Men of Mathematics

by E. T. Bell

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  Up till 1672 Leibniz knew but little of what in his time was modern mathematics. He was then twenty six when his real mathematical education began at the hands of Huygens, whom he met in Paris in the intervals between one diplomatic plot and another. Christian Huygens (1629-1695), while primarily a physicist, some of whose best work went into horology and the undulatory theory of light, was an accomplished mathematician. Huygens presented Leibniz with a copy of his mathematical work on the pendulum. Fascinated by the power of the mathematical method in competent hands, Leibniz begged Huygens to give him lessons, which Huygens, seeing that Leibniz had a first-class mind, gladly did. Leibniz had already drawn up an impressive list of discoveries he had made by means of his own methods—phases of the universal characteristic. Among these was a calculating machine far superior to Pascal’s, which handled only addition and subtraction; Leibniz’ machine did also multiplication, division, and the extraction of roots. Under Huygens’ expert guidance Leibniz quickly found himself. He was a born mathematician.

  The lessons were interrupted from January to March, 1673, during Leibniz’ absence in London as an attaché for the Elector. While in London, Leibniz met the English mathematicians and showed them some of his work, only to learn that it was already known. His English friends told him of Mercator’s quadrature of the hyperbola—one of the clues which Newton had followed to his invention of the calculus. This introduced Leibniz to the method of infinite series, which he carried on. One of his discoveries (sometimes ascribed to the Scotch mathematician James Gregory, 1638-1675) may be noted: if π is the ratio of the circumference of a circle to its diameter,

  the series continuing in the same way indefinitely. This is not a practical way of calculating the numerical value of π (3.1415926 . . .), but the simple connection between π and all the odd numbers is striking.

  During his stay in London Leibniz attended meetings of the Royal Society, where he exhibited his calculating machine. For this and his other work he was elected a foreign member of the Society before his return to Paris in March, 1673. He and Newton subsequently (1700) became the first foreign members of the French Academy of Sciences.

  Greatly pleased with what Leibniz had done while away, Huygens urged him to continue. Leibniz devoted every spare moment to his mathematics, and before leaving Paris for Hanover in 1676 to enter the service of the Duke of Brunswick-Lüneburg, had worked out some of the elementary formulas of the calculus and had discovered “the fundamental theorem of the calculus” (see preceding chapter)—that is, if we accept his own date, 1675. This was not published till July 11, 1677, eleven years after Newton’s unpublished discovery, which was not made public by Newton till after Leibniz’ work had appeared. The controversy started in earnest, when Leibniz, diplomatically shrouding himself in editorial omniscience and anonymity, wrote a severely critical review of Newton’s work in the Acta Eruditorum, which Leibniz himself had founded in 1682 and of which he was editor in chief. In the interval between 1677 and 1704 the Leibnizian calculus had been developed into an instrument of real power and easy applicability on the Continent, largely through the efforts of the Swiss Bernoullis, Jacob and his brother Johann, while in England, owing to Newton’s reluctance to share his mathematical discoveries freely, the calculus was still a relatively untried curiosity.

  One specimen of things that are now easy for beginners in the calculus, but which cost Leibniz (and possibly also Newton) much thought and many trials before the right way was found, may indicate how far mathematics has travelled since 1675. Instead of the infinitesimals of Leibniz we shall use the rates discussed in the preceding chapter. If u, v are functions of x, how shall the rate of change of uv with respect to x be expressed in terms of the respective rates of change of u and v with respect to x? In symbols, what is in terms of and Leibniz once thought it should be which is nothing like the correct

  The Elector died in 1673 and Leibniz was more or less free during the last of his stay in Paris. On leaving Paris in 1676 to enter the service of the Duke John Frederick of Brunswick-Lüneburg, Leibniz proceeded to Hanover by way of London and Amsterdam. It was while in the latter city that he engineered one of the shadiest transactions in all his long career as a philosophic diplomat. The history of Leibniz’ commerce with “the God-intoxicated Jew” Benedict de Spinoza (1632-1677) may be incomplete, but as the account now stands it seems that for once Leibniz was grossly unethical over a matter—of all things—of ethics. Leibniz seems to have believed in applying his ethics to practical ends. He carried off copious extracts from Spinoza’s unpublished masterpiece Ethica (Ordina Geometrica Demonstrata)—a treatise on ethics developed in the manner of Euclid’s geometry. When Spinoza died the following year Leibniz appears to have found it convenient to mislay his souvenirs of the Amsterdam visit. Scholars in this field seem to agree that Leibniz’ own philosophy wherever it touches ethics was appropriated without acknowledgment from Spinoza.

  It would be rash for anyone not an expert in ethics to doubt that Leibniz was guilty, or to suggest that his own thoughts on ethics were independent of Spinoza’s. Nevertheless there are at least two similar instances in mathematics (elliptic functions, non-Euclidean geometry) where all the evidence at one time was sufficient to convict several men of dishonesty grosser than that attributed to Leibniz. When unsuspected diaries and correspondence were brought to light years after the death of all the accused it was seen that all were entirely innocent. It may pay occasionally to believe the best of human beings instead of the worst until all the evidence is in—which it can never be for a man who is tried after his death.

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  The remaining forty years of Leibniz’ life were spent in the trivial service of the Brunswick family. In all he served three masters as librarian, historian, and general brains of the family. It was a matter of great importance to such a family to have an exact history of all its connections with other families as highly favored by heaven as itself. Leibniz was no mere cataloguer of books in his function as family librarian, but an expert genealogist and searcher of mildewed archives as well, whose function it was to confirm the claims of his employers to half the thrones of Europe or, failing confirmation, to manufacture evidence by judicious suppression. His historical researches took him all through Germany and thence to Austria and Italy in 1687-90.

  During his stay in Italy Leibniz visited Rome and was urged by the Pope to accept the position of librarian at the Vatican. But as a prerequisite to the job was that Leibniz become a Catholic he declined—for once scrupulous. Or was he? His reluctance to throw up one good post for another may have started him off on the next application of his “universal characteristic,” the most fantastically ambitious of all his universal dreams. Had he pulled this off he could have moved into the Vatican without leaving his face outside.

  His grand project was no less than that of reuniting the Protestant and Catholic churches. It was then not so long since the first had split off from the second, so the project was not so insane as it now sounds. In his wild optimism Leibniz overlooked a law which is as fundamental for human nature as the second law of thermodynamics is for the physical universe—indeed it is of the same kind: all creeds tend to split into two, each of which in turn splits into two more, and so on, until after a certain finite number of generations (which can be easily calculated by logarithms) there are fewer human beings in any given region, no matter how large, than there are creeds, and further attenuations of the original dogma embodied in the first creed dilute it to a transparent gas too subtle to sustain faith in any human being, no matter how small.

  A quite promising conference at Hanover in 1683 failed to effect a reconciliation as neither party could decide which was to be swallowed by the other, and both welcomed the bloody row of 1688 in England between Catholics and Protestants as a legitimate ground for adjourning the conference sine die.

  Having learned nothing from this farce Leibniz immediately organized another. His attempt to unite merely th
e two Protestant sects of his day succeeded only in making a large number of excellent men more obstinate and sorer at one another than they were before. The Protestant Conference dissolved in mutual recriminations and curses.

  It was about this time that Leibniz turned to philosophy as his major consolation. In an endeavor to assist Pascal’s old Jansenist friend Arnauld, Leibniz composed a semi-casuistical treatise on metaphysics destined to be of use to Jansenists and others in need of something more subtle than the too subtle logic of the Jesuits. His philosophy occupied the remainder of Leibniz’ life (when he was not engaged on the unending history of the Brunswick family for his employers), in all about a quarter of a century. That a mind like Leibniz’ evolved a vast cloud of philosophy in twenty five years need hardly be stated. Doubtless every reader has heard something of the ingenious theory of monads—miniature replicas of the universe out of which everything in the universe is composed, as a sort of one in all, all in one—by which Leibniz explained everything (except the monads) in this world and the next.

  The power of Leibniz’ method when applied to philosophy cannot be denied. As a specimen of the theorems proved by Leibniz in his philosophy, that concerning the existence of God may be mentioned. In his attempt to prove the fundamental theorem of optimism—“everything is for the best in this best of all possible worlds”—Leibniz was less successful, and it was only in 1759, forty three years after Leibniz had died neglected and forgotten, that a conclusive demonstration was published by Voltaire in his epoch-making treatise Candide. One further isolated result may be mentioned. Those familiar with general relativity will recall that “empty space”—space totally devoid of matter—is no longer respectable. Leibniz rejected it as nonsensical.

  The list of Leibniz’ interests is still far from complete. Economics, philology, international law (in which he was a pioneer), the establishment of mining as a paying industry in certain parts of Germany, theology, the founding of academies, and the education of the young Electress Sophie of Brandenburg (a relative of Descartes’ Elisabeth), all shared his attention, and in each of them he did something notable. Possibly his least successful ventures were in mechanics and physical science, where his occasional blunders show up glaringly against the calm, steady light of men like Galileo, Newton, and Huygens, or even Descartes.

  Only one item in this list demands further attention here. On being called to Berlin in 1700 as tutor to the young Electress, Leibniz found time to organize the Berlin Academy of Sciences. He became its first president. The Academy was still one of the three or four leading learned bodies in the world till the Nazis “purged” it. Similar ventures in Dresden, Vienna, and St. Petersburg came to nothing during Leibniz’ lifetime, but after his death the plans for the St. Petersburg Academy of Sciences which he had drawn up for Peter the Great were carried out. The attempt to found a Viennese Academy was frustrated by the Jesuits when Leibniz visited Austria for the last time, in 1714. Their opposition was only to have been expected after what Leibniz had done for Arnauld. That they got the better of the master diplomat in an affair of petty academic politics shows how badly Leibniz had begun to slip at the age of sixty eight. He was no longer himself, and indeed his last years were but a wasted shadow from his former glory.

  Having served princes all his life he now received the usual wages of such service. Ill, fast ageing, and harassed by controversy, he was kicked out.

  Leibniz returned to Brunswick in September, 1714, to learn that his employer the Elector George Louis—“the honest blockhead,” as he is known in English history—having packed up his duds and his snuff, had left for London to become the first German King of England. Nothing would have pleased Leibniz better than to follow George to London, although his enemies at the Royal Society and elsewhere in England were now numerous and vicious enough owing to the controversy with Newton. But the boorish George, now socially a gentleman, had no further use for Leibniz’ diplomacy, and curtly ordered the brains that had helped to lift him into civilized society to stick in the Hanover library and get on with their everlasting history of the illustrious Brunswick family.

  When Leibniz died two years later (1716) the diplomatically doctored history was still incomplete. For all his hard labor Leibniz had been unable to bring the history down beyond the year 1005, and at that had covered less than three hundred years. The family was so very tangled in its marital adventures that even the universal Leibniz could not supply them all with unblemished scutcheons. The Brunswickers showed their appreciation of this immense labor by forgetting all about it till 1843, when it was published, but whether complete or expurgated will not be known until the rest of Leibniz’ manuscripts have been sifted.

  Today, over three hundred years after his death, Leibniz’ reputation as a mathematician is higher than it was for many, many years after his secretary followed him to the grave, and it is still rising.

  As a diplomat and statesman Leibniz was as good as the cream of the best of them in any time or any place, and far brainier than all of them together. There is but one profession in the world older than his, and until that is made respectable it would be premature to try any man for choosing diplomacy as his means to a livelihood.

  CHAPTER EIGHT

  Nature or Nurture?

  THE BERNOULLIS

  These men certainly accomplished much and admirably attained the goal they had set themselves.—JOHANNES BERNOULLI

  SINCE THE GREAT DEPRESSION began deflating western civilization eugenists, geneticists, psychologists, politicians, and dictators—for very different reasons—have taken a renewed interest in the still unsettled controversy of heredity versus environment. At one extreme the hundred-percenter proletarians hold that anyone can be a genius given the opportunity; while at the other, equally positive Tories assert that genius is inborn and will out even in a London slum. Between the two stretches a whole spectrum of belief. The average opinion holds that nature, not nurture, is the determining factor in the emergence of genius, but that without deliberate or accidental assistance genius perishes. The history of mathematics offers abundant material for a study of this interesting problem. Without taking sides—to do so at present would be premature—we may say that the evidence furnished by the life histories of mathematicians seems to favor the average opinion.

  Probably the most striking case history is that of the Bernoulli family, which in three generations produced eight mathematicians, several of them outstanding, who in turn produced a swarm of descendants about half of whom were gifted above the average and nearly all of whom, down to the present day, have been superior human beings. No fewer than 120 of the descendants of the mathematical Bernoullis have been traced genealogically, and of this considerable posterity the majority achieved distinction—sometimes amounting to eminence—in the law, scholarship, science, literature, the learned professions, administration, and the arts. None were failures. The most significant thing about a majority of the mathematical members of this family in the second and third generations is that they did not deliberately choose mathematics as a profession but drifted into it in spite of themselves as a dipsomaniac returns to alcohol.

  As the Bernoulli family played a leading part in developing the calculus and its applications in the seventeenth and eighteenth centuries, they must be given more than a passing mention in even the briefest account of the evolution of modern mathematics. The Bernoullis and Euler were in fact the leaders above all others who perfected the calculus to the point where quite ordinary men could use it for the discovery of results which the greatest of the Greeks could never have found. But the mere volume of the Bernoulli family’s work is too vast for detailed description in an account like the present, so we shall treat them briefly together.

  The Bernoullis were one of many Protestant families who fled from Antwerp in 1583 to escape massacre by the Catholics (as on St. Bartholomew’s Eve) in the prolonged persecution of the Huguenots. The family sought refuge first in Frankfort, moving on presently to Switzerl
and, where they settled at Basle. The founder of the Bernoulli dynasty married into one of the oldest Basle families and became a great merchant. Nicolaus senior, who heads the genealogical table, was also a great merchant, as his grandfather and great-grandfather had been. All these men married daughters of merchants, and with one exception—the great-grandfather mentioned—accumulated large fortunes. The exception showed the first departure from the family tradition of trade by following the profession of medicine. Mathematical talent was probably latent for generations in this shrewd mercantile family, but its actual emergence was explosively sudden.

  Referring now to the genealogical table we shall give a very brief summary of the chief scientific activities of the eight mathematicians descended from Nicolaus senior before continuing with the heredity.

  Jacob I mastered the Leibnizian form of the calculus by himself. From 1687 to his death he was professor of mathematics at Basle. Jacob I was one of the first to develop the calculus significantly beyond the state in which Newton and Leibniz left it and to apply it to new problems of difficulty and importance. His contributions to analytic geometry, the theory of probability, and the calculus of variations were of the highest importance. As the last will recur frequently (in the work of Euler, Lagrange, and Hamilton), we may describe the nature of some of the problems attacked by Jacob I in this subject. We have already seen a specimen of the type of problem handled by the calculus of variations in Fermat’s principle of least time.

  The calculus of variations is of very ancient origin. According to one legend,I when Carthage was founded the city was granted as much land as a man could plow a furrow completely around in a day. What shape should the furrow be, given that a man can plow a straight furrow of a certain length in a day? Mathematically stated, what is the figure which has the greatest area of all figures having perimeters of the same length? This is an isoperimetrical problem; the answer here is a circle. This seems obvious, but it is by no means easy to prove. (The elementary “proofs” sometimes given in school geometries are rankly fallacious.) The mathematics of the problem comes down to making a certain integral a maximum subject to one restrictive condition. Jacob I solved this problem and generalized it.II