Descartes versus Roberval

Pascal preferred the solution of Roberval of the roulette over those of Fermat and Descartes.

adds that Mr. de Roberval did not stop there; and at the same time, that is to say in 1635 according to his calculation, but

In 1638, Roberval gave 2 other solutions,

1. The dimension of the solid of the roulette around the base
2. The invention of the tangents of this line by a method that he found which is so general that it extends to the tangents of all the curves, and consists in the composition of movements.

But Pascal only reported this long after the death of his father, and on the faith of the single Roberval, who was not always proof against dissimulation and boasting, as those living today who had the honor of knowing him still testify.

Descartes only treated this whole question because he was asked to.

Fermat had asked Father Mersenne to send on his behalf to Descartes what he had done on the roulette to know his opinion.

Mersenne had acquitted himself of his commission from July.

Descartes wrote back directly to Mr. de Fermat in the following month of August, to mark to him that the tangent of the curved line that the movement of a roulette describes, which he had demonstrated very well, was a very assured proof of the profound knowledge he had of geometry.

For, he says,

“as it seems to depend on the relationship that is between a straight line and a circular line, it is not easy to apply the rules that serve for the others. Roberval is one of the first geometricians of our century. He confessed not to know it, and even not to know any means to reach it. It is true that since that time he also said that he had found it, but that was exactly the day after having known that you and I were sending it to him. And a certain mark that he was mistaken, is that he said he had found at the same time that your construction was false, when the base of the curved line was more or less large than the circumference of the circle. What he could have said of mine in the same way, except that he had not yet seen it, because it agrees entirely with yours.”

For the rest, in order not to depart too lightly from the opinion of Mr. Pascal, one could say that Mr. de Roberval, after having charged Father Mersenne to know from Mr. Descartes and Mr. de Fermat, if they could teach him what he admitted not to know, would have meditated deeply on these questions while waiting for their answers, and would have found the tangents or touchers of which he was in pain, before receiving anything from them. Be that as it may, Mr. Desc. testified in his answer to Father Mersenne that he was very happy to see the questions, that Mr. de Roberval and the other geometricians had declared to him that they did not know, because in seeking them he would have the opportunity to experience if his analysis was better than that which they used. The first of these questions was to find “the tangents of the curves described by the movement of a roulette.”

to which Mr. Descartes replied that the straight line that passes through the point of the curve of which one wants to find the tangent, and by that of the base which the roulette touches while it describes it, always cuts this tangent “at right angles.” He also replied to all the other things for the instruction of Mr. de Roberval in a way that would have satisfied a more sincere or less difficult man. For him, it is not strange that he was not entirely satisfied with himself, because he had subjected himself to follow what had been prescribed to him, and that he would have been obliged to write too many things, if he had undertaken to demonstrate this tangent, and the other questions in a more beautiful and more geometric way. Which nevertheless did not diminish anything of the excellence of the answers that he sent to Father Mersenne for Mr. de Roberval, and the other mathematicians of Paris. He was so persuaded of it that he ended by saying to this father that if these geometricians were not content with these solutions, he could never succeed in contenting them, even if he had the gift of doing miracles; and that in that case he would no longer try it in his life.

Mr. de Roberval, not being able to persuade the public that his demonstration was as old as those of Mr. de Fermat and Mr. de Descartes, nor even that he had shown his before having seen the other two, applied himself only to looking for faults in these ones, to have a reason to prefer his own to them.

Descartes persisted in saying that Fermat had found the tangent of the roulette very well, and that it related to his; that Mr. de Roberval, whom he judged less skillful in geometry than Fermat, exposed himself to public ridicule, by claiming to have found the tangent of the roulette, only after having learned that he had sent it to Father Mersenne; and that he had mistaken himself by claiming by a pure quibble that the demonstration of Mr. de Fermat was not true.

Roberval to grant something to the movements of his jealousy, thought of saying that Mr. Descartes would not have found the space of his roulette, if Father Mersenne had not told him that it was “triple of the circle.” Mr. Descartes found this feint not very judicious, and he wrote back to Father Mersenne in these terms.

The space of his roulette is triple only in a single case; and the way I found it extends to all the others, even when the roulette is an ellipse or 2 hyperbolas.

Besides, I have not had a good enough opinion of him to stop at what he could say or think. Finally, the example of Mr. de Fermat, who after having known it like me of the circle, denied at the beginning that it was true, shows enough that this hardly helps to find the demonstration: as in fact, because it is true only in a single case, it can rather harm than serve, when one wants to seek generally what it is. As for the solid of the roulette, it is much larger than you send; and I believe that one can find the just size of it. But renouncing genuinely as I do geometry, I do not want to stop to seek it."

Roberval, believing that there would be confusion in being silent, reduced himself to saying that Mr. Descartes had changed his “medium” in his demonstration of the roulette. Mr. Descartes denied it, and showed him the wrong he had himself in boasting of having a “medium” to find the tangents of the roulette which applied to all cases. For the one he had sent him from Egmond was so general, that it did not serve only for all the cases of the circular roulette, but also for the lines described by such other bodies as it can be that one makes roll on a plane, whether curvilinear, or rectilinear. Mr. de Roberval alleged the difference of his demonstration from that of Messrs. Descartes and de Fermat, to show that he had found it without their help: and Father Mersenne to do him a service had not forgotten anything of what he had asked him to do to persuade these gentlemen.

These small disputes lasted until the month of November, where Mr. Descartes told this father, that although he had sent him “four or five times” the construction of Mr. de Roberval “for the tangent of the roulette,” he had not found that it was worth anything in any of the ways that this father had sent it to him; that it could be good elsewhere without nevertheless believing that he had found it by himself independently of that of Mr. de Fermat, and of his; that it was easy to disguise the same construction in a hundred ways; and that if it was true that he had found it, he would have made sure at least that his demonstration was consistent with his construction.

Mr. de Roberval found most of the mathematicians of Paris easier to persuade, than neither Mr. Descartes nor Mr. de Fermat. Mr. de Beaugrand, whom the bad success of his geostatics and of his speeches against Mr. Descartes had not entirely excluded from their number, believed that his reputation was at stake in taking some part in such a famous question.

The year 1638 was not yet finished when having gathered the solutions of the plane of the roulette of which Mr. de Roberval had taken care to have the copies multiplied by hand, with the excellent method of Mr. de Fermat his friend, “of maximis et minimis,” he sent one and the other to Galileo in Italy, without naming the authors. It is true that he did not say precisely that it was from him: but, according to the remark of Mr. Pascal, he wrote in such a way, that in not paying close attention, it seemed that it was only by modesty that he had not put his name on it. And to disguise things a little, he changed the first names of “roulette” and “trochoid,” of which one was from Father Mersenne, and the other from Mr. de Roberval, into that of “cycloid,” which was of his making. Which, according to Mr. de Roberval, was not very extraordinary for Mr. de Beaugrand, who had no difficulty in attributing to himself the inventions and works of others, by changing a few terms and suppressing their name.

But to make up for an omission of Mr. Pascal, we will say on the faith of the same Mr. de Roberval, that Mr. de Beaugrand having made himself the owner of the demonstration of the roulette made by Descartes, did nothing else than copy it by his hand as he had received it from Father Mersenne, and sent it at the same time to Galileo as if he had been the author of it; so that he became all at once a plagiarist of Roberval, of Fermat, and of Descartes, that is to say of three people indifferent enough for their own compositions, and who would not have had difficulty in making him a present of them, if he had humbled himself to ask them for them, without even excepting Mr. Descartes, although he had offended him inappropriately in various encounters.

Since that time, Descartes did not have a large part in all that happened concerning the roulette, if one excepts the occasions he had to discourse about it with Father Mersenne in private, and with Mr. Carcavi after the death of this father. From the end of the month of September, he had tried to get rid of it for good; and without claiming anything to the glory of this invention that he left with a good heart to Mr. de Roberval to apply himself to other things, he wrote to Father Mersenne to desist from it.

This is what he did by testifying that he was disgusted elsewhere with all the manners of Mr. de Roberval, and especially with his favorite way of concluding “ad absurdum,” which he practiced as much as possible, because he esteemed it more subtle than the other. In which Mr. Descartes seemed to tax his bad taste, alleging that this way of philosophizing had only been practiced by Apollonius and by Archimedes, when they had not been able to give better demonstrations.

He says to Father Mersenne,

“Roberval did not need any industry to find the figure of the roulette, since I had sent him the definition of it. His writing only serves to make me know that they have examined it very much, and that they have worked for a long time before being able to find the tangent of it. For it has been six or seven months since I had proposed it to them, and they have only begun to talk about it since a month. But I beg you not to mess me up with Mr. de Roberval anymore.”