Five-minute digest
Scientific and Mathematical Treatises
Summary
The treatises gathered here are the work of Pascal between his twenty-third and thirty-fifth years, before and just after the conversion that turned him toward theology. They fall into two groups: experimental writings on the vacuum, the weight of the air and the balance of liquids, and mathematical writings on the arithmetical triangle, on chance and on the method of geometry. In 1646-1647 at Rouen, Pascal repeated and varied the Italian experiment in which Evangelista Torricelli had filled a glass tube with mercury and inverted it, leaving an apparently empty space above the column. Pascal used tubes of glass forty feet long filled with water and red wine, siphons and syringes, and in the New Experiments concerning the Vacuum (1647) concluded that the space is apparently empty and that nature's resistance to a vacuum is limited and measurable. When the Jesuit Étienne Noël replied that the space must be filled with a subtle matter, Pascal answered that a hypothesis may be affirmed only when its contrary leads to absurdity, and denied only when it leads to absurdity itself; otherwise judgment must be suspended. The decisive test followed in September 1648, when his brother-in-law Florin Périer carried a mercury tube up the Puy de Dôme near Clermont and found the column more than three inches lower at the summit, as it would be if the air pressing on the reservoir had weight that diminishes with height. Pascal's Account of the Great Experiment on the Equilibrium of Fluids (1648) reports the result. The Treatises on the Equilibrium of Liquids and on the Weight of the Mass of the Air, written by about 1654 and printed only in 1663, generalise the findings. Pressure applied at any point of a confined liquid is transmitted throughout it, so a small piston can balance a great one, a 'new machine for multiplying forces'; and the effects once credited to nature's horror of a vacuum, from the rise of water in pumps to the adhesion of polished plates, are the work of the air's weight. A fragmentary Preface to a treatise on the vacuum distinguishes subjects ruled by authority, such as theology and history, from those ruled by reason and experiment, where later generations rightly surpass the ancients. In 1654 a gambler's question, how to divide the stakes fairly when a game is interrupted, set off an exchange of letters with Pierre de Fermat that laid the foundations of probability. Pascal's Treatise on the Arithmetical Triangle, printed that year but published in 1665, develops the table of numbers now named for him, applies it to combinations and to the division of stakes, and proves its properties by a method close to mathematical induction. On the Geometrical Spirit reflects on definition, demonstration and the art of persuading. Together these writings show an experimenter of unusual rigour and a mathematician who later set reason's power within limits. No public-domain English text of these treatises was available, so the record gives no quotations.
Top Lessons
Designing an experiment that can fail
The Puy de Dôme test was built so that the result could have gone against Pascal. If the air's weight held up the mercury, the column must fall with height; if nature's horror of a vacuum held it up, there was no reason the mountain should matter. Pascal also had a tube watched at the foot throughout the day to rule out a change in the weather. Students meet here an early, fully reported controlled experiment, designed in advance to decide between two explanations.
When may we affirm a hypothesis?
Answering Father Noël, Pascal distinguishes three cases: some claims must be affirmed because denying them leads to absurdity, some must be denied because affirming them does, and the rest must be held in suspense until evidence decides. Inventing an invisible matter to fill the space, he argues, explains nothing unless it can be shown. The same caution about the reach of reason runs through his later religious writing, where he insists both on reason's strength and on its limits.
Authority in theology, progress in physics
The Preface to the treatise on the vacuum does not set faith against reason. It assigns them different domains: history, law and theology depend on testimony and authority, so the ancients' word governs there; physics and mathematics depend on experiment and reasoning, so later learners may surpass their teachers without disrespect. Pascal compares the human race to one person learning across centuries. The distinction lets a Christian scientist honour Scripture and also correct Aristotle.
A fair share of an unfinished game
The problem of points asks how to divide the stakes when two players stop before either has won. Pascal reasons backward from the end of the game, giving each player what he could expect from the positions still possible; Fermat lists all the ways the remaining rounds could fall. Both arrive at the same division. Out of a gamblers' question came the idea of mathematical expectation, which Pascal would later turn to a very different use in the wager of the Pensées.