The Roulette Problem

In 1638, mathematicians were occupied with the famous question of the “roulette” an invention of de Roberval.

• He saw himself as among the first geometricians of his time.

It reached:

• Descartes in Holland
• Fermat in Languedoc

The name of “roulette” was from Father Mersenne who discovered it.

This first circumstance of its novelty must undoubtedly surprise us, to see that a line so common, which is hardly less frequent in the use of movement than the straight line and the circular line, and which is incessantly described before the eyes of everyone, was not “considered by the ancients,” in the writings of whom it is claimed that no trace of it is found.

This line is nothing other than the path that the nail of a wheel makes in the air, when it rolls by its ordinary movement, from the moment this nail begins to rise from the ground, until the continuous rolling of the wheel has brought it back to the ground after a complete turn is finished.

But in this definition it is necessary to suppose for the convenience of geometric operations that the wheel is a perfect circle; that the nail is a point marked in the circumference of this circle; and that the ground that this point touches at the beginning and at the end of its turn is perfectly smooth or flat.

It is claimed that Father Mersenne:

• was the first to notice it
• made the observation around 1615 by considering the rolling of wheels.

Without an authority of the weight of that of the Younger Mr. Pascal we would have trouble persuading ourselves that this observation was even so old by giving it to Father Mersenne.

Father Mersenne was then only 26 to 27 years old.

He lived in Nevers, far from the commerce of mathematicians.

He called it the “roulette” because it is turned as a wheel.

After having made the remark he wanted to recognize its nature and its properties. But as he was not as happy to solve the beautiful questions as to form them, he did not have enough penetration to succeed in this one.

This obliged him to propose it to others, and he exhorted all the skillful people of Europe whom he judged capable of it to seek the nature of this line, and among others the celebrated Galileo.

But none of them could succeed, and all seemed to lose hope of ever seeing the solution of this difficulty.

After 20 years, he asked de Roberval in 1634, newly a professor in the chair of Ramus to solve the roulette.

De Roberval demonstrated that the space of the roulette is triple that of the wheel that forms it. He thought then of calling it in Latin “trochoides” rather than “rotula,” from a name drawn from the Greek corresponding to the French word of “roulette.”

He made Father Mersenne know that the question was solved.

He even declared to him this “triple” reason, nevertheless demanding of him that he would keep it secret for the space of a year that he would take to propose this question again to all the geometricians.

The father, delighted with this success, wrote to all of them, if we believe Mr. Pascal, and he pressed them to think about it all over again, by declaring to them that De Roberval had solved it without telling them how. The year and more, according to the same author, passed, without any having found the solution.

Father Mersenne wrote to them for the third time in 1635, and he then discovered to them that the ratio of the roulette to the wheel was as three to one.

With this new help, continues Mr. Pascal, two were found who gave the demonstration.

Father Mersenne received their solutions almost at the same time, one from Mr. de Fermat, counselor to the parliament of Toulouse, the other from Mr. Descartes, both different from each other, and still from that of Mr. de Roberval. In such a way nevertheless that in considering all three together, it was not difficult to recognize which was that of the true author, that is to say of Mr. de Roberval, who had been the first to give the solution of the problem. For the demonstration of Mr. de Roberval had a very particular character to distinguish itself from the other two: it was taken by a way so beautiful and so simple, that it was easy to see that it was the natural one. It was indeed by this same way that Mr. de Roberval arrived since at much more difficult dimensions on this subject, to which neither the method of Mr. de Fermat nor that of Mr. Descartes could serve.

This account appears so well circumstantial, and it came to us from an author of such a great name, that it seems that nothing would remain for us to examine concerning the truth of this fact, mainly after Mr. Pascal, whom one should suppose to have been the best informed of men, and who “seems to have perfected the knowledge of all that can regard the roulette.” But as we should not show less love than him for the truth, we can take the liberty that he would have given us himself to retouch his account with all the less scruple, as he would have undoubtedly prevented us in what regards the part that Mr. Descartes may have had in the question of the roulette, if he had known the way in which Father Mersenne and Mr. Descartes lived together, and if he had been able to see what they had written to each other on this subject.

I pass the difficulty that I have already found in believing that Father Mersenne had thought of noticing the roulette from the year 1615, and that he had been twenty years since without being able to find anyone, not even Galileo who was capable of seeking the nature of this line. I want the almost infinite multitude of geometric operations that he had done for several years with Mr. Descartes, Mr. Mydorge, and Mr. Hardy before the retreat of the first to Holland, not to have produced anything on this subject, although Mr. Descartes was from then on in reputation for not being able to be at a loss on what can be of the resort of geometry. But if it was certain that this father in concert with Mr. de Roberval had written “to all the geometricians” from the year 1634 to propose to them the question of the roulette and to ask them for the solution, it is more than probable that he would not have forgotten Mr. Descartes, to whom for five years he had been in the habit of writing regularly every week in Holland, and very often from three days to three days on subjects of mathematics much less important. If he had written from then on to Mr. Descartes, he would have infallibly received some answer according to their conventions, by which Father Mersenne had obliged himself to tell him everything, and Mr. Descartes to answer him exactly to everything. Father Mersenne certainly would not have been reduced to writing to him on the roulette a second and third time to snatch an answer that it is claimed only came after more than a year, that is to say in 1635, which is a very manifest character of falsity. It is quite visible that the first time Father Mersenne wrote to Mr. Descartes concerning the roulette and the demonstration of Mr. Roberval only happened “three years after the time at which he is supposed to have written to him for the third time.” The letter of this father is of April 28, 1638. It taught Mr. Descartes that Mr. de Roberval had found a quantity of beautiful new speculations, as much geometric as mechanical; that among other things he had demonstrated that the space included by a curved line, whose extremities fall on the two ends of a straight line in a semicircle, is triple that of the roulette or circle that moves in this space from the first point of one extremity to the last point of the other on the plane or the straight line; that this space is made by the roulette itself which moves, when the straight line is equal to the circumference of this roulette, etc.

Mr. Descartes answered this letter towards the middle of the following month of May, in terms that it is necessary to report word for word, to serve as proof for what has just been marked. “I received,” he says to this father, “your letters of the twenty-eighth of April and of the first of May at the same time: and besides the letters of the others, I find there twenty-six pages of your writing to which I owe an answer. Truly it is an extreme obligation that I have to you, and I cannot think of the trouble that I give you without having a very great resentment of it. But ad rem. You begin with an invention of Monsieur de Roberval, concerning the space included in the curved line that a point of the circumference of a circle (or roulette) describes, that one imagines rolling on a plane; to which I admit that I have never thought before, and that the remark of it is quite beautiful. But I do not see that there is enough to make so much noise, to have found a thing which is so easy, that whoever knows a little geometry cannot fail to find it, provided that he seeks it.” Mr. Descartes then gives the demonstration of the roulette that Father Mersenne wished from him: and one will not doubt that this letter that Mr. Pascal had not seen any more than that of Father Mersenne, is of the year 1638, when one will notice that there is mention of his differences with Mr. de Fermat, Mr. de Roberval, Mr. Petit, and Mr. Morin, and of several other historical facts that happened this year and at the end of the previous one. Father Mersenne “did not fail to show to Mr. de Roberval the demonstration of the roulette that Mr. Descartes had sent him: but this one found it too short to be good,” in which he made it known that he had a taste very different from that of Mr. Descartes. This father wrote back to him in the following month of June to Mr. Descartes, who wanted to give him on this point some clarifications that he sent him in the month of July, by marking to him that he had not sent him the demonstration of the roulette in the last month of May as a thing of any value, but only in order to show to those who made a great noise about it, that it was very easy. “I had written it,” he says, “very succinctly, as much in order to save time, as because I thought that these gentlemen (that is to say Mr. de Roberval and perhaps the father Mr. Pascal) would not fail to recognize it as good, as soon as they would see the first words. But since I learn that they deny it, I will clarify it here in such a way, that it will be easy for everyone to judge of it.”

After this preamble, Mr. Descartes gave Father Mersenne a very ample explanation of his demonstration of the roulette, and warned him at the end that there was nothing to change in this demonstration, and that the clarification he had just added to it was only diffuse in order to be understood by those who did not use analysis, the others needing only “three strokes of the pen” to find it by calculation.

There were in various questions dependent on that of the roulette several things of which Mr. de Roberval testified to having no knowledge. He wrote about it to Father Mersenne, to beg him to inform himself about them to others, and to ask them for the explanation. The father addressed himself to Mr. Descartes, his ordinary resource, and he was satisfied with a letter written on August 23 of the same year. “I send you,” he told him, “solutions to everything that Mr. de Roberval says he does not know in the letter of which you sent me the copy. But I beg you to show them to several people before him, and even not to give him the original. For I have noticed so many indirect procedures in his conduct, that I believe one must not trust him too much. And if he had not been able to understand my first demonstration of the roulette, he will perhaps not understand either everything that is in these ones. But it would have cost me too much trouble, to explain and clarify all things by reducing them to the reach of children. I will be very happy to know what he will have said of my last explanation of the demonstration of the roulette: for I believe it is so clear, that if he denies it, the smallest schoolboys will be capable of mocking him.”