Blaise Pascal

From Example Problems
Jump to navigation Jump to search
Blaise Pascal
Blaise Pascal
June 19, 1623
Clermont, France
August 19, 1662
Paris, France

Blaise Pascal (June 19, 1623August 19,1662) was a French mathematician, physicist, and religious philosopher. Pascal was a child prodigy, who was educated by his father. Pascal's earliest work was in the natural and applied sciences, where he made important contributions to the construction of mechanical calculators and the study of fluids, and clarified the concepts of pressure and vacuum by expanding the work of Evangelista Torricelli. Pascal also wrote powerfully in defence of the scientific method.

He was a mathematician of the first order. In mathematics, Pascal helped create two major new areas of research. He wrote a significant treatise on the subject of projective geometry at the age of sixteen and corresponded with Pierre de Fermat from 1654 on probability theory, strongly influencing the development of modern economics and social science.

Following a mystical experience in late 1654, he left mathematics and physics and devoted himself to reflection and writing about philosophy and theology. His two most famous works date from this period: the Lettres provinciales and the Pensées. However, he had suffered from ill-health throughout his life and his new interests were ended by his early death two months after his 39th birthday.

Early life and education

Born in Clermont, in the Auvergne region of France, Blaise Pascal lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal (15881651), was a local judge and member of the petite noblesse, who also had an interest in science and mathematics. Blaise Pascal was brother to Jacqueline Pascal and two other sisters, only one of whom, Gilberte, survived past childhood.

In 1631, Étienne moved with his children to Paris. Étienne decided that he would educate his son, who showed extraordinary mental and intellectual abilities. Young Blaise showed immediate aptitude for mathematics and science, perhaps inspired by his father's regular conversations with Paris's leading geometricians, including Roberval, Mersenne, Desargues, Mydorge, Gassendi, and Descartes. At the age of eleven, he composed a short treatise on the sounds of vibrating bodies and Étienne responded by forbidding his son to further pursue mathematics until the age of fifteen, so as not to harm his study of Latin and Greek. One day Étienne found Blaise (now twelve) writing on the wall with a piece of coal an independent proof that the sum of the angles of a triangle is equal to two right angles. From then on, the boy was allowed to study Euclid.

Particularly of interest to the young Pascal was the work of Desargues. Following Desargues's thinking, at age sixteen Pascal produced a treatise on conic sections, Essai pour les coniques ("Essay on Conics"). Most of it has been lost, but an important original result has lasted, now known as Pascal's theorem. Pascal's work was so precocious that Descartes, when shown the manuscript, refused to believe that the composition was not by his father.

In 1638, Étienne's opposition to fiscal relations of Cardinal Richelieu caused the family to flee Paris. It was only when Jacqueline performed well in a children's play performed in front of Richelieu that Étienne was pardoned. By 1639, the family had moved to Rouen where Étienne became a tax collector.

At age eighteen Pascal constructed a mechanical calculator, called Pascal's calculator or the Pascaline, capable of addition and subtraction, to help his father with this work. The Zwinger museum, in Dresden, Germany, exhibits one of his original mechanical calculators. Though these machines stand near the head of the development of computer engineering, the calculator failed to be a great commercial success. Pascal continued to make improvements to his design through the next decade and built a total of fifty machines.

Contributions to mathematics

File:Blaise Pascal.jpeg
Portrait of Blaise Pascal

In addition to the childhood marvels recorded above, Pascal continued to influence mathematics throughout his life. In 1653 Pascal wrote his Traité du triangle arithmétique in which he described a convenient tabular presentation for binomial coefficients, the "arithmetical triangle", now called Pascal's triangle. (It should be noted, however, that Yang Hui, a Chinese mathematician of the Qin dynasty, had independently worked out a concept similar to Pascal's triangle four centuries earlier.)

In 1654, prompted by a friend interested in gambling problems, he corresponded with Fermat on the subject, and from that collaboration was born the mathematical theory of probabilities. The friend was the Chevalier de Méré, and the specific problem was that of two players who want to finish a game early and, given the current circumstances of the game, want to divide the stakes fairly, based on the chance each has of winning the game from that point. (This was the introduction of the notion of expected value.) Pascal later (in the Pensées) used a probabilistic argument, Pascal's Wager, to justify belief in God and a virtuous life. The work done by Fermat and Pascal into the calculus of probabilities laid important groundwork for Leibniz's formulation of the infinitesimal calculus. [1]

After a religious experience in 1654, Pascal mostly gave up work in mathematics. However, after a sleepless night in 1658 he offered, anonymously, a prize for the quadrature of a cycloid. Solutions were offered by Wallis, Huygens, Wren, and others; then Pascal, under a pseudonym published his own solution. A controversy followed in which the competitors, including Pascal, behaved less than philosophically.

Philosophy of mathematics

Pascal's major contribution to the philosophy of mathematics came with his De l'Esprit géométrique ("On the Geometrical Spirit"), originally written as a preface to a geometry textbook for one of the famous "Little Schools of Port-Royal" (Les Petites-Ecoles de Port-Royal). The work was unpublished until over a century after his death. Here Pascal looked into the issue of discovering truths, arguing that the ideal such method would be to found all propositions on already established truths. At the same time, however, he claimed this was impossible because such established truths would require other truths to back them up—first principles cannot be reached. Based on this, Pascal argued that the procedure used in geometry was as perfect as possible, with certain principles assumed and other propositions developed from them. Nevertheless, there was no way to know the assumed principles to be true.

In De l'Art de persuader, Pascal looked deeper into geometry's axiomatic method, specifically the question of how people come to be convinced of the axioms upon which later conclusions are based. Pascal agreed with Montaigne that achieving certainty in these axioms and conclusions through human methods is impossible. He asserted that these principles can only be grasped through intuition, and that this fact underscored the necessity for submission to God in searching out truths.

Pascal also used De l'Esprit géométrique to develop a theory of definition. He distinguished between definitions which are conventional labels defined by the writer and definitions which are within the language and understood by everyone because they naturally designate their referent. The second type would be characteristic of the philosophy of essentialism. Pascal claimed that only definitions of the first type were important to science and mathematics, arguing that those fields should adopt the philosophy of formalism as formulated by Descartes.

Contributions to the physical sciences

Pascal's work in the fields of the study of fluids (hydrodynamics and hydrostatics) centered on the principles of hydraulic fluids. His inventions include the hydraulic press (using hydraulic pressure to multiply force) and the syringe. By 1646 Pascal had learned of Evangelista Torricelli's experimentation with barometers. Having replicated an experiment which involved placing a tube filled with mercury upside down in a bowl of mercury, Pascal questioned what force kept some mercury in the tube and what filled the space above the mercury in the tube. At the time, most scientists contended that some invisible matter was present there—not a vacuum.

Following more experimentation in this vein, in 1647 Pascal produced Experiences nouvelles touchant le vide, which detailed basic rules describing to what degree various liquids could be supported by air pressure. It also provided reasons why it was indeed a vacuum above the column of liquid in a barometer tube.

In 1648 Pascal continued his experiments by having his brother-in-law carry a barometer to higher elevation, confirming that the level of mercury would change, a result which Pascal replicated by carrying a barometer up and down a church tower in Paris. The experiment was hailed throughout Europe as finally establishing the principle and value of the barometer.

In the face of criticism that some invisible matter existed in Pascal's empty space, Pascal delivered in his reply to Estienne Noel one of the seventeenth century's major statements on the scientific method: "In order to show that a hypothesis is evident, it does not suffice that all the phenomena follow from it; instead, if it leads to something contrary to a single one of the phenomena, that suffices to establish its falsity." His insistence on the existence of the vacuum also led to conflict with a number of other prominent scientists, including Descartes.

Mature life, religion, philosophy, and literature

Pascal's statue at the Louvre.

Religious conversion

Biographically, we can say that two basic influences led him to his conversion: sickness and Jansenism. As early as his eighteenth year he suffered from a nervous ailment that left him hardly a day without pain. In 1647 a paralytic attack so disabled him that he could not move without crutches. His head ached, his bowels burned, his legs and feet were continually cold, and required wearisome aids to circulation of the blood; he wore stockings steeped in brandy to warm his feet. Partly to get better medical treatment, he moved with Jacqueline to Paris. His health improved, but his nervous system had been permanently damaged. Henceforth he was subject to deepening hypochondria, which affected his character and his philosophy. He became irritable, subject to fits of proud and imperious anger, and he seldom smiled. [2]

In 1646, Pascal's father was wounded in the thigh and was consequently looked after by a Jansenist physician. Blaise spoke with the doctor frequently, and upon his successful treatment of Étienne, borrowed works by Jansenist authors through him. In this period, Pascal experienced a sort of "first conversion" and began in the course of the following year to write on theological subjects.

Pascal fell away from this initial religious engagement and experienced a few years of what he called a "worldly period" (1648–54). His father died in 1651, and Pascal gained control over both his inheritance and that of his sister Jacqueline. In the same year Jacqueline moved to become a nun at Port-Royal, despite her brother's opposition. When the time came for her to make her ultimate vows, he refused to return to her enough of her inheritance to pay her dowry as a bride of Christ; without money she would attain a less desirable position in the convent hierarchy. Eventually, however, he relented on this point. [3]

When this was settled, Blaise found himself both rich and free. He took a sumptuously furnished home, staffed it with many servants, and drove about Paris in a coach behind four or six horses. His leisure was spent in the company of wits, women, and gamblers (as evidenced by his work on probability). For an exciting while he pursued in Auvergne a lady of beauty and learning, whom he referred to as the "Sappho of the countryside." [4] About this time he wrote a Discours sur les passions de l'amour, and apparently he contemplated marriage—which he was later to describe as "the lowest of the conditions of life permitted to a Christian." [5]

His sister Jacqueline reproached him for his frivolity and prayed for his reform. During visits to his sister at Port-Royal in 1654, he displayed contempt for affairs of the world but was not drawn to God. [6] Until...

Upon brink of death

In late 1654 he was involved in an accident at the Neuilly bridge where the horses plunged over the parapet and the carriage nearly followed them. Fortunately, the reins broke and the coach hung half over the edge. Pascal and his friends emerged, but the sensitive philosopher, terrified by the nearness of death, fainted away, and remained unconscious for some time. Upon recovering fifteen days later, on November 23, 1654, between ten thirty and twelve thirty at night, Pascal had an intense religious vision and immediately recorded the experience in a brief note to himself, which began: "Fire. God of Abraham, God of Isaac, God of Jacob, not of the philosophers and the scholars…" and concluded by quoting Psalms 119:16: "I will not forget thy word. Amen." He seems carefully to have sewn this document into his coat and always transferred it when he changed clothes; a servant discovered it only by chance after his death.[7] During his lifetime, Pascal was often mistakenly thought to be a libertine, and was later dismissed as an individual who had only a deathbed conversion.

His belief and religious commitment revitalized, Pascal visited the older of two convents at Port-Royal for a two-week retreat in January 1655. For the next four years, he regularly traveled between Port-Royal and Paris. It was at this point immediately after his conversion when he began writing his first major literary work on religion, the Provincial Letters.

The Provincial Letters

Main article: Lettres provinciales

Template:French literature (small) Beginning in 1656, Pascal published his memorable attack on casuistry, a popular ethical method used by Catholic thinkers in the early modern period (especially the Jesuits). Pascal denounced casuistry as the mere use of complex reasoning to justify moral laxity. His method of framing his arguments was clever: the Provincial Letters pretended to be the report of a Parisian to a friend in the provinces on the moral and theological issues then exciting the intellectual and religious circles in the capital. Pascal, combining the fervor of a convert with the wit and polish of a man of the world, reached a new level of style in French prose. The 18-letter series was published between 1656 and 1657 under the pseudonym Louis de Montalte and incensed Louis XIV, who ordered in 1660 that the book be shredded and burnt. In 1661, the Jansenist school at Port-Royal was condemned and closed down; those involved in it had to sign a 1656 papal bull condemning the teachings of Jansen as heretical. The final letter defied the Pope himself, provoking Alexander VII to condemn the letters (September 6, 1657). But that didn't stop all of educated France from reading them. Even Pope Alexander, while publicly opposing them, nonetheless was persuaded by Pascal's arguments. He condemned "laxism" in the church and ordered a revision of casuistical texts just a few years later (1665–66).

Aside from their religious influence, the Lettres provinciales were popular as a literary work. Pascal's use of humor, mockery, and vicious satire in his arguments made the letters ripe for public consumption, and influenced the prose of later French writers like Voltaire and Jean-Jacques Rousseau. The first few letters promote major principles of Jansenist teaching, for instance the dogmas of "proximate power" (Letter I) and "sufficient grace" (Letter II), and explain why they are not heretical. The later letters find Pascal more on the defensive—pressure on the Port Royal Jansenists to renounce their teachings was constantly growing through this time—and contain the assault on casuistry. Letter XIV contains the unique apology, "I would have written a shorter letter, but I did not have the time."

Wide praise has been given to the Provincial Letters. Voltaire called the Letters "the best-written book that has yet appeared in France."[8] And when Bossuet was asked what book he would rather have written had he not written his own, he answered, the Provincial Letters of Pascal.[9]


When Pascal was back in Paris just after overseeing the publication of the last Letter, his religion was reinforced by the close association to an apparent miracle in the chapel of the Port-Royal nunnery. His 10-year-old niece, Marguerite Périer, was suffering from a painful fistula lacrymalis that exuded noisome pus through her eyes and nose—an affliction the doctors pronounced hopeless. Then on March 24, 1657, a believer had presented to Port-Royal what he and others claimed to be a thorn from the crown that had tortured Christ. The nuns, in solemn ceremony and singing psalms, placed the thorn on their altar. Each in turn kissed the relic, and one of them, seeing Marguerite among the worshipers, took the thorn and with it touched the girl's sore. That evening, we are told, Marguerite expressed surprise that her eye no longer pained her; her mother was astonished to find no sign of the fistula; a physician, summoned, reported that the discharge and swelling had disappeared. He, not the nuns, spread word of what he termed a miraculous cure. Seven other physicians who had had previous knowledge of Marguerite's fistula subscribed a statement that in their judgment a miracle had taken place. The diocesan officials investigated, came to the same conclusion, and authorized a Te Deum Mass in Port-Royal. Crowds of believers came to see and kiss the thorn; all Catholic Paris acclaimed a miracle. Later both Jansenists and Catholics used this well-documented miracle to their defense. In 1728, Pope Benedict XIII referred to the case as proving that the age of miracles had not passed.

Pascal made himself an armorial emblem of an eye surrounded by a crown of thorns, with the inscription Scio cui credidi—"I know whom I have believed." [10] His beliefs renewed, he set his mind to write his final, and alas, unfinished testament, the Pensées.

The Pensées

Main article: Pensées

Unfortunately, Pascal couldn't finish his most influential theological work, the Pensées, before his death. It was to have been a sustained and coherent examination of and defense of the Christian faith, with the original title Apologie de la religion Chrétienne ("Defense of the Christian Religion"). What was found upon sifting through his personal items after his death were numerous scraps of paper with isolated thoughts, grouped in a tentative, but telling, order. The first version of the detached notes appeared in print as a book in 1670 titled Pensées de M. Pascal sur la réligion, et sur quelques autres sujets ("Thoughts of M. Pascal on religion, and on other subjects") and soon thereafter became a classic. Because his friends and the scholars at Port-Royal were concerned that these fragmentary "thoughts" might lead to skepticism rather than to piety, they concealed the skeptical pieces and modified some of the rest, lest King or Church should take offense[11] for at that time the persecution of Port-Royal had ceased, and the editors were not interested in a renewal of controversy. Not until the nineteenth century were the Pensées published in their full and authentic text.

Pascal's Pensées is widely considered to be a masterpiece, and a landmark in French prose. When commenting on one particular section, Sainte-Beuve praised it as the finest pages in the French language.[12] Will Durant, in his 11-volume, comprehensive The Story of Civilization series, hailed it as "the most eloquent book in French prose."[13] In Pensées, Pascal surveys several philosophical paradoxes: infinity and nothing, faith and reason, soul and matter, death and life, meaning and vanity—seemingly arriving at no definitive conclusions besides humility, ignorance, and grace. Rolling these into one he develops Pascal's Wager.

Last works and death

T.S. Eliot described him during this phase of his life as "a man of the world among ascetics, and an ascetic among men of the world." Pascal's ascetic lifestyle derived from a belief that it was natural and necessary for man to suffer. In his last years of bad health, he frequently tried to reject the ministrations of his doctors, saying, "Sickness is the natural state of Christians." [14]

In 1659 Pascal, whose health had never been good, fell seriously ill. An autopsy performed after his death revealed grave problems with his stomach and other organs of his abdomen, along with damage to his brain. Despite the autopsy, the cause of his continual poor health was never precisely determined, though speculation focuses on tuberculosis, stomach cancer, or a combination of the two.[15] The headaches which afflicted Pascal are generally attributed to his brain lesion.

Louis XIV suppressed the Jansenist movement at Port-Royal in 1661. In response, Pascal wrote one of his final works, Écrit sur la signature du formulaire, exhorting the Jansenists not to give in. Later that year, his sister Jacqueline died, which convinced Pascal to cease his polemics on Jansenism. Pascal's last major achievement, returning to his mechanical genius, was inaugurating perhaps the first bus line, moving passengers within Paris in a carriage with many seats.

In 1662, Pascal's illness became more violent. Aware that he had little chance to survive, he sought a move to the hospital for incurable diseases, but his doctors declared that he was too unstable to be carried. In Paris on August 18, 1662, Pascal went into convulsions and received extreme unction. He died the next morning, his last words being "May God never abandon me," and was buried in the cemetery of Saint-Étienne-du-Mont.[16]


In honor of his scientific contributions, the name Pascal has been given to the SI unit of pressure, to a programming language, and Pascal's law (an important principle of hydrostatics), and as mentioned above, Pascal's triangle and Pascal's wager still bear his name.

In Canada, there is an annual math contest named in his honour. The Pascal Contest is open to any student in Canada that is 14 years or under and is in grade 9 or lower.

Pascal's development of probability theory was his most influential contribution to mathematics. Originally applied to gambling, today it is extremely important in economics, especially in actuarial science. John Ross writes, "Probability theory and the discoveries following it changed the way we regard uncertainty, risk, decision-making, and an individual's and society's ability to influence the course of future events." [17] However, it should be noted that Pascal and Fermat, though doing important early work in probability theory, did not develop the field very far. Christiaan Huygens, learning of the subject from the correspondence of Pascal and Fermat, wrote the first book on the subject. Later figures who continued the development of the theory include Abraham de Moivre and Pierre-Simon Laplace.

In literature, Pascal is regarded as one of the most important authors of the French Classical Period, and is read today as one of the greatest masters of French prose. His use of satire and wit influenced later polemicists. The content of his literary work is best remembered for its strong opposition to the rationalism of René Descartes and simultaneous assertion that the main countervailing philosophy, empiricism, was also insufficient for determining major truths.

A discussion of Pascal figures prominently in the movie My Night At Maud's by the French director Eric Rohmer.


  • Essai pour les coniques (1639)
  • Experiences nouvelles touchant le vide (1647)
  • Traité du triangle arithmétique (1653)
  • Lettres provinciales (1656–57)
  • De l'Esprit géométrique (1657 or 1658)
  • Écrit sur la signature du formulaire (1661)
  • Pensées (incomplete at death)


  • Broome, JH. Pascal. ISBN 0713150211
  • Muir, Jane. Of Men and Numbers. New York: Dover Publications, Inc, 1996. ISBN 0486289737
  • Encyclopedia of Philosophy, 1967 edition, s.v. "Pascal, Blaise."
  • Pascal, Blaise. Oeuvres complètes. Paris: Seuil, 1960.


  1. ^  Sainte-Beuve, Port-Royal, I, 89.
  2. ^  Muir, 93.
  3. ^  Pascal, Pensées, Havet ed. Introd., p. civ.
  4. ^  Mesnard, Pascal, 57.
  5. ^  Encyclopedia of Philosophy, 52.
  6. ^  Oeuvres complètes, 618.
  7. ^  Voltaire, Age of Louis XIV 424, 358.
  8. ^  Voltaire, Age of Louis XIV 359.
  9. ^  Sainte-Beuve, Port-Royal, III, 173f.; Beard, Charles, Port-Royal, I 84.
  10. ^  Pascal, Pensées, Introduction, p. xxviii; Mesnard, Pascal, 137-138.
  11. ^  Sainte-Beuve, Seventeenth Century, 174.
  12. ^  Durant, The Age of Louis XIV, 66.
  13. ^  Muir, 103.
  14. ^  Muir, 104.

See also

External links


bg:Блез Паскал ca:Blaise Pascal cs:Blaise Pascal cy:Blaise Pascal da:Blaise Pascal de:Blaise Pascal et:Blaise Pascal es:Blaise Pascal eo:Blaise PASCAL fr:Blaise Pascal ko:블레즈 파스칼 io:Blaise Pascal id:Blaise Pascal ia:Blaise Pascal it:Blaise Pascal he:בלז פסקל la:Blasius Pascalis lt:Blezas Paskalis hu:Blaise Pascal nl:Blaise Pascal ja:ブレーズ・パスカル nn:Blaise Pascal pl:Blaise Pascal pt:Blaise Pascal ro:Blaise Pascal ru:Паскаль, Блез sk:Blaise Pascal sl:Blaise Pascal fi:Blaise Pascal sv:Blaise Pascal tr:Blaise Pascal zh:布莱士·帕斯卡