Pascal

ropositions 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

e same time, carry on the main history of his life during these years. All that can be expected from the present writ

fter 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, bu

ttle treatise, 'Pour les Coniques,' still survives. It bears the date of 1640, and occupies on

ce 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. Aft

om 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 P

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 s

lace 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 produc

truction of such an instrument have seriously interfered with its success. It is said that Mr Babbag

d 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 line

wo 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 th

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 fiv

ose 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 r

nuity, 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

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 No?l in the same year. There can hardly be any doubt that this was the commencement of Pascal's hostile relations with t

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 D?me, which rises in the neighbourhood of Cler

nt 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 3? 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 D?me, 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 1? 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, wh

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vate house 90 steps high he found a difference of ? 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 sam

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

rse was otherwise free from offence, he was willing to attribute it to a "pardonable emulation among savants," rather than to any intention of ass

r 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

ant 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.

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,

hrough 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 cote

latter from doing full justice to his scientific position and suggestions; and having as yet heard nothing, in June 1649, of the speci

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 dou

ad 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 D?me"-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 wh

essels 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

ised 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

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 consist

tique,' 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 prin

the scientific studies which engrossed him at this time, and had become an inmate of Port Royal. But,

specially 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 a

dinate. 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 d

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 M