The Solid of the Roulette

Descartes did not seek the “solid of the roulette” because he genuinely renounced geometry.

This news did not please the geometricians of Paris who were expecting more and more extraordinary operations on this science from him.

Des Argues showed his displeasure to Father Mersenne, who made it seem good to Mr. Descartes as a testimony of the esteem he had for everything that could come from him.

Descartes took it well, and considered himself very obliged to Mr. Des Argues for his concern. In consideration of his care, he wrote back to Father Mersenne in the month of September of the year 1638 to let him know that he had only decided to quit abstract geometry, that is to say, the search for questions that only serve to exercise the mind: and that he had taken this side only to have so much more leisure to cultivate another kind of geometry, which has for its question the explanation of the phenomena of nature. That for the rest Mr. Des Argues would soon recognize that all his physics “was nothing else but geometry,” if he took the trouble to consider what he had written about salt, snow, the rainbow, etc. in his meteors.

Des Argues was of this small number of friends in favor of whom he had been willing to make an exception to the rule he had prescribed for himself 15 years before of no longer wasting his time giving the solution of geometry problems. He even did something more for the love of Mr. Des Argues in particular.

For having learned that the places of his printed geometry, where he had affected to be obscure, caused this friend pain, he wanted to give him the clarifications himself by a writing he made expressly, to let him know how far the zeal he had for his service went. He asked Father Mersenne to assure him of the recognition he had for all his good offices, and to testify to him that it was not for him that he had wished to make himself obscure, but for certain envious people who would have boasted of having known without his help the same things that he had written.

Besides these clarifications on some places proposed by Mr. Des Argues, he consented that a Dutch gentleman of his friends undertake a regular introduction of all his geometry to facilitate the understanding of it to all kinds of readers.

He sent it to Paris from the end of the month of May or the beginning of June, and it was found so excellent and so short, that it was believed that he was the author of it.

He judged that it was his duty to disabuse those who were in this opinion, and to make his friend fully enjoy the reward due to the author of this work.

Father Mersenne, at the solicitation of various people, asked Mr. Descartes for permission to have this introduction printed in Paris. Mr. Descartes having taken charge of speaking to the author about it wrote back to the father that this gentleman did not desire that it be printed, unless one wanted to be content to have only a dozen or two copies printed for those to whom this father would want to give copies; which would have been even more convenient than having it transcribed. But if it was a question of a public printing, the gentleman testified that he would prefer to have it done himself in Holland; and that in that case, he would still want to add many things to it; which he offered to do with time.

The most skillful complained of the brevity of this writing: and Mr. Descartes taking the defense of the author, believed he gave them satisfaction, by representing to them that it was not a commentary but an introduction. We must believe that it was rather the view of this writing than the bad disposition of his envious people who prevented him from mending his geometry, as he testified to having had the intention of doing in favor of the common readers, to make it more proportionate to their understanding. As for the introduction, we do not see that any other copies were made than handwritten ones. This is perhaps what led Mr. Bartolin to publish another of his own composition, in which he studied mainly to level the difficulties that algebra gives rise to, of which Mr. Descartes made the key to his geometry.

It is still to the year 1638 that the public is indebted for the excellent notes that Monsieur de Beaune, counselor to the presidial of Blois, made on the geometry of Mr. Descartes.

De Beaune did not see anyone before him on the ranks for mathematics; and he found himself side by side with Messrs. de Fermat, Mydorge, Hardy, de Roberval, and the other geometricians who passed for the first of the century. This is what gave a new luster to the treatise of Mr. Descartes, and which wonderfully increased the consideration of it with those, who could not understand it, or who could not estimate its price by themselves.

De Beaune sent his notes to Mr. Descartes towards the end of the same year. He read them with an attention mixed with an unspeakable pleasure, which increased until the end of the reading, all the more so as he did not meet anything there that was not perfectly in conformity with his thought. It is one of the rarest singularities that one has yet been able to notice in the republic of letters, which is filled with commentators, scholiasts and translators, but which is barely in a state to produce one of each kind who has been able to legitimately deserve until now from his author the honor that Mr. de Beaune received from Mr. Descartes on this occasion.

He wrote about it to Father Mersenne, to testify to him the satisfaction he had of having finally found the man that Providence “seemed to have prepared to strengthen his geometry against the rejections of the ignorant and the censures of the envious.” He told this father, to show him what were the foundations of his recognition, that there was not a single word in the notes of Mr. de Beaune that was not “entirely according to his intention” and that he had seen very well in his geometry the constructions and the demonstrations of the plane and solid places, of which the others said that he had only put a simple analysis.

He sent the same thing to Mr. de Beaune in a letter of thanks that he addressed to him the same day through the same father. Besides the joy he had of seeing that he had so precisely taken his thought and his sense everywhere, he still admired the penetration with which he had been able to recognize things that he had only put in his geometry in a very obscure way. The ordinary scholiasts who often think of their own glory rather than that of their authors are delighted to find in explaining them the opportunity to correct them and to straighten them, in order to be able to boast of having outbid them. Mr. de Beaune made it known in this encounter that he was very far from a similar passion, if it was not by compliment that Descartes made him believe, that he had noticed that he had had the intention of excusing in his notes the faults of his geometry, rather than discovering them.

This is what he took for a sincere testimony of his affection for which he thanked him, adding that he would not have thanked him less if he had noticed them, because of the utility that he could have derived from them. To show him that he did not flatter himself to the point of not recognizing any lack, he made him a detail of some places to which he could have made additions or retrenchments.

De Beaune had sent him at the same time some of his reflections on the curved lines with some difficulties of which he begged him to give him the solution.

Descartes had an interest in satisfying him on this point “better and sooner” than any other. This is what made him use a very extraordinary diligence to prevent those of France, to whom Mr. de Beaune could have proposed the same difficulties to solve. He therefore sent him what he had found concerning his “curved” lines: and he told him that the property of these lines of which he had sent him the demonstration had seemed so beautiful to him, “that he preferred it to the quadrature of the parabola found by Archimedes.”

Finally, to make him feel the effects of the most sincere friendship, he wanted Father Mersenne to give him not only the communication of the mathematical lawsuit he had with Mr. de Fermat, but also of the objections he had recently made on the new book of Galileo concerning mechanics and local movement, very recently printed in Leiden. These observations had been sent to Father Mersenne on the first day of October of the year 1638 in a long letter, where he also answered him on various subjects, and among others on the book of Mr. Bouilliaud concerning “the nature of light,” printed in Paris six or seven months before, and on the little solidity he claimed to have noticed in the judgment that this learned mathematician (whom he esteemed a lot elsewhere) made of his philosophy by confusing it with that of Epicurus and Democritus.

Descartes still had other friends of the same rank as Mr. de Beaune, to whom it was neither free nor honest to refuse the solution of the difficulties they could not apparently hope for from another. The main ones of those who gave him exercises of this nature during this year, were Monsieur de Sainte Croix, and Monsieur Frenicle, whom he was accustomed to calling Mr. de Bessy.

These two gentlemen had the reputation of being the first arithmeticians of the century, and Mr. Descartes did not consider their friendship less than their mind and their knowledge. Mr. de Sainte Croix above all seemed to apply himself to proposing to him only the questions of the finest subtlety:

Descartes seeing by the nature of these questions with whom he was dealing, considered himself very happy to be able to get out of all these difficulties to his liking.

So that he did not hesitate to often assure Father Mersenne that he considered the satisfaction that Mr. de Sainte Croix showed with his answers as a very particular grace, for which he claimed to be indebted to him all his life, because he was only half content himself with most “of the solutions he sent to Mr. de Sainte Croix.”

Their trade was not always limited to operations of arithmetic and algebra; it also extended to the care of their domestic affairs.

De Sainte Croix had testified to being in trouble with a boy who could relieve him in his studies of mathematics in the capacity of secretary or copyist.

Descartes seemed to have then on his hands the young Gillot who had been formerly to him, as much in the same capacity as in that of valet de chambre; and he was currently looking to establish him in Paris. On leaving Mr. Descartes, Gillot had gone to England, from where his relatives withdrew him, when he was beginning to succeed in the profession he was making of teaching mathematics privately. He had returned from there to Mr. Descartes in Holland, and he had set himself to teaching the same sciences to various officers of the army of the Prince of Orange.

But this employment being quite inconstant and caducous, Mr. Descartes gave without much deliberation into the opening that was made to him by Father Mersenne, who proposed to him the condition of Mr. de Sainte Croix, in which he followed the resolution he had taken not to neglect the first opportunity he would have to fix the state of Gillot, and to give to the city of Paris a man capable of teaching his method in general, and his geometry in particular. For he understood one and the other better than any of the other mathematicians, having had the leisure to study the very spirit of Mr. Descartes when he lived under him. He had even recently given marks of a more than ordinary capacity to Mr. de Fermat, when Mr. Descartes, indignant at the way in which this one dissimulated that he had been satisfied with his answer to the theorem of numbers that he had proposed to him, ordered him, by an appearance of contempt or indifference for Mr. de Fermat, to answer him concerning the broken numbers, in conformity with what he had demonstrated concerning the whole numbers; on the centers of gravity of various figures, and other most difficult questions. Mr. Gillot had succeeded so well that Mr. de Fermat was obliged to admit that he knew less in this point than “the schoolboy of Mr. Descartes.” (This is how he called Gillot; “or at least that he had received a complete satisfaction by his answer.”)

Gillot, according to the judgment that his master made of him, was a very faithful boy, of a very good mind, and of a very lovable nature. Although he had never been to college nor learned liberal arts, he did not fail to understand a little Latin and English. He knew French as if he had never left his country, and Flemish as if he had always lived in Holland. He possessed arithmetic and geometry perfectly, and he knew enough of the method of Mr. Descartes to learn alone and by himself everything that could be missing in the other parts of mathematics. Having acquired so many talents, he was no longer in a state or even of an age to reduce himself to a simple servitude. This is why Mr. Descartes who had always distinguished him a lot among his domestic staff, told Father Mersenne clearly that Mr. de Sainte Croix could take him with him as a man of letters or a secretary: but that he should not expect subjections from him as from a valet, because having always lived with people, who although above him, had not failed to tolerate him often “as a comrade,” he had never become accustomed to these subjections. He also had him advised not to demand from Gillot all the civilities that were practiced in Paris more than from a foreigner who would never have been raised there, and not to keep him too long on the difficult operations and calculations of numbers, for fear that he would be discouraged, because it is a very unfruitful work and which needed too much patience for a lively mind like that of Gillot.

Mr. Descartes judged no less advantageously of Mr. Frénicle than of Mr. de Sainte Croix. He testified to Father Mersenne by a letter of August 23 that what he had sent him was more than sufficient to let him know that his “arithmetic must be excellent, since it led him to things where analysis has a lot of trouble reaching.”

This judgment is of all the greater weight as Mr. Descartes was less prodigal of praises, especially writing to Father Mersenne, to whom he was accustomed to confiding his thoughts without “any other precaution than the discretion of the father.” Thus it was not by compliment that answering towards the same time to a letter from Mr. Frénicle, he testified to him some surprise to see that he was more learned in the science of numbers than he would have believed it possible without the help of algebra, which however Mr. Frénicle did not use. This is what would have excited in him the desire to be able to confer with this skillful arithmetician if he had considered himself capable of it at the time, or if it had been a study to which he had applied himself. But, he says, “I know so little of arithmetic (of this kind) that it is not yet a year since I was ignorant of what is called the aliquot parts of a number, and that I had to borrow a Euclid to learn it on the subject of a question that had been proposed to me.”

This declaration was no doubt an effect of this inviolable sincerity that reigned in the speeches and in the writings of Mr. Descartes, and which made him consequently admit that the lack of attention to some of the calculations of Mr. Frénicle had made him fall into some mistakes that he had recognized since. It was only the complaisance for his friends and the consideration for the merit of those of the rank where Messrs. de Sainte Croix, Frénicle, de Beaune, Des Argues, etc. were near him, who were capable of making him return to the operations of algebra and geometry, to which he had renounced to seek something that was more useful to man. Friendship did not allow him to shake this yoke, but he did not fail to secretly try with Father Mersenne the means of getting rid of it without displeasing them. The shortest of these means was to ask this father to break this trade.

The answer he made to the numerical questions of Mr. de Sainte Croix in the month of June 1638 had so tired him, that he begged this father not to send him any more of any nature whatsoever. For, he says, “when I have received them, it is difficult for me to abstain from seeking them, especially if I know that they come, like these, from some person of merit. And having proposed to myself a study for which all the time of my life, however long it may be, would not be enough, I would do very badly to use any part of it for things that do not serve it. But besides that, as for numbers, I never claimed to know anything about them; and I have exercised myself so little, that I can say with truth, that although I once learned division and the extraction of the square root, it has been more than eighteen years since I no longer know them: and if I had to use them, I would have to study them in some book of arithmetic, or try to invent them just as if I had never known them.”

He tried to get rid of the others with the same honesty and under similar pretexts; so that after having gradually weaned his main friends from proposing sterile problems and objections to him, he took little trouble to please or displease those who only sought to make a name for themselves with the skillful people by a mathematical trade with him of which they could boast.

Thus, tired of bearing the burdensome quality of oracle, he almost entirely dispensed himself from answering before the end of the year 1638; and he was content to make a triage of the best objections that had been made to him until then, and of the most beautiful problems that had been proposed to him to have them printed with his answers, when it would please him to whom it belongs to dispose of all things.