Men of Mathematics Read online

  It is very difficult to understand exactly what happened thereafter. Descartes was a gentleman with all the awe and reverence of a gentleman of those gallant, royalty-ridden times for even the least potent prince or princess. His letters are models of courtly discretion, but somehow they do not always ring quite true. One spiteful little remark, quoted in a moment, probably tells more of what he really thought of the Princess Elisabeth’s intellectual capacity than do all the reams of subtle flattery he wrote to or about his eager pupil with one eye on his style and the other on publication after his death.

  Elisabeth insisted upon Descartes giving her lessons. Officially he declared that “of all my disciples she alone has understood my works completely.” There is no doubt that he was genuinely fond of her in a fatherly, cat-looking-at-a-king’s-female-relative sort of way, but to believe that he meant what he said as a scientific statement of fact is to stretch credulity to the limit, unless, of course, he meant it as a wry comment on his own philosophy. Elisabeth may have understood too much, for it seems to be a fact that only a philosopher thoroughly understands his own philosophy, although any fool can think he does. Anyhow, he did not propose to her nor, so far as is known, did she propose to him.

  Among other parts of his philosophy which he expounded to her was the method of analytic geometry. Now there is a certain problem in elementary geometry which can be quite simply solved by pure geometry, and which looks easy enough, but which is a perfect devil for analytic geometry to handle in the strict Cartesian form. This is to construct a circle which shall touch (be tangent to) any three circles given at random whose centers do not all lie on one straight line. There are eight solutions possible. The problem is a fine specimen of the sort that are not adapted to the crude brute force of elementary Cartesian geometry. Elisabeth solved it by Descartes’ methods. It was rather cruel of him to let her do it. His comment on seeing her solution gives the whole show away to any mathematician. She was quite proud of her exploit, poor girl. Descartes said he would not undertake to carry out her solution and actually construct the required tangent circle in a month. If this does not convey his estimate of her mathematical aptitude it is impossible to put the matter plainer. It was an unkind thing to say, especially as she missed the point and he knew that she would.

  When Elisabeth left Holland she corresponded with Descartes to almost the day of his death. His letters contain much that is fine and sincere, but we could wish that he had not been so dazzled by the aura of royalty.

  In 1646 Descartes was living in happy seclusion at Egmond, Holland, meditating, gardening in a tiny plot, and carrying on a correspondence of incredible magnitude with the intellectuals of Europe. His greatest mathematical work lay behind him, but he still continued to think about mathematics, always with penetration and originality. One problem to which he gave some attention was Zeno’s of Achilles and the tortoise. His solution of the paradox would not be universally accepted today but it was ingenious for its era. He was now fifty and world-famous, far more famous in fact than he would ever have cared to be. The repose and tranquillity he had longed for all his life still eluded him. He continued to do great work, but he was not to be left in peace to do all that was in him. Queen Christine of Sweden had heard of him.

  This somewhat masculine young woman was then nineteen, already a capable ruler, reputedly a good classicist (of this, more later), a wiry athlete with the physical endurance of Satan himself, a ruthless huntress, an expert horsewoman who thought nothing of ten hours in the saddle without once getting off, and finally a tough morsel of femininity who was as hardened to cold as a Swedish lumberjack. With all this she combined a certain thick obtuseness toward the frailties of less thick-skinned beings. Her own meals were sparing; so were those of her courtiers. Like a hibernating frog she could sit for hours in an unheated library in the middle of a Swedish winter; her hangers-on begged her through their chattering teeth to throw all the windows wide open and let the merry snow in. Her cabinet, she noted without a qualm, always agreed with her. She knew everything there was to be known; her ministers and tutors told her so. As she got along on only five hours’ sleep she kept her toadies hopping through the hoop nineteen hours a day. The very hour this holy terror saw Descartes’ philosophy she decided she must annex the poor sleepy devil as her private instructor. All her studies so far had left her empty and hungering for more. Like the erudite Elisabeth she knew that only copious douches of philosophy from the philosopher himself could assuage her raging thirst for knowledge and wisdom.

  But for that unfortunate streak of snobbery in his make-up Descartes might have resisted Queen Christine’s blandishments till he was ninety and sans teeth, sans hair, sans philosophy, sans everything. Descartes held out till she sent Admiral Fleming in the spring of 1649 with a ship to fetch him. The whole outfit was generously placed at the reluctant philosopher’s disposal. Descartes temporized till October. Then, with a last regretful look round his little garden, he locked up and left Egmond forever.

  His reception in Stockholm was boisterous, not to say royal. Descartes did not live at the Palace; that much was spared him. Importunately kind friends, however, the Chanutes, shattered his last remaining hope of reserving a little privacy. They insisted that he live with them. Chanute was a fellow countryman, in fact the French ambassador. All might have gone well, for the Chanutes were really most considerate, had not the obtuse Christine got it into her immovable head that five o’clock in the morning was the proper hour for a busy, hardboiled young woman like herself to study philosophy. Descartes would gladly have swapped all the headstrong queens in Christendom for a month’s dreaming abed at La Flèche with the enlightened Charlet unobtrusively near to see that he did not get up too soon. However, he dutifully crawled out of bed at some ungodly hour in the dark, climbed into the carriage sent to collect him, and made his way across the bleakest, windiest square in Stockholm to the palace where Christine sat in the icy library impatiently waiting for her lesson in philosophy to begin promptly at five A.M.

  The oldest inhabitants said Stockholm had never in their memory suffered so severe a winter. Christine appears to have lacked a normal human skin as well as nerves. She noticed nothing, but kept Descartes unflinchingly to his ghastly rendezvous. He tried to make up his rest by lying down in the afternoons. She soon broke him of that. A Royal Swedish Academy of Sciences was gestating in her prolific activity; Descartes was hauled out of bed to deliver her.

  It soon became plain to the courtiers that Descartes and their Queen were discussing much more than philosophy in these interminable conferences. The weary philosopher presently realized that he had stepped with both feet into a populous and busy hornets’ nest. They stung him whenever and wherever they could. Either the Queen was too thick to notice what was happening to her new favorite or she was clever enough to sting her courtiers through her philosopher. In any event, to silence the malicious whisperings of “foreign influence,” she resolved to make a Swede of Descartes. An estate was set aside for him by royal decree. Every desperate move he made to get out of the mess only bogged him deeper. By the first of January, 1650, he was up to his neck with only a miracle of rudeness as his one dim hope of ever freeing himself. But with his inbred respect for royalty he could not bring himself to speak the magic words which would send him flying back to Holland, although he said plenty, with courtly politeness, in a letter to his devoted Elisabeth. He had chanced to interrupt one of the lessons in Greek. To his amazement Descartes learned that the vaunted classicist Christine was struggling over grammatical puerilities which, he says, he had mastered by himself when he was a little boy. His opinion of her mentality thereafter appears to have been respectful but low. It was not raised by her insistence that he produce a ballet for the delectation of her guests at a court function when he resolutely refused to make a mountebank of himself by attempting at his age to master the stately capers of the Swedish lancers.

  Presently Chanute fell desperately ill of inflammation of the lu
ngs. Descartes nursed him. Chanute recovered; Descartes fell ill of the same disease. The Queen, alarmed, sent doctors. Descartes ordered them out of the room. He grew steadily worse. Unable in his debility to distinguish friend from pest he consented at last to being bled by the most persistent of the doctors, a personal friend, who all the time had been hovering about awaiting his chance. This almost finished him, but not quite.

  His good friends the Chanutes, seeing that he was a very sick man, suggested that he might enjoy the last sacrament. He had expressed a desire to see his spiritual counsellor. Commending his soul to the mercy of God, Descartes faced his death calmly, saying the willing sacrifice of his life which he was making might possibly atone for his sins. La Flèche gripped him to the last. The counsellor asked him to signify whether he wished the final benediction. Descartes opened his eyes and closed them. He was given the benediction. Thus he died on February 11, 1650, aged 54, a sacrifice to the overweening vanity of a headstrong girl.

  Christine lamented. Seventeen years later when she had long since given up her crown and her faith, the bones of Descartes were returned to France (all except those of the right hand, which were retained by the French Treasurer-General as a souvenir for his skill in engineering the transaction) and were re-entombed in Paris in what is now the Pantheon. There was to have been a public oration, but this was hastily forbidden by order of the crown, as the doctrines of Descartes were deemed to be still too hot for handling before the people. Commenting on the return of Descartes’ remains to his native France, Jacobi remarks that “It is often more convenient to possess the ashes of great men than to possess the men themselves during their lifetime.”

  Shortly after his death Descartes’ books were listed in the Index of that Church which, accepting Cardinal Richelieu’s enlightened suggestion during the author’s lifetime, had permitted their publication. “Consistency, thou art a jewel!” But the faithful were not troubled by consistency, “the bugbear of little minds”—and the ratbane of inconsistent bigots.

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  We are not concerned here with the monumental additions which Descartes made to philosophy. Nor can his brilliant part in the dawn of the experimental method detain us. These things fall far outside the field of pure mathematics in which, perhaps, his greatest work lies. It is given to but few men to renovate a whole department of human thought. Descartes was one of those few. Not to obscure the shining simplicity of his greatest contribution, we shall briefly describe it alone and leave aside the many beautiful things he did in algebra and particularly in algebraic notation and the theory of equations. This one thing is of the highest order of excellence, marked by the sensuous simplicity of the half dozen or so greatest contributions of all time to mathematics. Descartes remade geometry and made modern geometry possible.

  The basic idea, like all the really great things in mathematics, is simple to the point of obviousness. Lay down any two intersecting lines on a plane. Without loss of generality we may assume that the lines are at right angles to one another. Imagine now a city laid out on the American plan, with avenues running north and south, streets east and west. The whole plan will be laid out with respect to one avenue and one street, called the axes, which intersect in what is called the origin, from which street-avenue numbers are read consecutively. Thus it is clear without a diagram where 1002 West 126 Street is, if we note that the ten avenues summarized in the number 1002 are stepped off to the west, that is, on the map, to the left of the origin. This is so familiar that we visualize the position of any particular address instantly. The avenue-number and street-number, with the necessary supplements of smaller numbers (as in the “2” in “1002” above) enable us to fix definitely and uniquely the position of any point whatever with respect to the axes, by giving the pair of numbers which measure its east or west and its north or south from the axes, this pair of numbers is called the coordinates of the point (with respect to the axes).

  Now suppose a point to wander over the map. The coordinates (x,y) of all the points on the curve over which it wanders will be connected by an equation, (this must be taken for granted by the reader who has never plotted a graph to fit data), which is called the equation of the curve. Suppose now for simplicity that our curve is a circle. We have its equation. What can be done with it? Instead of this particular equation, we can write down the most general one of the same kind (for example, here, of the second degree, with no cross-product term, and with the coefficients of the highest powers of the coordinates equal), and then proceed to manipulate this equation algebraically. Finally we put back the results of all our algebraic manipulations into their equivalents in terms of coordinates of points on the diagram which, all this time, we have been deliberately forgetting. Algebra is easier to see through than a cobweb of lines in the Greek manner of elementary geometry. What we have done has been to use our algebra for the discovery and investigation of geometrical theorems concerning circles.

  For straight lines and circles this may not seem very exciting; we knew how to do it all before in another, a Greek, way. Now comes the real power of the method. We start with equations of any desired or suggested degree of complexity and interpret their algebraic and analytic properties geometrically. Thus we have not only dropped geometry as our pilot; we have tied a sackful of bricks to his neck before pitching him overboard. Henceforth algebra and analysis are to be our pilots to the unchartered seas of “space” and its “geometry.” All that we have done can be extended, at one stride, to space of any number of dimensions; for the plane we need two coordinates, for ordinary “solid” space three, for the geometry of mechanics and relativity, four coordinates, and finally, for “space” as mathematicians like it, either n coordinates, or as many coordinates as there are of all the numbers 1, 2, 3, . . ., or as many as there are of all the points on a line. This is beating Achilles and the tortoise in their own race.

  Descartes did not revise geometry; he created it.

  It seems fitting that an eminent living mathematical fellow-countryman of Descartes should have the last word, so we shall quote Jacques Hadamard. He remarks first that the mere invention of coordinates was not Descartes’ greatest merit, because that had already been done “by the ancients”—a statement which is exact only if we read the unexpressed intention into the unaccomplished deed. Hell is paved with the half-baked ideas of “the ancients” which they could never quite cook through with their own steam.

  “It is quite another thing to recognize [as in the use of coordinates] a general method and to follow to the end the idea which it represents. It is exactly this merit, whose importance every real mathematician knows, that was preëminently Descartes’ in geometry; it was thus that he was led to what . . . is his truly great discovery in the matter; namely, the application of the method of coordinates not only to translate into equations curves already defined geometrically, but, looking at the question from an exactly opposite point of view, to the a priori definition of more and more complicated curves and, hence, more and more general . . . .

  “Directly, with Descartes himself, later, indirectly, in the return which the following century made in the opposite direction, it is the entire conception of the object of mathematical science that was revolutionized. Descartes indeed understood thoroughly the significance of what he had done, and he was right when he boasted that he had so far surpassed all geometry before him as Cicero’s rhetoric surpasses the ABC.”

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  I. Daughter of Frederick, Elector Palatine of the Rhine, and King of Bohemia, and a granddaughter of James I of England.

  CHAPTER FOUR

  The Prince of Amateurs

  FERMAT

  I have found a very great number of exceedingly beautiful theorems.

  —P. FERMAT

  NOT ALL OF OUR DUCKS can be swans; so after having exhibited Descartes as one of the leading mathematicians of all time, we shall have to justify the assertion, frequently made and seldom contradicted, that the greatest mathematician of the sev
enteenth century was Descartes’ contemporary Fermat (1601?–1665). This of course leaves Newton (1642–1727) out of consideration. But it can be argued that Fermat was at least Newton’s equal as a pure mathematician, and anyhow nearly a third of Newton’s life fell into the eighteenth century, whereas the whole of Fermat’s was lived out in the seventeenth.

  Newton appears to have regarded his mathematics principally as an instrument for scientific exploration and put his main effort on the latter. Fermat on the other hand was more strongly attracted to pure mathematics although he also did notable work in the applications of mathematics to science, particularly optics.

  Mathematics had just entered its modern phase with Descartes’ publication of analytic geometry in 1637, and was still for many years to be of such modest extent that a gifted man could reasonably hope to do good work in both the pure and applied divisions.

  As a pure mathematician Newton reached his climax in the invention of the calculus, an invention also made independently by Leibniz. More will be said on this later; for the present it may be remarked that Fermat conceived and applied the leading idea of the differential calculus thirteen years before Newton was born and seventeen before Leibniz was born, although he did not, like Leibniz, reduce his method to a set of rules of thumb that even a dolt can apply to easy problems.