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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 vivâ 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:—

Difference of altitude.	Changes in the height of the mercury.
Toises.	Lines.
500	37½
150	15½
27	2½
7	½

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.