Blaise Pascal

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Blaise Pascal
Blaise Pascal
Born June 19, 1623
Clermont-Ferrand, France
Died August 19, 1662
Paris, France
Occupation mathematician, physicist, philosopher

Blaise Pascal (pronounced [blez pɑskɑl]), ( June 19, 1623 – August 19, 1662) was a French mathematician, physicist, and religious philosopher. He 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, the study of fluids, and clarified the concepts of pressure and vacuum by generalizing the work of Evangelista Torricelli. Pascal also wrote powerfully in defense of the scientific method.

He was a mathematician of the first order. 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 and later on probability theory, strongly influencing the development of modern economics and social science.

Following a mystical experience in late 1654, he abandoned his scientific work and devoted himself to 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-Ferrand, in the Auvergne region of France, Blaise Pascal lost his mother, Antoinette Begon, at the age of three. His father, Étienne Pascal (1588–1651), 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 intellectual ability. Young Pascal showed immediate aptitude for mathematics and science, perhaps inspired by his father's regular conversations with Paris' 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, however, Étienne found Blaise (now twelve) writing an independent proof that the sum of the angles of a triangle is equal to two right angles with a piece of coal on a wall. From then on, the boy was allowed to study Euclid.

Particularly of interest to 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 the fiscal policies of Cardinal Richelieu eventually forced the family to flee Paris. It was only when Jacqueline performed well in a children's play with Richelieu in attendance 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 capable of addition and subtraction, called Pascal's calculator or the Pascaline, to help his father with this work. The Zwinger museum, in Dresden, Germany, exhibits one of his original mechanical calculators. Though these machines are early forerunners to 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 fifty machines in total.

Contributions to mathematics

Portrait of Blaise Pascal
Portrait of Blaise Pascal

In addition to the childhood marvels previously mentioned, Pascal continued to influence mathematics throughout his life. In 1653, Pascal wrote his Traité du triangle arithmétique ("Treatment of the Arithmetical Triangle") in which he described a convenient tabular presentation for binomial coefficients, now called Pascal's triangle. ( 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. From this discussion, the notion of expected value was introduced. 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.

After a religious experience in 1654, Pascal mostly gave up work in mathematics. However, after a sleepless night in 1658, he anonymously offered a prize for the quadrature of a cycloid. Solutions were offered by Wallis, Huygens, Wren, and others; Pascal, under a pseudonym, published his own solution. Controversy and heated argument followed after Pascal announced himself the winner.

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, therefore, 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.

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.

In De l'Art de persuader ("On the Art of Persuasion"), 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.

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, rather than a vacuum, some invisible matter was present.

Following more experimentation in this vein, in 1647 Pascal produced Experiences nouvelles touchant le vide ("New Experiments with the Vacuum"), 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 a 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 must exist in Pascal's empty space, Pascal, in his reply to Estienne Noel, gave 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.

Adult life, religion, philosophy, and literature

For after all what is man in nature? A nothing in relation to infinity, all in relation to nothing, a central point between nothing and all and infinitely far from understanding either. The ends of things and their beginnings are impregnably concealed from him in an impenetrable secret. He is equally incapable of seeing the nothingness out of which he was drawn and the infinite in which he is engulfed. -Blaise Pascal

Religious conversion

Pascal studying the cycloid, by Augustin Pajou, 1785, Louvre
Pascal studying the cycloid, by Augustin Pajou, 1785, Louvre

Biographically, we can say that two basic influences led him to his conversion: sickness and Jansenism. From as early as his eighteenth year, Pascal 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 to Paris with his sister Jacqueline. 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 seldom smiled.

In 1645, 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 as well as his sister Jacqueline's. In that 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 the money, she would attain a less desirable position in the convent hierarchy. Eventually, however, Pascal relented on this point.

With his sister's affairs settled, Pascal 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." Around this time, Pascal wrote Discours sur les passions de l'amour ("Conversation about the Passions of Love"), and apparently he contemplated marriage—which he was later to describe as "the lowest of the conditions of life permitted to a Christian."

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.

Brush with death

On November 23, 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 halfway over the edge. Pascal and his friends emerged unscathed, but the sensitive philosopher, terrified by the nearness of death, fainted away and remained unconscious for some time. Upon recovering fifteen days later, 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 Psalm 119:16: "I will not forget thy word. Amen." He seems to have carefully sewn this document into his coat and always transferred it when he changed clothes; a servant discovered it only by chance after his death. During his lifetime, Pascal was often mistakenly thought to be a libertine, and was later dismissed as an individual who had only made 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

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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. The king ordered that the book be shredded and burnt in 1660. In 1661, the Jansenist school at Port-Royal was condemned and closed down; those involved with the school had to sign a 1656 papal bull condemning the teachings of Jansen as heretical. The final letter from Pascal, in 1657, had defied the Pope himself, provoking Alexander VII to condemn the letters. 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 Provincial Letters 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, such as 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 XVI 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." 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.


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 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 signed 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 of 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." His beliefs renewed, he set his mind to write his final, and, alas, unfinished testament, the Pensées.

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Last works and death

Pascal's epitaph in Saint-Étienne-du-Mont, where he was buried
Pascal's epitaph in Saint-Étienne-du-Mont, where he was buried

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 1659, Pascal, whose health had never been good, fell seriously ill. During his last years, he frequently tried to reject the ministrations of his doctors, saying, "Sickness is the natural state of Christians."

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 ("Writ on the Signing of the Form"), 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 his health was fading quickly, 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.

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. The headaches which afflicted Pascal are generally attributed to his brain lesion.


In honour 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.

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." 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.


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

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

Roberto Rossellini directed a filmed biopic (entitled Blaise Pascal) which originally aired on Italian television in 1971. Pierre Arditi starred as Pascal.


  • 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)
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