Men of Mathematics Read online

  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.

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  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 becomi
ng 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.

  In connection with problems in combinatorial analysis and probability Pascal made extensive use of the arithmetical triangle in which the numbers in any row after the first two are obtained from those in the preceding row by copying down the terminal 1’s and adding together the successive pairs of numbers from left to right to give the new row; thus 5 = 1 + 4, 10 = 4 + 6, 10 = 6 + 4, 5 = 4 + 1. The numbers in the nth row, after the 1, are the number of different selections of one thing, two things, three things, . . . that can be chosen from n distinct things. For example, 10 is the number of different pairs of things that can be selected from five distinct things. The numbers in the nth row are also the coefficients in the expansion of (1 + x)n by the binomial theorem, thus for n = 4, (1 + x)4 = 1 + 4x + 6x2 + 4x3 + x4. The triangle has numerous other interesting properties. Although it was known before the time of Pascal, it is usually named after him on account of the ingenious use he made of it in probabilities.

  The theory which originated in a gamblers’ dispute is now at the base of many enterprises which we consider more important than gambling, including all kinds of insurance, mathematical statistics and their application to biology and educational measurements, and much of modern theoretical physics. We no longer think of an electron being “at” a given place at a given instant, but we do calculate its probability of being in a given region. A little reflection will show that even the simplest measurements we make (when we attempt to measure anything accurately) are statistical in character.

  The humble origin of this extremely useful mathematical theory is typical of many: some apparently trivial problem, first solved perhaps out of idle curiosity, leads to profound generalizations which, as in the case of the new statistical theory of the atom in the quantum theory, may cause us to revise our whole conception of the physical universe or, as has happened with the application of statistical methods to intelligence tests and the investigation of heredity, may induce us to modify our traditional beliefs regarding the “greatness and misery of man.” Neither Pascal nor Fermat of course foresaw what was to issue from their disreputable child. The whole fabric of mathematics is so closely interwoven that we cannot unravel and eliminate any particular thread which happens to offend our individual taste without danger of destroying the whole pattern.

  Pascal however did make one application of probabilities (in the Pensées) which for his time was strictly practical. This was his famous “wager.” The “expectation” in a gamble is the value of the prize multiplied by the probability of winning the prize. According to Pascal the value of eternal happiness is infinite. He reasoned that even if the probability of winning eternal happiness by leading a religious life is very small indeed, nevertheless, since the expectation is infinite (any finite fraction of infinity is itself infinite) it will pay anyone to lead such a life. Anyhow, he took his own medicine. But just as if to show that he had not swallowed the bottle too, he jots down in another place in the Pensées this thoroughly skeptical query, “Is probability probable?” “It is annoying,” as he says in another place, “to dwell upon such trifles; but there is a time for trifling.” Pascal’s difficulty was that he did not always see clearly when he was trifling, as in his wager against God, or when, as in the clearing up of the Chevalier de Méré’s gambling difficulties for him, he was being profound.

  * * *

  I. Authorities differ on Pascal’s age when this work was done, the estimate varying from fifteen to seventeen. The 1819 edition of Pascal’s works contains a brief résumé of the statements of certain propositions on conics, but this is not the completed essay which Leibniz saw.

  CHAPTER SIX

  On the Seashore

  NEWTON

  The method of Fluxions [the calculus] is the general key by help whereof the modern mathematicians unlock the secrets of Geometry, and consequently of Nature.—BISHOP BERKELEY

  I do not frame hypotheses.—ISAAC NEWTON

  “I DO NOT KNOW what I may appear to the world; but to myself I seem to have been only like a boy playing on the seashore, and diverting myself in now and then finding a smoother pebble or a prettier shell than ordinary, whilst the great ocean of truth lay all undiscovered before me.”

  Such was Isaac Newton’s estimate of himself toward the close of his long life. Yet his successors capable of appreciating his work almost without exception have pointed to Newton as the supreme intellect that the human race has produced—“he who i
n genius surpassed the human kind.”

  Isaac Newton, born on Christmas Day (“old style” of dating), 1642, the year of Galileo’s death, came of a family of small but independent farmers, living in the manor house of the hamlet of Woolsthorpe, about eight miles south of Grantham in the county of Lincoln, England. His father, also named Isaac, died at the age of thirty seven before the birth of his son. Newton was a premature child. At birth he was so frail and puny that two women who had gone to a neighbor’s to get “a tonic” for the infant expected to find him dead on their return. His mother said he was so undersized at birth that a quart mug could easily have contained all there was of him.

  Not enough of Newton’s ancestry is known to interest students of heredity. His father was described by neighbors as “a wild, extravagant, weak man”; his mother, Hannah Ayscough, was thrifty, industrious, and a capable manageress. After her husband’s death Mrs. Newton was recommended as a prospective wife to an old bachelor as “an extraordinary good woman.” The cautious bachelor, the Reverend Barnabas Smith, of the neighboring parish of North Witham, married the widow on this testimonial. Mrs. Smith left her three-year-old son to the care of his grandmother. By her second marriage she had three children, none of whom exhibited any remarkable ability. From the property of his mother’s second marriage and his father’s estate Newton ultimately acquired an income of about £80 a year, which of course meant much more in the seventeenth century than it would now. Newton was not one of the great mathematicians who had to contend with poverty.