Good notational innovations are a powerful help to progress in a subject such as mathematics; by their means Descartes was able to make his own contributions, and to facilitate that of others. And his own contributions were substantial. He is one of the independent founders of analytical geometry, he helped lay the foundations for differential calculus, he set out methods for solving equations up to the fourth degree (he believed his method would apply to higher degrees)—and more besides. Not all that he hoped for his mathematical work stood the test of subsequent examination; but these are substantial claims on posterity's notice nevertheless.
I dwell on Descartes' mathematical achievements because they illustrate the method he placed so much weight upon, and set out in detail in his Rules for the Direction of the Mind.14His view was that anything we wish to know about can be grasped by starting with the simplest elements, and then proceeding by small and careful steps from each element to the next, each step explaining the next. The simplest elements are distinguishable by their intuitive character, and can therefore be "clearly and distinctly" perceived. Truth in general requires a clear and distinct perception of its object; therefore given that the objects of knowledge constitute a series in which each can be understood in terms of the one before it, it is obvious—so Descartes argued—that enquiry must begin at the simplest first element and thence proceed carefully through the series, each connection being fully grasped.
The "I think therefore I am" dictum seemed to Descartes a paradigm case of a clear, distinct and directly intuitable starting truth of just the kind required as the first step in metaphysical enquiry. And this led to what he saw as the next truth: that he was essentially a "thinking thing"—a mind or soul—whose existence is independent of body. In geometry he solved the problem of giving a general treatment of curves by reducing them to straight lines, and of giving the locus of a point by determining its distance from two given straight lines, the axes of co-ordination. (Legend has it that the idea for this came to him as he watched a fly crawling on his ceiling; it was a moving point mappable from the straight edges of the ceiling where they met the supporting walls). In the case of physics, his prime idea was that the material universe can be exhaustively understood in terms of spatial extension and movement alone.
But "I think therefore I am," or indeed any of the clear and distinct perceptions from which enquiry can proceed according to this method, is not by itself enough, Descartes declared. A crucial element is missing: and that is the existence of a good God who will not allow us to be led astray in our reasonings if we use our intellectual powers responsibly and carefully. It is not merely the existence but the goodness of God which is crucial here; his goodness is what ensures that the right use of our minds will lead to truth, providing that we guard against our natural and sinful human propensity to error. In the Discourse Descartes merely invoked the idea of God's goodness as a guarantor of enquiry; a few years later, in the Meditations, he set out this aspect of his case more fully.
At the end of the Discourse Descartes explained that he wrote it in French instead of the more usual Latin because "I expect that those who use only their natural reason in all its purity will be better judges of my opinions than those who give credence only to the writings of the ancients."15 In sending a copy to one of the teachers at La Fleche he said that he had written in such a way that "even women might understand me."16 This is a little odd; in the seventeenth century Latin was the universal language of the literate and scholarly, and if Descartes wished to be understood everywhere, from England to Italy as well as in the Netherlands and the German states (at least those in which anyone was still reading, overwhelmed as they were by war and rapine during all these years), he would have done better to write in Latin. Yet he chose French, thus limiting his audience to his native land and those among the upper classes in other countries for whom French was a natural second language.
He sent copies of his book to King Louis XIII, Cardinal Richelieu, and the French ambassador to the Hague. Of course he did not expect any of them to read it. The remaining 197 copies—as indicated in his letter to Mersenne, he 'was determined to have 200 free copies from his publisher for his own use—were widely distributed. He sent three copies to La Fleche, as noted, hoping that the book would be adopted as a textbook, at least for the Meteorology section. He wrote to Etienne Noel, "There is no one, I think, who has a greater interest in examining the contents of this book than [the Jesuits] . . . I do not know how they will be able to teach these subjects from now on as they are taught year by year in most of their colleges, unless they either disprove what I have written or follow it."17
He sent a copy to Cardinal De Bagni, whom he had met in Paris in the late 1620s when De Bagni was Papal Nuncio there, and to Cardinal Barberini, nephew of Pope Urban VIII. These were intended to "test the waters," as he put it—not to see if his views were theologically acceptable, but to see if they were congenial; which is to say, he hoped for the Church's support, not merely its sufferance. A bookshop in Rome ordered a dozen copies from Jean le Maire in Leiden on condition that it said nothing about the movement of the earth; and since it happily did not, one can assume it was displayed and sold in the eternal city to French-reading persons of an enquiring turn.18
The book excited great attention, both positive and negative. Dozens of scholarly readers wrote responses to, and criticisms of, various parts of it. The most contentious part was the Discourse itself, for it touched (somewhat too lightly) on so many matters of philosophical importance that it was bound to provoke questions and disagreement. But each of the essays stimulated its quota of reactions, and Descartes replied to a number of his critics, even to those whose complaints were like the one from Libertus Fromondus of Louvain University, who said that Descartes' theory of vision had to be wrong because "noble actions like sight cannot result from so ignoble and brutish a cause as heat"—this being a paradigm of the kind of thinking that had hampered science since Aristotle.
But the adverse criticisms Descartes received from the Paris mathematicians to his Geometry and aspects of his Optics brought out his worst side—the same bitter and vituperative sentiments that he had displayed against Beeckman. Indeed, so angry was his response that it prompts a question about his relations with these mathematicians when he was in Paris; there is the flavour of bad history in it. Part of the problem was that Descartes did not fully work out the problems he presented in the Geometry: he skipped certain steps, and in some cases merely threw down a gauntlet to his readers to prove something for themselves if they could. But in Paris there were mathematicians of outstanding ability, not least among them Pierre de Fermat, and they understood Descartes' work very well—too well, as it happened, for there were aspects of the Geometry that invited criticism.
The problem began when Descartes sent a manuscript of the Discourse and Essays to Mersenne for the latter to secure a licence— a privilege—from Louis XIII for the book to be published in France. The government official responsible for sifting such applications was none other than Jean Beaugrand, a mathematician with a special interest in efforts to develop algebra, who had recently edited a new edition of the works of Francois Viete, and who was familiar with the work of the English mathematician Thomas Harriot, whose Artis Analyticae Praxis had appeared just six years before. Viete and Harriot were the two mathematicians principally responsible for developing algebra in the early modern period before Descartes. When he read the Discourse and Essays, Beaugrand noted Descartes' claim that he had constructed his mathematics wholly from his own resources, and disbelieved him, for he thought he saw a reliance on the work of both Viete and Harriot. He wrote to Mersenne accordingly, and published two pamphlets pursuing the allegations. Descartes denied them vigorously, but the imputation of unacknowledged borrowing—more bluntly, of plagiarism—was not good for Descartes' reputation, and it interfered with a proper assessment of his mathematical contribution.19
To add to this, one of the geometrical techniques devised b
y Descartes was shown by Beaugrand to be inferior to one already developed by Fermat (he was Fermat's patron and encourager). Descartes was chagrined by this; but he later quietly accepted an aspect of Fermat's technique in place of his own, for Fermat's method was indeed superior. But at the time the attack of the Paris mathematicians made him see red: he wrote a fulminating letter on the first day of March 1638, in response to three letters from Mersenne bringing news of the attacks by Beaugrand and others, together with what he thought was evidence of Beaugrand having misused his manuscript, failing to return it to Mersenne, and deliberately delaying the granting of a privilege:
I must first thank you very carefully [so Descartes' bitter reply to Mersenne began] for scrupulously drawing to my attention many things which it is important that I should know. I can assure you that so far from being bothered by the bad things that are said about me, I rather revel in it; indeed, the more extravagant and outrageous it is, the more I consider it to my advantage, the more it pleases me and the less troubled I am by it. I know that these spiteful people would not go to such lengths to speak ill of me if others did not speak well of me. Besides, truth sometimes needs to be contradicted in order to be better recognised. But those who speak without reason or justification deserve nothing but scorn.
As for M. Beaugrand, I am amazed that you condescend to speak of him, after the way he treated you . . . As for the discourses written by him and those of his ilk, please treat them with contempt, and make it clear to them that I have nothing but contempt for them. Above all, please do not agree to send on to me any writing, either of his or anyone else's, unless those who offer it to you add a note to the effect that they agree to my publishing it along with my reply . . . after seeing M. de Fermat's last letter, which he says he does not want published, I expressly asked you not to send me any more letters of that sort. Of course if a Jesuit or a priest of the Oratory, or anyone else who was incontestably honest and level-headed, wishes to send me something, we would have to be more cautious. I shall make myself freely available to people such as these, but not to those spiteful characters whose aim is anything but truth.20
Given that Fermat was the greatest mathematician of the day, and the Jesuits and priests of the Oratory did not always merit description as "incontestably honest," this reads oddly today. But there was more. Descartes refused to read one of Fermat's treatises, returning it unopened with the remark that everything it contained would already be found in his own work; and—not entirely consistently, given that he had just claimed that Fermat's insights aped his own—he described Fermat's work as "dung." In the same scatological vein, he described Beaugrand's letters as fit only for use as lavatory paper.21 Indeed, Beaugrand excited Descartes' special animus: his work was "so impertinent, so ridiculous, and so despicable" and the man himself "has as much impudence and effrontery as ignorance."22 No doubt the temperature of Descartes' remarks is explained by the fact that Beaugrand called him "the methodical impertinent," playing on the title of the book.23 Descartes dismissed other mathematicians who criticised his work as "flies"; he described Gilles Roberval as "less than a rational animal" and Pierre Petit as a "little dog who barks after me in the street." The great Thomas Hobbes, as posterity knows him to be, appeared to Descartes "extremely contemptible." Still smarting three years later when he wrote the Preface to his Meditations on First Philosophy, Descartes described his critics as "silly and weak," as holding "false and irrational" views, and as being "arrogant."24
Roberval was the professor of mathematics at the College de France, and one reason for his opposition to Descartes was the offence he understandably therefore took, as the country's top academic mathematician, at not being sent a copy of the Discourse, nor even of the separately printed pamphlet version of the Geometry. Whether this was an intended slight by Descartes, or (as Watson suggests) Mersenne's way of fostering controversy and dispute, the result was the same: Roberval attacked Descartes' work at every opportunity, and all Descartes' efforts to placate him, flatter him into compliance, and eventually insult him, could not avert the stream of hostility. Descartes ended by telling Mersenne that Roberval was as vain as a girl, had a head like a dwarf, and behaved like the fool in an Italian farce who "continues bragging and remains always victorious and invincible even after having his ears boxed and his face slapped with a slipper."25
In the light of Descartes' negative reaction to criticism of the Discourse, the motive for his endeavour in the later Meditations to work out his philosophical arguments in more detail—especially his arguments for the existence of God—becomes palpably clearer. In its Preface he provided an excuse for the shortcomings of the Discourse: "My purpose there was not to provide a full treatment, but merely to offer a sample, and learn from my readers how I should handle these topics at a later date." And he added that he thought it would not be helpful "to give a full account" of his views in a book written for the general reader in French, "in case weaker intellects might believe that they ought to set out on the same path."26
Even when the dispute between Descartes and a critic was friendly, as it was with another professor at the College de France called Jean-Baptiste Morin, any differences of opinion made Descartes eventually retreat from the field, asking Mersenne not to send him any more of the disputant's queries and comments. Morin was particularly flattering to Descartes; he described him as having "one of the most subtle and fertile minds of the century," a mind "capable of leaving something rare and excellent for posterity."27 Objections and questions from Morin went via Mersenne to Descartes, who collected them and replied by the same route; but eventually Descartes told Mersenne, "I will make no more reply to M. Morin . . . just between us, it seems to me that his thoughts are even more distant from mine than they were at the beginning, such that we could never come to an agreement."28
In sharp contrast to Descartes' bitter view of the mathematicians who attacked him was his reaction to those whose attitude was favourable. Girard Desargue and Florimond Debeaune liked his book, praised it, and in return bathed in his smiles. He told Mersenne that he thoroughly liked Desargue's "good mind" and the "curiosity and clarity of his language."29 Desargue helped Mersenne urge the granting of the privilege for the book, and on reading the essay on Optics immediately wished to have lenses ground according to its prescription. Debeaune proved helpful in another direction: he studied Descartes' book so thoroughly that he was able to point out a tiny error in the measurements of refraction, for which Descartes wrote in admiration to thank him. Debeaune saw that Descartes had deliberately obscured some of the workings in the Geometry in order not to reveal the principles underlying them, and argued that it would have been better for everyone if he had done otherwise. Remarkably, Debeaune had studied the Geometry in such depth that, he said, he could write it out from memory if ever the book were wholly lost, complete with all the necessary workings that Descartes had omitted. Soon afterwards Debeaune did indeed write out all the suppressed workings, and sent them to Descartes under the title "Notes sur la Geometrie." These notes were eventually translated into Latin and published as a supplement to the Latin version of the Geometry in 1649.30
The great interest generated by the Discourse persuaded Descartes of two things: that he had to leave mathematics behind him, and that he needed to write a more careful and thorough account of his philosophy than he had given in the Discourse. This latter task— the writing of the Meditations on First Philosophy—began to occupy him. And he made a strategic decision: that he would circulate the Meditations before publication, soliciting objections; and that he would publish the objections, together with his replies, along with the text of the Meditations itself, thus avoiding the messy sequel of correspondence that had attended the Discourse's appearance.
A final word on Descartes' mathematical contributions is in order, and properly belongs to the historians of the subject. "In terms of mathematical ability Descartes probably was the most able thinker of his day," said Carl Boyer, after describing the ideas in
the Geometry, "but he was at heart not really a mathematician. His geometry was only an episode in a life devoted to science and philosophy."31 This is true; but it was a significant enough episode. Speaking of Descartes' contribution, E. T Bell was more emphatic: "This one thing [analytic geometry] is of the highest order of excellence, marked by the sensuous simplicity of the half dozen or so greatest contributions of all time to mathematics. Descartes remade geometry and made modern geometry possible."32 As Boyer pointed out, most of the uses to which Cartesian co-ordinate geometry is now put were not anticipated by Descartes; he was working purely theoretically; his achievement was "a triumph of impractical theory as was the Conks of Apollonius in antiquity, despite the inordinately useful role that both were ultimately destined to play."33 But this is not of course a criticism; quite the contrary.
9
Descartes Contra Voetius
In the years following the publication of the Discourse, and despite the acrimonious feelings generated in him by some of his critics, Descartes was in good health, and enjoying respite from public quarrels past and to come in the company of his daughter. This was the period of Francine's brief lifetime, when he had both her and Helena with him at his seaside home at Santpoort near Harlem, under the immense open skies of the North Sea coast, with Amsterdam not far away, and nothing but a long stretch of grey water between him and Lowestoft in England. The Harlem coast consists of rolling sand dunes covered with tough, deep-green grass that undulates like the sea itself in the wind; a bracing, pleasing, peaceful conjunction of sea, sky and earth.
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