Leyden: Jean Maire, 1649.
First separate edition of of Descartes’s magnum opus (DSB), the invention of coordinate geometry and one of the key texts in the history of mathematics – this is an exceptional copy, uncut in the original interim boards. The Geometry was originally published in French as the third part of the Discours de la Méthode (1637). 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).
First separate edition of Descartes’s magnum opus (DSB), the invention of coordinate geometry and one of the key texts in the history of mathematics – this is an exceptional copy, uncut in the original interim boards. The Geometry was originally published in French as the third part of the Discours de la Méthode; the French text was not issued separately until 1664. 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). “Inspired by a specific and novel view of the world, Descartes produced his Géométrie, a work as exceptional in its contents (analytic geometry) as in its form (symbolic notation), which slowly but surely upset the ancient conceptions of his contemporaries. In the other direction, this treatise is the first in history to be directly accessible to modern-day mathematicians. A cornerstone of our ‘modern’ mathematical era, the Géométrie thus paved the way for Newton and Leibniz” (Serfati, p. 1). “Divided into three books, it opens with the claim that ‘Any problem in geometry can easily be reduced to such terms that a knowledge of the lengths of certain lines is sufficient for its construction.’ In this spirit, Book I is concerned with ‘Problems which can be constructed by the aid of circles and straight lines.’ The highlight of Book I is the solution, by algebraic means, of the problem, outstanding since the time of Euclid, of the four-line locus. Book II contains a little-used classification of curves, and Descartes’ method of drawing tangents. The final Book III deals with the solution of higher-order equations, as well as Descartes’ rule of signs” (Gjertsen, Newton Handbook, p. 170). It was through this Latin translation, with its extensive commentary by Frans van Schooten and Florimonde De Beaune, that Newton and other contemporary mathematicians acquired an understanding of Descartes’s work. It is also the most accessible edition for bibliophiles, the Discours now commanding a six-figure sum. We have never seen another copy uncut in original boards, and none is recorded on ABPC/RBH.
Descartes’ interest in geometry was stimulated when, in 1631, Jacob Golius (1596–1667), a professor of mathematics and oriental languages at Leyden, sent Descartes a geometrical problem, that of ‘Pappus on three or four lines’. It had originally been posed and solved shortly before the time of Euclid in a work called Five books concerning solid loci by Aristaeus, and was then studied by Apollonius and later by Pappus. But the solution was lost in the 17th century, and the problem became an important test case for Descartes. Claude Hardy, a contemporary at the time of its solution, later reported to Leibniz the difficulties that Descartes had met in solving it (it took him six weeks), which ‘disabused him of the small opinion he had held of the analysis of the ancients’. The Pappus problem is a thread running through the entire work.
Book One is entitled ‘Problems the construction of which requires only straight lines and circles,’ and it is in this opening book that Descartes details his geometrical analysis, that is, how geometrical problems are to be formulated algebraically. It begins with the geometrical interpretation of algebraic operations, which Descartes had already explored in the early period of his mathematical research. However, what we are presented in 1637 is a “gigantic innovation” both over Descartes’ previous work and the work of his contemporaries (Guicciardini, p. 38). On the one hand, Descartes offers a geometrical interpretation of root extraction and thus treats five arithmetical operations. Crucially, he also uses a new exponential notation (e.g. x3), which replaces the traditional cossic notation of early modern algebra, and allows Descartes to tighten the connection between algebra and geometry.
Descartes proceeds to describe how one is to give an algebraic interpretation of a geometrical problem:
‘If, then, we wish to solve any problem, we first suppose the solution already effected, and give names to all the lines that seem needful for its construction, to those that are unknown as well as to those that are known. Then, making no distinction between unknown and unknown lines, we must unravel the difficulty in any way that shows most naturally the relations between these lines, until we find it possible to express a single quantity in two ways. This will constitute an equation, since the terms of one of these two expressions are together equal to the terms of the other.’
Descartes applies his geometrical analysis to solve the four-line case of the Pappus problem, and shows how the analysis can be generalized to apply to the general, n-line version of the problem, which had not been solved by the ancients.
Book Two, entitled ‘On the Nature of Curved Lines,’ commences with Descartes’ famous distinction between ‘geometric’ and ‘mechanical’ curves. For Pappus, ‘plane’ curves were those constructible by ruler and compass, ‘solid’ curves were the conic sections, and ‘linear’ curves were the rest, such as the conchoids, the spiral, the quadratrix and the cissoid. The linear curves were also called ‘mechanical’ by the ancient Greeks because instruments were needed to construct them. Following Descartes, the supremacy of algebraic criteria became established: curves were defined by equations with integer degrees. Algebra thus brought to geometry the most natural hierarchies and principles of classification. This was extended by Newton to fractional and irrational exponents, and by Leibniz to ‘variable’ exponents (gradus indefinitus, or transcendental in modern terminology).
Book Three, entitled ‘The construction of solid, and higher than solid problems,’ is devoted to the theory of equations and the geometrical construction of their roots. “The abundance and variety of results in this section is remarkable. A number of the interesting results presented are not altogether new, some being due to Girolamo Cardano, Thomas Harriot and Albert Girard. The exposition is, however, clear and systematic, and expressed for the first time in history in modern notation... These results were taken up and extended by Newton in Arithmetica universalis (1707), in lectures between 1673 and 1683… Descartes is also interested in the number of real roots, and asserts without justification that the maximum number of positive or negative roots of an equation is that of the alternances or permanences of the signs ‘+’ and ‘−’ between consecutive coefficients. This is the celebrated ‘rule of signs’, which earned unfounded criticism for Descartes. Newton took up and extended the matter in the De limitibus aequationum, which concludes the Arithmetica universalis. The result was proved in the 18th century” (Landmark Writings, pp. 13-14). Book Three concludes with a discussion of the geometrical construction of roots of equations by means of intersecting curves, particularly cubic and quartic equations which Descartes treats using a circle and a parabola.
Descartes acknowledged that the mathematical language in which the Geometry was written would inhibit many readers, inserting a special ‘notice’ at the beginning of the text: “‘Up to this point I have tried to make myself intelligible to everyone. However, for this treatise, I fear that only those who already know what is in geometry books will be able to read it. The reason is that they contain many truths that are very well demonstrated, and I therefore thought it would be superfluous to repeat them and, for that reason, I have taken the liberty of using them.’ The caution was appropriate. Only those who were trained in mathematics could understand what the problems were” (Clarke, Descartes, p. 151).
The editor and translator of this edition, Frans van Schooten (1615-60), first saw the Géométrie at Leiden, as Descartes had come there to supervise the printing of the Discours. “After the death of his father in 1645, Schooten took over his academic duties. He also worked on a Latin translation of Descartes’ Géométrie. Although Descartes was not completely satisfied with Schooten’s version (1649), it found a broad and receptive audience by virtue of its more carefully executed figures and its full commentary. It was from Schooten’s edition of the Géométrie that contemporary mathematicians lacking proficiency in French first learned Cartesian mathematics. In this mathematics they encountered a systematic presentation of the material, not the customary, more classificatory approach that essentially listed single propositions, for the most part in unconnected parallel. Further, in the Cartesian scheme the central position was occupied by algebra, which Descartes considered to be the only “precise form of mathematics”” (DSB, under Schooten). Schooten included in the present edition the ‘Notae breves’ of Florimonde De Beaune (1601-52), a French jurist and amateur mathematician, which contains what became known as ‘De Beaune’s problem’, the important problem of determining a curve from the properties of its tangent. De Beaune’s notes evidently pleased Descartes, who wrote to him on 20 February 1639: “J’ai admiré que vous ayez pu reconnaître des choses que je n’y ai mises qu’obscurément comme en ce qui regarde la généralité de la méthode.”
“After 1649, the text became a long-lasting object of study for European mathematicians and a veritable bedside read for geometers, while the faithful Latin translation by the disciple van Schooten ensured its wide dissemination. The ample commentaries, painstakingly completed by De Beaune and much longer than the Géométrie itself, were indispensable in explaining Descartes’s ideas to his contemporaries, clarifying obscurities, reconstructing omitted calculations and also producing new constructions and loci … Leibniz, who knew the Géométrie in Paris no later than 1674, made full use of its methods in his arithmetic quadrature of the circle around 1674, his first discovery” (Serfati, pp. 17-18). “Newton possessed two copies of Descartes’ Geometry, both in Latin … Newton described his first contact with the work as: ‘in the year 1664 a little before Christmas … I bought Van Schooten’s Miscellanies and Descartes Geometry (having read this Geometry … above half a year before). He read it, according to Conduitt, slowly, and with some difficulty, before eventually making himself ‘Master of the whole without having the least light or instruction from anybody.’ His copy of Van Schooten has survived” (Gjertsen, p. 170). Newton’s annotated copy of the 1659 re-issue is held by Trinity College, Cambridge (NQ.16.203).
Cajori, A History of Mathematics, p. 174 (“Of epoch-making importance”); DSB IV, pp. 55-58; I. Grattan-Guinness (ed.), Landmark Writings in Western Mathematics 1640-1940 (2005), Ch. 1; Guibert 27; N. Guicciardini, Isaac Newton on Mathematical Certainty and Method (2009); PMM 129 (for the 1637 edition); Roller & Goodman I p. 314. Serfati, ‘René Descartes, Géométrie,’ Chapter 1 in Landmark Writings in Western Mathematics 1640-1940 (Grattan-Guinness, ed.), 2005.
4to (209 x 160 mm), pp. [xii], 336 [2, errata], title printed in red and black and with woodcut printer’s device, ornamental tailpiece at end, numerous diagrams in text. Contemporary interim boards, manuscript title to spine, lower part of spine very well repaired, entirely uncut. Cancelled library stamp at foot of final page of text.