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Geometria, à Renato Des Cartes anno 1637 Gallicè edita ; postea autem unà cum Notis Florimondi de Beavne, In Curia Blesensi Consiliarii Regii, Gallicè conscriptis in Latinam linguam versa, & commentariis illustrata ; operâ atque studio Francisci à Schooten, in Acad. Lugd. Batava Matheseos Professoris, Nunc demum ab eodem diligenter recognita, locupletioribus Commentariis instructa, multisque egregiis accessionibus, tam ad uberiorem explicationem, quàm ad ampliandam hujus Geometriæ excellentiam facientibus, exornata. Editio secunda, multis accessionibus exornata, & plus alterâ sui parte adaucta.
Amsterdam: Elzevir, 1659-1661. A fine copy of van Schooten's important second edition of the Geometria, Descartes's magnum opus (DSB), and one of the key texts in the history of mathematics. Descartes' "application of modern algebraic arithmetic to ancient geometry created the analytical geometry which was the basis of the post-Euclidean development of that science" (PMM). It "rendered possible the later achievements of seventeenth-century mathematical physics" (M. B. Hall, Nature and nature's laws (1970), p. 91). "The mathematical community learned about the wealth of Descartes's new ideas through the works of van Schooten ... In the second edition the commentaries were enlarged, and van Schooten included the work by his students van Heuraet, Hudde, Huygens and de Witt. This edition served as the basic textbook for the generation that, in the last quarter of the century, took the lead in introducing differential and integral calculus" (Jahnke). Newton, in particular learnt his Descartes from this edition: "There can be no doubt that Newton read the Géométrie in Schooten's second Latin edition" (Whiteside, Papers I, p. 7, n17). Newton's own heavily annotated copy of this edition is held in Cambridge University Library (NQ.16.203) The Géométrie first appeared (in French) as an appendix to Descartes' Discours de la Méthode (1637). Frans van Schooten (1615-60) first saw the Géométrie at Leiden, as Descartes had come there to supervise the printing of the Discours. Van Schooten published a Latin translation in 1649, adding his own extensive commentary. "The great success of Schooten's edition led him to prepare a second, much enlarged one in two volumes (1659-1661), which became the standard mathematical work of the period... In the second edition Schooten not only greatly expanded his commentary, but also added new material including an example of Fermat's extreme value and tangent method and a peculiar procedure for determining the center of gravity of parabolic segments. Since Fermat was not mentioned in the latter connection, it is likely that Schooten came upon the procedure independently, for he usually cited his sources very conscientiously. "In the first edition (1649) Schooten inserted Debeaune's rather insignificant Notae breves to the Géomérie. The commentary of the second edition contained valuable contributions by Huygens dealing with the intersections of a parabola with a circle and certain corollaries, as well as on an improved method of constructing tangents to the conchoid. Schooten also included longer contributions by his students: Jan Hudde's studies on equations and the rule of extreme values and Hendrik van Heuraet's rectification method. "Volume II of the second edition of the Geometria (1661) commences with a reprinting of Schooten's introductory lectures [Principia matheseos vniversalis, sev Introductio ad geometriæ methodvm Renati Des Cartes]. This material is followed by Debeaune's work on the limits of roots of equations [De aequationum natura, constitutione, & limitibus opuscula duo] and then by de Witt's excellent tract on conic sections [Elementa curvarum linearum]. The volume concludes with a paper by Schooten's younger half brother Pieter on the algebraic discussion of Descartes' data [Tractatus de concinnandis demonstrationibus geometricis ex calculo algebraïco]. This edition shows the great effort Schooten devoted to the training of his students and to the dissemination of their findings" (DSB, under Schooten). F. Cajori, A History of Mathematics, p. 174 ("Of epoch-making importance"); I. Grattan-Guinness (ed.), Landmark Writings in Western Mathematics 1640-1940 (2005), Ch. 1; N. Guicciardini, Isaac Newton on Mathematical Certainty and Method (2009); H. N. Jahnke, A history of analysis, 2003; PMM 129 (for the 1637 edition). 4to (196 x 153 mm), pp [xvi], 520, [2]; [xvi], 420, [4], with frontispiece portrait of Descartes in Vol. I with title printed in red and black and with woodcut printer's device, numerous diagrams in text. Contemporary limp vellum, manuscript title on spines, stamp from Parisian bookseller to 2nd fly leaf. An exceptionally fine and clean set.
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