Pascal: The Scientific Treatises

Blaise Pascal was a French boy who loved puzzles. When he was a teenager he built an adding machine with gears to help his father, who spent long nights doing tax sums. When he grew up he asked a big question: does air weigh anything?

In Italy, a scientist named Evangelista Torricelli had filled a long glass tube with mercury, a heavy liquid metal, and turned it upside down in a bowl of mercury. The mercury dropped a little and then stopped, about as high as your arm is long. Above it, at the top of the tube, was an empty space. What held the mercury up?

Pascal had an idea. Maybe the air outside, pressing down on the bowl, holds the mercury up, like a heavy blanket of air over the whole Earth. If that is true, then on top of a mountain, where there is less air above you, the mercury should stand lower.

In 1648 Pascal's brother-in-law, Florin Périer, carried the tube up the Puy de Dôme, a tall hill near Pascal's hometown. At the top, the mercury stood lower, just as Pascal had guessed. Coming back down, it rose again. Air has weight!

Pascal also loved a triangle of numbers. Start with 1 at the top. Each new number is the sum of the two numbers just above it. This triangle helps us count things, like how many ways you can pick two flavors from five.

Who, when and the problem

Blaise Pascal (1623–1662) was born in Clermont, in the Auvergne region of France, and was taught at home by his father, Étienne, a lawyer and tax official with a strong interest in mathematics. At 16 he wrote a short essay on conic sections. From 1642 he designed a gear-driven calculating machine, the Pascaline, to help with his father's tax work in Rouen.

The science volume of his work (volume 30 in the 1990 Great Books) gathers his short treatises on three subjects: the vacuum, the equilibrium of liquids and the weight of the air, and the arithmetical triangle. They were written between 1647 and 1654. Two of the biggest were printed only after his death: the liquids and air treatises in 1663 and the triangle in 1665.

The problem he faced was an old teaching that went back to Aristotle and was taught in the schools: nature abhors a vacuum. A pump lifts water, people said, because nature will not allow an empty space to open up. In 1643–1644 Torricelli and his friend Vincenzo Viviani in Florence turned a tube of mercury upside down in a dish and found the mercury always stopped at the same height, with an apparently empty space above it. Torricelli suggested that the weight of the air held the mercury up. Pascal heard of it in 1646 and set out to test the idea from every side.

The main ideas

• An empty space can exist. In Rouen in 1646–1647 Pascal repeated Torricelli's experiment with tubes, syringes and bellows, including tubes of water and of wine taller than a house. He reported the results in New Experiments Concerning the Vacuum (1647).
• The air has weight, and its weight presses on everything. If the mercury is held up by the air above it, there should be less push at the top of a mountain. The test on the Puy de Dôme (1648) showed exactly that.
• Liquids pass pressure on. In a closed vessel, a push on one part of the liquid is felt equally everywhere in it. A small piston can balance a large one: this is the idea behind the hydraulic press.
• Pressure depends on depth, not on the amount of liquid. A thin tube of water pushes on its base as hard as a wide one of the same height.
• Numbers in a triangle can count and divide fairly. His arithmetical triangle gives the counts of combinations, and he used it to settle the "problem of points": how to split the stakes when a game is stopped before it ends.

The great experiment on the mountain

Pascal could not travel much because of poor health, so in November 1647 he wrote to his brother-in-law, Florin Périer, who lived in Clermont, near the Puy de Dôme, a mountain about 1,465 meters high. On September 19, 1648, Périer and several witnesses filled two mercury tubes in the garden of a monastery in town. One stayed behind with a friar watching it. The other went up the mountain. At the summit the mercury stood a little over 8 centimeters lower than in town, and it rose again step by step on the way down, while the tube left in the garden did not change all day. Pascal published the account that year as the Story of the Great Experiment on the Equilibrium of Liquids, and checked it himself on a church tower in Paris.

Liquids in balance

In the Treatise on the Equilibrium of Liquids Pascal imagines a vessel full of water, closed on all sides, with two openings, one a hundred times as large as the other, each fitted with a piston. One man pushing the small piston balances a hundred men pushing the large one. He called this "a new machine for multiplying forces." Today we call it Pascal's principle, and it works every car's hydraulic brakes. The Flemish engineer Simon Stevin had already shown in 1586 that a liquid's push on the bottom depends only on its depth; Pascal made the ideas into one connected system.

The arithmetical triangle

Pascal's triangle is drawn as a grid of cells. The corner cell holds 1, the "generator." Every other cell is the sum of the cell just before it in its row and the cell just before it in its column. Read along the diagonals (Pascal's "bases") and you get the rows we usually draw today: 1; 1 1; 1 2 1; 1 3 3 1; and so on. From that one rule he proved nineteen "consequences"; several are proofs by what we now call mathematical induction, one of the first clear uses of that method.

In 1654 Pascal and Pierre de Fermat wrote letters to each other about the problem of points. Suppose two players each put 32 coins in a pot, and the first to win three rounds takes it all. They must stop when one player needs 1 more win and the other needs 2. What is a fair split? Pascal's answer: imagine the most rounds that could still be needed, and count, with the triangle, the ways each player could win them. Their letters are often counted as the start of probability theory.

Pascal in his own words

Notice how fair he is to the ancient thinkers. They were reasonable, given what they could see. But a general rule drawn from experiments is only as good as the cases it has met.

Setting the record straight

• Pascal did not climb the mountain. Florin Périer carried out the Puy de Dôme test at Pascal's request, with careful witnesses and a control tube left in town.
• Pascal did not invent "Pascal's triangle." It appears much earlier: in India in the 10th century (Halayudha's commentary on Pingala), in Persia (al-Karaji around 1000, and later Omar Khayyam), and in China, where it is called Yang Hui's triangle after a book of 1261 that credits Jia Xian (11th century). In Europe it was printed by Petrus Apianus in 1527 and used by Michael Stifel and Niccolò Tartaglia. Pascal's treatise gave it a full theory and new uses, and his name stuck in the West after the 18th century.
• Torricelli came first. The mercury tube is Torricelli's (1643–1644). Pascal's contribution was a long series of variations and the decisive test of the weight-of-air explanation.
• Who suggested the mountain? René Descartes later claimed in letters that he had suggested the altitude test to Pascal in 1647. Pascal never credited him. Historians still discuss it, and the claim cannot be settled from the surviving evidence.

What later science kept, and what it changed

• Kept: the weight of the air (atmospheric pressure), $P = \rho g h$ for liquids, Pascal's principle, the barometer for weather and the altimeter for height, and the triangle at the heart of combinatorics and the binomial theorem.
• Built on: Christiaan Huygens wrote the first printed book on probability (1657) after hearing of the Pascal–Fermat problems. Robert Boyle, with his air pump, measured how air's volume changes with pressure (1662).
• Changed: Pascal's empty space at the top of the tube is not quite empty: it holds a little mercury vapor. And in modern physics even the best vacuum is not simply "nothing," but Pascal's main point, that a space empty of air can exist, stands.

Authority and experiment. Pascal's Preface to the Treatise on Vacuum draws a line between subjects that rest on authority, such as history, law and, in his view, theology, and subjects that rest on reasoning and experiment, such as geometry and physics. In the second kind, he says, the ancients are to be respected but not followed blindly, because knowledge grows. This is one of the clearest early statements of the idea of scientific progress.

The debate with the plenists. Not everyone accepted an empty space. The Jesuit Étienne Noël wrote to Pascal in 1647 that the space at the top of the tube must be filled with a fine, invisible matter, and Descartes held a similar view: for him, extension itself was matter, so true emptiness was impossible. Pascal answered that an explanation which can never be tested by any experiment is no explanation. The question "what fills space?" returned later as the debate over the ether, which Huygens, Newton and Faraday all faced in their own ways.

Induction. In his triangle treatise Pascal proves a statement for the first base, then shows that if it holds for one base it holds for the next. That two-step pattern is mathematical induction. Earlier mathematicians, such as Francesco Maurolico (1575), argued in similar ways, but Pascal states the two steps clearly.

Pascal's other books. After a religious experience in November 1654, Pascal turned mostly to writing about faith and morals: the Provincial Letters (1656–1657) and the notes published after his death as the Pensées ("Thoughts"). They have their own rungs on this ladder. Readers of the Pensées will meet the same mind: there he writes about the infinitely large and the infinitely small, and about the limits of reason. His famous wager argument also uses the reasoning about chances that began with the problem of points.

Questions to carry forward. Does a rule drawn from many experiments ever become certain? Newton's Principia (next on the ladder) gives one answer in its "rules of reasoning." Is a theory that cannot be tested still science? Compare Pascal's reply to Noël with Newton's "I frame no hypotheses."