1.'30 Men of Mathematics

as he is known in English history-having packed up h :---_ his snuff, had left for London to become the first G IS duds aJtd

erman K England. Nothing would have pleased Leibniz better th lllg of George to London, although his enemies at the Roy 1 anSoct? follow 1 h · E a lety e sew ere m ngland were now numerous and vicious e h aJtd

h noug 0 . to t e controversy with Newton. But the boorish G W1ng

· ll eorge soc1a Y a gentleman, had no further use for Leibniz' d" 1 ' now

I d . 1p omacy curt y or ered the brams that had helped to lift h. · . ' and . . . 1m mto c1viJ· society to stick m the Hanover library and get on w"th h . !Zed 1 . h" I t elr e astmg !Story of the illustrious Brunswick family. Ver.

When Leibniz died two years later ( 1716) the d" 1 · d d h. 1p omatJcall

octore !story was still incomplete. For all his hard labor Leib .Y had been unable to bring the history down beyond th mz

d h e year 1005 an at t at had covered less than three hundred years Th r: . '

l . . . · e 1amt1y was so very tang ed m Its manta! adventures that even the · Le"b · 1 Universal

1 mz.cou d not supply ~em all with unblemished scutcheons. The ~runs"':'1ckers showed their appreciation of this immense labor b torgettmg all about it till 1843, when it was publisned, but wheth/r complete: or expurgated will not be known until the rest of Leibniz' manuscnpts 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 h1s secretary followed him to the grave and 1·t 1·s t"ll · · . , s 1 nsmg.

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 t~em 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?

TH E BERNOULLIS

'/'llest men certainly accomplislud much and admirably attained the goal they 1/JJ 11t themselves.-]oHANNES BERNOULLI

SINCE THE GREAT DEPRESSION began deflating western civilization eugenists, geneticists, psychologists, politicians, and dictators-for Yf!lY different reasons-have taken a renewed interest in the still un­.Wed 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 auert 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 1he emergence of genius, but that without deliberate or accidental aistance 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, Beveral of them outstanding, who in turn produced a swarm of descend­ants about half of whom were gifted above the average imd nearly all of whom, down to the present day, have been superior human beings. ~fewer than 120 of the descendants of the mathematical Bernoullis tbe e ~e~ traced genealogically, and of this considerable posterity

. hla.Jonty achieved distinction-sometimes amounting to eminence ;;:::. ~e law, scholarship, science, literature, the learned professions, thin lllistration, and the arts. None were failures. The most significant tbe g about a majority of the mathematical members of this family in

llecond and third generations is that they did not deliberately l S I

IS2 Men of Mathematics

choose mathematics as a profession but drifted into it m· ~ . . ~~~th selves as a dipsomaruac returns to alcohol. ell!..

As the Bernoulli family played a leading part in devel . I I d . ,. . oping th

ca cu us an Its app Icatwns in the seventeenth and eight h e . th . . eent centu nes, ey must be given more than a passmg mentio · -

b . fi n In even th ne est account of the evolution of modern math · e . emattcs. 'fh

Bernoulhs and Euler were in fact the leaders above all th e . o ers wh

perfected the calculus to the pomt where quite ordinary 0 . fi th . men could

use It or e discovery of results which the greatest of th G ld e reeks

cou never have found. But the mere volume of the Bemo II' r. . . . u I1anu!y's

work IS too vast for detailed description in an account like th e present so we shall treat them briefly together. '

I Jocoi.>I ~$~-/?'OS

Nicolov.s Sen/or /~ZJ-tros

I I

McoiO'V.ti

168T-17'.S9

I J0hon/7e\1 I

1"66T'"~8

McGV'ov.s .lJl ~-17'28

The Bernoullis were one of many Protestant families who fled from Antwerp in I58S 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 Switzerland, 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-grandfat~er 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 farnilY tradition of trade by following the profession of medicine. Mathe-

Nature or Nurture? ISS

. 1 talent was probably latent for generations in this shrewd .,atiCB tile family, but its actual emergence was explosively sudden. ~rring now to the genealogical table we shall give a very. ~rief

ary of the chief scienti~c activities of :he. eight_ mathematlcl~s IIJII1Ill ded from Nicolaus semor before contmumg With the heredity. ~b I mastered the Leibnizian form of the calculus. by himself.

I687 to his death he was professor of mathematics at Basle. ::: 1 was one of the first to develop the calculus significantly be­yond the state in which Newton a~d Leibniz left ~t and to. ap~ly it to oew problems of difficulty and Importance. Hts contnbutwns to palytic geometry, the th~ory of probability, and th.e calculus of varia­tions were of the highest Importance. As the last w1ll recur frequently {in the work of Euler, Lagrange, and Hamilton), we may describe tbe 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 tbe calculus of variations in Fermat's principle of least time.

The calculus of variations is of very ancient origin. According to cme legend,* 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 lbe 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 geome­bies 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. t . The discovery that the brachistochrone is a cycloid has been noted 111 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 ~e tautochrone. This struck Johannes I as something wonder­

bui;.Ac~aJly, here, I have combined two legends. Queen Dido was given a t!llc) 8 hide to "enclose" the greatest area. She cut it into one thong and

08~ a semicircle. J~storical notes on this and other problems of the calculus of variations 192s found in the book by G. A. Bliss, Calculus of Variations, Chicago,

· The Anglicized form of jacob I is james.

I

1S4 Men of Mathematics

ful 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 171 S. 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 ~he two and it is certain that he treated his brother with somethmg pre;ty close to dishonesty in the matter of isoperimetrical problems£ The Bemoullis took their mathematics in deadly earnest. Some ~ their letters about mathematics bristle with strong language that iS

usually reserved for horse thieves. For his part Johannes I not only=~ tempted to steal his brother's ideas but threw his own son out ?f ces house for having won a prize from the French Academy of Scten an for which Johannes himself had competed. After all, if rational hu;ow beings get excited about a game of cards, why should they not up over mathematics which is infinitely more exciting? . the

Jacob I had a mystical strain which is of some significan~ Jil jp­

study of the heredity of the Bernoullis. It cropped out once 1n an

Nature or Nurture? 1S5

resting way toward the end of his life. There is a certain spiral (the ~arithmic 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

1 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 :plied side Johannes I contributed extensively to optics, wrote on

e theory of the tides and the mathematical theory of ship sails, and enunciated the principle of virtual displacements in mechanics. Johan­n~ 1 was a man of w1Usual physical and intellectual vigor, remaining act~~ till within a few days of his death at the age of eighty. in lcolaus I, the brother of Jacob I and Johannes I, was also gifted &be lllathematics. Like his brothers he made a false start. At the age of aa:~n he took his doctor's degree in philosophy at the University of

e, and at twenty earned the highest degree in law. He was first

1S6 Men of Mathematics

a professor of law at Bern before becoming one of th r e mathe . raculty at the Academy of St. Petersburg. At the time f h' matical ~as so highly thought of that the Empress Catherine g~ve ~ildeath he he funeral at state expense. n a Pub..

. Heredity ca~e out curiously in the second generation. Joh tned to force his second son, Daniel, into business. But Da . 1 annes I h c d d' . me thou h e prererre me Ierne and became a physician before land· . g t fh. lf · th mg, m sp't

o Imse , m rna ematics. At the age of eleven Dan' 1 b 1 e 1 Ie egan taki essons in mathematics from his elder brother Nicolaus III ng

years older than himself. Daniel and the great Euler w ' ~nl~ five f . . . ere tnt1m t nends and at times fnendly rivals. Like Euler Daniel B 11. a e

th d' · · · ' ernou 1 has e Istmct10n of havmg won the prize of the French A d . ( . ca emy ten

times on a few occasions the prize was shared with other . successful competitors). Some of Daniel's best work went into hydrody ·

h · h h d . nam1cs w IC e eve loped umformly from the single principle that later cam; to be called the conservation of energy All who work tod ·

1. ft · . · ay m pure or app 1ed md mot10n know the name of Daniel Bernoulli.

In 172~ (at the age of twenty five) Daniel became professor of ma~em.atrcs a~ St. Petersburg, where the comparative barbarity of ~e hfe Irked h1m so greatly that he returned at the first opportunity, erght years later, to Basle, where he became professor of anatomy and botany, and finally of physics. His mathematical work included the cal~ulus, differential equations, probability, the theory of vibrating strmg.s, an at~empt at a kinetic theory of gases, and many other prob­lems m apphed mathematics. Daniel Bernoulli has been called the founder of mathematical physics.

From the stan.dpoint of heredity it is interesting to note that Daniel had a marke~ ve~ of speculative philosophy in his nature-possibly a r.efined subhmat10n of the Huguenot religion of his ancestors. The hke cropped out in numerous later descendants of the illustrious refu~ gees 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. Startin~ out .in law he became professor of eloquence at Basle before s~cceed~ng h1s father in the chair of mathematics. His work was prin~ Cipa.lly I.n physics and was sufficiently distinguished to capture the Pans pnz~ ~n three occasions (once is usually enough to satisfy a good mathematician- provided he is good enough).

Nature or Nurture? 1S7

~III, a son of Johannes II, repeated the family tradition of J~ a wrong start, and like his father began with law. At the age ~~een he took his doctor's degree in philosophy. By nineteen rJ res III had found his true vocation and was appointed astrono­~yal at Berlin. His interests embraced astronomy, geography, rre ro . ..-1 mathematics.

Jacob II, another son of johannes II, carried on the family ?lunder starting in law, only to change over at twenty one to expenmental

by ·cs He also turned to mathematics, becoming a member of the pbysi . St. Petersburg Academy in the section of .mathematics ~d physics. lfis early death (at the age of thirty) by accidental drownmg cut short avery promising career, and we do not know what 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 as­eerted 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 ma­

terial 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 another­unless we are the sort of bigots who insist that everything should be tnathematics, or biology, or sociology, or bridge and golf. It may be that the abandonment of mathematics as the family trade by the Ber­lloullis 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

ISS Men of Mathematics

violent in their language as the Bernoullis sometimes were. On these ripe old chestnuts may be retailed again as it is one of the e of

. l l h . . f h COilJ., parattve y ear y aut ennc mstances o a story whic must be at l as old as ancient Egypt, and of which we daily see variants p· east onto all sorts of prominent characters from Einstein down. Once mnhed

11. . . w en trave mg as a young man Dante! modestly mtroduced himself t . . h h oan mterestmg stranger wit w om he had been conversing: "I am D . B II. " "And I " "d th th · antel ernou 1. , sat e o er sarcastically, "am Isaac N ton." This delighted Daniel to the end of his days as the sm· ceew.

. . rest tnbute he had ever recetved.

CHAPTER NINE

Analysis Incarnate EULER

History shows that thou heads of empires who have encouraged the cultiva­titlll of mathematics, the com1Mn source of all the exact sciences, are also those wltM reigns have been the 1Mst brilliant and whose glory is the nwst durable.

-MICHEL CHASLES

"EULER CALCULATED WITHOUT APPARENT EFFORT, as men breathe, or 11 eagles sustain themselves in the wind" (as Arago said), is not an aaggeration of the unequalled mathematical facility of Leonard Euler {1707- 178S), the most prolific mathematician in history, and the man whom his contemporaries called "analysis incarnate." Euler wrote lis great memoirs as easily as a fluent writer composes a letter to an intimate friend. Even total blindness during the last seventeen years ofbis life did not retard his unparalleled productivity; indeed, if any­tbing, the loss of his eyesight sharpened Euler's perceptions in the illler world of his imagination.

The extent of Euler's work was not accurately known even in 19S6,

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 liohl many individuals and mathematical societies throughout the •orld-rightly claiming that Euler belongs to the whole civilized •orld and not only to Switzerland. The careful estimates of the prob­~e expe.nse (about $80,000 in the money of 1909) were badly upset "1 the dtscovery in St. Petersburg (Leningrad) of an unsuspected lllass of Euler's manuscripts. A Euler's mathematical career opened in the year of Newton's death. ~ lnore propitious epoch for a genius like that of Euler's could not

111 e be~n chosen. Analytic geometry ( made public in 16S7) had been Use runety years, the calculus about fifty, and Newton's law of uni-

1.'.19