Blaise Pascal was a French scientist and a philosopher, a precocious and influential mathematical writer.
Background
Blaise Pascal was born at Clermont-Ferrand, France on June 19, 1623. He was the son of Étienne Pascal, king's counselor and later president of the Court of Aids at Clermont.
Blaise's mother died in 1626, and he was left with his two sisters, Gilberte and Jacqueline. In 1631 the family moved to Paris.
Education
When Pascal was 12, he began attending meetings of a mathematical academy. His father taught him languages, especially Latin and Greek, but not mathematics. This ban on mathematics merely served to whet the boy's curiosity. He experimented with geometrical figures, inventing his own names for standard geometrical terms.
Career
Pascal proved himself no less precocious in mathematics. In 1640 he wrote an essay on conic sections, Essai pour les coniques, based on his study of the now classical work of Girard Desargues on synthetic projective geometry. The young man’s work, which was highly successful in the world of mathematics, aroused the envy of no less a personage than the great French Rationalist and mathematician René Descartes.
Pascal seemed to have made considerable progress by December 1640, having deduced from his theorem most of the propositions contained in the Conics of A pollonius. However, he worked only intermittently on completing the treatise. Although Desargues and Mersenne alluded to the work in November 1642 and 1644, respectively, it was apparently not until March 1648 that Pascal obtained a purely geometric definitive general solution to the celebrated problem of Pappus, which had furnished Descartes with the principal example for illustrating the power of his new analytic geometry (1637). Pascal’s success marked an important step in the elaboration of his treatise on conics, for it demonstrated that in this domain projective geometry might prove as effective as the Cartesian analytic methods. Pascal therefore reserved the sixth, and final, section of his treatise, “Des lieux solides” (geometric loci composed of conics), for this problems.
In 1654 Pascal indicated that he had nearly completed the treatise. He also mentioned some special geometric problems to which his projective method could usefully be applied: circles or spheres defined by three or four conditions; conics determined by five elements (points or tangents); geometic loci five elements (points or tangents); goe conics and a general method of perspective. Pascal made no further mention of this treatise, which was never published. It seems that only Leibniz saw it in manuscript, and the most precise details known about the work were provided by him.
In 1640 Pascal and his sisters joined their father, who since the beginning of that year had been living in Rouen as a royal tax official. Anxious to assist his father, whose duties entailed a great deal of accounting, Pascal sought to mechanize the two elementary operations of arithmetic, addition and subtraction. Toward the end of 1642 he began a project of designing a machine that would reduce these operations to the simple movements of gears. Having solved the theoretical problem of mechanizing computation, it remained for him to produce such a machine that would be convenient, rapid, dependable, and easy to operate. The actual construction, however, required relatively complicated wheel arrangements and proved to be extremely difficult with the rudimentary and inaccurate techniques available.
Supervising a team of workers, he constructed the first model within a few months but, judging it unsatisfactory, he decided to modify and improve it. The considerable problems he encountered soon discouraged him and caused him to interrupt his project. At the beginning of 1644 encouragement from several people, including the chancellor of France, Pierre Seguier, induced Pascal to resume the development of his “arithmetic machine.” After having constructed, in his words, “more than fifty models, all different," he finally produced the definitive model in 1645. He himself organized the manufacture and sale of the machine. Although its mechanism was quite complicated, Pascal’s machine functioned in a relatively simple fashion – at least for the two operations to which it was actually applied. Its high price, however, limited its sale and rendered it more a curiosity than a useful device. For a few years Pascal was actively involved in their manufacture and distribution, for which he had obtained a monopoly by royal decree (May 22, 1649). In 1652 he demonstrated his machine during a lecture before fashionable audience and presented one to Queen Christina of Sweden. For some time, however, he had been directing his attention to problems of a very different kind.
This activity was the context of Pascal’s second publication, an eighteen-page pamphlet consisting of a “Lettre dédicatoire” to Séguier and a report on the calculating machine – its purpose, operating principles, capabilities, and the circumstances of its construction (“Avis nécessaire à ceux qui auront curiosité de voir ladite machine et de s’en servir“). The text concluded with the announcement that the machine could be seen in operation and purchased at the residence of Roberval.
To understand and evaluate Pascal’s work in the statics of gases and liquids, it is necessary to trace the origins of the subject and the establish a precise chronology. In his Discorsi (1638) Galileo had noted that a suction pump cannot raise water to more than a certain height, approximately ten meters. This observation, which seemed to contradict the Aristotelian theory that nature abhors a vacuum, was experimentally verified about 1641 by R. Maggiotti and G. Berti. V. Viviani and E. Torricelli modified the experiment by substituting mercury for water, thereby reducing the height of the column to about seventy-six centimeters. Torricelli announced the successful execution of this experiment in two letters to M. Ricci of June 1644. Describing the experiment in detail, he gave a correct interpretation of it based on the weight of the external column of air and the reality of the existence of the vacuum. Mersenne, informed of the work of the Italian scientists, attempted unsuccessfully to repeat the experiment, which for some time fell into neglect.
Pascal wrote a report of his experiments at Rouen, a thirty-two-page pamphlet published in October 1647 as Experiences nouvelles touchant le vide. In this “abridgment” of a larger work that the planned to write, Pascal admitted that his initial inspiration derived from the Italian barometric experiment and stated that his primary goal was to combat the idea of the impossibility of the vacuum.
From his experiments he had deduced the existence of an apparent vacuum, but he asserted that the existence of an absolute vacuum was still an unconfirmed hypothesis. Consequently his pamphlet made no reference to the explanation of the barometric experiment by means of the weight of the air, proposed by Torricelli in 1644. In any case his concern was to convince his readers; he therefore proceeded cautiously, affirming only what had been irrefutably demonstrated by experiment.
The year 1654 was exceptionally fruitful for Pascal. He not only did the last refining of his treatises on geometry and physics but also conducted his principal studies on arithmetic, combinatorial analysis, and the calculus of probability. Pascal’s correspondence with Fermat between July and October 1654 marked the beginning of the calculus of probability. Their discussion focused on two main problems. The first concerned the probability that a player will obtain a certain face of the die in a given number of throws. The second, more complex, consisted in determining, for any game involving several players, the portion of the stakes to be returned to each player if the game is interrupted. Fermat succeeded in solving these problems by using only combinatorial analysis. Pascal, on the other hand, seemed gradually to have discovered the advantages of the systematic application of reasoning by recursion. This recourse to mathematical induction, however, was not clearly evident until the final section of the Traité du triangle arithmétique, of which Fermat received a copy before August 29, 1654.
Pascal’s contribution to the calculus of probability was much more direct and indisputable: indeed, with Fermat he laid the earliest foundations of this discipline. The Traité du triangle arithmétique contained only scattered remarks on the subject; in addition, only a part of the correspondence with Fermat had been preserved, and its late publication (1679 and 1779) certainly reduced its direct influence.
During 1658 and the first months of 1659 Pascal devoted most of his time to perfecting the “theory of indivisibles,” a forerunner of the methods of integral calculus. This new theory enabled him to study problems involving infinitesimals: calculations of areas and volumes, determinations of centers of gravity, and rectifications of curves.
Pascal first referred to the method of indivisibles in a work on arithmetic of 1654, “Potestatum numericarum summa.” He observed that the results concerning the summattion of numerical powers made possible the solution of certain quadrature problems. As an example he stated a known result concerning the integral of xn for whole n, in modern notation.
At the beginning of 1658 Pascal believed that he had perfected the calculus of indivisibles by refining his method and broadening its field of application. Persuaded that in this manner he had discovered the solution to several infinitesimal problems relating to the cycloid or roulette, he decided to challenge other mathematicians to solve these problems.
In December 1658 and January 1659 he brought out, under the pseudonym A. Dettonville, four letters setting forth the principles of his method and its applications to various problems concerning the cycloid, as well as to such questions as the quadrature of surfaces, cubature of volumes, determination of centers of gravity, and rectification of curved lines. In February 1659 these four pamphlets were collected in Lettres de A. Dettonville contenant quelques-unes de ses inventions de géométrie.
Early in 1659 Pascal again fell gravely ill and abandoned almost all his intellectual undertakings in order to devote himself to prayer and to charitable works.
Religion
Pascal's famous Provincial Letters deals with the great problems of Christian thought, faith versus reason, free will, and preknowledge. Pascal explains the contradictions and problems of the moral life in terms of the doctrine of the Fall and makes faith and revelation alone sufficient for their mutual justification.
Views
In most other respects, Pascal’s outlook is ahead of its time and admirable in its self-restraint and in its awareness of its own limitations. Unlike Bacon, he makes room for hypothesis and even imaginative insight and conjecture (l’esprit de finesse) and also allows a deductive component a la Descartes (l’esprit géométrique). He acknowledges that all hypotheses must be tested and confirmed by rigorous experiments, and even if he didn’t actually carry out his experiments exactly as described, he nevertheless accepts the necessity of such testing.
Pascal is frequently included in the ranks of “existentialist” philosophers, alongside names like Augustine, Kierkegaard, Nietzsche, Heidegger, and Sartre.
Personality
Pascal had a number of aristocratic friends and a little more money to spend from his patrimony.
His sister Gilberte tells of his asceticism, of his dislike of seeing her caress her children, and of his apparent revulsion from talk of feminine beauty.