Blaise Pascal

Chapter 1 · Educated Entirely by His Father

A calculating machine at nineteen

1623 – 1645 · Clermont · Paris · Rouen

Blaise Pascal was born at Clermont in Auvergne in 1623. His mother died when he was three. His father Étienne, a tax official and a competent mathematician himself, moved the family to Paris and educated all three children personally, refusing to send Blaise to school.

Étienne's plan was to teach languages first and keep mathematics back until the boy was fifteen. The story — told by Pascal's sister Gilberte — is that Blaise, forbidden the subject, worked out a good deal of elementary geometry alone with charcoal on the floor of a playroom, and was discovered at about twelve having reached the theorem that the angles of a triangle sum to two right angles. Étienne gave in and handed him Euclid. Family testimony is not neutral evidence, but the boy was certainly extraordinary: at sixteen he produced a theorem on conic sections that Descartes refused at first to believe was the work of a child.

In 1639 Étienne was appointed tax commissioner at Rouen, with an enormous quantity of arithmetic to do. Between 1642 and 1645 Blaise designed and built a mechanical calculator — the *Pascaline* — with geared wheels and an automatic carry mechanism, to do his father's sums.

About twenty were made. They were expensive, fragile and commercially unsuccessful. They also worked, and they are among the first mechanical calculators in history.

Why this matters

The Pascaline was built to solve a real administrative problem, and its automatic carry mechanism is a genuine ancestor of mechanical computing.

You have the boy on the playroom floor and the machine of gears. What would you ask?

Ask Pascal

• “Did you really work out geometry on the floor in charcoal?”
• “Why did your father keep mathematics from you?”
• “How does the carry mechanism in your machine work?”
• “Why did the calculator not sell?”
• “What did Descartes say about your work on conics?”

Chapter 2 · A Mountain Settles an Argument

The Puy-de-Dôme, 19 September 1648

1646 – 1648 · Rouen · Paris · Clermont · Puy-de-Dôme

Torricelli had inverted a tube of mercury in a dish and found the column standing at about twenty-nine inches, with an apparently empty space above it. The schools insisted the space could not be empty, because nature abhors a vacuum; some invisible subtle matter must be there.

Pascal held a different account: the mercury is held up by the *weight of the air* pressing down on the dish, and the space above is genuinely empty.

He then designed the experiment that must decide between them. If the air has weight, then carrying the barometer up a mountain — where there is less air above you — must make the column fall. If the vacuum-horror account is right, altitude should make no difference at all.

On 19 September 1648 his brother-in-law Florin Périer carried a barometer up the Puy-de-Dôme, about fourteen hundred yards above Clermont, in the company of witnesses. The mercury fell by more than three inches, and rose again on the descent. Meanwhile a second barometer was left at the foot under the eye of a monk who watched it all day; it did not move.

That control is what makes it a great experiment rather than a good observation. Périer also measured at intermediate heights, and got intermediate falls.

The design is the lesson. Pascal did not look for evidence supporting his view. He identified the one measurement on which the two accounts must differ, and arranged for it to be made.

“The heart has its reasons, of which reason knows nothing.”

— Blaise Pascal, Pensées (published posthumously, 1670)

Why this matters

The Puy-de-Dôme experiment is a model of experimental design: a single measurement on which two competing theories are forced to disagree, with a control left behind.

You have the mountain, the monk and the falling column. What would you ask?

Ask Pascal

• “Why must the mercury fall on a mountain if air has weight?”
• “Why leave a second barometer at the bottom?”
• “How do you design an experiment that must decide?”
• “What did the Jesuits say when the result came in?”
• “Why send Périer rather than go yourself?”

Chapter 3 · Bursting a Barrel With a Few Cupfuls

Pressure transmitted equally in all directions

1648 – 1654 · Paris · Rouen

Pascal went on to establish how pressure behaves in a fluid at rest, and two results follow from his work.

The first: a fluid transmits an applied pressure equally and undiminished throughout itself, in every direction. Push on a small piston in an enclosed liquid and the pressure appears everywhere. Since force is pressure times area, a small force on a small piston balances a very large weight on a large piston. That is the hydraulic press, and every car jack, hydraulic brake and excavator arm since works on it.

The second: the pressure at a point in a liquid depends only on the *depth*, not on the shape of the vessel or the quantity of liquid in it.

This second claim is counter-intuitive, and Pascal demonstrated it in the most memorable way available. He fixed a long, thin pipe into the lid of a strong barrel filled with water, and poured a few cupfuls into the pipe. The column of water in the thin tube was tall, so the pressure at the bottom was high — and the barrel burst. A few cupfuls of water destroyed a cask that would have held many gallons safely.

What he lacked was units. He had no way of stating pressure as a calculated quantity, and no molecular or kinetic account of air at all — for him it was simply a heavy fluid. The SI unit of pressure now carries his name.

Why this matters

The principle that a fluid transmits pressure equally in all directions is the basis of all hydraulics, from a car's brakes to an excavator.

You have the burst barrel and the thin pipe. What is your question?

Ask Pascal

• “How can a few cupfuls of water burst a full barrel?”
• “Why does the shape of the vessel not matter?”
• “How does a small force balance a large weight in a hydraulic press?”
• “What did you think air actually was?”
• “What could you not calculate without units of pressure?”

Chapter 4 · The Problem of the Points

A gambler's question, answered in letters with Fermat

1654 · Paris · Toulouse

In 1654 the Chevalier de Méré, a gambler and man of letters, put a problem to Pascal. Two players are partway through a game of chance for a stake, and it has to be abandoned. How should the stake be divided fairly?

The obvious answers are wrong. Splitting by rounds won so far ignores how close each player is to victory. Splitting evenly ignores the current position entirely.

Pascal wrote to Pierre de Fermat in Toulouse, and over a summer of letters the two of them worked it out. The correct division is in proportion to each player's probability of eventually winning from the current position — which means enumerating all the ways the remaining rounds could go, counting how many lead to each player winning, and dividing accordingly.

That is the beginning of the mathematical theory of probability. The key idea is *expectation*: a position part-way through an uncertain process has a definite present value, calculable from the possible futures and their likelihoods. Every insurance premium, actuarial table and financial derivative rests on it.

Pascal also worked out the arithmetical triangle of binomial coefficients — known long before him in China, India and Persia, but treated by him systematically and now bearing his name in the West — and used it to solve such problems.

And the *Pensées* contains the wager: an argument that treats belief in God as a decision under uncertainty, weighing finite costs against infinite outcomes. It is the first serious use of decision theory, applied to the most consequential question he could think of.

Why this matters

The idea that an unfinished uncertain process has a calculable present value is the foundation of insurance, finance and statistics.

You have the abandoned game and the letters to Toulouse. What would you ask?

Ask Pascal

• “Why is splitting the stake by rounds won so far unfair?”
• “What does it mean for a position to have a present value?”
• “Did you and Fermat approach it the same way?”
• “Is the wager an argument or a piece of arithmetic?”
• “Should mathematics be used to settle questions about belief?”

Chapter 5 · The Night of Fire

Port-Royal, the Provincial Letters, and thirty-nine years

1654 – 1662 · Paris · Port-Royal

On the night of 23 November 1654 Pascal had an intense religious experience lasting about two hours. He wrote an account of it on a scrap of parchment — beginning with the word *Fire* — and sewed it into the lining of his coat, where it was found after his death. He never showed it to anyone.

Afterwards he attached himself to Port-Royal and to Jansenism, an austere movement within French Catholicism which his sister Jacqueline had already joined as a nun. When the Jansenists came under attack from the Jesuits and the Sorbonne, Pascal wrote eighteen anonymous *Provincial Letters* in their defence between 1656 and 1657 — witty, savage, and written in a plain French that changed the language. Voltaire, who agreed with almost nothing in them, called them the best-written book in French. They were condemned and burned by order of the King.

He regarded most of his scientific work afterwards as vanity — a distraction from what mattered. He returned to mathematics only briefly, working on the cycloid in 1658, reportedly to take his mind off toothache.

He was ill for essentially his whole adult life, in more or less constant pain from causes that have never been established. He also, in his last year, organised the first public transport system in the world: a network of carriages running fixed routes across Paris at a fixed fare, with the profits going to the poor.

He died on 19 August 1662, aged thirty-nine. The *Pensées* — notes for an unfinished defence of Christianity — were published eight years later.

Why this matters

Pascal's renunciation of his own science is unusual and genuine: he did not lose interest in it, he decided it was less important than something else.

You have the parchment in the coat and the buses across Paris. What would you ask?

Ask Pascal

• “Why sew your account of that night into your coat and tell nobody?”
• “Do you really think your science was vanity?”
• “What made the Provincial Letters worth being burned for?”
• “You built a public transport system — why?”
• “What would you have done with another twenty years?”

What Pascal changed

The Puy-de-Dôme experiment remains a model of how to design a test that must decide between two theories, and the principle that a fluid transmits pressure equally in all directions is the basis of all hydraulics; the SI unit of pressure carries his name. With Fermat he founded the mathematical theory of probability, whose central idea — that an unfinished uncertain process has a calculable present value — underlies insurance, statistics and finance.

A debate that continues

Much of what is known about Pascal's childhood comes from his sister Gilberte's memoir and cannot be independently checked, and the cause of the illness that dominated his life has never been established.

Keep exploring — ask Pascal

• “Which of your experiments would you run again with modern instruments?”
• “Can a wager really be a reason to believe?”
• “What did the pain cost your work?”

Related lives

• Marin Mersenne — Who Counted A Musical Note
• Pierre de Fermat — The Margin Was Too Narrow
• John Venn — Who Drew The Empty Compartments
• Thomas Bayes — The Question Turned Round

Related themes

Pressure in fluids · Probability · Experimental design