Blaise Pascal

How to use this course

Blaise Pascal lived 39 years. In that time he proved a fundamental theorem of projective geometry at 16, built one of the first mechanical calculators at 19, settled the existence of the vacuum and the weight of the air at 25, founded the theory of probability with Pierre de Fermat at 31, and wrote two classics of French prose, the Provincial Letters and the Pensées. Most readers know the writer. This course concentrates on the mathematician and physicist, and places his science within a life increasingly governed by religion.

Each technical lesson opens with a paragraph for the general reader, then states results in modern notation, sketches a proof or derivation, and ends with exercises whose solutions are folded. Pascal wrote in words and diagrams, without algebraic symbols for the most part; where we use a modern proof in place of his, we say so.

Timeline

Clermont and Paris, 1623–1639

Blaise Pascal was born in Clermont on 19 June 1623, the third child and only son of Étienne Pascal, a senior judge in the tax court of Auvergne, and Antoinette Begon. His elder sister Gilberte later married Florin Périer and wrote the first life of her brother; his younger sister Jacqueline became a poet and a nun. Their mother died when Blaise was three. Blaise was a sickly child, and he suffered pain and illness for most of his life.

In 1631 Étienne sold his office and moved the family to Paris to oversee his son’s education himself. He was a capable mathematician, known today through the curve called the limaçon of Pascal. He planned a curriculum of languages first and mathematics later, and at first kept geometry books away from the boy. Gilberte tells how Blaise, on his own, worked out the elements of geometry and had reached the theorem that the angles of a triangle sum to two right angles before his father found out. The story is a family memory and is likely embellished, but Étienne then gave him Euclid.

From about the age of 14 Blaise went with his father to the weekly gatherings around the Minim friar Marin Mersenne, where Gilles Personne de Roberval, Claude Mydorge, Girard Desargues and others discussed mathematics and natural philosophy. Desargues’s projective treatment of conics, published in 1639, inspired Pascal’s first discovery.

The family’s fortunes turned in 1638. Étienne joined a protest against the government’s reduction of interest on state bonds and had to hide from Cardinal Richelieu. He was restored to favour, in part through Jacqueline’s performance in a play before the cardinal, and in 1639 he was appointed royal commissioner for taxes in Upper Normandy, based in Rouen.

Rouen: conics and the machine, 1639–1646

In June 1639, aged 16, Pascal presented to Mersenne’s circle a sheet of theorems on conic sections. In February 1640, after the family had moved to Rouen, it was printed as a broadsheet, the Essai pour les coniques. Its central result, now called Pascal’s theorem, concerns a hexagon inscribed in a conic (Lesson 04). Pascal planned a complete treatise on conics built on this theorem. Descartes, shown the work, at first refused to believe a boy had written it. The full treatise was never published, and survives only through notes Gottfried Wilhelm Leibniz made from the manuscript in Paris in the 1670s.

Étienne’s new post required endless arithmetic: assessing and reapportioning taxes across the province. To help him, Blaise began in 1642, at 19, to design an adding machine. He built about fifty prototypes over three years, working with craftsmen in Rouen, to solve the problem of carrying reliably from one wheel to the next. In 1645 he presented the machine publicly with a dedicatory letter to Chancellor Pierre Séguier, and in 1649 a royal privilege gave him exclusive rights to make calculating machines in France. Production was costly and sales few; around twenty machines were made, and nine survive (Lesson 05).

In January 1646 Étienne dislocated his hip on the ice. He was treated by two brothers, bone-setters who were followers of the abbé of Saint-Cyran, the leader of the French Jansenists. Jansenism was a rigorist movement within Catholicism stressing grace, sin and personal conversion. The whole family adopted it. Pascal’s religious commitment began here, eight years before his more famous conversion.

The vacuum, 1646–1651

In 1643 Evangelista Torricelli in Florence had filled a glass tube with mercury, inverted it in a bowl, and watched the column fall to about 76 cm, leaving an apparently empty space at the top. News reached France through Mersenne. Most philosophers, following Aristotle, held that nature abhors a vacuum, and that the space must contain some subtle matter.

In 1646 and 1647 Pascal repeated the experiment in Rouen on a large scale, with tubes of water and red wine many metres long, in front of hundreds of spectators. He published his results in October 1647 as Expériences nouvelles touchant le vide. A Jesuit, Étienne Noël, replied that the space was filled with a refined air; Pascal’s response is a classic statement of how a hypothesis must be tested. In September 1647 Descartes visited him twice in Paris. They disagreed about the vacuum, and Descartes later claimed he had suggested the mountain experiment.

Torricelli had proposed that the column is held up by the weight of the air. Pascal designed a decisive test: if so, the column should be shorter at altitude. On 19 September 1648 his brother-in-law Florin Périer carried a barometer to the summit of the Puy de Dôme, near Clermont, with a second instrument left at the bottom as a control. The mercury stood lower at the top by a little more than three French inches. Pascal published the account that year as Récit de la grande expérience de l’équilibre des liqueurs and confirmed the effect on the tower of Saint-Jacques in Paris.

He then wrote two treatises, on the equilibrium of liquids and on the weight of the mass of the air, completed around 1653–54 and published in 1663 after his death. They state the principle of hydrostatic pressure now called Pascal’s law (Lesson 06). Étienne Pascal died in September 1651.

The mystic hexagram

For the general reader. Draw any oval-shaped conic, an ellipse, parabola or hyperbola, and mark six points on it. Join them in order to make a hexagon, perhaps a self-crossing one. Extend the three pairs of opposite sides until they meet. The three meeting points always fall on a straight line. Pascal found this at 16; it underlies the whole projective theory of conics.

Working in the projective plane removes exceptions: if two opposite sides are parallel, their intersection is a point at infinity. Desargues had introduced points at infinity in 1639, and Pascal used his methods, projecting a circle to any conic. Since collinearity and incidence survive projection, it suffices to prove the theorem for a circle.

A modern proof

Let \(\ell_{AB}\) denote a linear form vanishing on line \(AB\), and so on. Consider the two cubic curves

Both pass through the nine points \(A,B,C,D,E,F,P,Q,R\). Pick a seventh point \(X\) on the conic \(\Gamma\), different from the six, and choose \(\lambda,\mu\) not both zero with \(\lambda K_1+\mu K_2\) vanishing at \(X\). The cubic \(K=\lambda K_1+\mu K_2\) meets the conic in at least seven points \(A,\dots,F,X\). By Bézout’s theorem a conic and a cubic with no common component meet in at most six points, so \(K\) contains \(\Gamma\) as a component: \(K=\Gamma\cdot L\) with \(L\) a line. (\(K\) is not identically zero, since \(K_1\) and \(K_2\) have no common line factor.) The points \(P,Q,R\) lie on \(K\) but not on \(\Gamma\), because a line meets a non-degenerate conic in at most two points and, for example, \(P\) lies on line \(AB\) other than at \(A\) or \(B\). Hence \(P,Q,R\) lie on the line \(L\).

The proof never uses the order of the six points, so all 60 hexagons formed from six points on a conic have Pascal lines. Jakob Steiner, Julius Plücker and others studied this configuration of 60 lines in the nineteenth century.

The Pascaline: arithmetic in brass

For the general reader. Pascal’s calculator was a box of numbered wheels, one for each digit. Turning a wheel past 9 had to advance the next wheel by one, and a long chain of 9s had to roll over all at once without jamming. Solving that carry problem was the real invention. Subtraction was done by a trick still used inside computers: adding complements.

Carries

Each wheel of the Pascaline shows a digit \(0\)–\(9\). Pascal’s carry device, the sautoir, was a weighted lever lifted gradually as a wheel turned from 0 toward 9 and dropped onto the next wheel as it passed from 9 to 0, advancing that wheel one step. Because each carry is driven by gravity rather than by the operator’s force on the first wheel, a carry can ripple through any number of wheels. Before each calculation the operator reset the machine, which displayed the maximum and then rolled to zero, testing the whole carry chain.

The machine computes the following, digit by digit from the right.

Proof. By definition \(x_i+y_i+c_i=s_i+bc_{i+1}\). Multiply by \(b^i\) and sum: \(x+y+\sum_i c_ib^i=\sum_i s_ib^i+\sum_i c_{i+1}b^{i+1}\). The carry terms cancel except \(c_nb^n\), since \(c_0=0\). Induction on \(i\) shows \(c_i\le1\).

Subtraction by complements

Turning wheels backwards would have disturbed the gravity-driven carry, so the Pascaline only added. Each wheel also carried a second set of numerals showing the nines’ complement \(9-d\), visible through a separate window.

Proof. \(\bar x+y=b^n-1-(x-y)\le b^n-1\), so no overflow occurs, and its complement is \(b^n-1-(b^n-1-(x-y))=x-y\). In practice the operator set the machine so that the complement window showed \(x\), added \(y\) on the wheels, and read \(x-y\) in the complement window. Modern processors subtract the same way, with two’s complement in base 2.

Pressure, the barometer, the Puy de Dôme

For the general reader. We live at the bottom of an ocean of air, and its weight presses on everything. Pascal showed this by an experiment on a mountain, and he explained why a small push on a narrow piston can lift a heavy weight on a wide one. The SI unit of pressure bears his name.

Combined with the weight of the fluid this gives the hydrostatic law. Consider a vertical column of fluid of density \(\rho\) with cross-section \(A\), between heights \(z\) and \(z+dz\). Balance of forces gives \(p(z)A-p(z+dz)A=\rho gA\,dz\), hence

Pressure depends only on depth, not on the shape of the vessel. This is the “hydrostatic paradox” Pascal illustrated with vessels of different shapes holding columns of equal height.

The hydraulic press

Pascal described a vessel closed by two pistons, one a hundred times larger than the other: one man pushing the small piston balances a hundred men on the large one. With areas \(A_1,A_2\), equal pressure gives \(F_2/F_1=A_2/A_1\). Energy is conserved: the large piston moves \(A_1/A_2\) times as far.

The barometer

In Torricelli’s tube the space above the mercury holds almost nothing, so the column of height \(h\) is supported by the air pressure \(p_0\) on the open bowl: \(p_0=\rho_{\text{Hg}}gh\). With \(\rho_{\text{Hg}}=13\,600\ \text{kg/m}^3\) and \(p_0=101\,325\ \text{Pa}\), \(h\approx0.76\ \text{m}\). Pascal’s long tubes of water and wine showed that the height scales inversely with density, as the weight-of-air explanation requires and the abhorrence-of-vacuum explanation does not.

The atmosphere

For air, \(\rho\) depends on \(p\). For an ideal gas at constant temperature \(T\), \(\rho=pM/(RT)\), so \(dp/dz=-p\,Mg/(RT)\) and

Pascal had no such formula. He argued only that the column must fall with height, and that the difference must be measurable on a mountain a thousand metres high.

The arithmetical triangle and induction

For the general reader. Write 1s down two sides of a triangle and fill each inner cell with the sum of the two above it. The numbers that appear count combinations, give the coefficients of powers of a sum, and solve problems of chance. The table was known for centuries in India, Persia, China and Europe. Pascal’s treatise gathered its properties, proved them carefully, and in doing so gave one of the first clear statements of proof by induction.

The treatise

The Traité du triangle arithmétique was printed in 1654 but not distributed until 1665. Pascal arranged the numbers in a square grid: the first row and first column are 1s, and each cell is the sum of the cell above and the cell to its left. The cells along each diagonal, which he called a base, form a row of the modern triangle. In modern notation the cell in base \(n\) at position \(k\) is \(\binom nk\).

Pascal’s induction

For the twelfth consequence Pascal wrote that although the proposition has infinitely many cases, he would prove it with two lemmas. The first is that it holds in the first base. The second is that if it holds in some base, it holds in the next. This is induction stated explicitly. In modern form: suppose \(T(n-1,j+1)/T(n-1,j)=(n-1-j)/(j+1)\) for all \(j\). Then

(The middle step uses the hypothesis twice, once with \(j=k\) and once with \(j=k-1\); edge cases \(k=0\) and \(k=n-1\) are checked directly.) Francesco Maurolico had used similar arguments in 1575; Pascal’s statement of the two steps is the cleaner of the two.

Pascal also gave, in a companion piece on sums of numerical powers, a method for \(\sum_{i=1}^{m} i^k\) for every \(k\) using the binomial expansion of \((i+1)^{k+1}-i^{k+1}\).

The rule of division: probability

For the general reader. A gambler asked Pascal why one bet that seemed equivalent to another kept losing money. Another old question asked how to split the stakes of an unfinished game. Pascal and Fermat answered both in letters during the summer of 1654. The idea that a future uncertain gain has a precise present value is the start of probability theory.

De Méré’s problem

Antoine Gombaud, the Chevalier de Méré, a writer and gambler in Pascal’s circle, knew that betting on at least one six in four throws of a die was profitable. He reasoned that betting on at least one double six in 24 throws of two dice should be equally good, since 24 is to 36 as 4 is to 6. Experience said otherwise. Pascal mentions the problem in his letter to Fermat of 29 July 1654, remarking that de Méré was clever but no geometer.

The problem of points

Two players of equal skill stake equal sums on a game won by whoever first wins a set number of rounds. Play stops when A needs \(a\) more rounds and B needs \(b\). Pascal’s principle: the value of a position is the average of the values of the two positions that can follow it. With \(E(a,b)\) the fraction of the stakes due to A,

Proof. The formula satisfies the boundary conditions. For the recursion, use \(\binom{n}{k}=\binom{n-1}{k-1}+\binom{n-1}{k}\): \(\sum_{k=0}^{b-1}\binom nk=\sum_{k=0}^{b-1}\binom{n-1}{k}+\sum_{k=0}^{b-2}\binom{n-1}{k}\), which is \(2^{n-1}\big(E(a-1,b)+E(a,b-1)\big)\). Divide by \(2^n\). Interpretation: in \(n\) further rounds, imagined played to the end, A wins the match exactly when B wins fewer than \(b\) of them. This is Fermat’s enumeration, and Pascal recognized that the two methods agree.

Expectation and the wager

For the general reader. Pascal’s solution of the problem of points contains a new idea: a chance of winning something has a definite value now, equal to the prize weighted by its probability. He later applied the same reasoning to the choice of whether to live as a believer, in the argument known as Pascal’s wager. It is the first piece of formal decision theory.

Christiaan Huygens learned of the Pascal–Fermat problems on a visit to Paris in 1655 and published the first printed treatise on probability, De ratiociniis in ludo aleae, in 1657. It takes expectation as its basic notion.

The wager

In a fragment of the Pensées headed “Infini rien”, Pascal argues that reason cannot settle whether God exists, yet one must choose how to live. He frames the choice as a game. Let \(p>0\) be the probability one assigns to God’s existence. Wagering for God costs at most a finite amount \(c\) of worldly pleasure, and if God exists gains an infinite happiness. Wagering against gains at most a finite amount. In modern terms:

Pascal argued that even with only one chance of gain against a finite number of chances of loss, an infinite stake decides the matter. The argument has drawn objections ever since: that belief cannot be chosen by calculation, which Pascal himself addressed by recommending practice; that many incompatible religions could each offer infinite rewards; and that infinite payoffs break ordinary expected-value reasoning, as in the St Petersburg paradox later posed by Nicolaus Bernoulli.

The cycloid and the sines of a quadrant

For the general reader. A point on the rim of a rolling wheel traces a curve of arches called the cycloid. In 1658 Pascal returned to mathematics to solve hard problems about its areas, centres of gravity and solids of revolution, and challenged Europe to do the same. One small lemma from this work, about a tiny triangle on a circle, gave Leibniz the key idea of his calculus.

The cycloid

A circle of radius \(r\) rolling along a line traces

for one arch. Roberval had found its area in the 1630s, and Torricelli had published a proof in 1644, which led to a priority dispute. In 1658, during a sleepless night of toothache according to his sister, Pascal found the solutions to a set of harder problems. He issued them as a prize challenge under the name Amos Dettonville, an anagram of Louis de Montalte, his pseudonym in the Provinciales. Only John Wallis and the Jesuit Antoine de Lalouvère submitted solutions, and Pascal’s judges, among them Roberval, awarded no prize. Christopher Wren, in response, found the arc length. Pascal published his own methods in the Lettres de A. Dettonville (1658–59) and a history of the curve.

Derivation. Area: \(\int y\,dx=\int_0^{2\pi}r^2(1-\cos t)^2\,dt=r^2\int_0^{2\pi}(1-2\cos t+\cos^2t)\,dt=r^2(2\pi+\pi)=3\pi r^2\). Length: \(ds=r\sqrt{(1-\cos t)^2+\sin^2t}\,dt=2r\sin\tfrac t2\,dt\), and \(\int_0^{2\pi}2r\sin\tfrac t2\,dt=8r\).

The sines of a quadrant

In the Traité des sinus du quart de cercle (1659), Pascal considered a quarter circle of radius \(R\). At a point \(D\) on the arc, take a small arc \(EE'\) through \(D\), with tangent segment \(ds\), and let \(dx\) be its horizontal projection. The small triangle with sides \(dx\), \(dy\), \(ds\) is similar to the triangle formed by the radius to \(D\), the ordinate \(y\) (the “sine”) and the horizontal from the centre.

With \(y=R\sin\theta\) and \(ds=R\,d\theta\), the lemma gives \(\int_0^{\alpha}\sin\theta\,d\theta=1-\cos\alpha\), the first integral of a trigonometric function. Leibniz wrote that reading this lemma in 1673 showed him that the same similar-triangle argument works for any curve, with the normal in place of the radius. The small triangle \((dx,dy,ds)\) became his characteristic triangle, and from it came the transformation of integrals that led him to the calculus. Pascal missed the generalization, Leibniz remarked, as if a blindfold had covered his eyes.

The night of fire and Port-Royal, 1654–1662

After his father’s death in 1651 and Jacqueline’s entry into Port-Royal in January 1652, Pascal spent a period in society. His friends included the Duke of Roannez, the Chevalier de Méré and Damien Mitton, and he toured the provinces with Roannez. The years 1653 and 1654 were also mathematically rich: the hydrostatics treatises, the arithmetical triangle, and the correspondence with Fermat.

On the night of 23 November 1654, from about half past ten until half past midnight, Pascal had an overwhelming religious experience. He recorded it on a sheet of paper, the Mémorial, which opens with the single word “Fire” and invokes the God of Abraham, Isaac and Jacob, not of the philosophers and scholars. He sewed it into the lining of his coat and moved it into each new coat; a servant found it after his death. From January 1655 he made retreats at Port-Royal des Champs, south-west of Paris.

The Provincial Letters

In 1655–56 the Sorbonne moved to condemn Antoine Arnauld, the leading theologian of Port-Royal. Between January 1656 and March 1657 Pascal published eighteen anonymous letters, the Lettres provinciales, signed Louis de Montalte. With irony and close quotation he attacked the moral casuistry of the Jesuits. They were a public sensation and are counted among the first masterpieces of modern French prose. The book was placed on the Index and ordered burned, but it shaped European ideas about the Jesuits for a century. In March 1656 his niece Marguerite Périer was healed of an eye condition after touching a relic held to be a thorn of the Crown of Thorns at Port-Royal, an event that deepened Pascal’s faith.

The Pensées

From 1656 to 1658 Pascal began collecting notes for an apology for the Christian religion, addressed to sceptics and free-thinkers. He never finished it. The notes, on scraps of paper sorted into bundles, were published by Port-Royal in 1670 as the Pensées. They contain the wager (Lesson 09), the reflections on the two infinities, and the famous image of the human being as a thinking reed.

Mathematics returned briefly in 1658–59 with the cycloid (Lesson 10). In 1660 Fermat proposed a meeting; Pascal, too ill to travel, answered with high praise for Fermat as a geometer, and added that he now regarded geometry as the finest of occupations but only an occupation. His last practical project was a public coach service in Paris, the carrosses à cinq sols, which began running in 1662 with the Duke of Roannez and others: fixed routes, fixed timetables and a fixed fare.

Death and legacy

Pascal’s health, poor all his life, collapsed in 1662. He gave much of his property to the poor and took into his house a family with a sick child, moving himself to Gilberte’s home in Paris. He died there on 19 August 1662, aged 39, after weeks of severe pain. An autopsy reported damage to the stomach and other organs and a lesion of the brain; the exact cause remains uncertain. He was buried in the church of Saint-Étienne-du-Mont in Paris.

Legacy

• Geometry: Pascal’s theorem became a cornerstone of projective geometry when Jean-Victor Poncelet and others revived the subject in the nineteenth century.
• Probability: with Fermat he founded the mathematical theory of chance, and his notion of expectation passed through Huygens to Jacob Bernoulli and Abraham de Moivre. The arithmetical triangle carries his name in Western languages because of Pierre Rémond de Montmort (1708) and de Moivre (1730).
• Calculus: his indivisible methods for the cycloid and the lemma on the sines of a quadrant led Leibniz to the characteristic triangle.
• Physics: the Puy de Dôme experiment established the weight of the air and the reality of the vacuum, and his hydrostatics underlies hydraulic machinery. The SI unit of pressure, the pascal, has borne his name since 1971.
• Computing: the Pascaline is among the earliest working mechanical calculators, and Leibniz’s stepped reckoner improved on it. Niklaus Wirth named the programming language Pascal after him.
• Letters and thought: the Provinciales and the Pensées are classics of French literature and of Christian apologetics, and the wager remains a standard topic in decision theory and philosophy of religion.

Sources and further reading

Biography

1. J. J. O’Connor and E. F. Robertson, Blaise Pascal, MacTutor History of Mathematics.
2. Blaise Pascal, Wikipedia.
3. G. Périer, La Vie de Monsieur Pascal, written 1662, first printed 1684.
4. J. Mesnard, Pascal, l’homme et l’œuvre, Boivin, 1951.
5. D. Adamson, Blaise Pascal: Mathematician, Physicist and Thinker about God, Macmillan, 1995.

Mathematics and physics

1. B. Pascal, Œuvres complètes, ed. J. Mesnard, Desclée de Brouwer, 1964–1992.
2. B. Pascal, Traité du triangle arithmétique, Paris, 1665.
3. B. Pascal, Traitez de l’équilibre des liqueurs et de la pesanteur de la masse de l’air, Paris, 1663.
4. Pascal’s triangle and Pascaline, Wikipedia (historical sections).
5. K. Brown, The Helen of Geometers, MathPages (the cycloid and the 1658 challenge).
6. A. W. F. Edwards, Pascal’s Arithmetical Triangle, Griffin and Oxford University Press, 1987.
7. K. Devlin, The Unfinished Game: Pascal, Fermat, and the Seventeenth-Century Letter that Made the World Modern, Basic Books, 2008.
8. I. Hacking, The Emergence of Probability, Cambridge University Press, 1975 (on the wager as decision theory).