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Rome: Bartholomeo Zannetti, 1608. First edition, author's presentation copy to the Jesuit College of Tivoli, of this rare and important text by "the most influential teacher of the Renaissance" (Sarton), "commonly considered one of the last and fullest syntheses of cossic algebra" (Feingold, Jesuit Science and the Republic of Letters, p. 63). Clavius "was one of the very first to use parentheses to express aggregation of terms... His Algebra marks the appearance in Italy of the German plus (+) and minus (-) signs and the algebraic symbols used by Stifel" (DSB). Only six copies located in auction records in the last 60 years, and no other author's presentation copy. It is from this book that many of the leading mathematicians of Europe learned their algebra over the next several decades. "Leibnitz wrote a letter to Bernoulli in 1703, indicating that he learned algebra from Clavius' book. As a child, I had studied the elementary algebra of one Lancius, and later that of Clavius" (Sigismondi, p. 232). Again, "Clavius was a key figure in Descartes' earliest (perhaps even initial) study of mathematics. For instance, from a letter of March 1646 written by John Pell to Charles Cavendish, we have good reason to believe that ca. 1616, while a student at La Flèche, Descartes read Clavius's Algebra (1608). Reporting on his meeting with Descartes in Amsterdam earlier that same year, Pell writes in particular that "[Descartes] says he had no other instructor for Algebra than ye reading of Clavy Algebra above 30 years ago"" (Stanford Encyclopedia of Philosophy). "Mersenne, Descartes' friend and confidant, would advise reading Clavius' Algebra, for he said that there one would always find interesting questions" (Udías, Jesuit Contribution to Science, p. 7). Clavius' work followed the textbook tradition of algebra of the Renaissance. Among his sources he explicitly mentions Diophantus, Jordanus of Nemore, Cardano, Tartaglia, Bombelli, and, perhaps most interestingly, Pedro Nuñes, who is known to have taught Clavius at Coimbra. "Clavius' Algebra may be placed historically at the intersection point of the tradition of cossist algebra and Diophantus' Arithmetica. Having adopted the algebraic symbolism of Michael Stifel, Clavius could represent by symbols more than one unknown quantity. Besides, he sometimes denoted certain numbers by signs, as Jordanus's De numeris datis had done in the thirteenth century. On the other hand, his Algebra was basically freed from the practical tradition of algebra. Clavius re-evaluated algebra as a respectable discipline, invoking the ancient authority of Diophantus and such Renaissance forerunners as Regiomontanus and Bombelli. "Nevertheless Clavius' treatise was a typical product of the sixteenth century textbook tradition of algebra even though it appeared at the beginning of the seventeenth century. It surely provided rather sophisticated techniques for solving both arithmetical and geometrical problems, but it is an open question whether or not it is a work leading straight to the idea of algebraic analysis, which made its full-scale appearance with Viète's In artem analyticem isagoge of 1591" (Sasaki, Descartes's Mathematical Thought, p. 81). Viète was mentioned in Chapter XII, 'On the extraction of roots which the rule of algebra is concerned with.' "Clavius was informed of the existing knowledge on the solution of the cubic equation through Cardano's Ars magna (1545) and Bombelli's Algebra (1572) ... Clavius also has certain information on Viète's method probably developed in the latter's De numerosa potestatum ad exegesin resolution. Viète's treatise on the numerical solution of the cubic and biquadratic equations was published in 1600 under the editorship of Marino Ghetaldi, a disciple of both Clavius and Viète" (ibid., p. 75). Cajori discusses Clavius' notations at some length (A History of Mathematical Notations, vol.1, pp. 151-154). He notes that it was Clavius who, in his Algebra of 1608, introduced the notations + and − for addition and subtraction, already used in Germany by Michael Stifel, to Italy. Clavius explains that he prefers these two signs, instead of the letters P and M commonly used at the time for plus and minus, to avoid confusion with the letters used to represent numbers. Clavius (p. 159) writes products of sums of integer multiples of square roots much as we would write the multiplication today, the only difference being that the square root was represented by the radical sign followed by a script letter (probably r). For an unknown quantity, Clavius used a cursive letter (apparently x) and for additional unknowns, he used A, B, etc. Thus, he writes 3x + 4A, 4B − 3A where we might write 3x + 4y, 4z − 3y. Sigismondi has framed Clavius' contribution within the context of an expanding early-modern world and the crucial role played by the Jesuits in the development of science in China: "Clavius' contributions to algebra, geometry, astronomy and cartography are enormous. He paved the way, with his texts and his teaching for 40 years in the Collegio Romano, for the development of these sciences ... all around the world, along the commercial paths of Portugal, which become also the missionary paths for the Jesuits. Clavius' books were translated into Chinese by one of his students, Matteo Ricci ..., and his influence for the development of science in China was crucial. The Jesuits became skilled astronomers, cartographers and mathematicians thanks to the example and the impulse given by Clavius. This success was possible also thanks to the contribution of Clavius in the definition of the Ratio Studiorum, the program of studies, in the Jesuit colleges, so influential for the whole history of modern Europe and all western World" (Sigismondi). The centrality of Clavius' influence on the curricula of Jesuit colleges worldwide makes our copy, a gift of the author to the Jesuit Collegium Tyburtinum, all the more desirable. Macclesfield 539; Seventeenth Century Italian Books in the British Library, I, p. 240; DSB III, 311; Smith, History I, 334. De Backer-Sommervogel II, 1221. C. Sigismondi, 'Christopher Clavius astronomer and mathematician,' 12th Italian-Korean Symposium on Relativistic Astrophysics, July 4-8, 2011, Pescara, Il Nuovo Cimento C, Issue 1, Suppl. 1 (2013), pp. 231-6. 4to (214 x 155 mm), pp. [xxxvi] 383 [1]; with woodcut Jesuit device on title, printer's device on final leaf verso, woodcut initials, head- and tailpieces, numerous woodcut diagrams in text (occasional light foxing). Ink inscription 'ex dono auctoris Collegio Tyburtino', with further inscriptions 'ex bibliotheca Collegii Tyburtini, Societatis Jesu', and 'coll. Tyburtino ex dono auctoris', followed by a later inscription recording the date for a catalogue entry, 1765, all on the title-page (the inscriptions showing through to verso, obscuring a few letters in the imprimatur). Contemporary limp vellum (rebacked, later endpapers).
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