Men of Mathematics (17 page)

Modern psychologists, no less than the ancients with ordinary common sense, have frequently remarked the high correlation between
sexual repression and morbid religious fervor. Pascal suffered from both, and his immortal
Pensées
is a brilliant if occasionally incoherent testimonial to his purely physiological eccentricities. If only the man could have been human enough to let himself go when his whole nature told him to cut loose, he might have lived out everything that was in him, instead of smothering the better half of it under a mass of meaningless mysticism and platitudinous observations on the misery and dignity of man.

Always shifting about restlessly the family returned to Paris in 1650. The next year the father died. Pascal seized the occasion to write Gilberte and her husband a lengthy sermon on death in general. This letter has been much admired. We need not reproduce any of it here; the reader who wishes to form his own opinion of it can easily locate it. Why this priggish effusion of pietistic and heartless moralizing on the death of a presumably beloved parent should ever have excited admiration instead of contempt for its author is, like the love of God which the letter in part dwells upon
ad nauseam,
a mystery that passeth all understanding. However, there is no arguing about tastes, and those who like the sort of thing that Pascal's much-quoted letter is, may be left to their undisturbed enjoyment of what is, after all, one of the masterpieces of self-conscious self-revelation in French literature.

A more practical result of Pascal senior's death was the opportunity which it offered Pascal, as administrator of the estate, of returning to normal intercourse with his fellow men. Encouraged by her brother, sister Jacqueline now joined Port Royal, her father being no longer capable of objecting. Her sweet concern over her brother's soul was now spiced by a quite human quarrel over the division of the estate.

A letter of the preceding year (1650) reveals another facet of Pascal's reverent character, or possibly his envy of Descartes. Dazzled by the transcendent brilliance of the Swedish Christine, Pascal humbly begged to lay his calculating machine at the feet of “the greatest princess in the world,” who, he declares in liquid phrases dripping strained honey and melted butter, is as eminent intellectually as she is socially. What Christine did with the machine is not known. She did not invite Pascal to replace the Descartes whom she had done in.

At last, on November 23, 1654, Pascal was really converted. According to some accounts he had been living a fast life for three years. The best authorities seem to agree that there is not much in this tradition
and that his life was not so fast after all. He had merely been doing his poor suffering best to live like a normal human being and to get something more than mathematics and piety out of life. On the day of his conversion he was driving a four-in-hand when the horses bolted. The leaders plunged over the parapet of the bridge at Neuilly, but the traces broke, and Pascal remained on the road.

To a man of Pascal's mystical temperament this lucky escape from a violent death was a direct warning from Heaven to pull himself up sharply on the brink of the moral precipice over which he, the victim of his morbid self-analysis, imagined he was about to plunge. He took a small piece of parchment, inscribed on it some obscure sentiments of mystical devotion, and thenceforth wore it next to his heart as an amulet to protect him from temptation and remind him of the goodness of God which had snatched him, a miserable sinner, from the very mouth of hell. Only once thereafter did he fall from grace (in his own pitiable opinion), although all the rest of his life he was haunted by hallucinations of a precipice before his feet.

Jacqueline, now a postulant for the nunnery at Port Royal, came to her brother's aid. Partly on his own account, partly because of his sister's persuasive pleadings, Pascal turned his back on the world and took up his residence at Port Royal, to bury his talent thenceforth in contemplation on “the greatness and misery of man.” This was in 1654, when Pascal was thirty one. Before forever quitting things of the flesh and the mind, however, he had completed his most important contribution to mathematics, the joint creation, with Fermat, of the mathematical theory of probability. Not to interrupt the story of his life we shall defer an account of this for the moment.

His life at Port Royal was at least sanitary if not exactly as sane as might have been wished, and the quiet, orderly routine benefited his precarious health considerably. It was while at Port Royal that he composed the famous
Provincial Letters,
which were inspired by Pascal's desire to aid in acquitting Arnauld, the leading light of the institution, of the charge of heresy. These famous letters (there were eighteen, the first of which was printed on January 23, 1656) are masterpieces of controversial skill, and are said to have dealt the Jesuits a blow from which their Society has never fully recovered. However, as a commonplace of objective observation which anyone with eyes in his head can verify for himself, the Society of Jesus still flourishes; so it may be reasonably doubted whether the
Provincial
Letters
had in them the deadly potency ascribed to them by sympathetic critics.

In spite of his intense preoccupation with matters pertaining to his salvation and the misery of man, Pascal was still capable of doing excellent mathematics, although he regarded the pursuit of all science as a vanity to be eschewed for its derogatory effects on the soul. Nevertheless he did fall from grace once more, but only once. The occasion was the famous episode of the cycloid.

This beautifully proportioned curve (it is traced out by the motion of a fixed point on the circumference of a wheel rolling along a straight line on a flat pavement) seems to have turned up first in mathematical literature in 1501, when Charles Bouvelles described it in connection with the squaring of the circle. Galileo and his pupil Viviani studied it and solved the problem of constructing a tangent to the curve at any point (a problem which Fermat solved at once when it was proposed to him), and Galileo suggested its use as an arch for bridges. Since reinforced concrete has become common, cycloidal arches are frequently seen on highway viaducts. For mechanical reasons (unknown to Galileo) the cycloidal arch is superior to any other in construction. Among the famous men who investigated the cycloid was Sir Christopher Wren, the architect of St. Paul's Cathedral, who determined the length of any arc of the curve and its center of gravity, while Huygens, for mechanical reasons, introduced it into the construction of pendulum clocks. One of the most beautiful of all the discoveries of Huygens (1629-1695) was made in connection with the cycloid. He proved that it is the
tautochrone,
that is, the curve (when turned upside down like a bowl) down which beads placed
anywhere
on it will all slide to the lowest point under the influence of gravity
in the same time.
On account of its singular beauty, elegant properties, and the endless rows which it stirred up between quarrelsome mathematicians
challenging one another to solve this or that problem in connection with it, the cycloid has been called “the Helen of Geometry,” after the Graeco-Trojan lady whose mere face is said to have “launched a thousand ships.”

Among other miseries which afflicted the wretched Pascal were persistent insomnia and bad teeth—in a day when such dentistry as was practised was done by the barber with a strong pair of forceps and brute force. Lying awake one night (1658) in the tortures of toothache, Pascal began to think furiously about the cycloid to take his mind off the excruciating pain. To his surprise he noticed presently that the pain had stopped. Interpreting this as a signal from Heaven that he was not sinning in thinking about the cycloid rather than his soul, Pascal let himself go. For eight days he gave himself up to the geometry of the cycloid and succeeded in solving many of the main problems in connection with it. Some of the things he discovered were issued under the pseudonym of Amos Dettonville as challenges to the French and English mathematicians. In his treatment of his rivals in this matter Pascal was not always as scrupulous as he might have been. It was his last flicker of mathematical activity and his only contribution to science after his entry to Port Royal.

The same year (1658) he fell more seriously ill than he had yet been in all his tormented life. Racking and incessant headaches now deprived him of all but the most fragmentary snatches of sleep. He suffered for four years, living ever more ascetically. In June, 1662, he gave up his own house to a poor family suffering from smallpox, as an act of self-denial, and went to live with his married sister. On August 19, 1662, his tortured existence came to an end in convulsions. He died at the age of thirty nine.

The post mortem revealed what had been expected regarding the stomach and vital organs; it also disclosed a serious lesion of the brain. Yet in spite of all this Pascal had done great work in mathematics and science and had left a name in literature that is still respected after nearly three centuries.

*  *  *

The beautiful things Pascal did in geometry, with the possible exception of the “mystic hexagram,” would all have been done by other men had he not done them. This holds in particular for the investigations on the cycloid. After the invention of the calculus all such things became incomparably easier than they had been before and in time
passed into the textbooks as mere exercises for young students. But in the joint creation with Fermat of the mathematical theory of probabilities Pascal made a new world. It seems quite likely that Pascal will be remembered for his part in this great and ever increasingly more important invention long after his fame as a writer has been forgotten. The
Pensées
and the
Provincial Letters,
apart from their literary excellences, appeal principally to a type of mind that is rapidly becoming extinct. The arguments for or against a particular point strike a modern mind as either trivial or unconvincing, and the very questions to which Pascal addressed himself with such fervent zeal now seem strangely ridiculous. If the problems which he discussed on the greatness and misery of man are indeed as profoundly important as enthusiasts have claimed, and not mere pseudo-problems mystically stated and incapable of solution, it seems unlikely that they will ever be solved by platitudinous moralizing. But in his theory of probabilities Pascal stated and solved a genuine problem, that of bringing the superficial lawlessness of pure chance under the domination of law, order, and regularity, and today this subtle theory appears to be at the very roots of human knowledge no less than at the foundation of physical science. Its ramifications are everywhere, from the quantum theory to epistemology.

The true founders of the mathematical theory of probability were Pascal and Fermat, who developed the fundamental principles of the subject in an intensely interesting correspondence during the year 1654. This correspondence is now readily available in the
Oeuvres de Fermat
(edited by P. Tannery and C. Henry, vol. 2, 1904). The letters show that Pascal and Fermat participated equally in the creation of the theory. Their correct solutions of problems differ in details but not in fundamental principles. Because of the tedious enumeration of possible cases in a certain problem on “points” Pascal tried to take a short cut and fell into error. Fermat pointed out the mistake, which Pascal acknowledged. The first letter of the series has been lost but the occasion of the correspondence is well attested.

The initial problem which started the whole vast theory was proposed to Pascal by the Chevalier de Mere, more or less of a professional gambler. The problem was that of “points”: each of two players (at dice, say) requires a certain number of points to win the game; if they quit the game before it is finished, how should the stakes be divided between them? The score (number of points) of each player
is given at the time of quitting, and the problem amounts to determining the probability which each player has at a given stage of the game of winning the game. It is assumed that the players have equal chances of winning a single point. The solution demands nothing more than sound common sense; the
mathematics
of probability enters when we seek a method for enumerating possible cases without actually counting them off. For example, how many possible different hands each consisting of three deuces and three other cards, none a deuce, are there in a common deck of fifty two? Or, in how many ways can a throw of three aces, five twos, and two sixes occur when ten dice are tossed? A third trifle of the same sort: how many different bracelets can be made by stringing ten pearls, seven rubies, six emeralds, and eight sapphires, if stones of one kind are considered as undistinguishable?

This detail of finding the number of ways in which a prescribed thing can be done or in which a completely specified event can happen, belongs to what is called
combinatorial analysis.
Its application to probability is obvious. Suppose, for example, we wish to know the probability of throwing two aces and one deuce in a single throw with three dice. If we know the
total
number of ways (6 × 6 × 6 or 216) in which the three dice can fall, and also the number of ways (say
n,
which the reader may find for himself) in which two aces and one deuce can fall, the required probability is n/216. (Here
n
is three, so the probability is 3/216.) Antoine Gombaud, Chevalier de Méré, who instigated all this, is described by Pascal as a man having a very good mind but no mathematics, while Leibniz, who seems to have disliked the gay Chevalier, dubs him a man of penetrating mind, a philosopher, and a gambler—quite an unusual combination.