Pascal’s scientific studies may be said to have begun with the remarkable incident of his youth already related, when he elaborated for himself, in a solitary chamber without books, thirty-two propositions of the first book of Euclid. On the other hand, these studies may be said to have extended to his closing years, when (in 1658 and 1659) he reverted to the abstruser mathematics, and made the cycloid a subject of special thought. But his scientific labours were in the main concentrated in the eight or ten years of his life which followed the removal of the family to Rouen. It will be convenient, therefore, to notice these labours and discoveries in a single chapter here, which will, at the same time, carry on the main history of his life during these years. All that can be expected from the present writer is a slight sketch of this part of the subject, which indeed is all that would be interesting to the general reader.

At the age of sixteen Pascal had already acquired a scientific reputation. He is spoken of by the Duchess d’Aiguillon, in the interview with Richelieu in which she pleaded the cause of the exiled father, as “very learned in mathematics;” and when his sister presented him after the dramatic representation on that occasion, the Duchess gave him “great commendation for his scientific attainments.” [26a] When allowed by his father to pursue the natural bent of his genius, he made extraordinary progress. He was still only twelve years of age, but Euclid’s Elements, as soon as put into his hands, were mastered by him without any explanation. By-and-by he began to take an active part in the scientific discussions which took place at his father’s house; and his achievement in Conic Sections has been already narrated.

Descartes’s incredulity was not without reason; but there is no room to doubt the fact. The little treatise, ‘Pour les Coniques,’ still survives. It bears the date of 1640, and occupies only six pages. [26b] After a very clear statement of his subject, the writer modestly concludes:—

“We have several other problems and theorems, and several consequences deducible from the preceding; but the mistrust which I have of my slight experience and capacity does not permit me to advance more till my present effort has passed the examination of able men who may oblige me by looking at it. Afterwards, if they think it has sufficient merit to be continued, we shall endeavour to push our studies as far as God will give the power to conduct them.”

It is interesting to notice the beginning of relations betwixt Descartes and Pascal, considering the jealousy that afterwards arose betwixt them. There is something of this feeling from the first in the older philosopher, who was now in the forty-fourth year of his age, and in the full zenith of his great reputation. He appears to have been greatly fascinated by Pascal’s peculiar powers; but the men were of too marked individuality of character, and too divergent in intellectual sympathy and personal aspiration, to appreciate each other fully.

Pascal’s next achievement was the invention of an arithmetical machine, chiefly prompted by a desire to assist his father in his official duties at Rouen. He has given us no description of this machine from his own pen. In the “Avis” addressed to all whose curiosity was excited by it, he excuses himself from this task by the natural remark that such a description would be useless without entering into a number of technical details unintelligible to the general reader; and that an actual inspection of it, combined with a brief viv voce explanation, would be far more satisfactory than any lengthened account in writing. There is an elaborate description, however, of the machine, by Diderot, in the first volume of the ‘EncyclopÉdie,’ which is reprinted in the collection of Pascal’s scientific works. Pascal’s main difficulties occurred, not in connection with the invention itself, which he seems to have very soon perfected according to his own conception, but with the construction of the instrument after he had mentally worked it out in all its details. These difficulties proved so great, and so many imperfect specimens of the instrument were made, that, in order to secure both his reputation and his interest, he acquired in 1649 a special “privilÉge du Roi,” which confined the manufacture of the machine to himself, and such workmen as he should employ and sanction. All others, “of whatever quality and condition,” were prohibited from “making it, or causing it to be made, or selling it.” But neither these precautions nor the merits of the invention itself, which were admitted by all competent judges, were of avail to make the instrument a practical success. Many men of mathematical and mechanical genius in different countries have applied themselves to the same task. The celebrated Leibnitz is said to have constructed a machine excelling Pascal’s in ingenuity and power. In our own time, Mr Babbage’s wonderful achievement in the same direction attracted wide attention, and has been lavishly eulogised by Sir David Brewster and others:—

“While all previous contrivances,” says Sir David, [28a] “performed only particular arithmetical operations, under a sort of copartnery between the man and the machine, the extraordinary invention of Mr Babbage actually substitutes mechanism in the place of man. A problem is given to the machine, and it solves it by computing a long series of numbers following some given law. In this manner it calculates astronomical, logarithmic, and navigation tables, as well as tables of the powers and products of numbers. It can integrate, too, innumerable equations of finite differences; and, in addition to these functions, it does its work cheaply and quickly; it corrects whatever errors are accidentally committed, and it prints all its calculations.”

Notwithstanding this brilliant picture, the great expense and the complications involved in the construction of such an instrument have seriously interfered with its success. It is said that Mr Babbage’s machine, much more his marvellous analytic engine, have never yet been properly constructed. [28b]

Pascal fortunately turned his thoughts into a new and more fruitful channel. We have now to contemplate him as one of an illustrious band associated in a great discovery in physical science. Before his time considerable progress had been made towards a knowledge of atmospheric pressure. Galileo and his pupil Torricelli had both been busy with the subject. To Pascal, however, remains the glory of carrying successfully to a conclusion the suggestion of Torricelli, and of verifying the results which he had indicated. Here, as in almost all such discoveries, it is found that different minds have been actively pursuing the same or similar lines of thought and observation, and controversy has arisen as to the exact merits of each; but Pascal has himself so candidly explained [29a] how far he was indebted to his great Italian predecessors, and how far he made original experiments of his own, that both his relation to them and his own work stand clearly apparent.

It had been found by the engineers engaged in the construction of fountains for Cosmo dei Medici in Florence that they could not raise water in an ordinary pump more than thirty-two feet above the reservoir. The water, having reached this height, would rise no higher. Galileo was appealed to for a solution of the difficulty. [29b] Imbued with the ancient notion that Nature abhors a vacuum, and that this was, as then prevalently believed, the explanation of the water following the elevation of the piston in the pump, the philosopher replied in effect that there were limits to the action of this principle, and that Nature’s abhorrence of a vacuum did not extend beyond thirty-two feet. He was himself, it need hardly be said, dissatisfied with such a reply, and accordingly he invited his pupil, Torricelli, to investigate the subject. The latter very soon found that the weight of the water was concerned in the result. He made experiments with a heavier fluid—mercury—and ascertained that a column of mercury enclosed in a tube three feet in length hermetically sealed at the lower end, and closed with the finger at the top, on being inserted in a basin of the same liquid and the finger withdrawn, stood at a height of about 28 inches in the basin. As the specific gravities of water and mercury were in the ratio of 32 feet and 28 inches, he was led to the conclusion that the water in the pump and the mercury in the tube at these respective heights exerted the same pressure on the same base, and that both were of course counterbalanced by a determinate force. But what was this force? He had learned from Galileo that the air was a heavy fluid, and he was carried, therefore, directly to the further conclusion that the weight of the atmosphere was the counteracting cause in both cases; in the one, pressing upon the reservoir from which the water was drawn—and in the other, on the surrounding mercury in the basin. He published his experiments and researches in 1645, but dying soon afterwards, his conclusions remained unverified.

The fame of Torricelli’s experiments had reached Paris as early as 1644, before their formal publication. Some one, Pascal says, had communicated them to Father Mersenne—both a religious and scientific intimate, as we have already seen, of the Pascal family. Mersenne had tried the experiments for himself, at first without success, but soon with better fortune, after he had been to Rome and had learned more fully about them. “The news of these having reached Rouen in 1646, where I then was,” says Pascal, [31] “I made the Italian experiment, founding on Mersenne’s account, with great success. I repeated it several times, and in this manner satisfying myself of its accuracy, I drew certain conclusions from it, for the proof of which I made new and very different experiments in presence of four or five hundred people of all sorts, and amongst others, five or six Jesuit fathers of the College of Rouen.” When his experiments became known in Paris, he adds, they were confounded with those which had been made in Italy, and the result was that some attributed to him a credit which was not his due, while others, “by a contrary injustice,” were disposed to take away the credit of what he had really done.

It was with the view of placing the matter in a clear light, and vindicating his own share in the train of experiments which had been made, that he published in 1647 his “Nouvelles ExpÉriences touchant le Vide,” the first of his hydrostatical treatises. He was at pains to explain the distinction betwixt his own experiments and those which had been made in Italy; and not content with this, he added in express words, in an “avis au lecteur,” that he “was not the inventor of the original experiment, but that it had been made in Italy four years before.” So little, indeed, did Pascal borrow directly from Torricelli, or seek to appropriate the fruits of his researches, that he was as yet ignorant of the explanation which the Italian had suggested of the phenomenon so fully established. He saw, of course, that the old maxim of Nature abhorring a vacuum had no solid foundation; but he tried to account for the vacuum above the water and the mercury by such a supposition as the following:—

“That it contained no portion of either of these fluids, or of any matter appreciable by the senses; that all bodies have a repugnance to separate from a state of continuity, and admit a vacuum between them; that this repugnance is not greater for a large vacuum than a small one; that its measure is a column of water about 32 feet in height, and that beyond this limit a great or small vacuum is formed above the water with the same facility, provided that no foreign obstacle interfere to prevent it.”

Pascal’s treatise, while still retaining so much of the old traditional physics, was made an object of lively attack by the Jesuit Rector of the College of Paris, Stephen NoËl. Pascal replied to him at first directly; and then in answer to a second attack—and so far also in answer to a treatise by the Jesuit, entitled “Le Plein du Vide,” published in 1648—he made a more elaborate statement in a letter addressed to M. le Pailleur, and in a further letter addressed to Father NoËl in the same year. There can hardly be any doubt that this was the commencement of Pascal’s hostile relations with the Jesuits. On their part, they failed not to remember in after years, and in a more serious struggle, that he was an old enemy; whilst he on his part probably drew something of the contemptuous scorn which he poured upon them from the recollection of their obstinate ignorance in matters of science.

Meanwhile, in defending himself from the attacks of ignorance, Pascal did not fail to open his own mind to fuller scientific light. As soon as the explanation of Torricelli was communicated to him, he accepted it without hesitation, and resolved to carry out a further series of experiments with the view of verifying this explanation, and of banishing for ever the scholastic nonsense of Nature’s abhorrence of a vacuum. If the weight of the air was really the cause which sustained the height of the mercury in the Torricellian tube, he saw at once that this height would vary at different elevations, according to the varying degree of atmospheric pressure at these elevations. He proceeded accordingly to test the result; but the higher levels around Rouen were too insignificant to enable him to draw any decisive inference. Accordingly, he communicated with his brother-in-law in Auvergne with the view of having an adequate experiment made during an ascent of the Puy de DÔme, which rises in the neighbourhood of Clermont to a height of about 3000 feet. The state of his own health prevented him from conducting the experiment personally, and M. PÉrier was detained by professional avocations from undertaking it immediately. But at length, in September 1648, the experiment was carried out successfully, and the results communicated to Pascal. I cannot do better than quote the account of this important event as rendered by an eminent scientific authority, [33] from M. PÉrier’s own recital of the facts in his letter to Pascal:—

“On the morning of Saturday, the 19th September, the day fixed for the interesting observation, the weather was unsettled; but about five o’clock the summit of the Puy de DÔme began to appear through the clouds, and PÉrier resolved to proceed with the experiment. The leading characters in Clermont, whether ecclesiastics or laymen, had taken a deep interest in the subject, and had requested PÉrier to give them notice of his plans. He accordingly summoned his friends, and at eight in the morning there assembled in the garden of the PÈres Minimes, about a league below the town, M. Bannier, of the PÈres Minimes; M. Mosnier, canon of the cathedral church; along with MM. la Ville and Begon, counsellors of the Court of Aides, and M. la Porte, doctor and professor of medicine in Clermont. These five individuals were not only distinguished in their respective professions, but also by their scientific acquirements; and M. PÉrier expresses his delight at having been on this occasion associated with them. M. PÉrier began the experiment by pouring into a vessel 16 lb. of quicksilver, which he had rectified during the three preceding days. He then took two glass tubes, four feet long, of the same bore, and hermetically sealed at one end and open at the other; and making the ordinary experiment of a vacuum with both, he found that the mercury stood in each of them at the same level and at the height of 26 inches 3½ lines. This experiment was repeated twice, with the same result. One of these glass tubes, with the mercury standing in it, was left under the care of M. Chastin, one of the Religious of the House, who undertook to observe and mark any changes in it that might take place during the day; and the party already named set out with the other tube for the summit of the Puy de DÔme, about 500 toises (a toise is about six feet in length) above their first station. Before arriving there, they found that the mercury stood at the height of 23 inches and 2 lines—no less than 3 inches and 1½ line lower than it stood at the Minimes. The party were ‘struck with admiration and astonishment at this result;’ and ‘so great was their surprise that they resolved to repeat the experiment under various forms.’ The glass tube, or the barometer, as we may call it, was placed in various positions on the summit of ‘the mountain’—sometimes in the small chapel which is there; sometimes in an exposed and sometimes in a sheltered position; sometimes when the wind blew, and sometimes when it was calm; sometimes in rain, and sometimes in a fog: and under all these various influences, which fortunately took place during the same day, the quicksilver stood at the same height of 23 inches 2 lines. During their descent of the mountain they repeated the experiment at Lafon-de-l’Arbre, an intermediate station, nearer the Minimes than the summit of the Puy, ‘and they found the mercury to stand at the height of 25 inches—a result with which the party was greatly pleased,’ as indicating the relation between the height of the mercury and the height of the station. Upon reaching the Minimes, they found that the mercury had not changed its height, notwithstanding the inconstancy of the weather, which had been alternately clear, windy, rainy, and foggy. M. PÉrier repeated the experiments with both the glass tubes, and found the height of the mercury to be still 26 inches 3½ lines. On the following morning M. de la Marc, priest of the Oratory, to whom M. PÉrier had mentioned the preceding results, proposed to have the experiment repeated at the top and bottom of the towers of Notre Dame in Clermont. He accordingly yielded to his request, and found the difference to be 2 lines. Upon comparing these observations, M. PÉrier obtained the following results, showing the changes in the altitude of the mercurial column corresponding to certain differences of altitude of position:—

When Pascal received these results, all the difficulties were removed; and perceiving from the two last observations in the preceding table that 20 toises, or about 120 feet, produce a change of 2 lines, and 7 toises, or 42 feet, a change of ½ a line, he made the observation at the top and bottom of the tower of St Jacques de la Boucherie, which was about 24 or 25 toises, or about 150 feet high, and he found a difference of more than 2 lines in the mercurial column; and in a private house 90 steps high he found a difference of ½ a line. . . . After this important experiment was made, Pascal intimated to M. PÉrier that different states of the weather would occasion differences in the barometer, according as it was cold, hot, dry, or moist; and in order to put this opinion to the test of experiment, M. PÉrier instituted a series of observations, which he continued from the beginning of 1649 till March 1651. Corresponding observations were made at the same time at Paris and at Stockholm by the French ambassador, M. Chanut, and Descartes; and from these it appeared that the mercury rises in weather which is cold, cloudy, and damp, and falls when the weather is hot and dry, and during rain and snow, but still with such irregularities that no general rule could be established. At Clermont the difference between the highest and the lowest state of the mercury was 1 inch 3½ lines; at Paris the same; and at Stockholm 2 inches 2½ lines.”

From the account here presented of these researches, there is no difficulty in determining the exact credit due to Pascal on the one hand, and his Italian predecessors on the other. He completed what they had begun, and verified what they had indicated. As the AbbÉ Bossut has expressed it, Galileo proved that air was a heavy fluid; Torricelli conceived that its weight was the cause of the suspension of the water in a pump and the mercury in a tube. Pascal demonstrated that this was the fact. No one was more anxious than Pascal himself that Torricelli should be acknowledged as the real discoverer of the principle which it was left to him to establish by the test of experiment. He claimed, however, his own definite share in the discovery, both as having carried on a series of independent experiments, and as having converted what he himself calls the “conjecture” of Torricelli into an established fact. It was painful to him, therefore, to have this share denied, and even open accusations made against him that he had appropriated, without acknowledgment, the results of Torricelli’s researches. This accusation was made in certain theses of philosophy maintained in the Jesuit College of Montferrand in 1651, and dedicated to Pascal’s own friend, M. de Ribeyre, first president at the Court of Aides at Clermont. Pascal’s name was not indeed mentioned in these theses; but there could be no doubt of the allusion made to “certain persons loving novelty” who claimed to be the inventors of a definite experiment of which Torricelli was the real author. It was this accusation which drew from Pascal his letter to M. Ribeyre, bearing the date of 12th July of the same year, in which he has described, with admirable lucidity and temper, his relations to the whole subject. In this letter he distinctly says that the Italian experiments were known in France from the year 1644; that they were repeated in France by several persons in several places during 1646; that he himself had made, as we have already seen, definite experiments in 1647, and published the results in the same year; and that he had then not mentioned the name of Torricelli, because, while he knew that the experiments were made in Italy four years before, he did not then know that the experimenter was Torricelli; but that so soon as he learned this fact—which he and his friends were so eager to know, that they sent a special letter of inquiry to Rome—he was “ravished with the idea that the experimenter was so illustrious a genius, whose mathematical writings, already well known, surpassed those of all antiquity.” He says, in conclusion, that it was only in the same year (1647), after the publication of his own researches, that he learned “the very fine thought” of Torricelli concerning the cause of all the effects which had been attributed to the horror of a vacuum. But “as this was only a conjecture as yet unverified,” he then, with the view of ascertaining the truth or falsehood of it, conceived the plan of the experiments carried out by M. PÉrier at the top and the foot of the Puy de DÔme. “It is true, sir,” he adds, “and I say it boldly, that this series of experiments was my own invention; and therefore I may say that the new knowledge thus acquired is entirely due to me.”

To this letter M. Ribeyre made a satisfactory and touching reply. He expresses disapproval of the allusion of the Jesuit father, but as the discourse was otherwise free from offence, he was willing to attribute it to a “pardonable emulation among savants,” rather than to any intention of assailing Pascal. He makes, in short, the best excuse he can for the Jesuit, and hastens to assure Pascal that his reputation needed no justification:—

“Your candour and your sincerity are too well known to admit any belief that you could do anything inconsistent with the virtuous profession apparent in all your actions and manner. I honour and revere your virtue more than your science; and as in both the one and the other you equal the most famous of the age, do not think it strange if, adding to the common esteem which all have of you, a friendship contracted many years ago with your father, I subscribe myself yours,” etc.

But Pascal had to sustain suspicion and attack in a quarter more formidable than that of the Jesuit fathers at Montferrand. We have already spoken of the rather unhappy commencement of relations between him and Descartes. Farther on we get a more pleasant glimpse of these relations, in a letter from Jacqueline Pascal to Madame PÉrier, dated 25th September 1647, and apparently shortly after Pascal had retired to Paris, along with his younger sister, leaving their father for some time still at Rouen. This letter is so interesting, both in its bearing on the question which arose between Descartes and Pascal, and in itself, as giving the only account we have of personal intercourse between these two illustrious men, that we present it almost entire:—

“I have delayed writing to you,” Jacqueline says, addressing her sister, [39a] “because I wished to tell to you at length of the interview of M. Descartes and my brother, and I had no leisure yesterday to say that on the evening of Sunday last M. Habert [39b] came, accompanied by M. de Montigny, a gentleman of Brittany, with the view of letting me know, in the absence of my brother, who was at church, that M. Descartes, his compatriot and good friend, had expressed a strong desire to see my brother, for the sake of the great esteem in which both he and my father were everywhere held, and that he begged to be allowed to wait upon him next day at nine o’clock in the morning, if this would not inconvenience him, whom he knew to be an invalid. When M. de Montigny proposed this, I felt hindered from giving a definite answer, because I knew that my brother was reluctant to force himself to conversation, especially in the morning. Nevertheless, I did not think it right to refuse, so we arranged that he should come at half-past ten next day. Along with M. Habert and M. de Montigny there were also a young man in the dress of a priest, whom I did not know, M. de Montigny’s son, and two or three other young people. M. de Roberval, whom my brother had informed of the intended visit, was also present. After some civilities, talk fell upon the instrument [probably that which Pascal had used in the experiments], which was very much admired, while M. de Roberval showed it. Then they spoke of the idea of a vacuum; and M. Descartes, on hearing of the experiments, and being asked what he thought was within the tube (dans la seringue), said with great seriousness that it was some subtle matter, to which my brother replied what he could. M. Roberval, believing that my brother had difficulty in speaking, took up the reply to M. Descartes with some heat, yet with perfect civility. M. Descartes answered with some harshness that he would talk to my brother as much as he wished, because he spoke with reason, but not to any one who spoke with prejudice. Thereupon, finding from his watch it was mid-day, he rose, being engaged to dine at the Faubourg Saint Germain. M. Roberval also rose, in such a way that M. Descartes conducted him to a carriage, where the two were alone, and battled at one another more strongly than playfully, as M. Roberval, who returned here after dinner, told us. . . . I have forgotten to tell you that M. Descartes, annoyed at seeing so little of my brother, promised to return next day at eight o’clock. . . . He desired this, partly to consult regarding my brother’s illness, as to which, however, he did not communicate anything of importance, only he counselled him to remain in bed every day as long as he could till he was tired, and to take plenty of soup. They spoke of many other things, for he was here till eleven o’clock, but I cannot tell you more particularly what they said, as I was not present on this occasion. We were prevented during the whole day from making him take his early bath. He had found it give him a little headache, but that was because he had taken it too late; and I believe the bleeding at the foot on Sunday had done him good, for on Monday he conversed freely and strongly all day—in the morning with M. Descartes, and after dinner with M. de Roberval, with whom he argued for a long time on many things, both belonging to theology and physics, and yet he took no further harm than perspiring much, and slept rather sound during the night.”

The revelations of this letter are very curious. The respectful desire of Descartes, already so distinguished, to make Pascal’s acquaintance, and to enter into conversation with him; his resentment of Roberval’s interference, and their earnest altercation, prolonged in the carriage after leaving Pascal’s house; the evidently serious character of Pascal’s maladies, and the watchful attention of his sister. It is clear through all that Descartes had been busily occupied with the same physical problems as Pascal, and that he was somewhat jealous of the results towards which Pascal and his friends were tending. Evidently there was a certain measure of unfriendliness between Roberval and Descartes. I am unable, however, to see any traces of a coterie surrounding Pascal and inimical to Descartes, as M. Cousin suggests. [41] If such a coterie existed at this time in Paris, of which the “hasty and jealous Roberval” was the centre, and which delighted in “abusing Descartes, and attacking him on all sides,” Jacqueline’s frank and lively letter seems enough to show that while Roberval was Pascal’s friend and Descartes’s disputant, there was nothing in the meantime between Descartes and Pascal but courteous friendliness and a cordial feeling of mutual respect.

Descartes, however, in his retirement at Stockholm, plainly cherished the impression that Roberval’s intimacy with Pascal prevented the latter from doing full justice to his scientific position and suggestions; and having as yet heard nothing, in June 1649, of the special results of Pascal’s experiments on the Puy de DÔme in the preceding year, he wrote to his friend Carcavi to let him know about these.

“I pray you, let me know of the success of an experiment which Pascal is said to have made on the mountains of Auvergne. . . . I had the right to expect this of him rather than of you, because it was I who advised him two years ago to make the experiment, and who assured him that, although I had not made it, I had no doubt of its success. But as he is the friend of M. Roberval, who professes not to be mine, I have some reason to think he follows the passions of his friend.” [42a]

That letter was immediately communicated to Pascal by Carcavi, who was his intimate associate no less than Roberval. But it seems to have elicited no reply. Bossut [42b] says that he despised it. On the other hand, Descartes’s biographer and eulogist, Baillet, blames Pascal for having carefully kept out of view Descartes’s name in all the accounts of his discoveries; and produces an array of passages from Descartes’s letters, showing plainly that his mind was in the line of discovery finally verified by the experiments in Auvergne. [43a] It may be granted beyond doubt this was the case. It would ill become any admirer of Pascal to detract from the glory of Descartes. But it must be held no less firmly, that in the personal question raised by Descartes’s letter, the balance of evidence is all in favour of Pascal. There are no indications that the two men ever met save on the occasion so frankly described by his sister Jacqueline. Before this Pascal had not only been busy with the subject, but says distinctly that he had meditated the experiment finally made on the Puy de DÔme from the time that he published his first researches. [43b] It was not, indeed, till about six weeks after Descartes’s visit, or on the 15th December 1647, that he communicated with M. PÉrier regarding these experiments, and his earnest desire that they should be made; and it was not till the following September, or about a year after Descartes’s visit, that they were actually made. But it is incredible that Pascal could have written as he did if he had really, for the first time, been indebted to Descartes for the suggestion. Descartes’s name is not mentioned in his correspondence with M. PÉrier, nor in any of his writings on the subject; and the delay in making the experiments is sufficiently explained by the facts stated by himself, that they could only be made effectually at some place of greater elevation than he could command—such as “Clermont, at the foot of the Puy de DÔme”—and by some person, such as M. PÉrier, on whose knowledge and accuracy he could rely. If we add to this the force of the statement already quoted from his letter to M. Ribeyre, four years afterwards, or in 1651, that he claimed the experiments as entirely “his own invention,” and that he did so “boldly,” the case seems put beyond all doubt—unless we are to suppose the author of the ‘Provincial Letters’ and the ‘Thoughts’ capable of wilful suppression of the truth. On the other hand, it is unnecessary to attribute to Descartes anything beyond a mistaken opinion of the value of certain statements which he had no doubt made to Pascal, and possibly some confusion of memory. And that this is not an unwarranted view appears from what he says in a subsequent letter to M. Carcavi, on the 17th August of the same year, 1649—that he was greatly interested in hearing of the success of the experiments, having two years before besought Pascal to make them, and assured him of success—because the supposed explanation was one, he adds, “entirely consistent with the principles of my philosophy, apart from which he [Pascal], would not have thought of it, his own opinion being quite contrary.” [44] This may or may not be true. Pascal certainly held as long as he could to the old maxim of “Nature’s abhorrence of a vacuum.” “I do not think it allowable,” he says in his letter to M. PÉrier, “to depart lightly from maxims handed down to us by antiquity, unless compelled by invincible proofs.” But the notions of Descartes on the subject of a vacuum were at least as confused as those originally held by Pascal. [45a] It is absurd, therefore, to suppose that the latter could have been indebted to the principles of the Cartesian philosophy—not to say that this is a very different suggestion from that of the former letter, that Descartes himself had advised the experiment to be made. Evidently the older philosopher wrote under vague and somewhat inflated ideas of the value of his labours and his conversation with Pascal; while the latter, again, absorbed in his own thoughts on the subject, and unconscious that he had received any special impulse from Descartes or his philosophy, naturally made no mention of his name. His silence when Descartes’s accusation was communicated to him indicates the same somewhat lofty reserve and confidence in the independence of his own researches, rather than any contempt. He felt too sure of his position to think of defending himself, or of repelling what he no doubt regarded as not so much a deliberate assault on the value of his own work, as an exaggerated estimate by the other of his share in that work.

Pascal’s researches regarding atmospheric pressure conducted him gradually to the examination of the general laws of the equilibrium of fluids. [45b] It had been already determined that the pressure of a fluid on its base is as the product of the base multiplied by the height of the fluid, and that all fluids press equally on all sides of the vessels enclosing them. But it still remained to determine exactly the measure of the pressure, in order to deduce the general conditions of equilibrium. With the view of ascertaining this, Pascal made two unequal apertures in a vessel filled with fluid, and enclosed on all sides. He then applied two pistons to these apertures, pressed by forces proportional to the respective apertures, and the fluid remained in equilibrio. “Having established this truth by two methods equally ingenious and satisfactory, he deduced from it the different cases of the equilibrium of fluids, and particularly with solid bodies, compressible and incompressible, when either partly or wholly immersed in them.”

“But the most remarkable part of his treatise on the ‘Equilibrium of Fluids,’” continues Sir David Brewster, from whose exposition we quote, [46a] “and one which of itself would have immortalised him, is his application of the general principle to the construction of what he calls the ‘mechanical machine for multiplying forces,’ [46b]—an effect which, he says, may be produced to any extent we choose, as one may by means of this machine raise a weight of any magnitude. This new machine is the Hydrostatic Press, first introduced by our celebrated countryman, Mr Bramah.

“Pascal’s treatise on the weight of the whole mass of air forms the basis of the modern science of Pneumatics. In order to prove that the mass of air presses by its weight on all the bodies which it surrounds, and also that it is elastic and compressible, a balloon half filled with air was carried to the top of the Puy de DÔme. It gradually inflated itself as it ascended, and when it reached the summit it was quite full and swollen, as if fresh air had been blown into it; or what is the same thing, it swelled in proportion as the weight of the column of air which pressed upon it diminished. When again brought down, it became more and more flaccid, and, when it reached the bottom, it resumed its original condition. In the nine chapters of which the treatise consists, he shows that all the phenomena or effects hitherto ascribed to the horror of a vacuum, arise from the weight of the mass of air; and after explaining the variable pressure of the atmosphere in different localities, and in its different states, and the rise of the water in pumps, he calculates that the whole mass of air round our globe weighs 8,983,889,440,000,000,000 French pounds.

“Having thus completed his researches respecting elastic and incompressible fluids, Pascal seems to have resumed with a fatal enthusiasm his mathematical studies: but, unfortunately for science, several of the works which he composed have been lost. Others, however, have been preserved, which entitle him to a high rank amongst the greatest mathematicians of the age. Of these, his ‘TraitÉ du Triangle ArithmÉtique,’ his ‘Tractatus de Numericis Ordinibus,’ and his ‘Problemata de Cycloide,’ are the chief. By means of the Arithmetical Triangle, an invention equally ingenious and original, he succeeded in solving a number of theorems which it would have been difficult to demonstrate in any other way, and in finding the coefficients of different terms of a binomial raised to an even and positive power. The same principles enabled him to lay the foundation of the doctrine of probabilities, an important branch of mathematical science, which Huyghens, a few years afterwards, improved, and which the Marquis la Place and M. Poisson have so greatly extended. These treatises, with the exception of that on the Cycloid, were composed and printed in the year 1654, but were not published till 1668, after the death of the author.”

Pascal’s discoveries as to the cycloid belong to a later period of his life, after he had long forsaken the scientific studies which engrossed him at this time, and had become an inmate of Port Royal. But, as we have already said, it is well to complete our view of his scientific labours in a single chapter.

During an access of severe toothache which, in 1658, deprived him of sleep, his thoughts fastened on certain problems connected with the cycloid. Fermat, Roberval, and Torricelli had all been occupied with the subject, and made some definite progress in ascertaining its properties. But much still remained to be done, and especially to resolve the problems connected with it in a “general and uniform manner.” “Pascal,” says Bossut, “devised within eight days, and in the midst of cruel sufferings, a method which embraced all the problems—a method founded upon the summation of certain series, of which he had given the elements in his writings accompanying his ‘TraitÉ du Triangle ArithmÉtique.’ From this discovery there was only a step to that of the Differential and Integral Calculus; and it may be confidently presumed that, if Pascal had proceeded with his mathematical studies, he would have anticipated Leibnitz and Newton in the glory of their great invention.”

Having communicated the result of his geometrical meditation to the Duc de Roannez and some of his other religious friends, they conceived the design of making it subservient to the triumph of religion. Pascal himself was an illustrious example that the highest mathematical genius and the humblest Christian piety might be united; but in order to give Éclat to such an example, his friends proposed to propound publicly the questions solved by the great Port Royalist in his moments of suffering, and to offer prizes for the best solutions given of them. This they did in June 1658. A programme was published making the offer of prizes of forty and twenty pistoles, for the best determination of the area and the centre of gravity of any segment of the cycloid, and the dimensions and centres of gravity of solids and half and quarter solids which the same curve would generate by revolving round an abscissa and an ordinate. The programme was put forth in the name of Amos Dettonville, the anagram of Pascal’s assumed name as the writer of the ‘Provincial Letters.’ Huyghens, Sluzsius, a canon of the Cathedral of LiÈge, and Wren, the architect of St Paul’s, sent in partial solutions of the problems—those of Wren especially attracting the interest of both Fermat and Roberval. But Wallis, of Oxford, and LallouÈre, a Jesuit of Toulouse, were the only two competitors who treated all the problems proposed. It was held that they had not completely succeeded in solving them; and Dettonville published his own solution in an elaborate letter addressed to M. Carcavi, and in a treatise on the subject. Carcavi was an old friend of Pascal’s father as well as of himself; and being a lawyer as well as a mathematician, the arrangement of the affair seems to have been intrusted to him. This did not save him, however, from attacks by the disappointed candidates, who accused him of unfairness; and Leibnitz has given his decision that both Wallis and LallouÈre, in the treatises which they published,—which did not, however, appear till after Pascal’s,—had succeeded in solving the problems. Upon such a point we cannot pretend to judge; but it may be safely said that the design of the Duc de Roannez was hardly realised in the issue. It was sufficiently proved, indeed, that Pascal, in the midst of all his austerities and devotional exercises, was the same Pascal who had held his own both with Descartes and with the Jesuits. But the life of thought which survived in him no sooner touched the outer world of intellectual ambition, than it flamed forth into something of the passion of controversy which his pen had already kindled in another direction. Religion is best vindicated, not in the strifes of science, but by the beauty of its own activities.

Pascal’s labours on the cycloid may be said to bring to a close his scientific career. There is still one invention, however, of a very practical kind, associated with the very last months of his life. Amongst the letters of Madame PÉrier, there is one of date March 24, 1662, addressed to M. Arnauld de Pompone [50]—a nephew of the great Arnauld—in which she gives a lively description of the success of an experiment “dans l’affaire des carrosses.” The affair was nothing less than the trial on certain routes in Paris of what is now known as an “omnibus;” and the idea of such conveyances for the public—“carrosses À cinq sols,” as they were called—is attributed to Pascal. It is certain that the privilege of running “carrosses À cinq sols” was granted to Pascal’s friend, the Duc de Roannez, and to other noblemen, by royal patent, in January 1662,—and that the experiment, as described by Madame PÉrier, was made with great success in the following March, and that Pascal had an active interest in the undertaking. His sister tells that he had mortgaged his share of its first year’s profits in order to provide for the poor at Blois; [51] and a note from his own hand, appended to his sister’s letter, shows with what eagerness he entered into the affair and hailed its success. It is singular to connect the name of Pascal, and that, too, during the last sad months of his life, with so world-wide a commonplace as the omnibus.