Dioptrics

The first essay on Descartes’s method is the treatise on dioptrics.

The intention for this treatise was to show us that one can go far enough in philosophy, to arrive by its means to the knowledge of arts that are useful to life.

He omitted nothing of what could be necessary to explain what is most important in optics and catoptrics. But he clarified all this matter in such a solid and new way, that public astonishment, which gave rise to admiration and recognition in minds desirous of learning, produced in some mathematicians a jealousy that only led to animosities and disputes.

They ignited between them a small war, whose consequences have been long and vexing, but nevertheless useful to the public, and glorious to Mr. Descartes.

If this treatise had its adversaries like the others, it also had its defenders and its commentators. Those among them who distinguished themselves during the lifetime of our philosopher will be able to provide matter for the history of his life in the rest of this work.

The treatise that constitutes the second essay of his method is that of the meteors, which he divided into as many parts or chapters as that of dioptrics. He deals there with terrestrial bodies; with vapors and exhalations; with salt; with winds; with clouds; with rain, snow, and hail; with storms, lightning, and other fires that ignite in the air; with the rainbow; with the color of clouds, and with the circles or crowns that sometimes appear around stars; with parhelia or the apparition of several suns. We have remarked elsewhere that this treatise mainly owes its origin to the observation of the parhelia that was made in Rome in March of the year 1629. This occasion had made him interrupt his other studies, to examine this phenomenon: and the satisfaction that he had received from himself on this point, had made him pass immediately to the search for meteors, of which he did not abandon the study until after having put himself in a state of being able to give a reason for it. But he did not subject himself to continuing the work to lead it to its end. The occasions that presented themselves since to make other observations on meteors, furnished him with the matter for some chapters that he only composed a few years later; and he only thought of incorporating them into the rest, when it was a question of putting the treatise under the press. The reading of this work produced in Mr. Descartes the effect that he had hoped for. This effect was none other than the persuasion that he intended to give to everyone of the total difference that was found between his manner of philosophizing, and that which was in use in the schools. In which one can say that he met fewer adversaries for his meteors than for all his other works.

The last of the essays of his method that he wanted to give to the public then is his treatise on geometry, which includes three books, where it is mainly a question of the construction of problems. The author’s intention in this work was to show by way of demonstration that he had found many things that had been ignored before him, and to insinuate at the same time that many others could still be discovered, in order to more effectively excite all men to the search for truth. He had not resolved at first to publish anything of his geometry among the essays of his method, and the printing of his meteors was already beginning, when he thought of working on it. The most skilled mathematicians could not persuade themselves that it was a work done in a hurry: but he did not want us to doubt this fact after having written to a Jesuit father in these terms. My geometry is a treatise that I almost only composed while my meteors were being printed, and I even invented a part of it during that time. But I did not fail to satisfy myself as much and more than I usually satisfy myself with what I write. One would be mistaken in believing that Mr. Descartes had the intention of giving the elements of geometry in this work, which requires other readers than students in mathematics. He had studied in the three treatises that precede this one, to make himself intelligible to everyone, because it was a question of making people understand things that had not yet been taught, or of which the true principles had not yet been given. But seeing that many works of geometry had been made before him, with which he found nothing to complain about, he did not believe he should repeat in his treatise what he had seen as good and very well demonstrated in the others. Far from wanting to make them useless by his work, he solidly contributed to making them necessary, since one must have read them to be able to understand his geometry. This is why he only began where they ended. He suppressed the principles of the greater part of his rules, and their demonstrations. He had even foreseen that many of those who would have read the other geometers, but who would have acquired only a common knowledge of this science, could with great difficulty reach the intelligence of his writing. I know, he said to the doctor Plempius, that the number of those who will be able to understand my geometry will be very small. For having omitted all the things that I judged not to be unknown to others, and having tried to include, or at least to touch upon several things in few words, (even all those that can ever be found in this science,) it does not only require very learned readers in all the things that have been known until now in geometry and in algebra, but also very laborious, very ingenious, and very attentive people.

After all, it was a little out of affectation and malice that he made himself difficult to understand in his geometry: and if it is vexing that he deserved on this point to be put in parallel with Aristotle on the subject of his studied obscurity, it is even more vexing that he finds today so many people who claim that he had more reason than Aristotle to use it in this way. One will judge by what he wrote about it eighteen months later to Mr. de Beaune in these terms. I have omitted in my geometry, he said, many things that could have been added there for the ease of practice. Nevertheless I can assure that I have omitted nothing but on purpose, except the case of the “asymptote” that I forgot. But I had foreseen that certain people who boast of knowing everything would not have failed to say that I had written nothing that they had not known before, if I had made myself intelligible enough for them: and I would not have had the pleasure of seeing the incongruity of their objections. Besides that what I have omitted harms no one. For the others, it will be more advantageous for them to make efforts to try to invent it by themselves, than to find it in a book. For me I do not fear that those who understand it, will impute any of these omissions to me as marks of my ignorance. For I have taken care to put in every encounter what is most difficult, and to leave only what is easiest.

The lack of solidity that appeared in this reason that Mr. Descartes was not ashamed to retail again to Father Mersenne, and to some other of his friends, made his enemies judge that his solitude and his philosophy had not yet entirely purified his passions. What also gave rise to such unfavorable judgments was the good opinion that he appeared to have for his geometry, and that they did not fail to attribute to movements of some secret vanity, at the very time that they joined their voices with those of his admirers to recognize that no greater geometer had been seen since the birth of the world. He would have apparently prevented this slander, if the complaisance for friends to whom he was not in a state to refuse anything, had not engaged him to ingenuously say his thought. I am not very happy, he said to one of them, to be obliged to speak advantageously of myself. But because there are few people who can understand my geometry, and that you desire that I tell you what my opinion is of it, I believe that it is appropriate that I tell you “that it is such that I wish nothing more from it.” I have tried by the dioptrics and by the meteors to persuade that my method is better than the ordinary method: but I claim to have demonstrated it by my geometry. For from the beginning I solve a question there that by the testimony of Pappus could not have been found by any of the ancients: and it can be said that it could not have been either by any of the moderns, since none has written about it, and that nevertheless the most skilled have tried to find the same things that Pappus says in the same place to have been sought by the ancients. This is what the authors of the Apollonius Redivivus, of the Apollonius Batavus, and the others, in the number of whom one must also put Mr. Your counselor, of Maximis et Minimis, have done. But none of these moderns knew how to do anything that the ancients had been ignorant of.

After that, what I give in the second book concerning the nature and properties of curved lines, and the way of examining them, is, it seems to me, as much beyond ordinary geometry, as the rhetoric of Cicero is beyond the a, b, c, of children. Mr. Descartes spoke thus of himself to friends who had his confidence, and whom he believed discreet, without thinking that what prudence holds hidden between friends during life, is often subject to becoming public after the death of one or the other. His envious ones who appeared much more ingenious in ruining his reputation than his friends were in managing it, tried to make him a new crime of the discernment that he had undertaken to make between those whom he believed capable of understanding his geometry, and those whom he did not judge capable of it. He put in the rank of the first Mr. de Méziriac, a gentleman from Bresse of the French academy, who was only three years older than he. He made a very particular case of his genius and his capacity, especially for arithmetic and algebra, which he possessed in a degree of depth that made him equal to Mr. Viète. He explained himself to Father Mersenne towards the month of February of the year 1638 in these terms. I expect a lot from Mr. Bachet to judge my geometry. I regret that Galileo has lost his sight, I persuade myself that he would not have despised my dioptrics. But he could not receive from Mr. de Méziriac for his geometry the satisfaction that he could not hope for from Galileo for his dioptrics: because Mr. de Méziriac lost his life around the same time in the greatest vigor of a man’s age, being barely forty-five years old when he died.

His work on Diophantus of Alexandria is more than enough to justify the esteem that Mr. Descartes had of him: but it is to be believed that the public would have still outbid on this esteem, if it had seen the treatise on algebra of Mr. de Méziriac, and some other manuscripts of this author, the most important of which is that of the thirteen books of the “elements of arithmetic serving for algebra,” written in Latin, and bought from the heirs of Mr. de Méziriac since about fifteen or sixteen years, by a person of the reformed religion, who has not failed to take it out of the kingdom at the time of the revolution of the state where the religionaries were before the revocation of the edict of Nantes.

Besides Mr. de Méziriac there were still in France some other mathematicians that Mr. Descartes esteemed very capable of understanding his geometry. He put in this number his friends Messieurs Mydorge and Hardy, and he did not exclude Mr. de Fermat from it, when he had recognized his skill. He also knew some people in the Low Countries, to the reach of whom he did not judge it disproportionate. Among those who understood it perfectly in Holland, he counted two individuals who made the profession of teaching mathematics to military people, and of whom one was Sieur Gillot who had been for some time with Mr. Descartes. He did not believe the Spanish Low Countries to be devoid of mathematicians skilled enough to understand it. He put in this number Sieur Vander Wegen, a Brabantian gentleman, and Godefroy Wendelin, a canon of Condé in Hainaut and a parish priest of Herk on the confines of Brabant and the country of Liège, a particular friend of Mr. Gassendi: and he wrote to the doctor Vopiscus Fort Plempius, to ask him to let him know the opinion that these gentlemen would have of it. But he preferred no one from any country whatsoever to Mr. de Beaune, counselor at the présidial of Blois, for the intelligence of his geometry. He recognized by a writing that Father Mersenne sent him from him, that he understood it “very well, and that he knew more than those who boasted more than he.” He confirmed himself more and more in this persuasion, and he explained himself to the same father the following year in these terms. The development that Mr. de Beaune made of my solutions serves to demonstrate two things; one, that Mr. de Beaune knows more than those who have not been able to manage it; the other, that the rules of my geometry are not useless, nor so obscure that they cannot be understood, nor so defective that they do not suffice for a man of spirit to do more than by the other methods. For he has understood them without any interpreter, and he uses them to do what your greatest geometers of Paris are ignorant of.

It was certainly a mark of great distinction among the first mathematicians of the century to find oneself without presumption in a state of being able to understand the geometry of Mr. Descartes. Those to whom he had been kind enough to render this testimony himself could be assured of being very profoundly in his esteem: but there was all the less confusion to fear for the others, as the matter was more difficult and more superior to the reach of common minds. Mr. Descartes himself did not claim to take away the title of mathematician from those who could not aspire to the intelligence of his geometry. It is nevertheless vexing for the reputation of the first university of Holland, that he did not find a professor of the public school in mathematics in Leiden who could understand it, not even Jacques Golius, who was his friend besides; but who seemed to distinguish himself more by the knowledge of oriental languages, and especially of Arabic, than by that of the mathematics that he professed. He did not have a better opinion of the professors of Amsterdam. Martin Hortensius of Delpht was undoubtedly the most famous and most skilled, even in the judgment of Mr. Gassendi who knew him very particularly. However, he is named by Mr. Descartes among those who did not understand his geometry. He did not know enough mathematics, and particularly enough algebra for that, and he understood it even less than Golius.

As for the other mathematicians and philosophers of Holland, most of them appeared so far from understanding anything there (if one excepts Sieur François Schooten who studied it since, and Sieur Jean Hudden who was not yet known) that they did not even find a word capable of opening their mouth, although they were excited besides to speak, whether by their own jealousy, or by the ill will of the ministers and other Protestant theologians who did not like Mr. Descartes, and who were not liked by him.

With regard to the mathematicians of Paris and of some provinces of France, whom he suspected of not being able to reach his geometry, it may be that he judged them a little too far. It may also be that he was not too hasty, when he said his thought to Father Mersenne in these terms. Your analysts do not understand anything in my geometry, “and I laugh at what they say. The constructions and the demonstrations of all the most difficult things are there: but I have omitted the easiest, so that their likes cannot bite into it.” But he had perhaps consulted only his resentments in the judgment he made since on the skill of those who found fault with his geometry. He would have undoubtedly not dared to say to another than to Father Mersenne to whom he discovered all his weaknesses, that he did not believe any of his adversaries capable of learning in all their life all that it contains, provided that he was not more skilled than Mr. de Roberval. He had taken sufficient assurances and measures not to be able to be surprised or convinced in his judgment and in his prediction. This is what appeared by the manner in which he treated the famous question of Pappus, a mathematician of Alexandria, living at the time of Theodosius the elder, to which he had testified four years before having employed five or six weeks to find the solution. The solution of this question, which requires a man consumed in the analysis of the ancients and in the algebra of the moderns had been attempted by Euclid, and pursued by Apollonius, without either Euclid, or Apollonius, or any of the mathematicians until Pappus, or finally those who had appeared in the world since Pappus until Mr. Descartes had come to the end of completing it. He did not believe he should lavish on the public the discovery that he had made of it, so as not to give occasion to the mathematicians of Paris who were envious of him, to snatch this little honor from him, and to boast after they would have owed it to him, of having learned it elsewhere, and already before, independently of him. The good thing about this affair, he said, concerning this question of Pappus is, that I have only put the construction and the demonstration, without putting all the analysis, which these gentlemen imagine that I have put alone, in which they show that they understand it very little. But what deceives them, is that I make the construction of it like architects make buildings, by prescribing only all that must be done, and leaving the work of the hands to the carpenters and masons. They do not also know my demonstration, because I speak there by A, B: which does not however make it in anything different from that of the ancients, except that in this way I can often put in one line what it took them to fill two or three pages. And for this cause it is incomparably clearer, easier, and less subject to error than theirs. For the analysis, I have omitted a part of it, in order to retain the ill-intentioned minds in their duty. For if I had given it to them, they would have boasted of having known it long before: whereas now they will not be able to say anything about it that does not make their ignorance known.