Fermat Versus Descartes

However, Fermat was starting to tire of the dispute.

• He feared that Roberval’s zeal would prolong it.

So he:

• left unanswered what Descartes had written against his last reply regarding optics
• wrote to Father Mersenne to ask him to make his peace with Descartes and meet him.

On the other hand, Mydorge and Hardy were suffering with “difficulty that a man of the merit and rank of Mr. de Fermat was getting so badly mixed up with Mr. Descartes,” were thinking of ways to reconcile them and to change their dispute into a perfect correspondence, the fruits of which could be tasted in a mutual communication of their lights.

They spoke of it to Father Mersenne, who wrote about it to Mr. Descartes even before he had received the last answer to the second writing of Mr. de Fermat’s two friends.

Descartes did not conceal from this father that this proposition was very agreeable to him, and he wrote back to him in these terms at the very time that Fermat was asking for his friendship through the mediation of the same father, without him yet knowing anything of his disposition.

“You say that these gentlemen who have taken cognizance of our conversation want to make Mr. de Fermat and me friends, you will assure them, if you please, that there is no one in the world who seeks nor cherishes the friendship of honest people more than I do; and that I do not believe that Mr. de Fermat can hold a grudge against me for having frankly said my opinion of his writing, after having provoked me to it like a gentleman. Nothing is more contrary to my humor than to take up others: but I could not avoid this occasion after his challenge, except by despising him: which would undoubtedly have offended him more than my answer.”

He wrote at the same time to Mr. Mydorge and Mr. Hardy, to thank them for the goodness with which they had supported his side regarding the rule “of maximis” of Mr. de Fermat, and for the happy turn they had given to this dispute to end it to his advantage and to the liking of both parties.

Fermat, in making these steps towards Mr. Descartes, did not consider himself as a vanquished and disarmed man who would have had no resource but in the clemency of the victorious. And Mr. Descartes on his side regarded the request that Mr. de Fermat made him for his friendship as a fruit, not of his victory, but of a peace that was equally glorious and useful to both. Although their peace was made without conditions, Mr. de Fermat who did not believe he should neglect the things that could serve for his justification, wrote to Father Mersenne to mark to him that he had been deceived “by the first answer that Mr. Descartes had made him”; and that having imagined to find some acrimony in his expressions, he had believed he should imitate his style to try to support himself against an adversary of this importance. Father Mersenne did not fail to send this letter of Mr. de Fermat to Mr. Descartes, who wrote back to this father in these terms.

**“I have seen what you were pleased to communicate to me of the letters that Mr. de Fermat wrote to you. And first for what he says of having found more acrimonious words in my first paper than he had expected, I most humbly beg him to excuse me, and to think that I did not know him. But his writing “de maximis” coming to me in the form of a cartel from a man who had already tried to refute my optics even before it was published, as if to stifle it before its birth, having had a copy of it that I had not sent to France for this subject: it seems to me that I could not answer him with softer words than I did, without showing some cowardice or some weakness. And as those who disguise themselves at carnival are not offended that one laughs at the mask they wear, and that one does not greet them when they pass by the street, as one would do if they were in their usual clothes: so he should not, it seems to me, find it bad that I answered his writing quite differently than I would have done to his person, which I esteem and honor as his merit obliges me to. I was not surprised that he approved the reasons of Messrs. Pascal and de Roberval, for civility did not allow him to use them otherwise: and in fact I do not know that one could have given better ones for the subject in question.

But I was astonished that Mr. de Fermat, not adding other reasons to those of these gentlemen, wanted to suppose that those ones have fully persuaded me; and to use this pretext to abstain from sending the tangent of the curved line that I had proposed to him. For I have testified enough by all my letters that they had not answered any of my objections directly; and that it is not a mark of the goodness of his rule “de maximis”, to say that it does not succeed in the example that I gave, which is the only reason that they have brought.

For all the other examples that you have told me at various times were sent to you by Mr. de Fermat, even if they were true, which I suppose since I have not seen them, they cannot prove that his method is generally good, but only that it succeeds in certain cases, which I have never had the intention of denying.

Civility would oblige me to no longer speak of this matter after having tacitly given me the hands, if he did not assure notwithstanding that, that his method is incomparably simpler, shorter, and easier than that which I have used to find the tangents. To which I am obliged to answer that in my first writing, and in the following ones, I have given reasons which show the contrary; and that neither he nor his defenders having answered anything to them, they have confirmed them enough by their silence. Although one can receive his rule for good being corrected, it is not a proof that it is so simple or so easy as that which I have used, unless one takes the words of “simple and easy,” for the same thing “as industrious” in which it is certain that it “excels”, because it follows only the way of proving which reduces “ad absurdum”, as I have warned from my first writing. But if one takes them in a contrary sense, one must also judge the contrary by the same reason. As for “being shorter”, one can refer to the experience that it will be easy to make of it in the example of the tangent that I had proposed to him. If I do not add anything more it is by the desire I have not to continue this dispute: and if I have put here something that is not agreeable to Mr. de Fermat, I most humbly beg him to excuse me, and to consider that it is the necessity of defending myself that has constrained me to it, and not any design to displease him. He will also have the goodness to excuse me if I do not answer his other questions, it is an exercise to which I entirely renounce.”**

Mr. de Fermat having received from Father Mersenne all the assurances that he could wish for on the part of Mr. Descartes, finally gave himself the satisfaction of writing to him directly to offer him his friendship and his services. One can put the acquisition of such a friend in the number of the best fortunes of Mr. Descartes. He knew perfectly the price of such an important friendship, and “he was so sensitive to it that he had no terms passionate enough to thank him for it.” It is all to say that he believed he had conquered a Bradamante, without thinking that it was presuming too much of himself to compare himself tacitly to a Roger.

It was not enough for Mr. de Fermat to have been paid with the friendship of Mr. Descartes in reward for his own: he still wanted to assure himself of his esteem, knowing what it could be worth to him in the world.

He wrote to him again to ask him to mark him precisely how far he could carry the opinion he should have of himself. And to engage him not to use flattery in his judgment, he assured him that he would make it a rule for himself to measure himself next to others. Mr. Descartes answered him towards the end of July in these terms. “I know well that my approval is not necessary to make you judge what opinion you should have of yourself: but if it can contribute something to it, as you do me the honor of writing to me, I believe I am obliged to admit here frankly that I have never known anyone who has made me appear that he was so learned in geometry as you… I beg you to believe that if I have heretofore testified not to quite approve of certain particular things that came from you, that does not prevent the declaration that I have just made from being very true. But as one more carefully notes the small straws of diamonds than the largest spots of common stones, so I believed I had to look more closely at what came from you, than if it had come from a less esteemed person. The same reason consoles me to see that good minds study to take up the things that I have written, so that instead of holding a grudge against them for it, I feel obliged to thank them for it: and this consideration alone would be enough to make me what I am to you besides.”

Mr. Descartes, in order not to be half satisfied with his reconciliation, wanted it to extend also to the two friends of Mr. de Fermat, who had taken the defense of his geometric writing “de maximis et minimis.” From the month of April he had believed he could hope for something from it, on the ground that “Father Mersenne had told him that these gentlemen did not have such a particular connection with Mr. de Fermat as he had been led to believe.” This being the case, he had not doubted that they were not disposed to prefer truth to the personal interests of Mr. de Fermat, and that they would not surrender to it as soon as they would recognize it. That’s why his conscience having nothing to reproach him for on their subject, and not believing to have put a syllable in his answer that could disoblige them, he asked Father Mersenne to testify to them that he sought nothing so much as the friendship of honest people, and that by this consideration he made a lot of case of theirs.

These gentlemen, that is to say, Mr. de Roberval in the name of the two, because Mr. Pascal had withdrawn, already regarding him as a friend whom they intended to treat with honesty, seemed to want to establish the trade of their friendship in the proposition of various geometric questions, which they could not solve, and which they believed could not be solved by his method. Mr. Descartes found that this side was not advantageous for him. For there is a kind of law established among geometricians, which forbids proposing to others questions that they cannot solve themselves, since there are impossible ones, like the quadrature of the circle, etc. In addition, there were questions which, although possible, nevertheless went beyond the columns he had posed, not that it required other rules and more mind, but because it demanded more work. Of this kind were those of which he had spoken in his answer to Mr. de Fermat on his writing “de maximis et minimis,” to warn him that if he wanted to go further than him, it was by there that he had to pass. Besides, there are some that belong to arithmetic rather than to geometry, like those of Diophantus, and two or three of those that Mess. Pascal and de Roberval had mentioned in their writing, which he did not promise to solve all. It is not that these last ones were more difficult than those of geometry: but it was enough to take away the thought of working on them that they were useless, or that they were not the share of a mind of his kind, but of those who, not being able to take a superior flight, subject themselves by an obstinate work to examine the sequence of numbers.

Father Mersenne who made it a pleasure to reconcile minds after having excited them against each other, asked Mr. Descartes to suppress in these favorable circumstances of reunion, a writing made by one of his zealous partisans against Messrs. de Roberval and de Fermat for his defense, because he feared that this would distance and sour minds so well disposed to reconciliation. Mr. Descartes replied to this father that he had great reason to give him this advice; that when the author of this writing would not have allowed him to suppress it he would not have failed to do so; that otherwise he would have participated in the fault of this author; that for the rest, he had no right to have slanders printed, except those of which he could be obliged to justify himself, or which it would be necessary to refute.

The heart of Mr. de Roberval did not seem made for that of Mr. Descartes, also they could never remain perfectly united. It was not the same for that of Mr. de Fermat, of whom one can say that Mr. Descartes was the master for the rest of his days. But what is quite ordinary in friends who have different lights, it is certain that their minds did not always follow the law of their hearts. Mr. de Fermat, persuaded as before of the excellence of his method, had not agreed to the exceptions that Mr. Descartes had made to it to make it such. He continued without prejudice to their new friendship to publish his complaisances for the invention of this method, and he even seemed to attribute to some lack of attention what Mr. Descartes judged that one could touch up on it. Father Mersenne did not fail to give notice of this conduct to Mr. Descartes, to whom it appeared quite incomprehensible. He wrote back to this father on the 23rd of August and made him a historical summary of their dispute to then make him the judge of this conduct. “You sent me,” he says, **“last winter on behalf of Mr. de Fermat a rule to find “the greatest and the smallest” in geometry. I believed it defective, and I verified it by the very example he had given. But I added that by correcting it one could make it good enough; although it was not as general as its author pretended. I showed nevertheless that one could not use it in the way it was dictated to find the tangent of a certain line that I named: and several reasons made me judge then that he had found it “by guesswork.” Having seen his writing, I judged by his steps that he wanted to test himself in geometry. But not believing that this subject was suitable enough for this design, because it was neither of the most difficult nor of the most important; I took the liberty of proposing to him three or four others, which are all things to which he would undoubtedly have answered since, if he had a way. Instead of that, someone in Paris who favored his party having seen my writing in your hands, tried to persuade you that I had mistaken, and asked you to delay sending it to Toulouse. You told me about it, and I assured you that I feared nothing on that side. You sent me some time later an answer made for Mr. de Fermat by this same man from Paris, in which finding nothing else except that he did not want a certain line to be named “the greatest”, he reminded me of these lawyers, who to make a trial last seek something to find fault with in the formalities which are of no use to the substance of the cause. I believed I had to warn you from then on that he only used this procedure to give more leisure to my party to think about answering me; and the event shows enough that my conjectures were true. Bored with the lengths of this small quibble, I have finally told them at length what had to be added to the rule in question to make it true, without for that changing the way it was conceived, and which had made me say that one could not use it to find the tangent that I had proposed.

Since that time, whether what I had corrected in this rule gave him more light, or whether he had more luck than before; finally after six months of delay, he found a way to turn it in a new bias, with the help of which he expresses in some way this tangent. Judge if that is worth the trouble of singing his victory so high. It was nothing easier than to find this new bias; and he could have drawn it from my geometry, where I use a similar means to avoid the embarrassment which makes his first rule “useless in this example.” But he has not satisfied what I had proposed to him, which was not to find this tangent, given that he could have it from my geometry, but to find it by only using his first rule, since he esteemed it so general and so excellent. It would undoubtedly have been more advantageous for him not to talk about this tangent, because the great noise he makes about it gives rise to believe that he had a lot of trouble finding it, and to note that his silence on the other things that I have objected to him is a testimony that he had nothing to answer to them, and that he does not yet know the foundation of his rule. I admit that since he saw what I have said that one should correct there, he can no longer ignore the means of using it. But if he has not had communication of what I have since told Mr. Hardy regarding the cause of the elision of certain terms which seems to be made there gratuitously, he will allow me to still doubt that he knows how to demonstrate it.”**

This is what Mr. Descartes told Father Mersenne secretly regarding the conduct of Mr. de Fermat, but without pretending that it should cause the slightest alteration in their friendship. Instead of insisting more on a subject of such little importance, he preferred to rely on the truth, to the force of which he did not despair of one day seeing the mind of Mr. de Fermat, and that of Mr. de Roberval, yield.

However, Mr. de Fermat did not want to diminish the good opinion he had once conceived of his rule and his method. He was right no doubt to esteem it after having corrected it on the reflections that Mr. Descartes made him do: but he made it known that he was a man by pretending that it was the same as before, as if he had not brought any change to it. This produced from time to time light disputes, not with Mr. Descartes who owed his time and his talents to something else than dispute, but with the young Gillot whom Mr. de Fermat called his schoolboy; with Mr. Chauveau his old classmate at the college of La Flèche; and with other mathematicians of Paris, who since this outburst declared themselves Cartesians from day to day, in spite of the jealousy of Mr. de Roberval. Mr. de Fermat still sought something else to wish for in the geometry of Mr. Descartes. Mr. Chauveau who did not have the indifference of Mr. Descartes on these matters believed he had to stop these liberties in Mr. de Fermat, against whom he wrote without consulting Mr. Descartes, who would undoubtedly not have allowed it. He nevertheless had the consideration of not having his answer printed, and he was content to note to several of his friends in private the faults with which he charged Mr. de Fermat, and the excellence of the writings of Mr. Descartes.

Mr. Des Argues whose skill was generally recognized by the geometricians of the time, also took the defense of Mr. Descartes against Mr. de Fermat in a rather long dissertation that he addressed to Father Mersenne in the form of a letter written on April 4 of the year 1638.

But as he seemed to be the common friend of all the illustrious scholars who had entered into this famous quarrel, one is not surprised to see that he says a lot of good, not only of Mr. Mydorge and the other partisans of Mr. Descartes, but also of Mr. de Fermat, of Mr. Pascal, and of Mr. de Roberval his adversaries, of whom he wished with all his heart that the merit was finally rewarded with the friendship of Mr. Descartes.

The state of the dispute having then made itself known in foreign countries, it is claimed that there has been almost no skillful geometrician who has not entered into the party of Mr. Descartes. This is what was noted mainly in Holland, where one even saw the learned Mr. Jean Hudde write expressly on this subject several years after the thing seemed to have subsided between Mr. Descartes and Mr. de Fermat. But we cannot conceal what the late Father Prestet of the Oratory, one of the most skillful mathematicians of our days, has done in these last years for the defense of Mr. Descartes. If one believes this father, the general method he has given to determine what are “the greatest and the smallest quantities”, is the most beautiful and the best of all those that have been invented. He admits that it does not appear at first, and that it is only with a little attention that one can see the excellence and the simplicity of it, because he speaks of it rather lightly and without “giving it a name.” This is what had deceived Mr. de Fermat, who had inappropriately taken up Mr. Descartes for not having said anything on a subject of this importance, for lack of application or meditation on this part. Mr. de Fermat having proposed at the same time his method of “greatest and smallest quantities” as a rare and new invention, it had been received with applause by Messrs. Pascal and de Roberval. But Mr. Descartes having examined it more closely than them, had found it defective and false in various encounters. Although he had shown the means to correct it and to make it just, he had not however been able to approve it entirely, because it could not serve to conclude except by the imperfect way of proving “which reduces to the absurd.” But at the same time he had neglected to clarify his own, and it was perhaps with a little too much pride or indifference that he never wanted to produce other examples than those that were already in his geometry. Mr. de Fermat and his two defenders had known how to take advantage of this bad disposition; and they had so much made it worth, that although the good right was not entirely for them, they had not failed to swell their party. They had been supported for some time by their own capacity, and especially by the vivacity of their imagination with all the more address, that they had made the strong of the dispute roll on equivocations, since they had seen themselves too strongly pressed on the capital point. Mr. Descartes on his side, whose great heart sometimes despised certain small aids too much, although otherwise very legitimate and even necessary, to want to cut the knots of the difficulties without facilitating the untying of them, had neglected for a time to draw all the advantages that he was assured of winning in the sequel. This is what makes that there are still today skillful people who judge the victory doubtful between these two great men. But Father Prestet does not believe that one can reasonably award it to Mr. de Fermat, after one has examined and understood one and the other method, and that one has carefully conferred them together.

While the parties were getting heated on the question of geometry concerning the art of finding the greatest and the “smallest quantities,” Mr. de Fermat let his other dispute which concerned optics subside.

He did not think of waking it up during the lifetime of Mr. Descartes. But after his death he willingly spoke of this difference, insinuating that Descartes had never fully satisfied him on the difficulties that he had proposed to him.

Rohault, believing from his ways of speaking that Mr. Descartes had forgotten or neglected to answer him, took up the pen to close his mouth. This innocent error produced this answer to Mr. de Fermat, which we now have in the third volume of the letters of Mr. Descartes. Mr. de Fermat who did not yet know Mr. Rohault, and who only saw his writing a long time later, always dissimulated that Mr. Descartes had answered him, and even seemed to invite from time to time some of his friends to take up this old quarrel again. Mr. Clerselier offered himself, and he ended it to the glory of Descartes, and to the satisfaction of Mr. de Fermat, who died a short time later as a good Cartesian.