Discours de la Methode pour bien conduire sa Raison, & chercher la Verité dans les Sciences. Plus la Dioptrique, les Meteores, et la Geometrie. Qui sont des essais de cete Methode

Leiden: Jan Maire, 1637. First edition. PMM 129 - THE INVENTION OF ANALYTIC GEOMETRY.

First edition, a fine, large copy, of Descartes' first and most famous work. Following the Discours, now celebrated as one of the canonical texts of Western philosophy, are three 'Essais', the last of which, La Géométrie, contains the birth of analytical or co-ordinate geometry, "of epoch-making importance" (Cajori, History of Mathematics, p. 174), designated by John Stuart Mill as "the greatest single step ever made in the progress of the exact sciences". It "rendered possible the later achievements of seventeenth-century mathematical physics" (Hall, Nature and nature's laws (1970), p. 91). The first of the Essais, La Dioptrique, contains Descartes' discovery of 'Snell's law' of refraction of light (earlier than Snell); the second, Les Météores, contains Descartes' explanation of the rainbow, based on the optical theories developed in the first Essai. "It is no exaggeration to say that Descartes was the first of modern philosophers and one of the first modern scientists; in both branches of learning his influence has been vast ... The revolution he caused can be most easily found in his reassertion of the principle (lost in the middle ages) that knowledge, if it is to have any value, must be intelligence and not erudition. His application of modern algebraic arithmetic to ancient geometry created the analytical geometry which is the basis of the post-Euclidean development of that science. His statement of the elementary laws of matter and movement in the physical universe, the theory of vortices, and many other speculations threw light on every branch of science from optics to biology. Not least may be remarked his discussion of Harvey's discovery of the circulation of blood, the first mention of it by a prominent foreign scholar. All this found its starting point in the 'Discourse on the Method for Proper Reasoning and Investigating Truth in the Sciences'. Descartes's purpose is to find the simple indestructible proposition which gives to the universe and thought their order and system. Three points are made: the truth of thought, when thought is true to itself (thus cogito, ergo, sum), the inevitable elevation of its partial state in our finite consciousness to its full state in the infinite existence of God, and the ultimate reduction of the material universe to extension and local movement" (PMM).

Provenance: Lessing J. Rosenwald (small morocco monogram bookplate); given to the Library of Congress (bookplate and duplicate stamp); Richard Green (Christie's NY, 17 June 2008, The Richard Green Library, lot 87, $116,500).

In October 1629 Descartes began work on The World, which included not only his Treatise on Light, first published as Le Monde in 1664, and the Treatise on Man, first published two years earlier as Renatus Descartes de Homine, but also the material on the formation of colours in the Meteors and the material on geometrical optics in the Dioptrics, both subsequently published in 1637 along with the Discourse and the Geometry. Descartes sets out the details of the treatise he was working on from mid-1629 to 1633 in part 5 of the Discourse: "I tried to explain the principles in a Treatise which certain considerations prevented me from publishing, and I know of no better way of making them known than to set out here briefly what it contained. I had as my aim to include in it everything that I thought I knew before I wrote it about the nature of material things. But just as painters, not being able to represent all the different sides of a body equally well on a flat canvas, choose one of the main ones and set it facing the light, and shade the others so as to make them stand out only when viewed from the perspective of the chosen side; so too, fearing that I could not put everything I had in mind in my discourse, I undertook to expound fully only what I knew about light. Then, as the opportunity arose, I added something about the Sun and the fixed stars, because almost all of it comes from them; the heavens, because they transmit it; the planets, comets, and the earth, because they reflect light; and especially bodies on the earth, because they are coloured, or transparent, or luminous; and finally about man, because he observes these bodies" (quoted in Gaukroger (ed.), Rene Descartes: The World and Other Writings, p. xi).

But The World was never published in Descartes' lifetime. "During the years immediately following the condemnation of Galileo, Descartes held fast to his initial view that the cardinals had made a mistake, though one that was potentially dangerous for himself. His fundamental idea was that the decision involved a misunderstanding of the role of the Bible as a source of scientific knowledge. He also argued that he was not bound to accept the Roman decision as a matter of faith, and he hoped that it would be reversed in due course so that he could publish his World without fear of censure. He had to concede, however, that as long as there was no change of mind about Galileo by the church, the World would remain 'out of season' ... In these circumstances, the next-best option was to consider ways in which parts of his work that were not theologically sensitive could be released to the public. Accordingly, during the years from 1633 to 1637, Descartes spent most of his time on this project. His efforts came to fruition with the publication of the Discourse on the Method for Guiding one's Reason and Searching for Truth in the Sciences, together with the Dioptrics, the Meteors, and the Geometry, which are samples of this Method (1637) ... [It] omitted what Descartes called the 'foundations of my physics', that is, the controversial view of the universe that included heliocentrism. He offered instead some examples of the results that one could expect from his basic theory when applied to specific areas such as dioptrics. For good measure, he made sure that the book appeared anonymously.

"The standard practice among scholars in the seventeenth century was to write in Latin, since that was the normal language for instruction at universities, and thus to make their publications accessible to academic readers all over Europe ... [Descartes] preferred to write in French not only because he found it easier to do so, but also because he wished to dissociate his work from the scholastic tradition that it criticizes, and because he trusted readers (including women) who had not been contaminated by school learning more than academic readers. The relative openness of women to new ideas later became one of the central features of Cartesian arguments in favour of women's education ...

"[The] Discourse on method has been adopted as part of the canon in Western philosophy, as an independent text from which modern readers are expected to distill its author's seminal contribution to modern thought ... The relative accessibility of the Discourse, and the fact that the essays have lapsed with the passing of time into obsolete texts in the history of science, has even had the remarkable effect of transforming retrospectively the original core of the book into what are often referred to as its 'appendices'. Yet it is obvious that the scientific essays are the main text, and their relative size alone confirms that fact. More fundamentally, it is clear from the history of its publication that the Discourse was planned merely as a Preface to the scientific essays, and that it was written when the book was being printed partly as a concession to Mersenne's importunate requests for publication of the underlying physical theory on which the essays were based ...

"Descartes' Dioptrics is designed as a discussion of the extent to which the invention of telescopes or other lenses can assist human vision. In fact, however, it is just as much a philosophical discussion of sensation and a reworking of some of the themes about perception that had been presented in the initial chapters of The World. Since he still had no definite plans for that book in 1637, Descartes offers readers a glimpse of his fundamental rejection of scholastic theories of perception by including in the Dioptrics, in the fourth discourse, a discussion of 'The Senses in General' ...

"He presents a mathematical analysis of how light is reflected from smooth surfaces, such as mirrors, and how it is refracted when it travels from one medium to another (e.g., when it passes from the air into glass). In the case of reflection, Descartes argues that the angle of incidence is equal to the angle of reflection. In the case of refraction, he develops a mathematical analysis that concludes with the sine law of refraction, the same conclusion that resulted from independent work by the Dutch physicist Snellius. Having established these laws of reflection and refraction, Descartes needs to describe the anatomy of the eye.

"It is hard to avoid the conclusion that the description of a cross-section of the eye, presented in the third discourse of the Dioptrics, is based at least in part on anatomical dissections that Descartes had done during the previous years. Without using technical terms such as 'iris', 'pupil', or even 'optical lens', he describes the various parts of the eye, including what he calls the 'optical nerve', with only enough detail to make it possible for readers to follow the discussion in subsequent chapters. However, as promised to Huygens, he does provide many diagrams to illustrate the points being made in the text ...

"The fourth, fifth, and sixth discourses of the Dioptrics provide a summary of the Cartesian theory of vision and, by extension, of sensation in general. Descartes concedes at the outset that 'it is the mind which senses and not the body', or, more exactly, it is the mind insofar as it is 'in the brain, where it exercises the faculty called common sense'. This claim requires an account of how the nerves work and how they can transmit information from the external senses to the brain. Descartes adopts the common opinion of anatomists, that a cross-section of nerves shows three distinct parts: an outer membrane, an inner filament, and a very subtle matter (called animal spirits) that lubricates the gap between the outer and inner layers and thereby allows the inner tube to move smoothly within the outer membrane. If one assumes that information is transmitted from the external senses to the brain by the motion of the inner filament, one can exploit the analogy of the blind man to show that the mind can acquire reliable information without having an image of perceived objects ...

"The optical part of this discussion is presented in the fifth discourse, in which Descartes accepts that an optical image resembling an object of perception is formed on the back of the eye. However, the novelty of his contribution is in the sixth discourse, in which he constructs a theory about how the information presented in this optical form can be transmitted to the centre of the brain. He repeats the general principle that 'our soul is of such a nature that the force of the movements' of the optical nerves 'makes it have the sensation of light', although 'there is no need, in this whole process, for any resemblance between the ideas that the soul conceives and the movements that cause those ideas'. The Cartesian theory is that the brain receives information from many sources apart from the optical image on the retina − for example, from the muscles that move the eyes, from muscles that adjust the focal length of the ocular lenses, from the degree of brightness of images received, and from changes in the size of the pupil ...

"The remainder of the text discusses the ways in which lenses of different shapes are suitable for various optical instruments, and especially the merits of hyperbolical and elliptical lenses, which Descartes claims are 'preferable to any others that can be imagined' ...

"The Meteors is, like the Dioptrics, a somewhat eclectic collection of explanations of meteorological phenomena, most of which had been discussed by Descartes in correspondence during the previous nine years. In addition to discussing the disparate phenomena mentioned, it also had as a centerpiece a prominent new discovery, namely, the Cartesian explanation of the rainbow. It is clear from the outset − a point reiterated in the concluding sentence of the book − that Descartes' objective was to demystify the apparently strange phenomena that occur in the space between the Earth and the heavens and that had been interpreted by others as omens or prophetic messages. 'That makes me hope that, if I explain the nature of clouds here ... it would be easy to believe that it is possible, in the same way, to find the causes of everything that is most admirable above the earth'. The explanations that are suggested are contrasted with the 'superstition and ignorance' that compromises even the very description of many meteorological events ... Much of what is explained in the Meteors derives from specific experiences that are dated by Descartes himself. Thus he refers to the shapes of snowflakes that he observed in Amsterdam, during the winter of 1635; to the avalanches he observed when crossing the Alps in May 1625; to the coronas around a lighted candle that he had recently seen while crossing the Zuiderzee; and to parhelias observed in Rome in March 1629. It was not merely a coincidence that Descartes tried to explain various phenomena that he had witnessed himself. He often argued that observations made by others are unreliable; accordingly, he conceded in the Meteors that he could speak only conjecturally about 'things that happen on the oceans, which I have never seen and about which I have only very imperfect descriptions'.

"The focal point of the Meteors, however, was the explanation of the rainbow that presupposed the sine law of refraction presented in the Dioptrics. Descartes acknowledged some of the familiar features of rainbows that require explanation, for example, that they are bow-shaped, that they occur at a definite angle in the sky relative to the observer, that there is often a secondary bow that is less bright than the primary bow, that the colours appear in reverse order in the two bows, and so on. The explanation he offered relied on an assumption that light is a tendency to move imparted by the Sun to the subtle matter that fills up all the apparently empty spaces between larger parts of matter, and that our perception of colours results from the different ways in which those subtle particles modify their spins when they tend to pass from one medium to another. Even though this 'assumption' was very wide of the mark, Descartes could still successfully calculate the ways in which light is refracted as it passes through raindrops, because that depended only on the refractive index of the rain rather than on any more general theory of what light is. In an ingenious suggestion, Descartes explains how the two bows appear. If light strikes a raindrop, it can be refracted twice (on entering and exiting the drop), and reflected once at the back of the raindrop that faces the observer. Likewise, it may be refracted twice and reflected twice, and this explains how the secondary bow appears at a different height, why it is less vivid (since only some of the light rays are reflected in this way), and why the colours appear in the opposite order in which they are seen in the primary bow" (Clarke, Descartes, pp. 126-51).

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. Besides, it is clear that Descartes had been working on some of these problems for at least ten years, that he had shared his interim conclusions with various correspondents, and that he had written the Geometry in its published form 'only when the Meteors was being printed' and had 'even invented part of it at that time'" (Clarke, p. 151). Descartes' interest in geometry was stimulated when, in 1631, Jacob Golius, a professor of mathematics and oriental languages at Leyden, sent Descartes a geometrical problem, that of 'Pappus on three or four lines'. Although solved by the ancients, the solution was lost in the 17th century and the problem became an important test case for Descartes and served as a thread running throughout the Geometry.

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, Isaac Newton on Mathematical Certainty and Method, 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 in Western Mathematics 1640-1940, 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. 

"The Discourse was a rather hastily written Preface designed to introduce the three scientific essays on 1637 ... As in the case of the three scientific essays, Descartes borrowed some of its content from material that he had written over a number of years and that still remained unpublished. These sources included a draft autobiography that he had discussed with Balzac in 1628; the Rules, an early draft of a treatise on metaphysics; and, most of all, The World and the Treatise on Man, in which his general approach to natural philosophy had been developed. He may have written the final section (Part VI) first, because it is the only section that refers to the accompanying essays. Having done that, he then added autobiographical material, four rules that summarize his method, eight pages of a detailed description of blood circulation that seems oddly out of context, and a somewhat protracted account of his frequent changes of plan about publishing or not publishing The World.

"Descartes' correspondence with Mersenne and Huygens, while the scientific essays were in production, shows that he changed the content of the Discourse, and that he modified his original ambition from a 'project of a universal science' to a 'preface or advice about method'. This scaling down of plans is reflected in the text of the Discourse itself, when he writes in Part I: 'my plan here is not to teach the method that everyone must follow in order to guide their reason, but merely to explain how I have tried to guide my own' ... The same diffidence or defensiveness reappears in Part II, where he claims that his aim was never more ambitious than 'an attempt to reform my own thoughts and to build on a foundation that was entirely my own'. He does not advise 'anyone else to imitate it'. With all these disclaimers in place, he then summarizes four rules that are borrowed from the Rules, but that in fact give very little indication of how to realize the significant results of the three essays.

"Descartes thought that he could also illustrate the range of disciplines to which his new method applied by giving examples of the progress that he had made in metaphysics, physiology, and generally in natural philosophy. The brief digression into metaphysics, by which he meant arguments about the existence of God and the immortality of the soul, outlined the approach to which he returned later in the Meditations. Likewise, the claim that he had discovered explanations of many natural phenomena was supported by reference to the unpublished World. Such explanations were radically different from what was widely taught in schools, for, according to Descartes' concept of matter, it contained 'none of those forms or qualities about which they dispute in the schools'.

"In the final section of the Discourse (Part VI), Descartes eventually addresses one of the fundamental questions raised by his scientific essays, namely, the extent to which the theories presented there could be confirmed by evidence, and the kind of evidence that would be relevant to the task. The solution he offered was a compromise that was excused by the provisional status of the publication. In other words, Descartes claimed to have a fundamental theory of nature in The World that could 'demonstrate' all the theories outlined in the 1637 book. At the same time, he could not publish that work yet. He therefore had no option, he claimed, but to begin the Dioptrics and the Meteors with various 'assumptions' that, had The World been published, could have been established more certainly.

The publication process was protracted for several reasons. "By November 1635, Descartes' plans for his publication were beginning to come into focus. He decided to include the Meteors together with the Dioptrics and, for the first time, he mentioned his ambition to add a Preface to the book, which he thought was likely to delay completion by two or three months. It is clear that the Dioptrics was already near enough to completion, by December 1635, to refer to its contents as if the structure of the book were fixed ... [By March 1636] he had finalized enough of his manuscript to show it to various printers with a view to publication. Once arrived in Leiden, Descartes began to discuss publication of his book with Elzevier. The publisher raised many difficulties and assumed, since the prospective author had travelled so far to petition their services, that he had little choice but to accept his proposed terms. This bargaining strategy failed, and Descartes went to another printer in the same town, Jan Maire ... By 13 July 1636, the printer was promising to have all the illustrations ready within three weeks and to begin printing immediately. Frans van Schooten, the son of a professor of mathematics at Leiden of the same name, had agreed to make the woodcuts. The printer so treasured his expertise that he induced him to lodge in his house, partly to hasten completion of the project and partly 'for fear that he would escape'. The printing of the Dioptrics was completed by 30 October, but since the engravings for the Meteors and the Geometry had not been done, the projected publication date was deferred to Easter 1637" (Clarke). Publication was further delayed by the difficulty in getting a French privilege (since the book was written in French, this was necessary to safeguard the interests of the Dutch publisher). The Dutch privilege was granted without difficulty on 20 December 1636, but it was only after much negotiation, requiring the intervention of Huygens and Mersenne, that the French privilege was finally granted on 4 May 1637. Maire printed the last folio on 8 June, using an abbreviated version of the privilege that omitted the author's name.

Dibner, Heralds of Science 81; Evans 5; Grolier/Horblit 24; Guibert, Bib. Descartes 1; Norman 621; PMM 129; Sparrow 54.

4to (203 x 152mm), pp. 78 [Discours]; [2], 413 [Essais], [1, Avertissement]; [34, tables, errata and French and Dutch privileges], woodcut printer's device on title, woodcut initials, numerous woodcut text illustrations and diagrams (some light spotting and browning, tiny associated hole on b4). Contemporary calf, red sprinkled edges (spine ends and corners restored, lightly rubbed, small split at head of upper joint); morocco pull-off case.