The Younger Pascal

Roberval imitated Descartes and Fermat.

He seemed to have let the question languish in France, and Mr. de Beaugrand had in a way entrusted his fortune to the Italians. He received no news from them for the rest of his life, which was only eighteen or nineteen months.

We do not know what use Galileo made of it, to whom Mr. de Beaugrand had addressed it; and his old age, combined with the loss of his sight, gives us reason to believe that he died without having worried much about stirring up this question.

He had as his successor in the profession of mathematics the sieur Evangelista Torricelli: and all his papers having come into his hands, he found among others these solutions of the roulette under the name of “cycloid,” written by the hand of Mr. de Beaugrand. Torricelli believed that he was the author of it, and having learned that he had been dead for a few years, he judged that enough time had passed for the memory of it to be lost.

This is what made him think of profiting from it.

He took the opportunity through the publication of various works of geometry that he had printed in a quarto volume in Florence in the year 1644.

The cycloid was not forgotten there.

But he attributed to Galileo, what was due to Father Mersenne, to have formed the question of the roulette; and to himself, what was due to Mr. de Roberval, and to Descartes, to have been the first to give the solution and the demonstration. In this he was not only suspected of bad faith, and appeared inexcusable for the theft he believed he had made from the late Mr. de Beaugrand, but also unfortunate for not having been able to maintain himself for a long time in such an unjust possession.

Few people were deceived there outside of Italy: and it is annoying that Mr. Descartes, who had not known the trick that Beaugrand had played on him, was one of this small number. Perhaps there was in this less surprise on his part than that pleasure we feel in seeing those we are not happy with humiliated. Mr. Descartes had believed until then that Mr. de Roberval was truly the author of the first solution or demonstration of the roulette, and that he had found the area or the space of the line it describes: he had only denied that he had found the tangents which he believed to have taught him, either alone, or jointly with Mr. de Fermat. But without knowing the wrong that was done to him in this claim, he wanted to do this little injustice to Mr. de Roberval on the faith of Torricelli, whom he did not suspect of being a plagiarist. This is what led Mr. Carcavi a few years later to pull him out of this error, and to make him know the conduct of Mr. de Beaugrand and the Sieur Torricelli, in a manner nevertheless that showed confusion in his mind or in his memory for the times and the order of things, and which by this place would have left Mr. Descartes a new subject to doubt the truth of the fact, if he had been concerned about it.

Be that as it may, the Sieur Torricelli gave matter for laughter in France to those who saw that he attributed to himself in 1644 an invention which had been recognized for almost eight years to be of Mr. de Roberval, and of which Mr. Des Argues, a particular friend of Mr. Descartes, had had an authentic testimony printed in a writing he had published from the month of August of the year 1640, with the privilege of the king. Mr. de Roberval was not insensitive to the usurpation of Torricelli. He complained to him himself by a letter he wrote to him from the same year that his book appeared: and Father Mersenne did the same, but in a still more severe style. Torricelli, touched by the proofs of this father, and not wanting the confusion of this enterprise to remain attached to his memory in the mind of posterity, felt obliged to give him his hands; and without losing his judgment he ceded the invention of the roulette to Mr. de Roberval. The letter he wrote about it in Paris dated from the year 1646, has been preserved in original until now by passing from the hands of Father Mersenne to those of Mr. de Roberval, and from those of Mr. de Roberval to those of Mr. Carcavi. He declares there without detour that this cycloid line or the roulette did not belong to him, and that until the death of Galileo, that is to say, in 1642, nothing was known about it in Italy. But there is no mention of the restitution due to Mr. Descartes, because Mr. de Roberval had not judged it appropriate to warn Torricelli of what belonged to him in the papers that Mr. de Beaugrand had sent to Galileo.

However, as the book of Torricelli was public, and his disavowal was not, the error did not fail to slip, especially in the mind of those who were not in mathematical commerce with Father Mersenne, or Messrs. Des Argues, de Fermat, Descartes, de Roberval. The Younger Mr. Pascal, although the son of a mathematician very instructed in all that had happened on this subject, and very united with Mr. de Roberval, admits that he was one of those who were deceived; and that in his first writings he had spoken of this line as being of Torricelli, because Mr. de Roberval had little cared elsewhere to attribute this invention to himself, and that he had neglected to have anything printed about it.

Torricelli after this little disgrace (according to the thought of the mathematicians of Paris) no longer being able to pass with those who knew the truth, for the author of the “dimension of the space of the roulette,” nor even of that “of the solid around the base” that Mr. de Roberval had already sent him; he tried to solve that “around the axis.” But he could not succeed; and he died a short time later having received before from Mr. de Roberval the conviction of his error, and the true and geometric solution of what he was looking for.

Mr. de Roberval did not stop at the only dimension of the first and simple roulette and of the solids, but he extended his discoveries to all kinds of “elongated or shortened” roulettes, of which Mr. Descartes had touched on something in advance in the explanation of his demonstration. He used for this purpose a general method which gave with an equal ease the tangents, the dimension of the planes and of their parts, their centers of gravity and the solids; as much around the base as around the axis.

The knowledge of the roulette had reached this point, when the Younger Mr. Pascal, who had renounced geometry for a few years, and who was meditating a great work on the truth of the Christian religion, was solicited by his friends to give first an essay of the force of his mind in mathematics to prevent the strong minds, the libertines, and the atheists, in favor of the treatise on religion that he was preparing against them. He believed them: and to show that he did not pretend to lead the minds of those he hoped to convince and persuade of our religion by the ordinary ways of those who had preceded him in this career, he resumed his old thoughts of geometry. He formed methods for the dimension and the centers of gravity of solids, of plane and curved surfaces and of curved lines, from which it appeared to him that few things could escape. His intention was not to use it to then give geometric proofs and demonstrations of the Christian faith in his work on religion, but to show only that being moreover capable of all that can be humanly on that side, it would be neither by ignorance, nor by weakness of mind that he would have recourse to moral proofs which had to go more to the heart than to the mind.

To make the essay of the methods he formed on one of the most difficult subjects, he proposed what remained to be known of the nature of the “roulette,” namely, “the centers of gravity of its solids and of the solids of its parts; the dimension and the centers of gravity of the surfaces of all these solids; the dimension and the centers of gravity of the very curved line of the roulette and of its parts.” He began with the centers of gravity of the solids and “of the half-solids,” which he found by means of his method, and which appeared to him so difficult by any other way, that to know if they were indeed as much as he had imagined, he resolved to propose the search for them to all the geometricians, and even with prizes for those who would succeed.

It was then that he made his Latin writings on this subject, and that he sent them everywhere to execute his plan without naming the author.

While they were looking for these problems concerning the solids, he applied himself to solving all the others, until he had received the answers of the geometricians on the subject of his writings. There were two kinds of them. Some imagined they had solved the proposed problems, and won the prizes: that is why it was necessary to make the examination of their writings. The others not claiming anything to these solutions were content to give their first thoughts on this line. He found very beautiful things in their letters, and very subtle ways of measuring the plane of the roulette, and among others in those of Mr. Sluze then canon of the cathedral of Liège, brother of the learned cardinal of this name; of Mr. Ricci of Rome, a disciple of Torricelli, who died a cardinal under Innocent XI; of Mr. Huyghens, son of the friend of Mr. Descartes Mr. de Zuytlichem of Holland, one of the ornaments of the royal academy of sciences in Paris, and still living today in Holland; and of the English Mr. Wren, pensioner of Vadham college who had distinguished himself in the knowledge of mathematics from his first youth.

He also received towards the same time the dimension of the roulette and of its parts and of their solids around the base only from Father Lallouëre, a Jesuit from Toulouse who sent it all printed. But he found that the problems of which he gave the solution there were none other than those that Mr. de Roberval had solved so long ago. It is true that his method was different: but it was easy to disguise propositions already found, and to resolve them in a new way by the knowledge one had of the first solution. Mr. Pascal had this father advised of it by Mr. Carcavi in the most obliging and most civil manner possible; and the father replied to it, to serve “as a prelude to the seven books” of cycloid that he had printed two years later in quarto in Toulouse.

But among all the writings of this nature, nothing seemed more beautiful to Mr. Pascal than what had been sent by Mr. Wren. For besides the beautiful way he gave of measuring the plane of the roulette, he had given the comparison of the curved line and of its parts with the straight line. His proposition was that the line of the roulette is four times its axis, of which he had sent the statement without demonstration. And as he was the first who had produced it, Mr. Pascal did not hesitate to award him the honors of the first invention, although there had been geometricians in France, and among others Mr. de Fermat, and Mr. de Roberval, who had found the demonstration as soon as the statement had been communicated.

This is what was found most remarkable in the writings sent by those who claimed nothing to the prizes proposed by Mr. Pascal. As for the others who found themselves reduced to two, the examination was to begin from the following first of October in the presence of Mr. Carcavi, into whose hands the prizes had been deposited. The first of the two after having communicated his writing in private and recognized his fault, prevented the day of the examination, and gave his desistance. The other persisted in maintaining that he had found “an entire method for the resolution of all the problems with the solutions and the demonstrations in fifty-four articles.” Nothing of all that appeared to the judges established for this affair. It was judged that neither in his writing nor in the corrections he had sent afterwards, he had found “neither the true dimension of the solids around the axis, nor the center of gravity of the half-roulette, nor of its parts,” (which had been solved for a long time by Mr. de Roberval) nor any “of the centers of gravity of the solids, nor of their parts, as much around the base as around the axis,” which were properly the only problems proposed by Mr. Pascal, with the condition of the prizes, as having not yet been solved by anyone. So that it was concluded that Mr. Carcavi would remit into the hands of Mr. Pascal the prizes, which had been entrusted to him in deposit, as not having been won by anyone; and that Mr. Pascal “finally revealing himself would give the true solutions of these problems, of which all the other mathematicians had not been able to succeed.” This is what he did before the end of the year 1658, and having collected the letters and the other writings concerning this matter, he made a quarto volume of it that he published at the beginning of the following year under the assumed name of the Sieur A d’Ettonville and under the title of “treatise on the roulette.”

Since that time we do not see that anyone has made any new discovery on the nature of the roulette, whose history consists entirely in knowing; 1 that the first who noticed this line in nature, but without penetrating its properties, was Father Mersenne who gave it the name of “roulette” 2 that the first who knew its nature, and who demonstrated its space, was Mr. de Roberval who called it by a name drawn from the Greek “trochoid” 3 that the first who found its tangent was Mr. Descartes; and almost at the same time Mr. de Fermat, although in a defective way: after which Mr. de Roberval was the first to measure the planes and the solids, and to give the center of gravity of the plane and of its parts; 4 that the first who named it “cycloid” was Mr. de Beaugrand without contributing anything of his own; that the first who attributed it to himself before the public and who brought it to light was the Sieur Torricelli; 5 that the first who measured the curved line and its parts, and who gave the comparison with the straight line was Mr. Wren, without demonstrating it. 6 that the first who found the center of gravity of the solids, and half-solids of the line and of its parts, both around the base and around the axis was the Younger Mr. Pascal; that the same also found the first the center of gravity of the line and of its parts; the dimension and the center of gravity of the surfaces, half-surfaces, quarter-surfaces, etc. described by the line and by its parts turned around the base and around the axis; and finally the dimension of all the curved lines of the elongated or “shortened” roulettes.

The small number of copies that the so-called Sieur d’Ettonville had been content to have printed of his book did not prevent the history of this whole affair from spreading in foreign countries. It was important above all that it pass the Alps, and that it penetrate at least to the city of Florence, where Galileo and Torricelli enjoyed in peace the honors of the roulette that one believed in France had to be rendered to Father Mersenne and to Mr. de Roberval. When the history of the roulette written in French and Latin was seen there in the manner reported above, trouble set in the minds of most of the literary people of the city. The friends and disciples of Galileo and Torricelli found themselves offended by the disobliging twist that the author of this writing had given to the conduct that the latter had held in this affair; and one of the most zealous among them took up the pen to avenge his master and to have his reputation restored to him. This zealous man was the Sieur Charles Dati, an academician of the Crusca, who had printed in Florence in the year 1663 in quarto an Italian writing addressed to the philalèthes or lovers of truth under the mask of Timauro Antiate and under the title “della vera storia della cicloide,” etc. There this author, after a magnificent protestation of saying only the simple truth, without prejudice and without passion, first has recourse to the plausible, to say that it is probable that Galileo having thought of this line around the year 1600 will have communicated it to Father Mersenne. It is a pity that his proofs are later than Torricelli, on the faith of whom they appear founded. They should at least have been anterior to the time, where we noted that Mr. de Beaugrand had sent to Galileo what had been done in France on the roulette.

But although there is nothing convincing in the writing of the Sieur Dati for the justification of Torricelli, one can grant to the merit of this celebrated mathematician what the mediocre skill of his lawyer could not have obtained for him. One can therefore absolve him of the crime of plagiarist, all the more willingly as the theft was of little consequence, and that Galileo and he can very naturally have found without the help of Mersenne and “of Roberval a thing on which they would have only remembered to work after having seen the observations of these ones.”

This is in what consists almost all the reasoning of the English Sieur John Wallis, who took the defense of Torricelli against Mr. Pascal in more than one encounter.

But Sieur Wallis will be only a very weak adversary of Roberval as long as he will have only possibilities to oppose to a fact as well circumstantial as is that which Mr. Pascal reports in his history of the roulette.