Men of Mathematics

by E. T. Bell

  The discovery that the brachistochrone is a cycloid has been noted in previous chapters. This fact, that the cycloid is the curve of quickest descent, was discovered by the brothers Jacob I and Johannes I in 1697, and almost simultaneously by several others. But the cycloid is also the tautochrone. This struck Johannes I as something wonderful and admirable: “With justice we may admire Huygens because he first discovered that a heavy particle falls on a cycloid in the same time always, no matter what the starting-point may be. But you will be petrified with astonishment when I say that exactly this same cycloid, the tautochrone of Huygens, is the brachistochrone we are seeking.” (Bliss, loc. cit., p. 54.) Jacob also waxes enthusiastic. These again are instances of the sort of problem attacked by the calculus of variations. Lest they seem trivial, we repeat once more that a whole province of mathematical physics is frequently mapped into a simple variational principle—like Fermat’s of least time in optics, or Hamilton’s in dynamics.

  After Jacob’s death his great treatise on the theory of probability, the Ars Conjectandi, was published in 1713. This contains much that is still of the highest usefulness in the theory of probabilities and its applications to insurance, statistics, and the mathematical study of heredity.

  Another research of Jacob’s shows how far he had developed the differential and integral calculus: continuing the work of Leibniz, Jacob made a fairly exhaustive study of catenaries—the curves in which a uniform chain hangs suspended between two points, or in which loaded chains hang. This was no mere curiosity. Today the mathematics developed by Jacob I in this connection finds its use in applications to suspension bridges and high-voltage transmission lines. When Jacob I worked all this out it was new and difficult; today it is an exercise in the first course in the calculus or mechanics.

  Jacob I and his brother Johannes I did not always get on well together. Johannes seems to have been the more quarrelsome of the two, and it is certain that he treated his brother with something pretty close to dishonesty in the matter of isoperimetrical problems. The Bernoullis took their mathematics in deadly earnest. Some of their letters about mathematics bristle with strong language that is usually reserved for horse thieves. For his part Johannes I not only attempted to steal his brother’s ideas but threw his own son out of the house for having won a prize from the French Academy of Sciences for which Johannes himself had competed. After all, if rational human beings get excited about a game of cards, why should they not blow up over mathematics which is infinitely more exciting?

  Jacob I had a mystical strain which is of some significance in the study of the heredity of the Bernoullis. It cropped out once in an interesting way toward the end of his life. There is a certain spiral (the logarithmic or equiangular) which is reproduced in a similar spiral after each of many geometrical transformations. Jacob was fascinated by this recurrence of the spiral, several of whose properties he discovered, and directed that a spiral be engraved on his tombstone with the inscription Eadem mutata resurgo (Though changed I shall arise the same).

  Jacob’s motto was Invito patre sidera verso (Against my father’s will I study the stars)—in ironic memory of his father’s futile opposition to Jacob’s devoting his talents to mathematics and astronomy. This detail favors the “nature” view of genius over the “nurture.” If his father had prevailed Jacob would have been a theologian.

  Johannes I, brother of Jacob I, did not start as a mathematician but as a doctor of medicine. His dispute with the brother who had generously taught him mathematics has already been mentioned. Johannes was a man of violent likes and dislikes: Leibniz and Euler were his gods; Newton he positively hated and greatly underestimated, as a bigoted champion of Leibniz was almost bound to do from envy or spite. The obstinate father attempted to cramp his younger son into the family business, but Johannes I, following the lead of his brother Jacob I, rebelled and went in for medicine and the humanities, unaware that he was fighting against his heredity. At the age of eighteen he took his M.A. degree. Before long he realized his mistake in choosing medicine and turned to mathematics. His first academic appointment was at Groningen in 1695 as professor of mathematics; on the death of Jacob I in 1705 Johannes I succeeded to the professorship at Basle.

  Johannes I was even more prolific than his brother in mathematics and did much to spread the calculus in Europe. His range included physics, chemistry, and astronomy in addition to mathematics. On the applied side Johannes I contributed extensively to optics, wrote on the theory of the tides and the mathematical theory of ship sails, and enunciated the principle of virtual displacements in mechanics. Johannes I was a man of unusual physical and intellectual vigor, remaining active till within a few days of his death at the age of eighty.

  Nicolaus I, the brother of Jacob I and Johannes I, was also gifted in mathematics. Like his brothers he made a false start. At the age of sixteen he took his doctor’s degree in philosophy at the University of Basle, and at twenty earned the highest degree in law. He was first a professor of law at Bern before becoming one of the mathematical faculty at the Academy of St. Petersburg. At the time of his death he was so highly thought of that the Empress Catherine gave him a public funeral at state expense.

  Heredity came out curiously in the second generation. Johannes I tried to force his second son, Daniel, into business. But Daniel thought he preferred medicine and became a physician before landing, in spite of himself, in mathematics. At the age of eleven Daniel began taking lessons in mathematics from his elder brother Nicolaus III, only five years older than himself. Daniel and the great Euler were intimate friends and at times friendly rivals. Like Euler, Daniel Bernoulli has the distinction of having won the prize of the French Academy ten times (on a few occasions the prize was shared with other successful competitors). Some of Daniel’s best work went into hydrodynamics, which he developed uniformly from the single principle that later came to be called the conservation of energy. All who work today in pure or applied fluid motion know the name of Daniel Bernoulli.

  In 1725 (at the age of twenty five) Daniel became professor of mathematics at St. Petersburg, where the comparative barbarity of the life irked him so greatly that he returned at the first opportunity, eight years later, to Basle, where he became professor of anatomy and botany, and finally of physics. His mathematical work included the calculus, differential equations, probability, the theory of vibrating strings, an attempt at a kinetic theory of gases, and many other problems in applied mathematics. Daniel Bernoulli has been called the founder of mathematical physics.

  From the standpoint of heredity it is interesting to note that Daniel had a marked vein of speculative philosophy in his nature—possibly a refined sublimation of the Huguenot religion of his ancestors. The like cropped out in numerous later descendants of the illustrious refugees from religious intolerance.

  The third mathematician in the second generation, Johannes II, brother of Nicolaus III and Daniel, also made a false start and was pulled back into line by his heredity—or possibly by his brothers. Starting out in law he became professor of eloquence at Basle before succeeding his father in the chair of mathematics. His work was principally in physics and was sufficiently distinguished to capture the Paris prize on three occasions (once is usually enough to satisfy a good mathematician—provided he is good enough).

  Johannes III, a son of Johannes II, repeated the family tradition of making a wrong start, and like his father began with law. At the age of thirteen he took his doctor’s degree in philosophy. By nineteen Johannes III had found his true vocation and was appointed astronomer royal at Berlin. His interests embraced astronomy, geography, and mathematics.

  Jacob II, another son of Johannes II, carried on the family blunder by starting in law, only to change over at twenty one to experimental physics. He also turned to mathematics, becoming a member of the St. Petersburg Academy in the section of mathematics and physics. His early death (at the age of thirty) by accidental drowning cut short a very promising career, and we do not know w
hat Jacob II really had in him. He was married to a granddaughter of Euler.

  The list of Bernoullis who showed mathematical talent is not yet exhausted, but the rest were less distinguished. It is sometimes asserted that the strain had worn thin. Quite the contrary seems to be the case. When mathematics was the most promising field for superior talent to cultivate, as it was immediately after the invention of the calculus, the gifted Bernoullis cultivated mathematics. But mathematics and science are only two of innumerable fields of human endeavor, and for gifted men to swarm into either when both are overcrowded with high ability indicates a lack of practical sense. The Bernoulli talent was not expended; it merely spent itself on things of equal—or perhaps greater—social importance than mathematics when that field began to resemble Epsom Downs on Derby Day.

  Those interested in the vagaries of heredity will find plenty of material in the history of the Darwin and Galton families. The case of Francis Galton (a cousin of Charles Darwin) is particularly interesting, as the mathematical study of heredity was founded by him. To rail at the descendants of Charles Darwin because some of them have achieved eminence in mathematics or mathematical physics rather than in biology is slightly silly. The genius is still there, and one expression of it is not necessarily “better” or “higher” than another—unless we are the sort of bigots who insist that everything should be mathematics, or biology, or sociology, or bridge and golf. It may be that the abandonment of mathematics as the family trade by the Bernoullis was just one more instance of their genius.

  Many legends and anecdotes have grown up round the famous Bernoullis, as is only natural in the case of a family as gifted and as violent in their language as the Bernoullis sometimes were. One of these ripe old chestnuts may be retailed again as it is one of the comparatively early authentic instances of a story which must be at least as old as ancient Egypt, and of which we daily see variants pinned onto all sorts of prominent characters from Einstein down. Once when travelling as a young man Daniel modestly introduced himself to an interesting stranger with whom he had been conversing: “I am Daniel Bernoulli.” “And I,” said the other sarcastically, “am Isaac Newton.” This delighted Daniel to the end of his days as the sincerest tribute he had ever received.

  * * *

  I. Actually, here, I have combined two legends. Queen Dido was given a bull’s hide to “enclose” the greatest area. She cut it into one thong and enclosed a semicircle.

  II. Historical notes on this and other problems of the calculus of variations will be found in the book by G. A. Bliss, Calculus of Variations, Chicago, 1925. The Anglicized form of Jacob I is James.

  CHAPTER NINE

  Analysis Incarnate

  EULER

  History shows that those heads of empires who have encouraged the cultivation of mathematics, the common source of all the exact sciences, are also those whose reigns have been the most brilliant and whose glory is the most durable.

  —MICHEL CHASLES

  “EULER CALCULATED WITHOUT APPARENT EFFORT, as men breathe, or as eagles sustain themselves in the wind” (as Arago said), is not an exaggeration of the unequalled mathematical facility of Leonard Euler (1707-1783), the most prolific mathematician in history, and the man whom his contemporaries called “analysis incarnate.” Euler wrote his great memoirs as easily as a fluent writer composes a letter to an intimate friend. Even total blindness during the last seventeen years of his life did not retard his unparalleled productivity; indeed, if anything, the loss of his eyesight sharpened Euler’s perceptions in the inner world of his imagination.

  The extent of Euler’s work was not accurately known even in 1936, but it has been estimated that sixty to eighty large quarto volumes will be required for the publication of his collected works. In 1909 the Swiss Association for Natural Science undertook the collection and publication of Euler’s scattered memoirs, with financial assistance from many individuals and mathematical societies throughout the world—rightly claiming that Euler belongs to the whole civilized world and not only to Switzerland. The careful estimates of the probable expense (about $80,000 in the money of 1909) were badly upset by the discovery in St. Petersburg (Leningrad) of an unsuspected mass of Euler’s manuscripts.

  Euler’s mathematical career opened in the year of Newton’s death. A more propitious epoch for a genius like that of Euler’s could not have been chosen. Analytic geometry (made public in 1637) had been in use ninety years, the calculus about fifty, and Newton’s law of universal gravitation, the key to physical astronomy, had been before the mathematical public for forty years. In each of these fields a vast number of isolated problems had been solved, with here and there notable attempts at unification; but no systematic attack had yet been launched against the whole of mathematics, pure and applied, as it then existed. In particular the powerful analytical methods of Descartes, Newton, and Leibniz had not yet been exploited to the limit of what they were then capable, especially in mechanics and geometry.

  On a lower level algebra and trigonometry were then in shape for systematization and extension; the latter particularly was ready for essential completion. In Fermat’s domain of Diophantine analysis and the properties of the common whole numbers no such “temporary perfection” was possible (it is not even yet); but even here Euler proved himself the master. In fact one of the most remarkable features of Euler’s universal genius was its equal strength in both of the main currents of mathematics, the continuous and the discrete.

  As an algorist Euler has never been surpassed, and probably never even closely approached, unless perhaps by Jacobi. An algorist is a mathematician who devises “algorithms” (or “algorisms”) for the solution of problems of special kinds. As a very simple example, we assume (or prove) that every positive real number has a real square root. How shall the root be calculated? There are many ways known; an algorist devises practicable methods. Or again, in Diophantine analysis, also in the integral calculus, the solution of a problem may not be forthcoming until some ingenious (often simple) replacement of one or more of the variables by functions of other variables has been made; an algorist is a mathematician to whom such ingenious tricks come naturally. There is no uniform mode of procedure—algorists, like facile rhymesters, are born, not made.

  It is fashionable today to despise the “mere algorist”; yet, when a truly great one like the Hindu Ramanujan arrives unexpectedly out of nowhere, even expert analysts hail him as a gift from Heaven: his all but supernatural insight into apparently unrelated formulas reveals hidden trails leading from one territory to another, and the analysts have new tasks provided for them in clearing the trails. An algorist is a “formalist” who loves beautiful formulas for their own sake.

  Before going on to Euler’s peaceful but interesting life we must mention two circumstances of his times which furthered his prodigious activity and helped to give it a direction.

  In the eighteenth century the universities were not the principal centers of research in Europe. They might have become such sooner than they did but for the classical tradition and its understandable hostility to science. Mathematics was close enough to antiquity to be respectable, but physics, being more recent, was suspect. Further, a mathematician in a university of the time would have been expected to put much of his effort on elementary teaching; his research, if any, would have been an unprofitable luxury, precisely as in the average American institution of higher learning today. The Fellows of the British universities could do pretty well as they chose. Few, however, chose to do anything, and what they accomplished (or failed to accomplish) could not affect their bread and butter. Under such laxity or open hostility there was no good reason why the universities should have led in science, and they did not.

  The lead was taken by the various royal academies supported by generous or farsighted rulers. Mathematics owes an undischargeable debt to Frederick the Great of Prussia and Catherine the Great of Russia for their broadminded liberality. They made possible a full century of mathematical progr
ess in one of the most active periods in scientific history. In Euler’s case Berlin and St. Petersburg furnished the sinews of mathematical creation. Both of these foci of creativity owed their inspiration to the restless ambition of Leibniz. The academies for which Leibniz had drawn up the plans gave Euler his chance to be the most prolific mathematician of all time; so, in a sense, Euler was Leibniz’ grandson.

  The Berlin Academy had been slowly dying of brainlessness for forty years when Euler, at the instigation of Frederick the Great, shocked it into life again; and the St. Petersburg Academy, which Peter the Great did not live to organize in accordance with Leibniz’ program, was firmly founded by his successor.

  These Academies were not like some of those today, whose chief function is to award membership in recognition of good work well done; they were research organizations which paid their leading members to produce scientific research. Moreover the salaries and perquisites were ample for a man to support himself and his family in decent comfort. Euler’s household at one time consisted of no fewer than eighteen persons; yet he was given enough to support them all adequately. As a final touch of attractiveness to the life of an academician in the eighteenth century, his children, if worth anything at all, were assured of a fair start in the world.