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.

Plot or Argument

• New Experiments concerning the Vacuum (1647)

  Eight experiments at Rouen with mercury, water and wine in tubes, siphons, syringes and bellows show that a space apparently empty of any known matter can be produced, and that the force by which nature resists it is finite: it can hold up a column of water about thirty-one feet high and no more.

• The exchange with Father Noël (1647-1648)

  Against the Jesuit's claim that the space contains a purified air or subtle matter, Pascal sets out rules of judgment: a hypothesis is to be held only when its denial leads to absurdity, and invisible entities brought in to save a theory carry no weight unless evidence shows them.

• Account of the Great Experiment (1648)

  Letters between Pascal and Florin Périer describe the ascent of the Puy de Dôme on 19 September 1648, with tubes measured at the foot, at stages up the mountain and at the summit, and a control tube watched below throughout the day. The steady fall of the mercury with height is taken as proof that the air's weight supports the column.

• Treatises on the Equilibrium of Liquids and the Weight of the Air (c. 1654; printed 1663)

  The first treatise states that liquids weigh according to their height, not their quantity, and that pressure is transmitted equally in all directions, so a vessel of water can act as a press multiplying force. The second shows that the mass of the air has weight, that its pressure varies with altitude and weather, and that every effect traditionally ascribed to the horror of a vacuum follows from it.

• Preface to the Treatise on the Vacuum (c. 1651)

  A surviving fragment on authority and progress: in theology the ancients' testimony governs, but in physics and mathematics experiment and reasoning govern, and the whole succession of human beings may be considered as one person who keeps learning.

• The Arithmetical Triangle and the letters to Fermat (1654)

  Pascal constructs the triangular table in which each number is the sum of the two above it, derives nineteen consequences, applies it to combinations, binomial powers and the fair division of stakes, and proves general properties by passing from one row to the next. The letters with Fermat, from July to October 1654, compare his recursive method with Fermat's enumeration of cases and reach the same answers.

• On the Geometrical Spirit

  An unfinished reflection, probably from after 1654, on the ideal method of defining every term and proving every proposition, why geometry cannot fully attain it, and how one persuades both the understanding and the will.

Characters

• Evangelista Torricelli, originator of the mercury experiment

  Galileo's successor at Florence, whose 1643-1644 experiment with an inverted tube of mercury reached France through Marin Mersenne and set Pascal's work in motion.

• Florin Périer, experimenter on the Puy de Dôme

  Pascal's brother-in-law, a magistrate at Clermont, who carried out the mountain experiment Pascal had designed and reported every reading.

• Étienne Noël, opponent

  Jesuit rector of the Collège de Clermont in Paris and former teacher of Descartes, who defended the plenum against Pascal's conclusions.

• Pierre de Fermat, correspondent

  Councillor at Toulouse and mathematician whose letters on the division of stakes, together with Pascal's, are counted the beginning of probability theory.

• The Chevalier de Méré, source of the problem

  Courtier and gamester usually credited with putting to Pascal the questions about dice and interrupted games that prompted the Fermat correspondence.

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.

  • What result on the mountain would have counted against Pascal's view?
  • Why did Périer leave a tube at the foot of the mountain with an observer all day?
  • Pascal had the test repeated at the top and bottom of a tall church tower in Paris. What does a repetition add to a single experiment?

• 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.

  • State Pascal's rule for affirming a hypothesis in your own words.
  • Was Noël's subtle matter unreasonable, or simply untested? What would have been needed to test it?
  • Where in your science classes are you asked to suspend judgment, and how do you decide when enough evidence is in?

• 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.

  • Which subjects does Pascal place under authority, and which under reason and experiment?
  • Why does Pascal think that surpassing the ancients in physics is not ingratitude?
  • Is Pascal's line between the two domains always easy to draw? Give an example that tests it.

• 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.

  • Two players need one and two more wins respectively, and each round is equally likely to go either way. How should a stake of 64 coins be divided? Show your reasoning.
  • How do Pascal's and Fermat's methods differ, and why do they agree?
  • Pascal's triangle was known earlier in China, Persia and Europe. What, then, did Pascal contribute?

Context

Pascal's experiments entered a dispute running through the 1640s about whether a vacuum could exist. Aristotelian natural philosophy held that nature abhors a vacuum, and Descartes, from different premises, held that space and matter are the same, so that empty space is impossible. Torricelli's experiment of 1643-1644, publicised in France by Marin Mersenne, made the question experimental. Pascal's family circle in Rouen and Paris, his father's ties to Mersenne's academy, and the glassworks of Rouen gave him the means to test it on a large scale. The mathematical work belongs to the same Parisian network, in which problems circulated by letter among Mersenne, Roberval, Fermat, Desargues and others.

Composition

The New Experiments appeared in October 1647 and the Account of the Great Experiment in 1648. The Preface to a treatise on the vacuum survives as a fragment, usually dated about 1651. The treatises on liquids and air were complete by about 1654 but were published posthumously by Florin Périer in 1663. The Treatise on the Arithmetical Triangle was printed in 1654 and issued in 1665; the Fermat letters date from summer and autumn 1654. On the Geometrical Spirit is undated and probably later than Pascal's conversion of 23 November 1654.

Reception

Descartes, who visited Pascal in September 1647, later claimed to have suggested the mountain experiment, a claim that remains disputed. Pascal's results were extended by Otto von Guericke's air pump and Robert Boyle's experiments of the 1650s and 1660s. The principle that pressure is transmitted throughout a fluid bears Pascal's name, as does the unit of pressure. Christiaan Huygens's treatise on games of chance (1657) drew on the problems discussed by Pascal and Fermat. The treatises, in Richard Scofield's translation, were chosen for the Great Books of the Western World (2nd ed., 1990), volume 30.

Comparisons

• Pensees

  The Pensées were written by the same hand within a few years, and the two books illuminate each other: the experimenter's insistence on suspending judgment reappears in the fragments on the limits of reason, and the expectation reckoned in the Fermat letters returns, transformed, in the wager on God's existence.

• Galileo: The Starry Messenger, Letter to the Grand Duchess Christina, and Two New Sciences (First Day)

  Galileo's Letter to the Grand Duchess Christina also separates the authority of Scripture from the evidence of the senses in natural philosophy. Pascal's Preface makes a similar division, and his vacuum experiments test a problem Galileo had left open when he noted that pumps cannot raise water more than about eighteen cubits.

• The Geometry

  Descartes and Pascal both came out of Mersenne's circle but disagreed about method and about the vacuum itself, which Descartes held impossible. Where Descartes reduced geometry to algebra, Pascal's work on conics and on the triangle stayed closer to synthetic and combinatorial reasoning.

• The Sceptical Chymist

  Boyle carried Pascal's line of inquiry into the air pump and the spring of the air, and shared his preference for reported experiments over speculative principles. Reading the two shows the experimental temper spreading from Paris to Oxford and London within a decade.

• The Provincial Letters

  The Provincial Letters (1656-1657) turn the controversialist of the Noël exchange against Jesuit moral theology. The weapons are the same, clarity, irony and insistence on what an opponent's words actually entail, but the subject has moved from the vacuum to grace and casuistry.

Quotes

Creative Prompts

• Classroom

  Using a clear straw, a glass of water and your finger, show how the air's weight holds a column of liquid. Then write Pascal's argument about the Puy de Dôme as a step-by-step lab report with hypothesis, prediction, control and result.

• Dialogue

  Write a short exchange of letters between Pascal and Father Noël in which each defends his explanation of the empty space at the top of the mercury tube, keeping to the arguments each could actually have made in 1647.

• Essay

  Pascal said that in physics later generations may surpass the ancients, while in theology the ancients' testimony governs. Write an essay testing whether this division is sound, using one example from science and one from Scripture or church history.

Further Reading

• Pascal's Arithmetical Triangle, A. W. F. Edwards

  A short history of the triangle before and after Pascal, with a careful account of his proofs and the problem of points.

• Pascal: The Life of Genius, Morris Bishop

  A readable full biography that gives the Rouen experiments, the Puy de Dôme and the Fermat correspondence their place in Pascal's life before and after 1654.

• Pensées, Blaise Pascal

  Read the fragments on the two infinities and on the wager after the scientific treatises.