cifoletti, 'The algebraic art of discourse,' in Chemla, p. 125). Peletier believed that French was the perfect instrument for the sciences and "wrote L'Algèbre in French in his own orthographic style. He adopted several ingenious ideas from Stifel's Arithmetica integra (1544) and showed himself to have been strongly influenced by Cardano. He was the first mathematician to recognize relations between the coefficients and roots of equations" (DSB). Peletier's innovative mathematical symbolism can be seen as anticipating the introduction of symbolic algebra by François Viète (1540-1603) in his In artem analyticem isagoge (1591) (see below). Peletier's "principal innovation resides in the introduction of as many symbols as there are unknowns in the problem, as well as in the fact that the unknowns in the problem correspond to the unknowns in the equations, in contrast to what was being suggested by, for example, Cardan and Stifel" (Cifoletti, p. 1396). "The introduction of arbitrary representations for the several unknowns in a problem is indeed part of Peletier's central idea of elaborating an "automatic procedure" to tackle the problems under consideration. Instead of having recourse to sophisticated artifices like those used by Diophantus several centuries before the Renaissance, the symbolic representation of several unknowns offered the basis for a clear and efficient method ... Peletier's immense genius led him to see that the key concept of our contemporary school algebra is the equation. For sure, Arab algebraists classified equations before abacists such as Pacioli or della Francesca and humanists like Peletier or Gosselin, but these equations referred to 'cases', distinguished according to the objects related by the equality. For Peletier, the equation belongs to the realm of [abstract] representation" (Radford, The cultural-epistemological conditions of the emergence of algebraic symbolism). Only three copies have appeared at auction in the past fifty years: Macclesfield (18th century calf, "a few headlines shaved"), Honeyman (a reimboitage with new endpapers), and the Norman copy (this copy). The only other copy we know of in commerce was offered by Amélie Sourget in 2014, bound with a later edition of Peletier's L'Arithmétique (1584) (Cat. 5, no. 3, €29,000). Provenance: Bibliothèque de Picpus (stamp on title and at end); Haskell F. Norman (his sale Christie's New York, 18 March 1998, lot 153). Peletier (1517-82) published L'art poëtique d'Horace, traduit en vers Françios in 1541, the preface of which pleaded for a national language. He also studied Greek, mathematics, and later medicine, always as an autodidact. In the winter of 1547/8 he joined the 'salon' of humanists composed of Jean Martin, Denis Sauvage, Théodore de Bèze and Jean-Paul Dauron (a mathematician of Provençal origin), who discussed mathematics, the primacy of the French language over Latin, and spelling reform. Peletier shared with the Pléiade, a group of seven poets whose leader was Pierre de Ronsard, a desire to create a French literature. He also stated that French was the perfect instrument for the sciences and planned to publish mathematical books in the vernacular. In 1549 he published at Poitiers his L'Aritmétique, one of the first arithmetics written in French. In the following year his Dialogue de l'orthographe et prononciacion françoese appeared, in which one side (represented by de Bèze) advocates the use of etymological spelling, the other (represented by Dauron) advocates of phonetic spelling. The entire dialogue is written in a new orthography, based upon pronunciation, invented by Peletier which he continued to use in his writings for the rest of his life. In 1554 he moved to Lyon, where he worked as a correcteur in the printing house of Jean de Tournes. De Tournes accepted Peletier's spelling reform and published, under his supervision, Peletier's books on orthographe réformé, beginning with his L'Algèbre, and an updated and expanded version of his arithmetic. "Stifel's example of explicitly using a symbolic equation was followed in mid-sixteenth-century France by Jacques Peletier and Jean Borrel, both working in a mid-century liberal arts context characterized by an emphasis on rhetoric and the honing of reasoning skills and both focused mathematically on solving systems of linear equations. Peletier, in his Algèbre of 1554, adopted Stifel's notation 1A and 1B for the second and third unknowns, although he used 1R for the first (standing for 'radix') and reverted to using p. and m. for plus and minus. Still, he treated a system of three equations as a single object, identified the equations by number as he worked on them, and explicitly added and subtracted equations (and their multiples) together in order to eliminate two of the unknowns" (Katz & Hunger Parshall, p. 212). Peletier's innovations were the beginnings of modern symbolic algebra created by Viète at the end of the sixteenth century and Descartes at the beginning of the seventeenth: "... by the end of the sixteenth century, the plurality of algebras that had existed earlier in the century had begun not only to take on a whole new look but also to coalesce -- as Peletier had advocated and foreseen -- into a general problem-solving technique, the objective of which was "to solve every problem" ... It was René Descartes (1596-1650), in his text La Géométrie, published in 1637, who truly realized the grand ambitions for algebra that Viète shared with his French predecessors Peletier and Gosselin" (ibid., pp. 245-8). Viète became aware of the Arithmetica of Diophantus through references by Peletier and Peter Ramus (see DSB, under Viète). Inspired by Peletier, Guillaume Gosselin published his De Arte magna at Paris in 1577. Peletier's famous two-leaf manifesto for the French language, 'Jacques Peletier aus Francoes. De Lion ce 28 juillet 1554,' printed in his new characters, is usually found at the beginning of the work but is in this copy bound at the end. Subsequent editions were published by Jean de Tournes at Cologne in 1609 and at Geneva in 1620. The work was translated into Latin and published by Cavellat at Paris in 1560 under the title De occulta parte numerorum. Brunet IV, 471; BM/STC French p. 343; Cartier, De Tournes 284; Macclesfield 216; Norman 1677 (this copy); Smith p. 245 note; Tchemerzine V, 148 (with incorrect collation, omitting the two-leaf apologia 'Jacques Peletier aus Francoes'). Chemla (ed.), History of Science, History of Text, 2006; Cifoletti, 'La question de l'algèbre. Mathématiques et rhétorique des hommes de droit dans la France du 16e siècle,' Annales Histoire, sciences sociales, 50e année, No. 6 (1995), pp. 1385-1416. Katz & Hunger Parshall, Taming the Unknown, 2015. Two parts in one volume, 8vo (152 x 94 mm), pp. [xvi], 229, [13]. Printer's woodcut device on title, some woodcut diagrams in text (without final blank, a few headlines shaved, some minor marginal dampstains). Eighteenth-century French speckled vellum. [Bookseller: SOPHIA RARE BOOKS]
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