De maximis et minimis

Descartes thought that Fermat didn’t need to send him his treatise de maximis et minimis for examination.

But he took it as a challenge.

The manner of the summons, joined with the merit and dignity of the person who sent him the cartel, prevented him from avoiding this encounter.

He responded to the challenge with the writing he sent to Father Mersenne against the treatise de maximis.

Father Mersenne made Paris the place where their differences were to be examined by giving to Pascal and Roberval in Paris the writing of Descartes that Mersenne was supposed to send to Toulouse for Fermat.

Seeing that Monsieur de Fermat had important friends who were so interested in his defense, he could not doubt that he had amiable qualities that invited them to do so.

Moreover, he extremely valued the loyalty these friends showed him; and because it is a virtue as rare as it is precious, he assures them that it alone was sufficient to oblige him to be their most humble servant. But since these gentlemen had judged it appropriate to make themselves “the lawyers of his party” in a cause that seemed to him to be of little merit, he expressed the hope that they would not wish to be his judges; and that they would not find it amiss that he recused them along with some of Monsieur de Fermat’s other friends.

There were many other mathematicians who could have known about this matter in Paris.

But some were not in a state to understand Descartes’s geometry perfectly enough.

Others were not known enough to him, with the exception of 2 illustrious geometers:

1. Mydorge
2. Hardy

They were Descartes’ intimate friends.

This consideration made them no less recusable to Monsieur de Fermat than Monsieur Pascal and Monsieur de Roberval were to Monsieur Descartes for their friendship with Monsieur de Fermat.

It was therefore necessary to resolve to choose them not as his judges, but as his lawyers; or to speak in the terms of the cartel presented by Monsieur de Fermat, Monsieur Mydorge and Monsieur Hardy were retained by Monsieur Descartes to be his seconds, and to be opposed to Monsieur Pascal and Monsieur de Roberval, who had offered themselves to Monsieur de Fermat to second him in the combat.

Father Mersenne was asked to remain neutral.

Descartes wished it this way in this encounter as in all others, according to the regards and courtesies he ordinarily used with Father Mersenne, whose friendship he did not intend to abuse; and he took care above all things never to commit his person or the sanctity of his robe inappropriately.

“You should not fear,” he told this father, “that the advice you have the goodness to give me concerning what is said against me will ever turn to your prejudice. For there is nothing I would not suffer sooner than to involve you in my quarrels. But I am also sure that for your part you would not want to hold my hands while I am being beaten, to prevent me from defending myself: and those who give you objections against me cannot reasonably blame you for the answers I will make to them, nor be angry that you send them to me. For knowing the affection you have for me, they cannot give them to you for any other purpose than to make me see them: and all the civility I have believed I could use so far towards Monsieur de Fermat, has been that I have pretended to be ignorant of his name, so that he knows that I only respond to his writing, and that you have only sent me his objections, without engaging his reputation in them.”

What determined Monsieur Descartes to thus regulate the state of his dispute was first a letter from Father Mersenne dated February 8, 1638, which was followed by another that the same father wrote to him four days later concerning the movements that Messrs. Pascal and de Roberval were making in favor of Monsieur de Fermat.

On the very day he received this last one, he wrote to Monsieur Mydorge and Monsieur Hardy to inform them of what was happening, and to interest them in his cause. He enclosed these letters in the packet he addressed to Father Mersenne: but he wanted to insert the response he made to the first writing of Pascal and Roberval in the letter he wrote to Monsieur Mydorge, so that if this father feared that these gentlemen would find it amiss that he had shown this response to Monsieur Mydorge rather than to them, he could excuse himself by this means. We have lost the letter he wrote to Monsieur Hardy: but the one that was to Monsieur Mydorge has been preserved for us, and which contains, besides the necessary instructions for the knowledge of his mathematical lawsuit, a response to the last writing of Monsieur de Fermat, which was his reply to the response that Monsieur Descartes had made to the objections he had proposed against his Dioptrique.

He told Monsieur Mydorge that, having learned from Father Mersenne that he had supported his party for some time in his presence, he counted on his ordinary affection for all the other occasions where it would be a question of rendering him similar services, and which could be all the more frequent in the future as he learned that he was often put on the spot in good company. To imitate those who, finding themselves obliged to borrow money, always address themselves more freely to the people they already owe than they do to others, he wanted to add to all the other obligations for which he was indebted to him, that of still owing him the success of this matter. He therefore begged him to see the documents of “his lawsuit” and at the same time recommended to him to forget or suspend the sentiments of his friendship, to follow only the rules of justice and truth. The first of the documents he asked him to see was the letter of Monsieur de Fermat to Father Mersenne, containing the objections of that author against his Dioptrique.

The second was his response to this letter from Monsieur de Fermat. The third was the Latin writing of Monsieur de Fermat, “de maximis et minimis, et de inventione tangentium linearum curvarum” that he had had sent to him, to make him see that he had forgotten this matter in his geometry; and that he also had a better way of finding the tangents of curved lines than the one that Monsieur Descartes had given. The fourth was the response to this writing de maximis.

The fifth was the writing of some friends of Monsieur de Fermat in reply to his response against the Latin writing of Monsieur de Fermat, and which Monsieur Descartes attributed to Monsieur de Roberval alone. The sixth was Monsieur Descartes’s response to these friends of Monsieur de Fermat, that is, to Messrs. Pascal and de Roberval. The seventh was the reply of Monsieur de Fermat to the first response of Monsieur Descartes concerning his Dioptrique.

As for the eighth document, which was Monsieur Descartes’s response to this reply from Monsieur de Fermat on the subject of Dioptrique, it was contained in the very letter he was sending him, and it composed the greater part of it.

These were the documents of the lawsuit that Monsieur de Fermat had filed against Monsieur Descartes, and which Father Mersenne was to provide to Monsieur Mydorge, except for the sixth which Monsieur Descartes sent him directly with the eighth in the same packet, and of which he asked him to keep a copy, before the original, which was for the two friends of Monsieur de Fermat, was put into their hands by Father Mersenne, to whom Monsieur Mydorge was commissioned to return it. After having responded to the main points of Monsieur de Fermat’s reply, Monsieur Descartes ended his letter to Monsieur Mydorge by asking him that Monsieur Hardy also have communication of all these documents of his lawsuit, so that both of them could examine his case thoroughly. For it was just, according to him, that since two of Monsieur de Fermat’s friends had presented themselves to support his cause, he also employ for the defense of his own two of his friends in whom he had the most confidence, and whom he considered the most skilled for the matter in question. Monsieur Descartes wrote at the same time to Father Mersenne, to ask him to be willing to keep copies of all the documents he was to communicate both to Monsieur Mydorge and to the two friends of Monsieur de Fermat, and to show them to all those who would have the curiosity, but particularly to Monsieur des Argues, if he would take the trouble. But he judged that it was very important that one paper should not be seen without the other; and he would have wished for this that all these documents were written in a row in the same notebook. He sent back to this father in the same packet the copy of the treatise of Monsieur de Fermat, de locis planis et solidis, which was being requested from him without having had the convenience of reading it: and he asked him to keep copies of everything he would send him to Holland, or that he would wish to have sent back to him, another time.

Monsieur Mydorge was not long in handing over to Father Mersenne the original of the response that Monsieur Descartes had made to the writing of Messrs. Pascal and de Roberval on the subject of the treatise de maximis et minimis and this father immediately took it to Monsieur de Roberval, at the college of master Gervais. The latter, without letting the heat in which the reading of this response had put him subside, immediately composed a reply under the name of the two friends of Monsieur de Fermat, that is, of Monsieur Pascal, and his own. It is a name they had legitimately acquired by the first service they had rendered to Monsieur de Fermat, whom they claimed to have known until then only by reputation, no more than Monsieur Descartes. But it was necessary for Monsieur de Roberval to impose on Monsieur Pascal, or that he had his word to continue the dispute of Monsieur de Fermat in his name against Descartes.

Pascal was no longer in Paris at that time.

Roberval had enough good faith to indicate his absence by subscribing to their common reply alone.

Pascal could not have much part in this difference afterwards.

• He had believed himself obliged a few days before to move away from the city, and to withdraw far from public commerce, for fear that his presence might irritate some offended powers, and that it might lead them to do something to the prejudice of his liberty.

The disgrace into which he believed he had fallen was only the consequence of that of one of his intimate friends who had been arrested and taken to the Bastille for some troubles excited at the city hall.

Pascal, persuaded of the rectitude of his friend’s heart, had remarked that there was more misfortune than crime in the manner in which he had given rise to the trouble. He had not contented himself with speaking in favor of his friend, he had also dared to take the defense of various people unjustly treated by the harassment of some interested officers. He had also learned that this matter had been reported with very odious circumstances to Monsieur the Chancellor Séguier. That is why the fear of having displeased this first magistrate of the kingdom had made him step aside to prevent the effects of his resentment. He remained about a year in his absence, until Monsieur the Cardinal de Richelieu, informed of his merit and the reason for his retreat by Madame the Duchess d’Aiguillon and by Monsieur the Chancellor himself, had him return in 1639, and established him a short time later as intendant of Normandy in Rouen.

Descartes got the reply of the friends of Fermat to the response he had made to them, had no trouble recognizing the style of Monsieur de Roberval. The harshness of the manners and the unbecoming expressions of “absurdity, ignorance, and bad faith,” made him judge that Monsieur Pascal was truly absent or that he had no part in the composition of this new writing. Also, he only attributed to Monsieur de Roberval the precipitation with which, at the end of this writing, they undertook to judge generally of his method, his Dioptrique, and his Météores, when it was only a question of some omissions that were imputed to his geometry.

He wrote back to Father Mersenne about it at the end of March, and he told him that he was not resolved to make a response to this second writing, because he noticed that the one who had composed it “was getting piqued.”

But he begged this father that when he saw Monsieur de Roberval’s anger appeased, he should let him know “the little reason he had to get heated,” and the little conformity that the passion he had to censure everything that came from him could have with the moderation that Messrs. de Fermat and Pascal otherwise used towards him. Notwithstanding his off-putting manners and his prejudices, he had Father Mersenne assure him that he was his most humble servant, and that he was no more offended by everything that was in his writing than one usually is in a game, by the anger of those who lose. But that, as there is no pleasure in playing against those who get angry, he would henceforth take the side of no longer responding to any writing in which he would notice more passion than love for the truth.

Father Mersenne had a talent for:

• setting scholars against each other
• prolonging the disputes he had excited

He was not satisfied with the resolution that Descartes had made not to respond to Roberval.

He wrote to him about it on March 16: and Monsieur Descartes, to provide him with some rest on that side, felt obliged to nevertheless send him the response he was waiting for; but he took care not to let anything slip into it that could still stir up Monsieur de Roberval’s bile.