Fermat Gets Dioptrique

The printer in Leiden had, with his delays, tested the patience of the Parisian mathematicians whom Father Mersenne had notified of the printing of his essays as early as the start of 1636.

The dioptrics alone had groaned for more than a year under the press.

De Beaugrand, one of the most curious and impatient, had posted someone in Leiden to send him the sheets as they were being printed.

By this means he found himself provided with a copy before Descartes could have one delivered to his friends of the first rank.

De Beaugrand, having browsed through it, hastened to send it to Toulouse by way of Bordeaux, to have it read by Fermat, a counselor at the parliament of Languedoc, who had shown more than ordinary passion to see what would come from Mr. Descartes’s pen. Father Mersenne having learned what Mr. De Beaugrand had done, wrote to Mr. De Fermat, to make him know the intentions of Mr. Descartes with regard to those who would read his works, and who would be capable of forming difficulties and objections there to clarify the truths. He added that he would not be dispensed from rendering this service to Mr. Descartes, since he was very capable of it; and he asked him in particular for his opinion on his dioptrics; in return for which he promised him the other treatises of Mr. Descartes that were to appear incessantly.

This father had known “the merit of Mr. De Fermat for a few years, and after the various proofs he had already received of it, he was almost no longer in a state to be mistaken in the judgment he made of his ability.”

Fermat was one of those happy subjects that nature makes fit for everything. He was not only one of the witty people of his time for the delicacy and taste of the true beauty of things.

He also had a genius of such a vast scope, that having embraced the knowledge of several sciences very far from each other, he possessed them as perfectly as if he had only applied himself to one in particular. He was a great humanist, a delicate and successful poet in dead and vernacular languages, very versed in all antiquity; skillful and sure to draw the sense and thought from the most impenetrable places of difficult and obscure authors.

He was in addition very skilled in jurisprudence, and he filled the duties of his office with an application and a sufficiency, which made him pass for one of the great jurisconsults of his time. But what shows that his mind was of a force and a depth equal to its scope, is that he had become so great a mathematician, that after Mr. Descartes, and the son of President Pascal his friend, the public has found no one to prefer to him among the first men of this profession. He excelled in all the parts of mathematics, but particularly in the science of numbers, in beautiful geometry, and in optics.

This is what has appeared not only by the beautiful works he has given to the public, but above all by the occasions he had to measure his forces with Mr. Descartes, who would have perhaps known his merit less perfectly, if Father Mersenne had not thought of committing them together. This father had already sent questions from Mr. De Fermat to Mr. Descartes before the publication of his essays: but he had not worried about even declaring to him the name of this magistrate, and he had been content to make him know him only by the appellative term of “counselor of Toulouse.” This was not however an obstacle to the penetration and discernment of Mr. Descartes, who did not fail to write back to Father Mersenne as early as May of the year 1637 in these terms. You send me a proposition “of a geometer counselor of Toulouse, which is very beautiful, and which has rejoiced me very much. As it will be resolved very easily by what I have written in my geometry, and as I generally give the way, not only to find all the plane loci, but also all the solid ones: I hope that if this counselor is a frank and ingenuous man, he will be one of those who will make the most of it, and that he will be one of the most capable of understanding it. For I will tell you that I greatly fear that there will be found only very few people who can understand it.”

Mr. De Fermat, assured by Father Mersenne of Mr. Descartes’s favorable dispositions with regard to those who would take the trouble to examine his writings, began reading his dioptrics; and he sent to this father as early as November of the same year, as many remarks or objections as a letter of four or five pages could contain. He excused himself for not having been able to send more on the little time that Mr. De Beaugrand had given him to browse through the treatise. The necessity of promptly returning the copy to Paris was not the only pretext: he also rejected the cause on the fact that the matter was in itself very subtle, and very thorny. So that time failed him to digest his reflections, and to make his thoughts less obscure and less embarrassed. Father Mersenne sent Mr. Descartes the letter from Mr. De Fermat such as he had received it, without even touching the places too freely expressed, to be seen by others than the one to whom it was written. Mr. Descartes wrote back to this father to thank him for it as early as the 10th or 12th of December, and made a separate response for Mr. De Fermat, but addressed nevertheless to this father, to whom he left the freedom to send it or not to send it to Mr. De Fermat. He asked him at the same time to continue always to let him know everything that would be said and written against him, and even to invite those whom he would see disposed to it to send him objections, promising them to send them the responses without fail, and to have their very objections printed as soon as he would have received enough to make a just volume of them, provided that they gave their consent.

Mr. De Fermat, persuaded that something was missing from his objections on Mr. Descartes’s dioptrics to put them out of reach, had no doubt that he would not use his advantage to respond to them. This is what made him from then on put his resource in the hope of a reply, where what he would have to say would be better digested than the first time. But in the interval of time that it had taken for his objections to go from Toulouse to Paris and from Paris to Egmond in North Holland, he received the geometry of Mr. Descartes by the care of Father Mersenne: and having read this treatise, he sent him in diligence by the same father his writing “de maximis et minimis” under the name of Mr. De Carcavi, who was then his colleague at the parliament of Toulouse, who had been until then the confidant of his studies, who was after his death the depositary of his writings, and who has been since a counselor to the grand council and guardian of the king’s library until the death of Mr. Colbert. This present that Mr. De Fermat was making to Mr. Descartes was not only a mark of his esteem and his gratitude, but also a warning of what he believed that Mr. Descartes had forgotten without thinking, or omitted inappropriately in his geometry. Mr. Descartes was asked on behalf of the author to examine it with as much freedom as Mr. De Fermat had taken concerning his dioptrics. This made a new incident in the quarrel that Mr. De Fermat had innocently excited, and which he believed to be in a state of ending in a few days. But it was not easy for him to extinguish these first sparks.

The fire of the dispute took great increases by the zeal of those who wanted to enter it; and it all rolled in the suite on two important points, of which one concerned dioptrics, and the other geometry.

Here is the subject of this famous quarrel, which lasted even beyond the death of Mr. Descartes. Here is what Mr. De Fermat called his “little war against Mr. Descartes” and what Mr. Descartes called “his little lawsuit of mathematics against Mr. De Fermat.”

The Latin writing of Fermat was **“de maximis et minimis, et de tangentibus”.

It was for:

• the determination of plane and solid problems
• the invention of tangents or touching and of curved lines, of the centers of gravity of solids; and even for numerical questions.

He was awaiting Descartes’s remarks on all these things.

He had not even yet received his response to the objections he had made to his dioptrics, when he learned that a new combatant had presented himself against Descartes’s dioptrics.

This brave man was Mr. Petit, who was then the provincial commissioner of artillery and royal engineer, and who was since intendant of fortifications.

He was a young man provided with a lot of genius for mathematics, who excelled particularly in astronomy, and who had a particular passion for things of which the knowledge depends on experiments.

He had had his chronological discourses for the defense of Scaliger, Temporarius, and Father Petau against Sieur De La Peyre printed the previous year; and he had newly returned from a trip to Italy, where Cardinal De Richelieu had sent him for the service of the king, when he heard about Mr. Descartes’s dioptrics in Paris. He read it, and made objections to it at the same time that Father Mersenne received those of Mr. De Fermat. Mr. Petit, who had from then on a great correspondence with this father for experiments and researches, was curious to see the objections of Mr. De Fermat before this father sent them to Descartes.

He then wrote to this father, both to thank him, and to mark the judgment he made of Mr. De Fermat’s objections with his own. Father Mersenne sent the letter from Petit to Fermat, who found it “very excellent,” whether for the matter or for the style. It left him a very ardent desire to make the acquaintance of its author, and he asked Father Mersenne that it be by his means. He also solicited him to procure for him the reading of a discourse that Mr. Petit promised concerning “refraction” in his beautiful letter; and he asked him (as by a presumptive privilege of their future friendship) the communication of the experiments he had made, adding that he could well mix geometry with it, if he found them in conformity with his sentiment.

He took a copy of the letter and of Mr. Petit’s objections on Mr. Descartes’s dioptrics; “and he sent back the original to Father Mersenne, having taken the liberty of erasing on the end some words that marked that Mr. Petit’s objections against Mr. Descartes’s dioptrics were stronger and less subject to reply than his own.” It was not because he wanted to doubt it, he said, since he had conceived a very great opinion of Mr. Petit’s mind, but because he wished “to be put aside, and to see all these beautiful disputes rather as a witness than as a party.”

But this disposition lasted at most only until he had received news from Mr. Descartes: after which he was no longer the master of his heart. Although he believed he was then in a perfect indifference, he did not fail to show great impatience to see Mr. Descartes’s response to his objections of dioptrics, and the remarks he was to make on his treatise “de maximis et minimis.” He feared that Father Mersenne would have difficulty in sending them to him in case there were some words not very obliging for him. It is on this that he wanted to prevent him, in order to remove all the obstacles that could deprive him of this satisfaction. If there is, he said to this father, some small bitterness in these responses or in these remarks, as it is difficult that there is not, given the contradiction that is found between our sentiments, that should not turn you away from showing them to me. For I protest to you that that will not make any effect in my mind, which is so far from vanity, that Mr. Descartes could not estimate me so little that I do not estimate myself even less. This is not that politeness could oblige me to recant a truth that I will have known: but I let you know my humor by that.