Men of Mathematics Read online

  In his outlook on experimental science Pascal had a far clearer vision than Descartes—from a modern point of view—of the scientific method. But he lacked Descartes’ singleness of aim, and although he did some first-rate work, allowed himself to be deflected from what he might have done by his morbid passion for religious subtleties.

  It is useless to speculate on what Pascal might have done. Let his life tell what he actually did. Then, if we choose, we can sum him up as a mathematician by saying that he did what was in him and that no man can do more. His life is a running commentary on two of the stories or similes in that New Testament which was his constant companion and unfailing comfort: the parable of the talents, and the remark about new wine bursting old bottles (or skins). If ever a wonderfully gifted man buried his talent, Pascal did; and if ever a medieval mind was cracked and burst asunder by its attempt to hold the new wine of seventeenth-century science, Pascal’s was. His great gifts were bestowed upon the wrong person.

  At the age of seven Pascal moved from Clermont with his father and sisters to Paris. About this time the father began teaching his son. Pascal was an extremely precocious child. Both he and his sisters appear to have had more than their share of nature’s gifts. But poor Blaise inherited (or acquired) a wretched physique along with his brilliant mind, and Jacqueline, the more gifted of his sisters, seems to have been of the same stripe as her brother, for she too fell a victim to morbid religiosity.

  At first everything went well enough. Pascal senior, astonished at the ease with which his son absorbed the stock classical education of the day, tried to hold the boy down to a reasonable pace to avoid injuring his health. Mathematics was taboo, on the theory that the young genius might overstrain himself by using his head. His father was an excellent drillmaster but a poor psychologist. His ban on mathematics naturally excited the boy’s curiosity. One day when he was about twelve Pascal demanded to know what geometry was about. His father gave him a clear description. This set Pascal off like a hare after his true vocation. Contrary to his own opinion in later life he had been called by God, not to torment the Jesuits, but to be a great mathematician. But his hearing was defective at the time and he got his orders confused.

  What happened when Pascal began the study of geometry has become one of the legends of mathematical precocity. In passing it may be remarked that infant prodigies in mathematics do not invariably blow up as they are sometimes said to do. Precocity in mathematics has often been the first flush of a glorious maturity, in spite of the persistent superstition to the contrary. In Pascal’s case early mathematical genius was not extinguished as he grew up but stifled under other interests. The ability to do first-class mathematics persisted, as will be seen from the episode of the cycloid, late into his all too brief life, and if anything is to be blamed for his comparatively early mathematical demise it is probably his stomach. His first spectacular feat was to prove, entirely on his own initiative, and without a hint from any book, that the sum of the angles of a triangle is equal to two right angles. This encouraged him to go ahead at a terrific pace.

  Realizing that he had begotten a mathematician, Pascal senior wept with joy and gave his son a copy of Euclid’s Elements. This was quickly devoured, not as a task, but as play. The boy gave up his games to geometrize. In connection with Pascal’s rapid mastery of Euclid, sister Gilberte permits herself an overappreciative fib. It is true that Pascal had found out and proved several of Euclid’s propositions for himself before he ever saw the book. But what Gilberte romances about her brilliant young brother is less probable than a throw of a billion aces in succession with one die, for the reason that it is infinitely improbable. Gilberte declared that her brother had rediscovered for himself the first thirty two propositions of Euclid, and that he had found them in the same order as that in which Euclid sets them forth. The thirty second proposition is indeed the famous one about the sum of the angles of a triangle which Pascal rediscovered. Now, there may be only one way of doing a thing right, but it seems more likely that there are an infinity of ways of doing it wrong. We know today that Euclid’s allegedly rigorous demonstrations, even in the first four of his propositions, are no proofs at all. That Pascal faithfully duplicated all of Euclid’s oversights on his own account is an easy story to tell but a hard one to believe. However, we can forgive Gilberte for bragging. Her brother was worth it. At the age of fourteen he was admitted to the weekly scientific discussions, conducted by Mersenne, out of which the French Academy of Sciences developed.

  While young Pascal was fast making a geometer of himself, old Pascal was making a thorough nuisance of himself with the authorities on account of his honesty and general uprightness. In particular he disagreed with Cardinal Richelieu over a little matter of imposing taxes. The Cardinal was incensed; the Pascal family went into hiding till the storm blew over. It is said that the beautiful and talented Jacqueline rescued the family and restored her father to the light of the Cardinal’s countenance by her brilliant acting, incognito, in a play presented for Richelieu’s entertainment. On inquiring the name of the charming young artiste who had captivated his clerical fancy, and being told that she was the daughter of his minor enemy, Richelieu very handsomely forgave the whole family and planted the father in a political job at Rouen. From what is known of that wily old serpent, Cardinal Richelieu, this pleasing tale is probably a fish story. Anyhow, the Pascals once more found a job and security at Rouen. There young Pascal met the tragic dramatist Corneille, who was duly impressed with the boy’s genius. At the time Pascal was all mathematician, so probably Corneille did not suspect that his young friend was to become one of the great creators of French prose.

  All this time Pascal was studying incessantly. Before the age of sixteen (about 1639)I he had proved one of the most beautiful theorems in the whole range of geometry. Fortunately it can be described in terms comprehensible to anyone. Sylvester, a mathematician of the nineteenth century whom we shall meet later, called Pascal’s great theorem a sort of “cat’s cradle.” We state first a special form of the general theorem that can be constructed with the use of a ruler only.

  Label two intersecting straight lines l and l’ On l take any three distinct points A, B, C, and on l any three distinct points A’, B’, C’. Join up these points by straight lines, crisscross, as follows: A and B’, A’ and B, B and C’, B’ and C, C and A’, C’ and A. The two lines in each of these pairs intersect in a point. We thus get three points. The special case of Pascal’s theorem which we are now describing states that these three points lie on one straight line.

  Before giving the general form of the theorem we mention another result like the preceding. This is due to Desargues (1593-1662). If the three straight lines joining corresponding vertices of two triangles XYZ and xyz meet in a point, then the three intersections of pairs of corresponding sides lie on one straight line. Thus, if the straight lines joining X and x, Y and y, Z and z meet in a point, then the intersections of XY and xy, YZ and yz, ZX and zx lie in one straight line.

  In Chapter 2 we stated what a conic section is. Imagine any conic section, for definiteness say an ellipse. On it mark any six points, A, B, C, D, E, F, and join them up, in this order, by straight lines. We thus have a six-sided figure inscribed in the conic section, in which AB and DE, BC and EF, CD and FA are pairs of opposite sides. The two lines in each of these three pairs intersect in a point; the three points of intersection lie on one straight line (see figure in Chapter IS, page 217). This is Pascal’s theorem; the figure which it furnishes is what he called the “mystic hexagram.” He probably first proved it true for a circle and then passed by projection to any conic section. Only a straightedge and a pair of compasses are required if the reader wishes to see what the figure looks like for a circle.

  There are several amazing things about this wonderful proposition, not the least of which is that it was discovered and proved by a boy of sixteen. Again, in his Essai pour les Coniques (Essay on Conics), written around his great theorem by
this extraordinarily gifted boy, no fewer than 400 propositions on conic sections, including the work of Apollonius and others, were systematically deduced as corollaries, by letting pairs of the six points move into coincidence, so that a chord became a tangent, and other devices. The full Essai itself was never published and is apparently lost irretrievably, but Leibniz saw and inspected a copy of it. Further, the kind of geometry which Pascal is doing here differs fundamentally from that of the Greeks; it is not metrical, but descriptive, or projective. Magnitudes of lines or angles cut no figure in either the statement or the proof of the theorem. This one theorem in itself suffices to abolish the stupid definition of mathematics, inherited from Aristotle and still sometimes reproduced in dictionaries, as the science of “quantity.” There are no “quantities” in Pascal’s geometry.

  To see what the projectivity of the theorem means, imagine a (circular) cone of light issuing from a point and pass a flat sheet of glass through the cone in varying positions. The boundary curve of the figure in which the sheet cuts the cone is a conic section. If Pascal’s “mystic hexagram” be drawn on the glass for any given position, and another flat sheet of glass be passed through the cone so that the shadow of the hexagram falls on it, the shadow will be another “mystic hexagram” with its three points of intersection of opposite pairs of sides lying on one straight line, the shadow of the “three-point-line” in the original hexagram. That is, Pascal’s theorem is invariant (unchanged) under conical projection. The metrical properties of figures studied in common elementary geometry are not invariant under projection; for example, the shadow of a right angle is not a right angle for all positions of the second sheet. It is obvious that this kind of projective, or descriptive geometry, is one of the geometries naturally adapted to some of the problems of perspective. The method of projection was used by Pascal in proving his theorem, but had been applied previously by Desargues in deducing the result stated above concerning two triangles “in perspective.” Pascal gave Desargues full credit for his great invention.

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  All this brilliance was purchased at a price. From the age of seventeen to the end of his life at thirty nine, Pascal passed but few days without pain. Acute dyspepsia made his days a torment and chronic insomnia his nights half-waking nightmares. Yet he worked incessantly. At the age of eighteen he invented and made the first calculating machine in history—the ancestor of all the arithmetical machines that have displaced armies of clerks from their jobs in our own generation. We shall see farther on what became of this ingenious device. Five years later, in 1646, Pascal suffered his first “conversion.” It did not take deeply, possibly because Pascal was only twenty three and still absorbed in his mathematics. Up to this time the family had been decently enough devout; now they all seem to have gone mildly insane.

  It is difficult for a modern to recreate the intense religious passions which inflamed the seventeenth century, disrupting families and hurling professedly Christian countries and sects at one another’s throats. Among the would-be religious reformers of the age was Cornelius Jansen (1585-1638), a flamboyant Dutchman who became bishop of Ypres. A cardinal point of his dogma was the necessity for “conversion” as a means to “grace,” somewhat in the manner of certain flourishing sects today. Salvation, however, at least to an unsympathetic eye, appears to have been the lesser of Jansen’s ambitions. God, he was convinced, had especially elected him to blast the Jesuits in this life and toughen them for eternal damnation in the next. This was his call, his mission. His creed was neither Catholicism nor Protestantism, although it leaned rather toward the latter. Its moving spirit was, first, last and all the time, a rabid hatred of those who disputed its dogmatic bigotries. The Pascal family now (1646) ardently—but not too ardently at first—embraced this unlovely creed of Jansenism. Thus Pascal, at the early age of twenty three, began to die off at the top. In the same year his whole digestive tract went bad and he suffered a temporary paralysis. But he was not yet dead intellectually.

  His scientific greatness flared up again in 1648 in an entirely new direction. Carrying on the work of Torricelli (1608-1647) on atmospheric pressure, Pascal surpassed him and demonstrated that he understood the scientific method which Galileo, the teacher of Torricelli, had shown the world. By experiments with the barometer, which he suggested, Pascal proved the familiar facts now known to every beginner in physics regarding the pressure of the atmosphere. Pascal’s sister Gilberte had married a Mr. Périer. At Pascal’s suggestion, Périer performed the experiment of carrying a barometer up the Puy de Dôme in Auvergne and noting the fall of the column of mercury as the atmospheric pressure decreased. Later Pascal, when he moved to Paris with his sister Jacqueline, repeated the experiment on his own account.

  Shortly after Pascal and Jacqueline had returned to Paris they were joined by their father, now fully restored to favor as a state councillor. Presently the family received a somewhat formal visit from Descartes. He and Pascal talked over many things, including the barometer. There was little love lost between the two. For one thing, Descartes had openly refused to believe the famous Essai pour les coniques had been written by a boy of sixteen. For another, Descartes suspected Pascal of having filched the idea of the barometric experiments from himself, as he had discussed the possibilities in letters to Mersenne. Pascal, as has been mentioned, had been attending the weekly meetings at Father Mersenne’s since he was fourteen. A third ground for dislike on both sides was furnished by their religious antipathies. Descartes, having received nothing but kindness all his life from the Jesuits, loved them; Pascal, following the devoted Jansen, hated a Jesuit worse than the devil is alleged to hate holy water. And finally, according to the candid Jacqueline, both her brother and Descartes were intensely jealous, each of the other. The visit was rather a frigid success.

  The good Descartes however did give his young friend some excellent advice in a truly Christian spirit. He told Pascal to follow his own example and lie in bed every day till eleven. For poor Pascal’s awful stomach he prescribed a diet of nothing but beef tea. But Pascal ignored the kindly meant advice, possibly because it came from Descartes. Among other things which Pascal totally lacked was a sense of humor.

  Jacqueline now began to drag her genius of a brother down—or up; it all depends upon the point of view. In 1648, at the impressionable age of twenty three, Jacqueline declared her intention of moving to Port Royal, near Paris, the main hangout of the Jansenists in France, to become a nun. Her father sat down heavily on the project, and the devoted Jacqueline concentrated her thwarted efforts on her erring brother. She suspected he was not yet so thoroughly converted as he might have been, and apparently she was right. The family now returned to Clermont for two years.

  During these two swift years Pascal seems to have become almost half human, in spite of sister Jacqueline’s fluttering admonitions that he surrender himself utterly to the Lord. Even the recalcitrant stomach submitted to rational discipline for a few blessed months.

  It is said by some and hotly denied by others that Pascal during this sane interlude and later for a few years discovered the predestined uses of wine and women. He did not sing. But these rumors of a basely human humanity may, after all, be nothing more than rumors. For after his death Pascal quickly passed into the Christian hagiocracy, and any attempts to get at the facts of his life as a human being were quietly but rigidly suppressed by rival factions, one of which strove to prove that he was a devout zealot, the other, a skeptical atheist, but both of which declared that Pascal was a saint not of this earth.

  During these adventurous years the morbidly holy Jacqueline continued to work on her frail brother. By a beautiful freak of irony Pascal was presently to be converted—for good, this time—and it was to be his lot to turn the tables on his too pious sister and drive her into the nunnery which now, perhaps, seemed less desirable. This, of course, is not the orthodox interpretation of what happened; but to anyone other than a blind partisan of one sect or the other—Christian
or Atheist—it is a more rational account of the unhealthy relationship between Pascal and his unmarried sister than that which is sanctioned by tradition.

  Any modern reader of the Pensées must be struck by a certain something or another which either completely escaped our more reticent ancestors or was ignored by them in their wiser charity. The letters, too, reveal a great deal which should have been decently buried. Pascal’s ravings in the Pensées about “lust” give him away completely, as do also the well-attested facts of his unnatural frenzies at the sight of his married sister Gilberte naturally caressing her children.

  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.