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the nature of geometry, of which he had often heard the associates speak. His father told him generally, that it related to the measurement of bodies, and showed how to construct figures with accuracy, and to ascertain their relations to each other. More ioformation was refused ; but a promise was given, that he should study the subject after he had learned enough Latin and Greek. The importunate curiosity of the boy could not tolerate this delay. During his leisure hours, he shut himself up in a chamber, and with a piece of charcoal traced figures upon the floor, such as parallelograms, triangles, and circles, seeking to find their relative dimensions. He knew not even the names of these figures, but called a circle a round, and a line a bar. Definitions and axioms he framed to suit himself, and in this way proceeded by degrees, as we are told, till he came to a knowledge of the thirty-second proposition of Euclid, that the three angles of a triangle are equal to two right angles. While thus engaged, he was one day surprised by his father, who was naturally amazed at the progress made under such circumstances, and ran immediately to communicate the fact to his intimate friend, Le Pailleur. After this discovery, no further restraint was put upon the boy's genius. Euclid's “ Elements” were given to him, and he read the book by himself, without asking any aid, before he was thirteen

years old.

This account is given by the elder sister, who was in the family at the time, and must have known the facts ; and as her character does not allow her veracity to be questioned, there seems no room to doubt its substantial accuracy. It was published, also, when some of the associates of the elder Pascal were still alive, who could have refuted any misstatement. Yet the story seems so marvellous, that many have considered it a mere fable. The only part of the statement that is really incredible, however, is the explanation of the process, or method, by which the boy arrived at such astonishing results. The order in which geometry is taught in the books is surely the very reverse of that in which the great truths of this science were first discovered. Instead of beginning with axioms and definitions, and advancing through the more simple propositions to the more complex, the process must have begun with the discovery, either by accident or measurement, of some advanced the

orem, and, in seeking to demonstrate this, subsidiary truths came to light as the media of proof. Pythagoras certainly was acquainted with the famous proposition about the square of the hypothenuse, before he was able to demonstrate it. Euclid teaches the elements synthetically ; he discovered them by analysis. Now, if we suppose that Pascal, in the scientific meetings at his father's house, bad overheard mention of the fact that the three angles of a triangle are equal to two right angles, and endeavoured to discover the proof of this theorem, the story ceases to be incredible, or even very remarkable. If we consider the astonishing acuteness and vigor of his mind, as subsequently displayed in other ways, it seems quite probable, that he succeeded in inscribing a triangle in a circle, and in ascertaining that an angle at the centre is twice as great as one at the circumference standing upon the same arc, whence the passage to the truth be was seeking to demonstrate is obvious. He may have found out more or less than this ; the account on which we rely being quite indefinite as to the particulars of his achievement. The only thing really marvellous about it is, that a boy twelve years of age, without advice or instigation, should have troubled himself at all about the matter.

His aptitude for mathematical investigations soon appeared in a nianner that admits no doubt nor cavil. He now took an active share in the discussions that were held by his father with his scientific associates, and when he was but sixteen years old, he composed a short treatise on conic sections, which was considered as a prodigy of genius. It was published in 1640, and astonished Descartes himself, who persisted in maintaining that it was the work of Pascal's instructers, as he could not believe it was the production of a child. But the progress of his studies was now interrupted by domestic misfortunes. His father incurred the resent. ment of Richelieu, by offering some opposition to an arbitrary plan for cutting short the income attached to the Hôtel de Ville. An order was made out for committing bim to the Bastille ; but obtaining seasonable notice of ii, he fed from Paris, and concealed himself in his native province of Auvergne. A singular circumstance aided the ialents and filial piety of his children, 10 which he was at last indebted for restoration from exile. The cardinal, it is well known, had a passion for dramatic performances, and even wrote a

play himself, which was quite bad enough to be worthy of a prime-minister. He took a fancy about this time, ibat a iragi-comedy by Scudéri, called “L'Amour Tyrannique,” should be represented in his presence by a party of young girls. The Duchess d’Aiguillon, who had charge of the affair, selected Jacqueline Pascal, then about thirteen years old, the younger sister of Blaise, to be one of the performers. The permission of the elder sister, afterwards Madame Perier, who was the head of the family during the absence of the father, was asked ; but she coldly answered, that they did not consider themselves under any great obligation at that time to please the cardinal. The duchess persisted in her request, and hinted that the pardon of the father might reward them for compliance. They yielded to this suggestion, and the representation took place on the 3d of April, 1639. Jacqueline acted her part like a little fairy, and her grace and spirit quite captivated the spectators, and excited all the good feelings of Richelieu. It had been arranged, that the little actress should approach the minister at the close of the piece, and recite some verses pleading for the restoration of her father. She did so with a degree of simplicity and earnestness that delighted the cardinal, who embraced her as soon as she had finished, and exclaimed, “Yes, my child, I grant all that you ask for ; write to your father, that he may immediately return with safety.” The kind duchess then spoke with strong commendation of the merits of the family, and added, pointing to Blaise, who was standing near, “ There is the son, who is but fifteen years old, and is already a distinguished mathematician.”

The elder Pascal relurned to Paris, and was received with great kindness by Richelieu, who soon afterwards appointed him to an honorable and lucrative office in the government of Rouen. He removed his family to that city, and the numerous accounts and calculations that were necessary in his official business were confided to his son. Weary of the prolix and monotonous processes of arithmetic, the young man endeavoured to invent some mechanical means of executing the work. After two years of intense application, he produced the celebrated arithmetical machine which bears his name. It was a marvellous effort for a boy of nineteen. Leibnitz speaks of it with admiration, and made some attempts to improve it ; and in our own day, the magnificent

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project of Mr. Babbage, which seems fated never to be any ihing more than a project, is a mere revival and amplification of the ingenious contrivance of the young Frenchman. Pascal's machine consists of a kind of framework, supporting several parallel bars, which turn on their axes, each one having two series of numbers inscribed upon it. A combination of wheels and pinions behind directs the revolution of these bars, and after the machine is set for a particular process, the numbers forming the result appear through the opening in the face of the instrument. The complexity of the work prevents us from giving a more detailed description of it. It is enough to say, that it executes all the lower processes of arithmetic with quickness and certainty, and performs some of the more complex and difficult operations. The arithmetical triangle, invented by Pascal in 1654, is a natural complement to this machine. It gives the coefficients of a binomial raised to any power denoted by an integer, so that it is in part an anticipation of Newton's beautiful theorem. It was applied, also, to the theories of combinations and probabilities, facilitating the calculations in each, and indicating certain results in them not before known.

Pascal was proud of these inventions, and with good reason, considering their fertility and the originality of the ideas on which they rest. He says, that the operation of his machine resembles, far more than the instinct of animals, the workings of the human intellect. In 1650, he sent one of the instruments to Queen Christina of Sweden, with a letter which is a perfect masterpiece of tact and delicacy in complimentary address, and shows that the writer was not more a man of science than an accomplished French gentleman. But the cost of the machine, and its liability to get out of repair, prevented it from coming into extensive use ; and the invention of logarithms renders all contrivances of this class in a great degree unnecessary. In speaking of the mechanical skill of Pascal, his biographers uniformly attribute to him the invention of the wheel sedan-chair and the truck, though it is difficult to believe that these simple instruments were not in use long before his time.

He probably made some marked improvements in the common mode of constructing them. The intense and continued exercise of his mind, during two years, upon his arithmetical contrivance proved a permanent injury to his physical constitution,

which was naturally frail and sensitive ; and ever after this period he suffered under the complication of maladies which finally caused his death.

It would be tedious to dwell upon the history of Pascal's discoveries in mathematical science. They were conspicuous and important enough to attract the attention and envy of Descartes, who seemed to arrogate to himself at this period the whole province of pure mathematics as his particular domain. The researches upon the theory of the cycloid have been already mentioned ; inserior as they are to the results since obtained so easily by the use of ihe infinitesimal calculus, they must be regarded as almost miraculous achievements of the geometry of Pascal's time. The calculation of chances, various problems in which are so complex and far-reaching as to tax the utmost resources of the improved science of our own day, owes its earliest development, and the establishment of some of its most important principles, to the genius of this youthful mathematician. Huygens, to whom the praise of originating the true theory of games of chance is sometimes awarded, frankly avows, in the preface to his work on this subject, that the invention does not belong to him, as “ all these questions have already been discussed by the greatest geometers of France.” In truth, the work of Huygens appeared in 1657, while the solutions of Pascal were well known in 1654, when he was but thirtyone years of age. The subject was proposed to him by a celebrated gamester, who wished to know in what proportions the stake should be divided between two players, if they agreed to separate without finishing the game. Pascal solved the problem in its most general form, so as to divide the sum equitably among any number of players who might be engaged. Roberval and Fermat, two of the most distinguished mathematicians in France, attempted to answer these questions at the same time ; the former failed entirely ; the latter succeeded by applying the theory of combinations. Pascal, who had solved the problem by another method, believed at first that the solution by Fermat was not correct, although the result agreed with his own ; but on further examination he retracted this opinion, and acknowledged that the process was equally accurate and elegant.

Passing over Pascal's other mathematical labors, though many of them are of considerable note, we come to his con

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