Fermat Versus Descartes

The response that Mr. Descartes made to Mr. De Fermat’s objections on his dioptrics, and that he had sent to Father Mersenne as early as mid-December of the year 1637, had nothing in its style or manners that could cause the slightest trouble to Mr. De Fermat, or give the slightest scruple to this father. So he did not hesitate to send it to him from the same hand that he had received it: and a few days later he sent to Mr. Descartes another treatise by Mr. De Fermat who was beginning to fear that Mr. Descartes only knew half of what he knew how to do in mathematics. This new treatise was entitled “de locis planis ac solidis” (on plane and solid loci). It was an analytical writing concerning the solution of plane and solid problems: and Mr. De Fermat was happy that Father Mersenne addressed it as if on his own initiative to Mr. Descartes, without testifying that it was on the part of the author, so that it would not appear that he would have only worked on plane and solid problems, after having seen what Mr. Descartes had written about them in his geometry.

Mr. Descartes told Father Mersenne as early as January of the following year that he had received this new writing; and he sent him back at the same time the original of Mr. De Fermat against his dioptrics, because this father had indicated to him that it was without the author’s knowledge that he had sent it to him.

He did not use the same with regard to his writing “de maximis et minimis”, that is to say, of the greatest and the smallest of all quantities, under the pretext that it was a counselor of his friends, and not Mr. De Fermat himself who had given it to this father to send to him. I believed, he said in his letter to this father, that I should retain the original of this writing, and be content to send you a copy of it, given mainly that it contains faults that are so apparent, that he would perhaps accuse me of having supposed them, if I did not retain his hand to defend myself from it. In effect, according as I have been able to judge by what I have seen of him, he is a lively mind, full of invention and boldness, who has in my opinion precipitated himself a little too much, and who having acquired all at once the reputation of knowing a lot in algebra for having perhaps been praised for it by people who did not take the trouble, or who were not capable of judging it, has become so bold, that he does not bring, it seems to me, all the attention that would be needed to what he does.

Mr. Descartes accompanied this letter with the response he had made to Mr. De Fermat’s treatise “de maximis et minimis”, and he told Father Mersenne that he would be very happy to know what this author would say both about this response and about the one he had previously addressed to him concerning the objections where the demonstration against his dioptrics. It is true that both responses seemed to be only for Father Mersenne, if one refers to their address: but Mr. Descartes would have been very annoyed if Mr. De Fermat had not seen them. He therefore asked this father to send them to him incessantly, adding that he had not wanted to name Mr. De Fermat there, so that he would have less confusion of the faults that he had been engaged to remark there, not in the design of doing anything that was shocking or disagreeable to Mr. De Fermat, but only to defend himself.

And because, he said, that Mr. De Fermat could boast to my prejudice in his writings or in his discourses, I believe that it is appropriate that several also see my defenses. This is why I ask you not to send them to him without retaining a copy. If he speaks to you of sending you still other writings to have me see them, please ask him “to better digest them than the preceding ones.” Otherwise, you would oblige me not to take the trouble to address them to me. For between us, if, when he wants to do me the honor of proposing objections to me, he does not want to give himself more trouble than he took the first time, I would be ashamed to see myself reduced to the trouble of responding to so little; and on the other hand I could not honestly dispense with it, when it would be known that you would have sent them to me. I will be very happy that those who will want to make me objections do not hurry, and that they try to understand all that I have written before judging a part. For the whole holds together, and the end serves to prove the beginning. But I promise myself that you will always continue to frankly tell me what will be said of me, whether good or bad. Besides, everyone knowing that you do me the favor of loving me as you do; nothing is said of me in your presence that one does not presuppose that you warn me of it: and thus you can no longer abstain from it without doing me wrong.

Whatever defects Mr. Descartes found then in Mr. De Fermat’s first writings, he did not fail to already perceive marks of the ability of this illustrious unknown: and the esteem that he conceived for his merit grew as their dispute increased. He still believed himself then dispensed from the regards and the ménagements that it would have been necessary to take if they had known each other, or if they had written immediately to each other. This is what Mr. De Fermat was obliged to excuse in the suite, when they came to clarifications on their conduct on both sides.

While Mr. De Fermat in the midst of the occupations of the palace and his domestic affairs was applying himself to make a reply to the response that Mr. Descartes had made to his objections on the dioptrics, Father Mersenne received Mr. Descartes’s remarks on the treatise “de maximis et minimis.” But instead of sending it straight to Mr. De Fermat according to the intention of Mr. Descartes who had asked him since he had learned that this treatise was from him, he judged it appropriate to show them to two of Mr. De Fermat’s particular friends, who were in Paris. One was Mr. Pascal, president in the court of aids of Auvergne, the other was Mr. De Roberval, professor of mathematics “in the chair of Ramus.”

These gentlemen having learned that Mr. De Fermat was occupied with the composition of his reply to Mr. Descartes on matters of dioptrics, and fearing that Mr. Descartes would want to take advantage of the embarrassments and delays of Mr. De Fermat, believed they should espouse the quarrel of their friend. They dispensed him for his relief from the care of pursuing the quarrel of geometry, and they took charge of responding to Mr. Descartes in favor of his treatise “de maximis et minimis” against the response or the remarks that Mr. Descartes had made to it. They sent (but always through the channel of Father Mersenne) their response to Mr. Descartes, before the reply of Mr. De Fermat on the dioptrics had come. Mr. Descartes read this response of the two friends with enough surprise. He praised their zeal, approved the dispositions of their heart, and judged Mr. De Fermat happy to have been prevented by such help in so great a need. He could not even prevent himself from conceiving esteem for the capacity of which he saw marks in the writing of these two characters: but he found that if they had well filled the duties of friendship with regard to Mr. De Fermat, they had acquitted themselves quite badly of the commission they had taken to discharge and defend him. We have lost this writing of Messieurs Pascal and De Roberval: at least it has not been possible for Mr. Clerselier to recover it, to be able to insert it among the pieces serving this famous lawsuit that he has thrown pell-mell in the third volume of Mr. Descartes’s letters. It is annoying that we can only judge the goodness of this piece on the testimony of Mr. Descartes, that is to say, of the interested and suspect party: but the inconvenience will not appear irreparable to those who will want to examine the pieces, or treatises that followed it.

It will be enough to remark that the piece although written in the name of two friends of Mr. De Fermat, was all in the style of Mr. De Roberval, and that Mr. Pascal had no other part in it than that of consent and communication. At least it was the opinion of Mr. Descartes, who attributed it all to Mr. De Roberval alone.

To tell the truth the politeness and the other advantages of the education that Mr. Pascal had on Mr. De Roberval did not allow that neither Mr. Descartes, nor those who had the honor of knowing this illustrious magistrate, had this thought of him. They knew enough that the style of the language or of the pen being only the expression of the soul, Mr. Pascal would have chosen to write against Mr. Descartes manners more in conformity with himself. The merit of this man was already making itself known then by many other places than by that of mathematics. The qualities that compose and that perfect the magistrate and the good man, were already making him consider as a person whose services should not be limited to his province: and Mr. Descartes who did not have bad discernment did not hesitate to flatter himself with his friendship in the very time that he saw him engaged in the party of his adversaries. Mr. Pascal was from Clermont in Auvergne and from one of the good houses of the province. His father had been treasurer of France in Riom; and his mother who similarly bore the surname of Pascal was the daughter of the seneschal of Auvergne in Clermont. He was eight years older than Mr. Descartes, and he died a year after him. He had a son who was still only in the fifteenth year of his life, who was already distinguished among the old mathematicians, and who then had a part in the esteem and friendship of Mr. Descartes.

The education of this son had served as a motive for the father to leave the province after having had his charge of president passed to one of his brothers, and to retire to Paris as in a place favorable to his designs. They succeeded so well for him, that after having put this son in a state to erase the others, he was erased himself.

Descartes supposed that Father Mersenne would have sent his response on the treatise “de maximis et minimis” to Mr. De Fermat: and he was surprised to learn by a letter from this father dated February 8 that he had delayed sending it to him, on the fact that two of his friends had told him that he had erred in some place. In which he saw a new trait of the ordinary credulity of the father, who had been good enough to let himself be persuaded by the friends of his party to his prejudice; and who had not noticed that they were only diverting him to gain time, and to prevent him from letting his response be seen “by others.” Be that as it may, the writing that the two friends of Mr. De Fermat had made against this response to defend the geometric treatise “de maximis et minimis”, was refuted by Mr. Descartes before the end of February: and having finally received the reply of Mr. De Fermat concerning the dioptrics, he made various responses to it in the same month, which he addressed to his main friends, one to Mr. Mydorge, another to Mr. Hardy, a third to Father Mersenne.

This reply of Mr. De Fermat to the response that Mr. Descartes had made against his objections on his dioptrics was addressed to Father Mersenne as the other pieces that had preceded it, and it is found printed among Mr. Descartes’s letters. The author protested at the entrance that it was not by envy or by emulation “that he continued this little dispute,” but only to discover the truth.

Of which he presumed that Mr. Descartes would not bear him ill will, all the more so as he knew “his very eminent merit.” This is, he said to this father, what I wanted to make a very express declaration of to you at the head of my reply; and I will add, before entering into the matter, that I do not desire that my writing be exposed to a greater light than that that a familiar conversation can suffer, of which I confide in you.

This restriction thought to put Mr. Descartes in anger after the request he had made to Father Mersenne not to receive any writing from anyone to send to him, if those who would present it to him did not write at the bottom that they consented that he had it printed with his response. He had made an exception to this rule only for the Jesuits, the priests of the Oratory, and the honest people who would be recognized as having no other passion than that of seeking the truth. And if he had not resisted his bad humor, he would have excluded Mr. De Fermat from the number of these last, despite the characters of honest man with which his writings were marked.

He was already closing the packet where the response to Messieurs Pascal and De Roberval was on the geometric treatise “de maximis et minimis” of Mr. De Fermat, when the reply of this one concerning the second discourse of his dioptrics was given to him. He first read the first article, and he was put off by its reading by the condition that the author seemed to require of Father Mersenne, not to have it printed. But having then reflected on himself, he resumed the reading of it with a more settled sense.

The fruit of this reading that prevented the responses he then made, was that he did not find in this writing a single word that could excuse the faults he had remarked in the preceding objections of Mr. De Fermat, or that had any force against what he had responded to him. He pretended that in each article of what he was objecting anew, he was making a paralogism, or that he was corrupting the sense of the reasons he had alleged to him, or finally that he had not understood them. This is what he obliged himself to show as clear as day (to use his terms) provided that Mr. De Fermat found it good that the public and posterity were the judge of it, according as he had marked in the discourse of his method. For his leisure was not intended to respond to the objections of individuals, or even to read them, unless by making them public jointly with his responses, they could serve for all those who would have the same doubts.