MEN OF MATHEMATICS E. T. BELL

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E. T. B ELL

MEN OF MATHEMATICS

323

VOLUME TWO

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CONTENTS OF VOLUME TWO

PREFACE

16. THE COPERNICUS OF GEOMETRY
    Lobatchewsky (1793-1856)

The widow^s mite, Kazan. Appointed professor and spy.
Universal ability, Lobatchewsky as an administrator,
Beason and incense combat the cholera. Russian gralittide.
Humiliated in his prime. Blind as MiUon, Lobatchewsky
dictates his masterpiece. His advance beyond Euclid. Non-
Euclidean geometry, A Copernicm of the intellect.

17. GENIUS AND POVERTY
    Abeb (1802-29)

Norway in 1802. Smothered by clerical fecundity. AbeTs
awakening. Generosity of a teacher. A pupil of the masters.
His lucky blunder, Abel and the quiniic. The GovernmerU
to the rescue. AbeVs grand tour of mathematical Europe
not so grand. French civility and German cordiality,
Crelle and his Journal. Cauchy^ s unpardonable sin. ‘’AbeVs
Theorem.”* SomeMng to keep mathematicians busy 500
‘ years. Crowning a colpse.

18. THE GREAT ALGORIST
    Jacobi (1804-51)

Galvanoplastics verstbs mathematics. Born rich. Jacobi* s
philological ability. Dedicates himself to mathematics.
Early work,’ Cleaned out, A goose among foxes. Hard
times. Elliptic functions. Their place in the general
development. Inversion. Work in arithmetic^ dynamics^
algebra, and Abelian functions. Fouriers pontiflcation, Jacobi s retort.

19. AN IRISH TRAGEDY
    Hamilton (1805-65)

IrekmcVs greatest. Elabofate miseducation. Discoveries
seventeen, A unique university career. Disappointed
love, Hamilton and the poets. Appointed at Dvrt^0*

.§• ft

COXTEKTS

FcrnMl’s Last Theorem storied. Theory ideal nunAers,
Kummer’s indention compareAle to Ldbatchexsky’s. Wave
surface in four dimensions. Big of body, mind, and hearL
DedtJani, last pupil of Gauss. First expositor of Galois.

Early interest in science. Turns to maOiemMixs. Dede-
land’s mark on continuity. His ciedtion of the theory of
ideals.

2b. THE LAST UNIVERSALIST S80

POIN’CABE (1354-1912)

Poincare’s iinivcrsdiiy and methods. Childmd setbacks.

Seiad by mathematics. Keeps Ms sanity in Franco-Prus-
sian ’ixar. Starts as mining engineer. First great mrk.
Aulomorphic functions, ‘The keys of the algebraic cosmos.’

The problem of n bodies. Is Finland ekiUzed^ PoincarFs
nes metfiods in celestial mechanics. Cosmogony. How
mathematical discocerm are made. PoincarFs account.
Forebodings and premedure death.

29. PARADISE LOST? 612

Can-toe (1845-1918)

Old foes with new faces. EoUing creeds. Cantor’s artistic
inheritance and father-fixation. Escape, but too late. His
trcohdionary work grts him nowhere. Academic pettiness.
Disastrous consequences of ‘safety pst’. An epochal result.
Paradox or truth? Infinite existence o/ trcmscenderdeds.
Aggtessiteness adcances, timidity retires. Further specta-
cular claims. Two types of malhematkiems. ImsrmeS
Counter-recoklion. The haUle grows fiercer. Cursing &e
enemy. Universal loss of temper. Where stands mathe-
saaties to-day? And where will it stand to-morrow? Invictus.

ISDEX TO VOUTUES OXE AXD TwO

640

PREFACE TO VOLUME TWO

• The general introduction in the first volume of this book need
  not be repeated here. However, a few points may be briefly
  recalled, as they are relevant for both volumes. We shall also
  take a quick glance ahead beyond the last of the men mentioned.

Mathematics as understood by mathematicians is based on
deductive reasoning applied to sets of outright assumptions
called axioms or postulates. It is sufficient here to describe
deductive reasoning as the rules of common logic, although
mathematical logic goes far beyond that. The postulates under-
lying a particular division of mathematics, such as elementary
algebra or school geometry, may have been suggested by every-
day observation of the world as it presents itself to our senses.
Many of the propositions of geometry, for instance, such as that
gem attributed to Thales in the sixth century b.c., ‘The angle
inscribed in a semicircle is a right angle’, axe evident to the eye.
But however obvious and sensible they may seem, they are not
a part of mathematics until they have been deduced from a set
of postulates accepted without argument as self-consistent. The
great hut (to us) nameless mathematicians of Babylonia dis-
covered, or invented, many beautiful things in both algebra and
geometry, but, so far as is known, they proved none of them.

It remained for the Greeks of about 600 b.c. to invent proof –
deductive reasoning. With that epochal invention mathematics
was bom. But logic and proof are by no means the whole story.
Intuition and insight axe as freely used to-day in mathematics
as they must have been by the Babylonians.

It was many centuries before the full significance of what
those old Greeks had done was understood and applied to all
mathematics, and thence to all reasoning. A notable instance is
school algebra, first thoroughly understood and rigorouriy
developed only in the 1830’s by the British School, of whom –
George Peacock (1791-1858) is especially memorable. Unfortu-
nately there is not space here to tell the lives of these little-

xxiii

PEEFACE

knoTm men -who helped to prepare the way for the vast develop-
ment of mathematics in the nineteenth and twentieth centuries.

As we pass from the eighteenth to the nineteenth century
we are overwhelmed by a tidal wave of free inventiveness.
Xew departments of mathematics were created and developed
in bewildering profusion. The great mathematicians of the nine-
teenth century, some of whom are presented iiere, seem
to be almost of a different species from their predecessors. The
new men were not content with special problems, but attacked
and solved general problems whose solutions yielded those of a
multitude of problems which, in the eighteenth century, would
have been considered one by one. A striking example has often
been noted in the contrast between Gauss (1777-1855) and Abel
(1802-29) in the theory of algebraic equations. There is a
similar distinction in the matter of geometry between Gauss
and his pupil Riemann (1825-66). It is no disparagement of
Gauss, but merely a statement of historical fact, to say that he
was content with the problem of finding the algebraic solution
of binomial equations, and did not even mention the general
problem, solved by Abel and Galois (1811—32), of determining
necessary and sufficient conditions that any given algebraic
equation be solvable by radicals. The nature of the general
problem is explained in the accoimts given here of Abel and
Galois. Of course there is a certain loose continuity in all
mathematics, clear back to Babylon and Egypt, but the
interesting and fruitful points on the curve of progress are the
discontinuities that appear when the curve is closely analysed
as in that just noted of Gauss, Abel, and Gabis. One such from
the 1930 s must suffice here as a current example.

The paradoxes of Zeno and the repeated attempts to establish
the differential and integral calculus on a firm logical foundation
exercised mathematicians as early as the seventeenth century,
continued to worry them aU through the second half of the
nineteenth. Among those who struggled at the task were three
whom we shall meet later. Cantor, Dedekind, and Weieistrass.
Dedckind admitted failure. But failure or not to achieve the
desired end, the work of all three gave a tremendous impulse
to the study of all mathematical reasoning. How was it to he

PEEFACE

decided that a certain theorem had really been proved? Might
not there be concealed inconsistencies in the very foundations
and postulate systems on which the whole elaborate structure
had been reared? It began to appear that an exhaustive re-
examination of everything from the ground up was demanded.
The capital problem was to prove the self-consistency of
mathematical analysis – the calculus and its numerous modern
offshoots. Presently this programme turned out to be far more
difficult than had been anticipated, and David Hilbert (1862—
1943), the last of the giants from the nineteenth century, in
1898 proposed the more modest problem of proving the consis-
tency of arithmetic. This led to the like for mathematical logic.

All was going well till 1931, when Kurt Gddel (1906- )

showed that in any well-defined system of mathematical axioms
there exist mathematical questions which cannot be settled on
the basis of these axioms. But suppose we go to a more inclusive
system in which, perhaps, the questions can be settled. The
same difficulty appears in the new system, and so on indefi-
nitely. There are thus specific purely mathematical ‘yes-no’
questions which will be forever undecidable by human beings.

This whoUy unexpected conclusion has been called the most
significant advance in logic since Aristotle. It does not mean
that mathematics has gone to smash, but it does suggest that
some of the claims made for mathematics in the past will have
to he moderated. One philosophical die-hard who thoroughly
misunderstood what Gddel had done, proudly proclaimed, ‘I
am an Aristotelian. The old logic is good enough for me’, which
sounded like an echo of the revivalist hymn ‘The old-time
religion, the old-time religion is good enough for me.’ Aristo-
telian logic may be good enough for the old-timers, hut it is
not good enough for mathematics, nor has it been for at least
three centuries. As one detail, Aristotle’s logic makes no pro-
vision for variables and functions as they occur in mathematics.
There is not space here to elaborate any of this, but those
interested will find an elementary and lucid account by Alfred
Tarski in his Introduction to Logic and the Methodology of Deduc^
five Sciences (Oxford University Press, 2nd Edition, 1941).

1953 E . T. B E n n

MEN OF MATHEMATICS

VOLUME TWO

CHAPTER SIXTEEN

THE COPERNICUS OF GEOMETRY
Lobatchewsky

• Granting that the commonly accepted estimate of the im-
  portance of what Copernicus did is correct, we shall have to
  admit that it is either the highest praise or the severest con-
  demnation humanly possible to call another man the ‘Coper-
  nicus’ of anything. When we understand what Lobatchewsky
  did in the creation of non-Euclidean geometry, and consider
  its significance for all human thought, of which mathematics is
  only a small if important part, we shall probably agree that
  Clifford (1845-79), himself a great geometer and far more than
  a ‘mere mathematician’, was not overpraising his hero when he
  called Lobatchewsky ‘The Copernicus of Geometry’.

Nikolas Ivanovitch Lobatchewsky, the second son of a minor
government official, was bom on 2 November 1793 in the
district of Makarief, government of Nijni Novgorod, Russia.
The father died when Nikolas was seven, leaving his widow,
Praskovia Ivanovna, the care of three young sons. As the
father’s salary had barely sufficed to keep his family going
while he was alive the widow found herself in extreme poverty.
She moved to Kazan, where she prepared her boys for school as
best she could, and had the satisfaction of seeing them accepted,
one after the other, as free scholars at the Gymnasium. Nikolas
was admitted in 1802 at the age of eight. His progress was
phenomenally rapid in both mathematics and the classics. At
the age of fourteen he was ready for the university. In 1807 he
entered the University of Kazan (founded in 1805), where he

828

MEN OF MATHEMATICS

was to Spend the next forty years of his life as student, assistant
professor, professor, and finally rector,

Hoping to make Kazan ultimately the equal of any university
in Exxrope, the authorities had imported several distinguished
professors from Germany. Among these was the astronomer
Littrow, who later became director of the Observatory at
Vienna, whom Abel mentioned as one of his excuses for seeing
something of ‘the south’. The German professors quickly recog-
nized Lobatchewsky’s genius and gave him every encourage-
ment.

In 1811, at the age of eighteen, Lobatehewsky obtained his
master’s degree after a short tussle with the authorities, whose
ire he had incurred through his youthful exuberance. His
German friends on the faculty took his part and he got his
degree with distinction. At this time his elder brother Alexis
was in charge of the elementary mathematical courses for the
training of minor government officials, and when Alexis
presently took a sick-leave, Nikolas stepped into his place as
substitute. Two years later, at the age of twenty-one, Lobat-
chewsky received a probationary appointment as ‘Extra-
ordinary Professor’ or, as would be said in America, Assistant
Professor.

Lobatehewsky *s promotion to an ordinary professorship came
in 1816 at the unusually early age of twenty-three. His duties
were heavy. In addition to his mathematical work he was
charged with courses in astronomy and physics, the former to
substitute for a colleague on leave. The fine balance with which
he carried his heavy load made him a conspicuous candidate for
yet more work, on the theory that a man who can do much is
capahfe of doing more, and presently Lobatehewsky found
himself University Librarian and curator of the chaotically
disordered University Museum.

Students are often an unruly lot before life teaches them that
generosity of spirit does not pay in the cut-throat business of
earning a living. Among Lobatchewsky’s innumerable duties
from 1819 tiH the death of the Tsar Alexander in 1825 was that
of supervisor of all the students in Kazan, from the elementary
schools to the men taking post-graduate courses in the Univer-

324

THE COPERNICUS OF GEOMETRY

sity. The supervision was primarily over the political opinions
of his charges. The difficulties of such a thankless job can easily
be imagined. That Lobatchewsky contrived to send in his
reports day after day and year after year to his suspicious
superiors without once being called on the carpet for laxity in
espionage, and without losing the sincere respect and affection
of all the students, says more for his administrative ability than
do all the gaudy orders and medals which a grateful GJovern-
ment showered on him and with which he delighted to adorn
himself on state occasions.

The collections in the University Museum to all appearance
had been tossed in with a pitchfork. A similar disorder made the
extensive library practically unusable. Lobatchewsky was
commanded to clean up these messes. In recognition of his
signal services the authorities promoted him to the deanship of
the Faculty of Mathematics and Physics, but omitted to appro-
priate any funds for hiring assistance m straightening out the
library and the museum. Lobatchewsky did the work with his
own hands, cataloguing, dusting, and casing, or wielding a mop
as the occasion demanded.

With the death of Alexander in 1825 things took a turn for
the better. The particular official responsible for the malicious
persecution of the University of Kazan was kicked out as being
too corrupt for even a government post, and his successor
appointed a professional curator to relieve Lobatchewsky of his
endless task of cataloguing books, dusting mineral specimens,
and deverminizing stuffed birds. Needing political and moral
support for his work in the University, the new curator did
some high politics on his own account and secured the appoint-
ment in 1827 of Lobatchewsky as Rector. The mathematician
was now head of the University, but the new position was no
sinecure. Under his able direction the entire staff was reorga-
nized, better men were brought in, instruction was liberalized
in spite of official obstruction, the library was built up to a
higher standard of scientific sufficiency, a mechanical workshop
was organized for making the scientific instnnnents required in
research and instruction, an observatory was founded and
equipped – a pet project of the energetic Rector’s – and the vast

325

yLBS OF MATHEMATICS

mineralogical collection, representative of the whole of Russia,
was put in order and constantly enriched.

Even the new dignity of his rectorship did not deter Lobat-
chewsky from manual labour in the library and museum when
he felt that Ms help was necessary. The University was his life
and he loved it. On the slightest provocation he would take oft
his collar and coat and go to work. Once a distinguished
foreigner, taking the coatless Rector for a janitor or workman,
asked to be shown through the libraries and museum collec-
tions. Liobatchewsky showed him the choicest treasures,
explaining as he eshibited. The visitor was charmed and greatly
impressed by the superior intelligence and courtesy of this
obliging Russian worker. On partiDg from his guide he tendered
a hand^me tip. Lobatchewsky, to the foreigner’s bewilder-
ment, froze up in a cold rage and indignantly spurned the
proffered coin. Thinking it but just one more eccentricity of the
bi^-minded Russian Janitor, the visitor bowed and pocketed
his money. That evening he and Lobatchewsky met at the
Gkjvemor’s dinner table, where apologies were offered and
accepted on both sides.

Lobatchewsky was a strong believer in the philosophy titiat
in order to get a thing done to your own liking you must either
do it yourself or understand enough about its execution to be
able to criticize the work of another intelligently and construe-
tiveiy. As has been said, the University was his life. When the
Government decided to modernize the buildings and add new
ones, Lobatchewsky made it his business to see that the work
was done properly and the appropriation not squandered. To
fit himself for this task he learned arcMtecture. So practical was
his mastery of the subject that the buildings were not only
handsome and suited for their purposes but, what must be
almost unique in the history of governmental building, were
constructed for less money than had been appropriated. Some
years latex (in 1842) a disastrous fire destroyed half and
took with it Lobatchewsky’s finest buildings, the

barely completed observatory —the pride of his heart. But owing
to his energetic cool-headedness the instruments and the
hbrary were saved. After the fire he set to work immediately to

THE COPERNICUS OP GEOMETRY

rebuild. Two years later not a trace of the disaster remained.

We recall that 1842, the year of the fibre, was also the year in
which, thanks to the good offices of Gauss, Lobatehewsky was
elected a foreign correspondent of the Royal Society of Gottin-
gen for his creation of non-Euclidean geometry. Although it
seems incredible that any man so excessively burdened with
teaching and administration as Lobatehewsky was, could fiind
the time to do even one piece of mediocre scientific work, he had
actually, somehow or another, made the opportunity to create
one of the great masterpieces of all mathematics and a land-
mark in human thought. He had worked at it off and on for
twenty years or more. EQs first public communication on the
subject, to the Physical-Mathematical Society of Kazan, was
made in 1826. He might have been speaking in the middle of the
Sahara Desert for all the echo he got. Gauss did not hear of the
work till about 1840.

Another episode in Lobatehewsky ’s busy life shows that it
was not only in mathematics that he was far ahead of his time.
The Russia of 1830 was probably no more sanitary than that of
a century later, and it may be assumed that the same disregard
of personal hygiene which filled the German soldiers in
World War I with an amazed disgust for their unfortunate
Russian prisoners, and which to-day causes the industrious
proletariat to use the public parks and playgrounds of Moscow
as vast and convenient latrines, distinguished the luckless inha-
bitants of Kazan in Lobatchewsky’s day when the cholera
epidemic foimd them richly prepared for a prolonged visitation*
The germ theory of disease was still in the future in 1830^
although progressive minds had long suspected that filthy
habits had more to do with the scourge of the pestilence than
the anger of the Lord.

On the arrival of the cholera in Kazan the priests did what
they could for their smitten people, herding them into the
churches for united supplication, absolving the dying and
burying the dead, but never once suggesting that a shovel
might be useful for any purpose other than digging graves.
Realizing that the situation in the town was hopeless, Lobat-
chewsky induced his faculty to bring their families to the

m

MEK OF ^lATHEMATICS

UniTersity and prevailed upon – practically ordered – some of
the students to join him in a rational, human fight against the
cholera. The mndoTVS were kept closed, strict sanitary regula-
tions were enforced, and only the most necessary forays for
replenishing the food supply were permitted. Of the 660 men,
women and children thus sanely protected, only sixteen died, a
mortality of less than 2.6 per cent. Compared to the losses under
the traditionalremedies practised in the town this was negligible.

It might be imagined that after all his distinguished services
to the State and his European recognition as a mathematician,
Lobatchewsky would be in line for substantial honours from
his Government, To imagine anjrthing of the kind would not
only be extremely naive hut would also traverse the scriptural
injunction ‘Put not your trust in princes’. As a reward for all
his sacrifices and his unswerving loyalty to the be^ in Russia,
Lobatchewsky was brusquely relieved in 1846 of his Professor-
ship and his Rectorship of the university. No explanation of
this singular and unmerited double insult was made public.
Lobatchewsky was in his fifty-fourth year, vigorous of body
and mind as ever, and more eager than he had ever been to
continue with his mathematical researches. His colleagues to a
man protested against the outrage, jeopardizing their own
security, but were curtly informed that they as mere professors
were constitutionally incapable of comprehending the higher
mysteries of the science of government.

The ill-disguised disgrace broke Lobatchewsky. He was still
peiinitt€td to retain his study at the University. Hut when his
successor, hand-picked by the Government to discipline the
disaffected faculty, arrived in 1847 to take up his ungracious
task, Lobatchewsi^ abandoned all hope of ever being anybody
again in the University which owed its intellectual eminence
almost entirely to his efforts, and he appeared thereafter only
occasionally to assist at examinations. Although his eyesight
was f a iling rapidly he was still capable of intense mathematical
thinking.

He still loved the University. His health broke when his son
died, but he lingered on, hoping that he might still be of some
use. In 1855 the University celebrated its semi-centennial anni-

828

THE COPERNICUS OF GEOMETRY

versary. To do honour to the occasion, Lobatchewsky attended
the exercises in person to present a copy of his Pangeometry ^ the
completed work of his scientific life. This work (in French and
Russian) was not written by his own hand, but was dictated, as
Lobatchewsky was now blind. A few months later he died, on
24 February 1856, at the age of sixty-two.

To see what Lobatchewsky did we must first glance at
Euclid’s outstanding achievement. The name Euclid until quite
recently was practically synonymous with elementary school
geometry. Of the man himself very little is known beyond his
doubtful dates, 330-275 b.c. In addition to a systematic
account of elementary geometry his Elements contain all that
was known in his time of the theory of numbers. Geometrical
teaching was dominated by Euclid for over 2,200 years. TTis
part in the Elements appears to have been principally that of a
co-ordinator and logical arranger of the scattered results of his
predecessors and contemporaries, and his aim was to give a
connected, reasoned account of elementary geometry such that
every statement in the whole long book could be referred back
to the postulates. Euclid did not attain this ideal or anything
even distantly approaching it, although it was assumed for
centuries that he had.

Euclid’s title to immortality is based on something quite
other than the supposed logical perfection which is still some-
times erroneously ascribed to liim. This is his recognition that
the fifth of his postulates (his Axiom XI) is a pure assumption.

I

The fifth postulate can be stated in many equivalent fomas,
each of which is deducible from any one of the others by means
of the remaining postulates of Euclid’s geometry. Possibly the
simplest of these equivalent statements is the following: Given
any straight line I and a point P not on Z, then in the plane
determined by I and P it is possible to drsLW precisely one straight

329

HEN OF mathematics

line I through P such that V never meets Z no matter how far V
and I are extended (in either direction). Merely as a nominal
definition we say that two straight lines lying in one plane
which never meet aveparalleL Thus the fifth postulate of Euclid
asserts that through P there is precisely one straight line
parallel to L Euclid’s penetrating insight into the nature of
geometry convinced him that this postulate had not, in his
time, been deduced from the others, although there had been
many attempts to prove the postulate. Being unable to deduce
the postulate himself from his other assumptions, and wishing
to use it in the proofs of many of his theorems, Euclid honestly
set it out with his other postulates.

There are one or two simple matters to be disposed of before
we come to Lobatchewsky’s Copemican part in the extension
of geometry. We have alluded to ‘equivalents’ of the parallel
postulate. One of these, ‘the hypothesis of the right angle’, as it
is called, will suggest two possibilities, neither equivalent to
Euclid’s assumption, one of which introduces Lohatchewsky’s
geometiv’, the other, Riemann’s,

Consider a figure which ‘looks like’ a rectangle, con-

sisting of four straight lines AX, XY, YB, BA, in which BA
(or AB) is the base, AX and YB (or BY) are drawn equal and
perpendicular to AB, and on the same side of AB. The essential
things to be remembered about this figure are that each of the
angles XAB^ YBA (at the base) is a right angle, and that the
sides AX, BY are equal in length. Without using the parcdlel
postulate, it can be proved that the angles AXY, BYX^ are
8S0

THE COPERNICUS OF GEOMETRY

equal, but, without using this postulate, it is impossible to prove
that AXY, BYX are right angles, although they look it. If we
assume the parallel postulate we can prove that AXY, BYX are
right angles and, conversely, if we assume that AXY, BYX are
right angles, we can prove the parallel postulate. Thus the
assumption that AXY, BYX are right angles is equivalent to the
parallel postulate. This assumption is to-day called the hypothesis
of the right angle (since both angles are right angles the singular
instead of the plural ‘angles’ is used).

It is known that the hypothesis of the right angle leads to a
consistent, practically usefiil geometry, in fact to Euclid’s
geometry refurbished to meet modem standards of logical
rigour. But the figure suggests two other possibilities: each of
the equal angles AXY, BYX is less than a right angle – the
hypothesis of the acute angle*, each of the equal angles AXY,
BYX is greater than a right angle – the hypothesis of the obtuse
angle. Since any angle can satisfy one, and only one, of the
requirements that it be equal to, less than, or greater than a right
angle, the three hypotheses – of the right angle, acute angle,
and obtuse angle respectively – exhaust the possibilities.

Common experience predisposes us in favour of the first
hypothesis. To see that each of the others is not as unreasonable
as might at first appear we shall consider something closer to
actual human experience than the highly idealized ‘plane’ in
which Euclid imagined his figtires drawn. But first we observe
that neither the hypothesis of the acute angle nor that of the
obtuse angle will enable us to prove Euclid’s parallel postulate,
because, as has been stated above, Euclid’s postulate is equi-
valent to the hypothesis of the right angle (in the sense of inter-
deducibility; the hypothesis of the right angle is both necessary
and sufficient for the deduction of the parallel postulate)*
Hence if we succeed in constructing geometries on either of the
two new hypotheses, we shall not find in them parallels in
Euclid’s sense.

To make the other hypotheses less unreasonable than they
may seem at first sight, suppose the Earth were a perfect sphere
(without irr^ularities due to mountains, etc.). A plane drawn
through the centre of this ideal Earth cuts the surface in a great

331

MEX OF MATHEMATICS

circle. Suppose we wisli to go from one point A to another B on
the surface of the Earth, keeping always on the surface in
passing from A to B, and suppose further that we wish to make
the journey by the shortest way possible. This is the problem of
\great circle sailing’. Imagine a plane passed through A, B, and
the centre of the Earth (there is one, and only one, such plane).
This plane cuts the surface in a great circle. To make our
shortest journey we go from Ato B along the shorter of the two
arcs of this great circle joining them. If .4, B happen to lie at
the extremities of a diameter, we may go by either arc.

The preceding example introduces, an important dejSnition,
that of a geodesic on a surface^ which will now be explained. It
has just been seen that the shortest distance joining two points
on a sphere, the distance itself being measured on the surface,
is an arc of the great circle joining them. \Ye have also seen that
the longest distance joining the two points is the other arc of the
same great circle, except in the case when the points are ends of
a diameter, when shortest and longest are equal. In the chapter
on Fermat ‘greatest’ and ‘least’ were subsumed under the
common name “extreme’, or ‘ extremum’. We recall now one
usual definition of a straight-line segment joining two points in
a plane – ‘the shortest distance between two points’. Trans-
ferring this to the sphere, we say that to straight line in the
plane corresponds great circle on the sphere. Since the Greek
word for the Earth is the first syllable ge (y^) of geodesic we
call all extrema joining any two points on any surface the
geodesics of that surface. Thus in a plane the geodesics are
Euclid’s straight lines; on a sphere they are great circles. A
geodesic can be visualized as the position taken by a string
stretched as tight as possible between two points on a surface.

Now, in navigation at least, an ocean is not thought of as a
flat surface (Euclidean plane) if even moderate distances are
concerned ; it is taken for what it very approximately is, namely,
a part of the surface of a sphere, and the geometry of great
circle sai l i ng is not Euclid’s. Thus Euclid’s is not the only
geometry of human utility. On the plane two geodesics inter-
sect in exactly one point unless they are parallel, when they do
not intersect (in Euclidean geometry); but on the sphere any

832

THE COPERNICUS OF GEOMETRY

two geodesics always intersect in precisely two points. Again,
on a plane, no two geodesics can enclose a space – as Euclid
assumed in one of the postulates for his geometry; on a sphere,
any two geodesics always enclose a space.

N

S

Imagine now the equator on the sphere and two geodesics
drawn through the north pole perpendicular to the equator. In
the northern hemisphere this gives a triangle with curved sides,
two of which are equal. Each side of this triangle is an arc of a
geodesic. Draw any other geodesic cutting the two equal sides
so that the intercepted parts between the equator and the cut-
ting line are equal. We now have, on the sphere^ the four-sided
figure corresponding to the AXYB we had a few moments ago
in the plane. The two angles at the base of this figure are right
angles and the corresponding sides are equal, as before, bid each
of the equal angles at X^Y is now greater than a right angle. So,
in the highly practical geometry of great circle sailing, which is
closer to real human experience than the idealized diagrams of
elementary geometry ever get, it is not Euclid’s postulate
which is true – or its equivalent in the hypothesis of the right
angle – but the geometry which follows from the hypothesis of
the obtuse angle.

In a similar manner, inspecting a less familiar surface, we can

838

ilEN OF MATHEMATICS

make reasonable the hypothecs of the acute angle. The surface
looks like two infini tely long trumpets soldered together at their
largest ends. To describe it more accurately we must introduce
the plane curve called the tractrixt which is generated as follows.

Let two lines XOX\ YOY’ be drawn in a horizontal plane
intersecting at right angles in 0, as in Cartesian geometry.
Imagine an inextensible fibre lying along YOF’, to one end of
which is attached a small heavy pellet; the other end of the
fibre is at O. Pull this end out along the line 0X» As the pellet

Y

y’

follows, it traces out one half of the tractrix; the other half is
traced out by drawing the end of the fibre along OX’, and of
course is merely the reflection or image in OY of the first half.
The drawing out is supposed to continue indefinitely – Ho

THE COPERNICUS OF GEOMETRY

infinity’ – in each instance. Now imagine the tractrix to be
revolved about the line XOX\ The double-trumpet surface is
generated; for reasons we need not go into (it has constant
negative curvature) it is called a pseudo-sphere. If on this
surface we draw the four-sided figure with two equal sides and
two right angles as before, using geodesics, we find that the
hypothesis of the acute angle is realized.

Thus the hypotheses of the right angle, the obtuse angle, and
the acute angle respectively are true on a Euclidean plane, a
sphere, and a pseudosphere respectively, and in all cases
‘straight lines’ are geodesics or extrema. Euclidean geometry is a
limiting, or degenerate, case of geometry on a sphere, being
attained when the radius of the sphere becomes infinite.

Instead of constructing a geometry to fit the Earth as human
beings now know it, Euclid apparently proceeded on the as-
sumption that the Earth is flat. If Euclid did not, his prede-
cessors did, and by the time the theory of ‘space’, or geometry,
reached him the bald assumptions which he embodied in his
postulates had already tal^en on the aspect of hoary and im-
mutable necessary truths, revealed to mankind by a higher
intelligence as the veritable essence of all material things. It
took over 2,000 years to knock the eternal truth out of geome-
try, and Lobatchewsky did it.

To use Einstein’s phrase, Lobatchewsky challenged an axiom.
Anyone who challenges an ‘accepted truth’ that has seemed
necessary or reasonable to the great majority of sane men for
2,000 years or more takes his scientific reputation, if not his
life, in his hands. Einstein himself challenged the axiom that
two events can happen in different places at the same time, and
by analysing this hoary assumption was led to the invention of
the special theory of relativity. Lobatchewsky challenged the
assumption that Euclid’s parallel postulate or, what is eqid-
valent, the hypothesis of the right angle, is necessary to a con-
sistent geometry, and he backed his challenge by producing a
system of geometry based on the hypothesis of the acute angle
in which there is not one parallel through a fixed point to a
given straight line but two. Neither of Lobatchewsky’s parallels
meets the line to which both are parallel, nor does any straight

S35

MEN OF MATHEMATICS

line drawn through the feed point and lying within the angle
formed by the two parallels. This apparently bizarre situation
is ‘realized’ by the geodesies on a pseudosphere.

For any everyday purpose (measurements of distances, etc.),
the differences between the geometries of Euclid and Lobat-
chewsky are too small to count, but this is not the point of
importance; each is self-consistent and each is adequate for
human experience. Lobatchewsky abolished the necessary
*truth’ of Euclidean geometry. His geometry was but the first
of several constructed by his successors. Some of these substi-
tutes for Euclid’s geometry – for instance the Riemannian
geometry’ of general relativity – are to-day at least as important
in the still living and growing parts of physical science as
Euclid’s was, and is, in the comparatively static and classical
parts. For some purposes Euclid’s geometry is best or at least
sufficient, for others it is inadequate and a non-Euclidean
geometry is demanded,

Euclid in some sense was believed for 2,200 years to have
discovered an absolute truth or a necessary mode of human
perception in his system of geometry. Lobatchewsky’s creation
was a pragmatic demonstration of the error of this belief. The
boldness of his challenge and its successful outcome have
inspired mathematicians and scientists in general to challenge
other ‘axioms’ or accepted ‘truths’, for example the ‘law’ of
cau^ty, which, for centuries, have seemed as necessary to
straight t hinkin g as Euclid’s postulate appeared till Lobat-
chewsky discarded it.

pie full impact of the Lobatchewskian method of
axioms has probably yet to be felt- It is no exaggeration to call
Lobatchewsky the Copernicus of Geometry, for geometry is
only a part of the vaster domain which he renovated; it might
even be just to designate him as a Copernicus of all thought.

CHAPTER SEVENTEEN

GENIUS AND POVERTY
Abel

• An astrologer in the year 1801 might have read in the stars that
  a new galaxy of mathematical genius was about to blaze forth
  inaugurating the greatest century of mathematical hjLstory. In
  all that galaxy of talent there was no brighter star than Niels
  Henrik Abel, the man of whom Hermite said, Tie has left
  mathematicians something to keep them busy for five hundred
  years’.

Abel’s father was the pastor of the little village of Findo, in
the diocese of Kristiansand, Norway, where his second son,
Niels Henrik, was bom on 5 August 1802. On the father’s side
several ancestors had been prominent in the work of the church
and all, including Abel’s father, were men of culture. Anne
Marie Simonsen, Abel’s mother, was chiefly remarkable for her
great beauty, love of pleasure, and general flightiness – quite an
exciting combination fox a pastor’s helpmeet. From hex Abel
inherited his striking good looks and a very human desire to get
something more than everlasting hard work out of life, a desire
he was seldom able to gratify.

The pastor was blessed with seven children in all at a time
when Norway was desperately poor as the result of wars with
England and Sweden, to say nothing of a famine thrown in for
good measure between wars. Nevertheless the family was a
happy one. In spite of pinching poverty and occasional empty
stomachs they kept their chins up. There is a charming picture
of Abel after his mathematical genius had seized him sitting by
the fireside with the others chattering and laughing in the room
while he researched with one eye on his mathematics and the
other on his brothers and sisters. The noise never distracted him
and he joined in the badinage as he wrote.

387

MEN OF MATHEMATICS

Like several of the first-rank mathematieians Abel discovered
his talent early. A brutal schoolmaster unwittingly threw
opportunity Abel’s way. Education in the first decades of the
nineteenth century was virile, at least in Norway. Corporal
punishment, as the simplest method of toughening the pupils’
characters and gratifying the sadistic inclinations of the
masterful pedagogues, was generously administered for every
trivial offence. Abel was not awakened through his own skin,
as Newton is said to have been by that thundering kick donated
by a playmate, but by the sacrifice of a fellow student who had
been flogged so unmercifully that he died. This was a bit too
thick even for the rugged school board and they deprived the
teacher of his job. A competent but by no means brilliant
mathematician filled the vacancy, Bemt Mchael Holmboe
(1795-1850), who was later to edit the first edition of Abel’s
collected works in 1839.

Abel at the time was about fifteen. Up till now he had shown
no marked talent for anything except taking his troubles with
a sense of humour. Under the Mndly, enlightened Holmboe’s
teaching Abel suddenly discovered what he was. At sixteen he
began reading privately and thoroughly digesting the great
works of his predecessors, including some of those of Newton,
Euler, and Lagrange. Thereafter real mathematics was not only
his serious occupation but his fascinating delight. Asked some
years later how he had managed to forge ahead so rapidly to
the front rank he replied, ‘By studying the masters, not their
pupils’ — a prescription some popular writers of textbooks might
do well to mention in their prefaces as an antidote to the
poisonous mediocrity of their uninspired pedagogics.

Holmboe and Abel soon became close ffiends. Although the
teacher was himself no creative mathematician he knew and
appreciated the masterpieces of mathematics, and his
eager suggestions Abel was soon mastering the toughest of the
classics, including the Disquisitmm Arithmeticae of Gauss.

To-day it is a commonplace that many fine things the old
masters thought they had proved were not really proved at all.
Paiticulariy is this true of some of Euler’s work on infinite
scries and some of Lagrange’s on analysis. Abel’s keen mind

GENIUS AND POVERTY

was one of the first to detect the gaps in his predecessors’
reasoning, and he resolved to devote a fair share of his lifework
to caulking the cracks and making the reasoning watertight.
One of his classics in this direction is the proof of the general
binomial theorem, special cases of which had been stated by-
Newton and Euler. It is not easy to give a sound proof in the
general case, so perhaps it is not astonishing to find alleged
proofs still displayed in -the schoolbooks as if Abel had never
lived. This proof, however, was only a detail in Abel’s vaster
programme of cleaning up the theory and application of infinite
series.

Abel’s father died in 1820 at the age of forty-eight. At the
time Abel was eighteen. The care of his mother and six children
fell on his shoulders. Confident of himself Abel assumed his
sudden responsibilities cheerfully. Abel was a genial and opti-
mistic soul. With no more than strict justice he foresaw himself
as an honoured and moderately prosperous mathematician in a
university chair. Then he could provide for the lot of them in
reasonable security. In the meantime he took private pupils
and did what he could. In passing it may be noted that Abel
was a very successful teacher. Had he been footloose poverty
would never have bothered him. He could have earned enough
for his own modest needs, somehow or other, at any time. But
with seven on his back he had no chance. He never complained,
but took it an in his stride as part of the day’s work and kept at
his mathematical researches in every spare moment.

Convinced that he had one of the greatest mathematicians of
all time on his hands, Holmboe did what he could by getting
subsidies for the young man and digging down generously into
his own none too deep pocket. But the country was poor to the
point of starvation and not nearly enough could be done. In
those days of privation and incessant work Abel immortalized
himself and sowed the seeds of the disease which was to kill
him before he had half done his work,

Abel’s first ambitious venture was an attack on the general
equation of the fifth degree (the ‘quintic’). All his great pre-
decessors in algebra had exhausted their efforts to produce a
solution, without success. We can easily imagine Abel’s exulta-

839

MEN OF MATHEMATICS

tion when he mistakenly imagined he had succeeded. Through
Holmboe the supposed solution was sent to the most learned
mathematical scholar of the time in Denmark who, fortunately
for Abel, asked for further particulars without committing him-
self to an opinion on the correctness of the solution. Abel in the
meantime had found the flaw in his reasoning. The supposed
solution was of course no solution at alL This failure gave him
a most salutary jolt; it jarred him on to the right track and
caused him to doubt whether an algebraic solution was possible.
He proved the impossibility. At the time he was about nineteen.
But he had been anticipated, at least in part, in the whole
project.

As this question of the general quintic played a role in
algebra similar to that of a crucial experiment to decide the fate
of an entire scientific theory, it is worth a moment’s attention.
We shall quote presently a few things Abel himself says.

The nature of the problem is easily described. In early school
algebra we learn to solve the general equations of the first and
second degrees in the unknown r, say

ax -T =0, ax- -f 4- c = 0,

and a little later those of the third sud fourth degrees, say
aa!® + 6aj*4-aj4-d:s=o, a2!^ + 6a^-f-caj2-{-ifoj4-e = o.

That is, we produce finite (closed) formulae for each of these
general equations of the first four d^rees, expressing the
unknown x in terms of the given coefiELcients a,d,c,d,e. A solution
Euch as any one of these four which can be obtained by only a
fiinUe number of additions, multiplications, si 4 btractio 7 is, divisions,
and extractions of roots, all these operations being performed on
the given coefficients, is called algebraic. The important qualifi-
cation in this definition of an algebraic solution is ‘finite ’ ; there
is no difficulty in describing solutions for any algebraic equation
which contain no extractions of roots at all, but which do imply
an infinity of the other operations named.

After this success with algebraic equations of the first four
840

GENIUS AND POVERTY

degrees, algebraists struggled for nearly three centuries to pro-
duce a gimilar algebraic solution for the general quintic
ax^ + bx^ cos^ dx^ + ex + f 0,

They failed. It is here that Abel enters.

The following ejdxacts are given partly to show how a great
inventive mathematician thought and partly for their intrinsic
interest. They are from Abel’s memoir On ike algebraic resolution
of equations.

‘One of the most interesting problems of algebra is that of the
algebraic solution of equations. Thus we find that nearly all
mathematicians of distinguished rank have treated this subject.
We arrive without difficulty at the general expression of the
roots of equations of the first four degrees. A uniform method
for solving these equations was discovered and it was believed
to be applicable to an equation of any degree; but in spite of all
the efforts of Lagrange and other distinguished mathematicians
the proposed end was not reached. That led to the presumption
that the solution of general equations was impossible algebrai-
cally; but this is what could not be decided, since the method
followed could lead to decisive conclusions only in the case
where the equations were solvable. In effect they proposed to
solve equations without knowing whether it was possible. In
this way one might.indeed arrive at a solution, although that
was by no means certain; but if by ill luck the solution was
impossible, one might seek it for an eternity, without finding it.
To arrive infallibly at something in this matter, we must there-
. fore follow another road. We can give the problem such a form
that it shall always be possible to solve it, as we can always do
with any problem.* Instead of asking for a relation of which it
is not known whether it exists or not, we must ask whether
such a relation is indeed possible. . , . When a problem is posed
in this way, the very statement contains the germ of the solu-
tion and indicates what road must be taken; and I believe there

• , cfi gii’on peut ioujours faire d^un probUme qaelconque^ is what

Abel says. This seems a trifle too optimistic; at least for ordinary
mortals. How would the method be applied to Fermat’s Last
Theorem?

341

MEN OF MATHEMATICS

mil be few instances where we shall fail to arrive at propositions
of more or less importance, even when the complication of the
calculations precludes a complete answer to the problem.

He goes on to say that this, the true scientific method to be
followed, has been but little used owing to the extreme complin
cation of the calculations (algebraic) which it entails; ‘but’, he
adds, in many instances this complication is only apparent and
vanishes after the first attack.’ He continues :

‘I have treated several branches of analysis in this manner,
and although I have often set myself problems beyond my
powers, I have nevertheless arrived at a large number of general
results which throw a strong light on the nature of those quan-
tities whose elucidation is the ohject of mathematics. On
another occasion I shall ^ve the results at which I have
arrived in these researches and the procedure which has led me
to them. In the present memoir I shall treat the problem of the
algebraic solution of equations in all its generality.’

Presently he states two general inter-related problems which
he proposes to discuss:

‘1. To find all the equations of any given degree which are
solvable algebraically.

2. To determine whether a given equation is or is not solv-
   able algebraically.’

At bottom, he says, these two problems are the same, and
although he does not claim a complete solution, he does indicate
an infallible method (des moyem siirs) for disposing of them
fully.

Abel’s irrepressible inventiveness hurried him on to vaster
problems before he had time to return to these; their complete
solution – the explicit statement of necessary and sufficient
conditions that an algebraic equation be solvable algebraically
was to be reserved for Galois. When this memoir of Abel’s was
publi^ed in 1828, Galois was a boy of sixteen, already well
started on his career of fundamental discovery. Galois later
came to know and admire the work of Abel; it is probable that
Abel never heard the name of Galois, although when Abel
visited Paris he and his brilliajit successor could have been only
a few miles apart. But for the stupidity of Galois’ teachers and

842

GENIUS AND POVEUTY

the loftiness of some of Abel’s mathematical ‘superiors’,
it is quite possible that he and Abel might have met.

Epoch-making as Abel’s work in algebra was, it is over-
shadowed by his creation of a new branch of analysis. This, as
Legendre said, is Abel’s ‘time-outlasting monument’. If the
stor}^ of his life adds nothing to the splendour of his accom-
plishment it at least suggests what the world lost when he died.
It is a somewhat discouraging tale. Only Abel’s unconquerable
cheerfulness and unyielding courage under the stress of poverty
and lack of encouragement from the mathematical princes of
his day lighten the story. He did, however, find one generous
friend in addition to Holmboe.

In June 1822 when Abel was nineteen, he completed his
required work at the University of Kristiania. Holmboe had
done everything possible to relieve the young man’s poverty,
convincing his colleagues that they too should subscribe to
make it possible for Abel to continue his mathematical re-
searches. They were immensely proud of him but they were also
poor themselves. Abel quickly outgrew Scandinavia, He longed
to visit France, then the mathematical queen of the world,
where he could meet his great peers (he was in a class far above
some of them, but he did not know it). He dreamed also of
touring Germany and meeting Gauss, the undisputed prince of
them all.

Abel’s mathematical and astronomical friends persuaded the
University to appeal to the Norwegian Government to subsidize
the young man for a grand mathematical tour of Europe. To
impress the authorities with his worthiness, Abel submitted an
extensi’re memoir which, from its title, was probably connected
with the fields of his greatest fame. He himseK thought highly
enough of it to believe its publication by the University would
bring Norway honour, and Abel’s opinion of his own work,
never more than just, was probably as good as anyone’s. Unfor-
tunately the University was having a severe financial struggle
of its own, and the memoir was finally lost. After undue deli-
beration the Government compromised ~ does any Government
ever do anything else? – and instead of doing the only sensible
thing, namely sending Abel at once to France and Germany,

V0L.I1.

B

843

MEN OF MATHEMATICS

granted him a subsidy to continue his university studies at
Kristiania in order that he might brush up his French and
Crerman, That is exactly the sort of decision he might have
expected from any body of officials conspicuous for their good
hearts and common sense. Common sense, however, has no
business dictating to genius.

Abel dallied a year and a half at Ejistiania, not wasting his
time, but dutifully keeping his part of the contract by wrestling
(not too successfully) with German, getting a fair start on
French, and working incessantly at his mathematics. With his
incurable optimism he had also got himself engaged to a young
woman – Crelly Kemp. At last, on 27 August 1825, when Abel
was twenty-three, his friends overcame the last objection of the
Government, and a royal decree granted him si^cient funds
for a year’s travel and study in France and Germany. They did
not give him much, but the fact that they gave him anything
at all in the straitened financial condition of the country says
more for the state of civilization in Norway in 1825 than could
a whole encyclopaedia of the arts and trades. Abel was grateful.
It took him about a month to straighten out his dependents
before leaving. But thirteen months before this, innocently
believing that all mathematicians were as generous-minded as
himself, he had burned one of his ladders before ever setting
foot on it.

Out of his own pocket – God only knows how – Abel had paid
for the printiDg of his memoir in which the impossibility of
solving the general equation of the fifth degree algebraically
is proved. It was a pretty poor job of printing but the best back-
ward Norway could manage. This, Abel naively believed, was
to be bis scientific passport to the great mathematicians of the
Continent. Gauss in particular, he hoped, would recognize the
signal merits of the work and grant him more than a formal
interview. He could not know that Hhe prince of mathemati-
dans^ sometimes exhibited anything but a princely generosity
to young matliematiciaiis struggling for just recognition.

Gauss duly received the paper. Through unimpeachable
witnesses Abel heard how Gauss welcomed the offering. With-
out deigning to read it he tossed it aside with the disgusted

S44

GENIUS AND POVERTY

exclamation ‘Here is another of those monstrosities!’ Abel
decided not to call on Gauss. Thereafter he disliked Gauss
intensely and nicked liim whenever he could. He said Gauss
wrote obscurely and hinted that the Germans thought a little
too much of him. It is an open question whether Gauss or Abel
lost more by this perfectly understandable dislike.

Gauss has often been censured for his ‘haughty contempt’ in
this matter, but those are hardly the right words to describe his
conduct. The problem of the general equation of the fifth degree
had become notorious. Cranks as well as reputable mathemati-
cians had been burrowing into it. Now, if a mathematician to-
day receives an alleged squaring of the circle, he may or may
not write a courteous note of acknowledgement to the author,
but he is almost certain to file the author’s manuscript in the
waste-basket. For he knows that lindemann in 1882 proved
that it is impossible to square the circle by straight-edge and
compass alone – the implements to which cranks limit them-
selves, just as Euclid did. He knows also that Lmdemann’s
proof is accessible to anyone. In 1824 the problem of the general
quintic was almost on a par with that of squaring the circle.
Hence Gauss’ impatience. But it was not quite as bad; the
impossibility had not yet been proved. Abel’s paper supplied
the proof; Gauss might have read something to interest him
intensely had he kept his temper. It is a tragedy that he did
not, A word from him and Abel would have been made. It is
even possible that his life would have been lengthened, as we
shall admit when we have his whole story before us.

After leaving home in September 1825, Abel first visited the
notable mathematicians and astronomers of Norway and Den-
mark and then, instead of hurrying to Gottingen to meet Gauss
as he had intended, proceeded to Berlin. There he had the great
good fortune to fall in with a man, August Leopold Crelle
(1780-1856) who was to be a scientific Holmboe to him and who
had far more weight in the mathematical world than the good
Holmboe ever had. If Crelle helped to make Abel’s reputation,
Abel more than paid for the help by making Crelle’s. Wherever
mathematics is cultivated to-day the name of Crelle is a house-
hold word, indeed more; for ‘Crelle’ has become a proper noun

B2 845

Z^E’S OF 3IATHEMATICS

signifying the great journal he founded, the first three volumes
of which contained twenty-two of Abel’s memoirs. The journal
made Abel, or at least made him more widely known to Conti-
nental mathematicians than he could ever have been without
it; Abel’s great work started the journal oft wdth a bang that
was heard round the mathematical world; and finally the
journal made Crelle. This self-effacing amateur of mathematics
deserves more than a passing mention. His business ability and
his sure instinct for picking collaborators who had real mathe-
matics in them did more for the progress of mathematics in the
nineteenth century than half a dozen learned academies.

Crelie himself was a self-taught lover of mathematics rather
than a creative mathematician. By profession he was a civil
engineer. He early rose to the top in his work, built the first
railroad in Germany, and made a comfortable stake. In Ms
leisure he pursued mathematics as something more than a
hobby. He himself contributed to mathematical research before
and after the great stimulus to German mathematics wMch Ms
Journal fur die reine und angemandte Mathematik (Journal for
pure and applied Mathematics) gave on its foundation in 1826.
This is CreHe’s greatest contribution to the advancement of
mathematics.

The Journal was the first periodical in the world devoted
exclusively to mathematical research. Expositions of old work
were not welcomed. Papers (except some of Crelle’s own) were
accepted from anyone, provided only the matter was new, true,
and of sufficient importance’ – an intangible requirement – to
merit publication. Regularly once every three months from
1826 to the present day ‘‘Creile’ has appeared with its sheaf of
new mathematics. In the chaos after the World War “Crelle’
tottered and almost went down, but was sustained by sub-
scribers from all over the world who were unwilling to see tMs
great monument to a more tranquil civilization our own
obliterated. To-day hundreds of periodicals are devoted either
wholly or in considerable part to the advancement of pure and
applied mathematics. How many of them will survive our next
outburst of epidemic insanity is anybody’s guess.

When Abel arrived in Berlin in 1825 CreUe had just about

646

GENIUS AND POYEETY

made up his mind to start his great venture with his own funds.
Abel played a part in clinching the decision. There are two
accounts of the first meeting of Abel and Crelle, both interest-
ing. Crelle at the time was holding down a government job for
which he had but little aptitude and less liking, that of examiner
at the Trade-School {Gewerbe-Institut) in Berlin. At third-hand
(Crelle to Weierstrass to Mttag-Leffler) Crelle’s account of that
historic meeting is as follows.

‘One fine day a fair young man, much embarrassed, with a
very youthful and very intelligent face, walked into my room.
Believing that I had to do with an examination-candidate for
admission to the Trade-School, I explained that several
separate examinations would be necessary. At last the young
man opened his mouth and explained [in poor German], ‘”Not
examination, only mathematics”.’

Crelle saw that Abel was a foreigner and tried him in French,
in which Abel could make hi m self understood with some diffi-
culty. Crelle then questioned him about what he had done in
mathematics. Diplomatically enough Abel replied that he had
read, among other things, Crelle’s own paper of 1823, then
recently published, on ‘analytical faculties’ (now called
‘factorials’ in English). He had found the work most interesting
he said, but Then, not so diplomatically, he proceeded to tell
Crelle that parts of the work were quite wrong. It was here that
Crelle showed his greatness. Instead of freezing or blowing up
in a rage at the daring presumption of the young man before
him, he pricked up his ears and asked for particulars, which he
followed with the closest attention. They had a long mathe-
matical talk, only parts of which were intelligible to Crelle.
But whether he understood all that Abel told him or not, Crelle
saw clearly what Abel was. Crelle never did understand a tenth
of what Abel was up to, but his sure instinct for mathematical
genius told him that Abel was a mathematician of the first
water and he did everything in his power to gain recognition
for his young prot^e. Before the interview was ended Crelle
had made up his mind that Abel must be one of the first
contributors to the projected Journal,

Abel’s account differs, but not essentially. Reading between

347

MEN OF MATHEMATICS

the lines we may see that the differences are due to Abel’s
modesty. At first Abel feared bis project of interesting Crelle
was fated to go on the rocks. Crelle could not make out what the
young man wanted, who he was, or anything about him. But at
Crelle’s question as to what Abel had read in mathematics
things brightened up considerably. When Abel mentioned the
works of the masters he had studied Crelle became instantly
alert. They had a long talk on several outs t a nding unsettled
problems, and Abel ventured to spring his proof of the impossi-
bility of sohdiig the general quintic algebraically on the unsus-
pecting Crelle. Crelle wouldn’t hear of it; there must he some-
thing wrong with any such proof. But he accepted a copy of the
paper, thumbed through it, admitted the reasoning was beyond
him – and finally published Abel’s amplified proof in his
Journal. Although he was a limited mathematician with no
pretensions to scientific greatness, CreUe was a broad-minded
man, in fact, a great man.

Crelle took Abel everywhere, showing him off as the finest
mathematical discovery yet made. The self-taught Swiss
Steiner – ihe greatest geometer since Apollonius’ – sometimes
accompanied Crelle and Abel on their rounds. When Crelle’s
Mends saw him coining with his two geniuses in tow they would
exclaim ‘Here comes Father Adam again with Cain and Abel.’

The generous sociability of Berlin began to distract Abel
firom his work and he fled to Freibuig where he could concen-
trate. It was at Freiburg that he hewed his greatest work into
shape, the creation of what is now called Abel’s Theorem. But
he had to be getting on to Paris to meet the foremost French
oaatheznatimns of the day — Legendre, Cauchy, and the rest.

It can be said at once that Abel’s reception at the hands of
the French mathematicians was as civil as one would expect
finom distinguished representatives of a very civil people in a
rery civil age. They were all very civil to Mm – damned civil,
in fact, and that was about all that Abel got out of the visit to
whk^ he had looked forward with such ardent hopes. Of course
they did not know who or what he was. They made only per-
functory efforts to find out. If Abel opened his mouth – when
he got within talking distance of them – about his own work,
848

GENIUS AND POVERTY

they immediately began lecturing about their own greatness.
But for his indifference the venerable Legendre might have
learned something about his own lifelong passion (for elliptic
integrals) which would have interested him beyond measure.
But he was just stepping into his carriage when Abel called and
had time for little more than a very civil good-day. Later he
made handsome amends.

Late in July 1826 Abel took up his lodgings in Paris with a
poor but grasping family who gave him two bad meals a day
and a vile room for a sufficiently outrageous rent. After four
months of Paris Abel writes his impressions to Holmboe:

PariSi 24 October 1862^
To tell you the truth this noisiest capital of the Continent
has for the moment the effect of a desert on me. I know
practically nobody; this is the lovely season when every-
body is in the country. . . . Up tiil now I have made the ‘
acquaintance of Mr LegendrCi Mr Cauchy and Mr Hackette,
and some less celebrated but very able mathematicians:

Saigey^ editor of the Bulletin des Sciences, and Mr
Lejeune’Dirickletf a Prussian who came to see me the other
day believing me to be a compatriot of his. He is a mathe-
matician of great penetration. With Mr Legendre he has
proved the impossibility of solving = z® in whole

numbers, and other very fine things. Legendre is extremely
polite, but unfortunately very old. Cauchy is mad. , . .
What he does is excellent, but very muddled. At first I
understood practically none of it; now I see spme of it more
clearly. , , . Cauchy is the only one occupied with pure
mathematics, Poisson, Fourier, Ampire, etc., busy them-
selves exclusively with magnetism and other physical sub-
jects. Mr Laplace writes nothing now, I believe. His last
work was a supplement to his Theory of Probabilities. I
have often seen him at the Institut. He is a very jolly little
chap, Poisson is a little fellow; he knows how to behave
with a great deal of dignity; Mr Fourier the same. Lacroix
is quite old. Mr Hachette is going to present me to several
of these men.

The French are much more reserved with strangers than
the Germans. It is extremely difficult to gain their inti-
macy, and I do not dare to urge my pretensions as far as

349

MEN or MATHEMATICS

that; finally every beginner has a great deal of difficulty in
getting noticed here. I have just finished an extensive
treatise on a certain class of transcendental functions [his
masterpiece] to present it to the Institut [Academy of
Sciences], which will be done next Monday. I showed it to
Mr Cauchy y but he scarcely deigned to glance at it. And 1
dare to say, without bragging, that it is a good piece of
work. I am curious to hear the opinion of the Institut on it.

I shall not fail to share it with you. . . .

He then tells what he is doing and continues with a rather
disturbed forecast of his prospects. ‘I regret having set two
years for my travels, a year and a half would have sufficed.’ He
has got all there is to be got out of Continental Europe
und is anxious to be able to devote his time to working up
what he has invented.

So many things remain for me to do, but so long as I am
abroad, all that goes badly enough. K I had my professor-
ship as IVIr Kielhau has hisl My position is not assured, it
is true, hut I am not uneasy about it; if fortime deserts me
in one quarter perhaps she will smile on me in another.

From a letter of earlier date to the astronomer Hansteen we
take two extracts, the first relating to Abel’s great project of
re-establishing mathematical analysis as it existed in his day
on a firm foundation, the second showing something of his
human side. (Both are free translations.)

In the higher analysis too few propositions are proved
with conclusive rigour. Everywhere we find the unfortu-
nate procedure of reasoning from the special to the general,
and the miracle is that after such a process it is only seldom
that we find what are called paradoxes. It is indeed exceed-
ingly interesting to seek the reason for this. This reason, in
my opinion, resides in the fact that the functions which
have hitherto occurred in analysis can be expressed for the
most part as powers. . . , When we proceed by a general
method, it is not too difficult [to avoid pitfalls] ; but I have
Imd to be very circfumspect, because propositions without
rigorous proof (i.e. without any proof) have taken root in
me to such an extent that I constantly run the risk of
using them without further examination. These trifles
will appear in the journal published by Mr Crelle,

350

GENIUS AND POYEETY

Immediately following tliis he expresses his gratitude for his
treatment in Berlin. ‘It is true that few persons are interested
in me, but these few are infinitely dear to me, because they have
shown me so much kindness. Perhaps I can respond in some
way to their hopes of me, for it must be hard for a benefactor to
see his trouble lost.’

He tells then how CreUe has been begging him to take up his
residence permanently in Berlin. Crelle was already using all his
human engineering skiU to hoist the Norwegian Abel into a
professorship in the University of Berlin. Such was the Ger-
many of 1826. Abel of course was already great, and the sure
promise of what he had in him indicated him as the likeliest
mathematical successor to Gauss. That he was a foreigner made
no difference; Berlin in 1826 wanted the best in mathematics.
A century later the best in mathematical physics was not good
enough, and Berlin quite forcibly got rid of Einstein. Thus do
we progress. But to return to the sanguine Abel.

At first I counted on going directly from Berlin to Paris,
happy in the promise that Mr Crelle would accompany me.
But Mx Crelle was prevented, and I shall have to travel
alone. Now I am so constituted that I cannot endure soH-
tude. Alone, I am depressed, I get cantankerous, and I
have little inclination for work. So I said to myself it
would be much better to go with IVIr Boeck to Vieima, and
this trip seems to me to be justified by the fact that at
Vienna there are men like Littrow, Burg^ and still others,
all indeed excellent mathematicians; add to this that I
shall make but this one voyage in my life. Could one find
anythingbut reasonableness in this wish of mine to see some-
of the life of the South? I could work assiduously enough
while travelling. Once in Vienna and leaving there for
Paris, it is almost a bee-line via Switzerland. Why shouldn’t
I see a little of it too? My Godt I, even I, have some taste
for the beauties of nature, like everybody else. This whole
trip would bring me to Paris two months later, that’s aU.

I could quickly catch up the tune lost. Don’t you thirik
such a trip would do me good?

So Abel went South, leaving his masterpiece in Cauchy’s care
to be presented to the Institut. The prolific Cauchy was so busy

351

HEN OE MATHEMATICS

l&yxng of own and cackling about them that he had no
time to examine the veritable roc’s egg which the modest Abel
had deposited in the nest. Hachette, a mere pot-washer of a
mathematician, presented Abel’s Memoir on a general property
of a very extensive doss of transcendental functions to the Paris
Academy of Sciences on 10 October 1826. This is the work
which Legendre later described in the words of Horace as
^tnonumentum acre perennius*, and the 500 years’ work which
Hennite said Abel had laid out for future generations of mathe-
maticians. It is one of the crowning achievements of modem
mathematics.

What happened to it? Legendre and Cauchy were appointed
as referees. Legendre was seventy-four, Cauchy thirty-nine.
The veteran was losing his edge, the captain was in bis self-
centred prime. Legendre complained (letter to Jacobi, 9 April
1829) that ‘we perceived that the memoir was barely legible; it
was written in ink almost white, the letters badly formed; it
was agreed between us that the author should be asked for a
neater copy to be read.’ What an alibi! Cauchy took the
memoir home, mislaid it, and foigot all about it.

To match this phenomenal feat of forgetfulness we have to
imagine an Egyptologist misla37ing the Rosetta Stone. Only by
a sort of miracle was the memoir unearthed after Abel’s death.
Jacobi heard of itfrom Legendre, with whom Abel corresponded
after returning to Norway, and in a letter dated 14 March 1829
Jacobi exclaims, ‘What a discovery is this of Mr AbeTsl . . .
Did anyone ever see the like? But how comes it that this discov-
ery, perhaps the most important mathematical discovery that
has been made in our Century, having been communicated to
your Academy two years ago, has escaped the attention of your
colleagues?’ The enquiry reached Norway. To make a long story
fihoxt, the Norwegian consul at Paris raised a diplomatic row
about the missing manuscript and Cauchy dug it up in 1830.
Finally it was printed, but not till 1841, in the Merrioires
pr^sente’s par divers savants d ? Academic royaJLe des sciences de
r Institut de Franc€f voL 7, pp. 176-264. To crown this epic in
pcwo of crass incompetence, the editor, or the printers, or both
between them, succeeded in losing the manuscript before the
852

GENIUS AND POVERTY

proof-sheets were read.*** The Academy (in 1830) made amends
to Abel by awarding him the Grand Prize in Mathematics
jointly with Jacobi. Abel, however, was dead.

The opening paragraphs of the memoir indicate its scope.

The transcendental functions hitherto considered by
mathematicians are very few in number. Practically the
entire theory of transcendental functions is reduced to
that of logarithmic functions, circular and exponential
functions, fimctions which, at bottom, form but a single
species. It is only recently that some other functions have
begun to be considered- Among the latter, the elliptic
transcendents, several of whose remarkable and elegant
properties have been developed by IVIr Legendre, hold the
first place. The author [Abel] has considered, in the memoir
which he has the honour to present to the Academy, a very
extended class of fxmctions, namely: all those whose deri-
vatives are expressible by means of algebraic equations
whose coefficients are rational functions of one variable,
and he has proved for these functions properties analogous
to those of logarithmic and elliptic functions . . . and he
has arrived at the following theorem:

If we have several functions whose derivatives can be
roots of one and the same cdgebraic equation, all of whose
coefficients are rational functions of one variable, we can
always express the sum of any number of such functions
by an algebraic and logarithmic function, provided that we
establish a certain number of algebraic relations between
the variables of the functions in question.

The number of these relations does not depend at all
upon the number of functions, but only upon the nature of
the particular functions considered. . . .

• Libri, a soi-disant mathematician, who saw the work through the
  press, adds, “by permission of the Academy’, a smug footnote acknow-
  ledging the genius of the lamented Abel. This is the last straw; the
  Academy might have come out with all the facts or have held its
  official tongue. But at all costs the honour and dignity of a stuifed
  shirt must be upheld. Finally it may be recalled that valuable manu-
  scripts and books had an unaccountable trick of vanishing when Libri
  was round.

85a

MEN OF MATHEMATICS

The theorem which Abel thus briefly describes is to-day
Imawn as Abel’s Theorem. His proof of it has been described as
nothing more than ‘a marvellous exercise in the integral
calculus’. As in his algebra, so in his analysis, Abel attained his
proof with a superb parsimony. The proof, it may be said with-
out exaggeration, is well within the purview of any seventeen-
year-old who has been through a good first course in the calcu-
lus. There is nothing high-falutin’ about the classic simplicity
of Abel’s own proof. The like cannot be said for some of the
nineteenth-century expansions and geometrical reworkings of
the original proof. Abel’s proof is like a statue by Phidias; some
of the others resemble a Gothic cathedral smothered in Irish
lace, Italian confetti, and French pastry.

There is ground for a possible misimderstanding in Abel’s
opening paragraph. Abel no doubt was merely being Mndly
courteous to an old man who had patronized him – in the bad
sense – on first acquaintance, but who, nevertheless, had spent
most of his long working life on an important problem without
seeing what it was all about. It is not true that Legendre had
discussed the elliptic functions^ as Abel’s words might imply;
what Legendre spent most of his life over was elliptic integrals
which are as different from elliptic functions as a horse is from
the cart it pulls, and therein precisely is the crux and the germ
of one of Abel’s greatest contributions to mathematics. The
matter is quite simple to anyone who has had a school course in
trigonometry; to obviate tedious explanations of elementary
matters this much will be as s u m ed in what follows presently.

For those who have forgotten all about trigonometry, how-
ever, the essence, the methodology^ of Abel’s epochal advance
can be analogized thus. “VVe alluded to the cart and the horse.
The frowsy proverb about putting the cart before the horse
describes what Legendre did; Abel saw that if the cart was to
move forward the horse should precede it. To take another
instance; Francis Galton, in his statistical studies of the relation
between poverty and chronic drunkenness, was led, by bis
impartial mind, to a reconsideration of all the self-righteous
pktitudes by which indignant moralists and economic crusaders
with an axe to grind evaluate such social phenomena. Instead

354t

GENIUS AND POVERTY

of assuming that people are depraved because they drink to
excess, Galton inverted this hypothesis and assumed temporarily
that people drink to excess because they have inherited no moral
guts from their ancestors, in short, because they are depraved.
Brushing aside all the vaporous moralizing of the reformers,
Galton took a firm grip on a scientific, unemotional, workable
hypothesis to which he could apply the impartial machinery of
mathematics. His work has not yet registered socially. For the
moment we need note only that Galton, like Abel, inverted his
problem – turned it upside-down and inside-out, back-end-to
and foremost-end-backward. Like Hiawatha and his fabulous
mittens, Galton put the skinside inside and the inside outside.

All this is far from being obvious or a triviality. It is one of
the most powerful methods of mathematical discovery (or
invention) ever demised, and Abel was the first human being to
use it consciously as an engine of research. ‘You must always
invert’, as Jacobi said when asked the secret of his mathema-
tical discoveries. He was recalling what Abel and he had done.
If the solution of a problem becomes hopelessly involved, try
turning the problem backwards, put the quaesita for the data
and vice versa. Thus if we find Cardan’s character incompre-
hensible when we think of him as a son of his father, shift the
emphasis, invert it, and see what we get when we analyse
Cardan’s father as the begetter and endower of his son. Instead
of studying ‘inheritance’ concentrate on ‘endowing’. To return
to those who remember some trigonometry.

Suppose mathematicians had been so blind as not to see that
sin X, cos X and the other direct trigonometric functions are
simpler to use, in the addition formulae and elsewhere, than the
inverse functions sin“^ x, cos~^ x. Recall the formula sin {x -f y)
in terms of sines and cosines of x and y, and contrast it with the
formula for shr^ {x–y) in terms of x and y. Is not the former
incomparably simpler, more elegant, more ‘natural’ than the
latter? Now, in the integral calculus, the inverse trigonometric
functions present themselves naturally as definite integrals ‘of
simple algebraic irrationalities (second degree); such integrals
appear when we seek to find the length of an arc of a circle by
means of the integral calculus. Suppose the inverse trigono-

355

MEX OF MATHEiMATICS

metric factions Iiad first presented ‘ themselves this way.
Would it not have been ‘more natural’ to consider the inverses
of these functions, that is, the faimhar trigonometric functions
themselves as the given functions to be studied and analysed?
Undoubtedly; but in shoals of more advanced problems, the
simplest of which is that of finding the length of the arc of an
ellipse by the integral calculus, the awkward inverse elliptic’
(not ‘circular’, as for the arc of a circle) functions presented
themselves It took Abel to see that these functions should
be ‘inverted* and studied, precisely as in the case of sin x, cos x
instead of sin”^ x, cos”^ x. Simple, was it not? Yet Legendre, a
great mathematician, spent more than forty years over his
“elliptic integrals’ (the awkward ‘inverse functions’ of his
problem) without ever once sui^ecting that he should invert*
This ejrtxemely simple, uncommonsensical way of looking at an
apparently simple but profoundly recondite problem was one
of the greatest mathematical advances of the nineteenth
centmv’.

All this however was but the beginning, although a suffi-
ciently tremendous beginning — like Kipling’s dawn coming up
like thunder – of what Abel did in his magnificent theorem and
in his work on elliptic functions. The trigonometric or circular
functions have h single real period, thus sin {x + 2ir) = sin a?,
etc. Abel discovered that his new functions provided by the
inversion of an elliptic integral have precisely two periods,
whose ratio is imaginary. After that, Abel’s followers in this
direction – Jacobi, Rosenhain, Weierstrass, Biemann, and
many more – mined deeply into Abel’s great theorem and by
carrying on and extending his ideas discovered functions of n
variables having 2n periods. Abel himself carried the exploita-
tion of his discoveries far. His successors have applied all this
work to geometry, mechanics, parts of mathematical physics,
and other tracts of mathematics, solving impctont problems

• In ascribing priority to Abel, rather than ‘Joint discovery’ to Abel
  and Jacobi, in this matter, I have followed Mittag-Leffler. From a
  thorough acquaintance with all the published evidence, J am con-
  vinced that Abel’s claim is indisputable, although Jacobi’s com-
  patriots ajgue otherwise.

3SG

GENIUS AND POVERTY

which^ Tsithout this work initiated by Abel, would have been
unsolvable.

While in Paris Abel consulted good physicians for what he
thought was merely a persistent cold. He was told that he had
tuberculosis of the lungs. He refused to believe it, wiped the
mud of Paris off his boots, and returned to Berlin for a short
visit. His funds were running low; about seven dollars was the
extent of his fortune. An urgent letter brought a loan from
Holmboe after some delay. It must not be supposed that Abel
was a chronic borrower on no prospects. He had good reason
for believing that he should have a paying job when he got
home. Moreover, money was still owed to him. On Holmboe’s
loan of about sixty dollars Abel existed and researched from
March till May 1827. Then, all his resources exhausted, he
turned homeward and arrived in Kristiania completely
destitute.

But all was soon to be rosy, he hoped. Surely the University
job would be forthcoming now. His genius had begun to be
recognized. There was a vacancy. Abel did not get it, Holmboe
reluctantly took the vacant chair which he had intended Abel
to fill only after the governing board threatened to import a
foreigner if Holmboe did not take it, Holmboe was in no way
to blame. It was assumed that Holmboe would be a better
teacher than Abel, although Abel had amply demonstrated his
ability to teach. Anyone familiar with the current American
pedagogical theory, fostered by professional Schools of Educa-
tion, that the less a man knows about what he is to teach the
better he will teach it, will understand the situation perfectly.

Nevertheless things did brighten up. The University paid
Abel the balance of what it owed on his travel money and
Holmboe sent pupils bis way. The professor of astronomy took
a leave of absence and suggested that Abel he employed to
cany part of his work. A well-to-do couple, the ScLjeldnips,
took him in and treated him as if he were their own son. But
with all this he could not free himself of the burden of his
dependents. To the last they clung to him, leaving him prac-
tically nothing for himself, and to the last he never uttered an
impatient word.

357

MEX OF MATHEMATICS

By tile middle of January 1829 Abel knew that he had not
long to Ive. The eTldence of a haemorrhage is not to be denied*
T. will fight for my life!* he shouted in his delirium. But in more
tranquil moments, exhausted and trying to work, he drooped
“like a sick eagle looking at the sun’, knowing that his weeks
were numbered.

Abel spent Ms last days at Froland, in the home of an English
family where his fiancee (Crelly Kemp) was . governess. His last
thoughts were for her future, and he wrote to his friend
Kielhaii, “She is not beautiful; she has red hair and freckles, but
she is an admirable woman.’ It was Abel’s wish that Crelly and
Eaelnau should marry after Ms death; and although the two
had never met, the^^ did as Abel had half-jokingiy proposed.
Toward the last Crelly insisted on taking care of Abel without
help, ‘to possess these last moments alone’. Early in the morn-
ing of 6 April 1829 he died, aged twenty-six years, eight
months.

Twro days after Abel s death Crelle wrote to say that Ms
negotiations had at last proved successful and that Abel would
he appointed to the professorsMp of mathematics in the
University of Berlin.

CHAPTER EIGHTEEN

THE GREAT ALGORIST
Jacobi

• The name Jacobi appears frequently in the sciences, not always
  meaning the same man. In the 1840’s one very notorious Jacobi
• M. H. – had a comparatively obscure brother, C. G. J.?
  whose reputation then was but a tithe of M. H.*s. To-day the
  situation is reversed: C. G. J. is immortal – or seemingly so,
  while M. H. is rapidly receding into the obscurity of limbo.
  M. H. achieved fame as the founder of the fashionable quackery
  of galvanoplastics; C. G. J.’s much narrower but also much
  higher reputation is based on mathematics. During his lifetime
  the mathematician was always being confused with his more
  famous brother, or worse, being congratulated for his involun-
  tary kinship to the sincerely deluded quack. At last C. G. J.
  could stand it no longer. ‘Pardon me, beautiful lady% he
  retorted to an enthusiastic admirer of M. H. who had compli-
  mented him on ha^dng so distinguished a brother, ‘but I am my
  brother.’ On other occasions C. G. J. would blurt out, ‘I am not
  his brother, he is mine\ There is where fame has left the rela-
  tionship to-day.

Carl Gustav Jacob Jacobi, bom at Potsdam, Prussia, Ger-
many, on 10 December 1804 was the second son of a prosperous
banker, Simon Jacobi, and his wife (family name Lehmann).
There were in ail four children, three boys, Moritz, Carl, and
Eduard, and a girl, Therese. Carl’s first teacher was one of his
maternal uncles, who taught the boy classics and mathematics,
preparing him to enter the Potsdam Gymnasium in 1816 in his
twelfth year. From the first Jacobi gave evidence of the
‘universal mind’ which the rector of the Gymnasium declared
him to be on his leaving the school in 1821 to enter the Univer-
sity of Berlin. like Gauss, Jacobi could easily have made a

859

MEK OF MATHEMATICS

high reputation in philology had not mathematics attracted
him more strongly. Having seen that the boy had mathematical
genius, the teacher (Heinrich Bauer) let Jacobi work by him self

• after a prolonged tussle in which Jacobi rebelled at learning
  mathematics by rote and by rule.

Young Jacobi’s mathematical development was in some
respects curiously parallel to that of his greater rival Abel.
Jacobi also went to the masters; the works of Euler and
Lagrange taught him algebra and the calculus, and introduced
him to the theory of numbers. This earliest self-instruction was
to give Jacobi’s first outstanding work – in elliptic functions –
its definite direction, for Euler, the master of ingenious devices,
found in Jacobi his brilliant successor. For sheer manipulative
ability in tangled algebra Euler and Jacobi have had no rival,
unless it be the Indian mathematical genius, Srinivasa Rama-
nujan, in our own century. Abel also could handle formulae
like a master when he wished, but his genius was more philo-
sophical, less formal than Jacobi’s. Abel is closer to Gauss in
his insistence upon rigour than J acobi was by nature – not that
Jacobi’s work lacked rigour, for it did not, but its inspiration
appears to have been formalistic rather t ha n rigoristic.

Abel was two years older than Jacobi Unaware that Abel
had attacked the general quintic in 1820, Jacobi in the same
year attempted a solution, reducing the general quintic to the

fotmflj* – —pandshowingthatthesolutiQnofthisequa-

tion would follow from that of a certain equation of the tenth
degree. Although the attempt was abortive it taught Jacobi a
great deal of algebra and he ascribed considerable importance
to it as a step in his mathematical education. But it does not
seem to have occurred to him, as it did to Abel, that the general
quintio might be unsolvable algebraicafly. This oversight, or
lack of imagination, or whatever we wish to call it, on Jacobi’s
part ia typical of the difference between him and Abel. Jacobi,
who had a magnificently objective mind mid not a particle of
envy or jeakni^ in his generous nature, himself said of one of
Abel s masterpieces, ‘It is above my praises as it is above my
own works.*

Jaoobfs student days at Berlin lasted from April 1821 to

m

THE GREAT ALGORIST

May 1825. During the first two years he spent his time about
equally betw^n philosophy, philology, and mathematics. In
the philological seminar Jacobi attracted the favourable atten-
tion of P. A. Boeckh, a renowned classical scholar who brought
out (among other works) a fine edition of Pindar. Boeckh,
luckily for mathematics, failed to convert his most promising
pupil to classical studies as a life interest. In mathematics not
much was offered for an ambitious student and Jacobi con-
tinued his private study of the masters. The university lectures
in mathematics he characterized briefly and sufficiently as
twaddle. Jacobi was usually blunt and to the point, although
he knew how to be as subservient as any courtier when trying
to insinuate some deserving mathematical friend into a worthy
position.

While Jacobi was diligently making a mathematician of
himself Abel was already weU started on the very road which
was to lead Jacobi to fame. Abel had written to Holmhoe on
4 August 1823 that he was busy with elliptic functions: ‘This
little work, you will recall, deals with the inverses of the
elliptic transcendents, and I proved something [that seemed]
impossible; I begged Degen to read it as soon as he could from
one end to the other, but he could find no false conclusion, nor
understand where the mistake was;’ God knows how I shall get
myself out of it.’ By a curious coincidence Jacobi at last made
up his mind to put his all on mathematics almost exactly when
Abel wrote this. Two years’ difference in the ages of young men
around twenty (Abel was twenty-one, Jacobi nineteen) count
for more than two decades of maturity. Abel got a tremendous
start but Jacobi, unaware that he had a competitor in the race,
soon caught up. Jacobi’s first great work was in Abel’s field of
elliptic functions. Before considering this we shall outline his
busy life.

Having decided to go into mathematics for all he was worth,
Jacobi wrote to his uncle Lehmann his estimate of the labour
he had undertaken. ‘The huge colossus which the works of
Euler, Lagrange, and Laplace have raised demands the most
prodigious force and exertion of thought if one is to penetrate
into its inner nature and not merely rummage about on its

861

MEN OF MATHEMATICS

surface. To dominate this colossus and not to fear being
crushed by it demands a strain which permits neither rest nor
peace till one stands on top of it and surveys the work in its
entirety. Then only, when one has comprehended its spirit, is
it possible to work justly and in peace at the completion of its
details.’

With this declaration of willing servitude Jacobi forthwith
became one of the most terrific workers in the history of mathe-
matics. To a timid friend who complained that scientific
research is exacting and likely to impair bodily health, Jacobi
retorted:

‘Of course! Certainly I have sometimes endangered my health
by overwork, but what of it? Only cabbages have no nerves,
no worries. And what do they get out of their perfect well-
being?’

In August 1825 Jacobi received his Ph.D. degree for a disser-
tation on partial fractions and allied topics. There is no need to
explain the nature of this – it is not of any great interest and is
now a detail in the second course of algebra or the integral
calculus. Although Jacobi handled the general case of his
problem and showed considerable ingenuity in manipulating
formulae, it cannot be said that the dissertation exhibited any
marked originality or gave any definite hint of the author’s
superb talent. Concurrently with his examination for the Ph.D.
degree, Jacobi roxmded off his training for the teaching pro-
fession.

After Ms degree Jacobi lectured at the University of Berlin
on the applications of the calculus to curved surfaces and
twisted curves (roughly, curves determined by the intersections
of surfaces). From the very first lectures it was evident that
Jacobi was a bom teacher. Later, when he began developing his
own ideas at an amazing speed, he became the most inspiring
mathematical teacher of his time.

Jacobi seems to have been the first regular mathematical
instructor in a university to train students in research by
lecturing on his own latest discoveries and letting the students
see the creation of a new subject taking place before them. He
believed in pitching young mmx into the icy water to learn to
S62

THE GREAT ALGORIST

swim or drown by themselves. Many students put off attempt-
ing anything on their own account till they have mastered
everything relating to their problem that has been done
by others. The result is that but few ever acquire the knack
of independent work. Jacobi combated this dilatory erudition.
To drive home the point to a gifted but diffident young man
who was always putting off doing anything imtil he had learned
something more, Jacobi delivered himself of the following
parable. ‘Your father would never have married, and you
wouldn’t be here now, if he had insisted on knowing all the girls
in the world before marrying one.’

Jacobi’s entire life was spent in teaching and research except
for one ghastly interlude, to be related, and occasional trips to
attend scientific meetings in England and on the Continent, or
forced vacations to recuperate after too intensive work. The
chronology of his life is not very exciting – a professional
scientist’s seldom is, except to himself.

Jacobi’s talents as a teacher secured him the position of
lecturer at the University of Konigsberg in 1826 after only half
a year in a similar position at Berlin. A year later some results
which Jacobi had published in the theory of numbers (relating
to cubic reciprocity; see chapter on Gauss) excited Gauss’
admiration. As Gauss was not an easy man to stir up, the
Ministry of Education took prompt notice and promoted
Jacobi over the heads of his colleagues to an assistant profes-
sorship – quite a step for a young man of twenty-three. Natu-
rally the men he had stepped over resented the promotion; but
two years later (1829) when Jacobi published his first master-
piece, Fundamenta Nova Theoriae Functionum Ellipticanim
(New Foundations of the Theory of Elliptic Functions) they
were the first to say that no more than justice had been done
and to congratulate their brilliant young colleague.

In 1832 Jacobi’s father died. Up till this he need not have
worked for a living. His prosperity continued about eight years
longer, when the family fortune went to smash in 1840. Jacobi
was cleaned out himself at the age of thirty-six and in addition
had to provide for his mother, also ruined.

Gauss all this time had been watching Jacobi’s phenomenal

863

HEX OF HATHEHATICS

activity with more than a mere scientific interest, as many of
Jacobi’s discoveries overlapped some of those of his own youth
which he had never published. He had also (it is said) met the
young man personally: Jacobi called on Gauss (no account of
the visit has survived) in September 1839, on his return trip to
Konigsbeig after a vacation in Marienbad to recuperate from
overwork. Gauss appears to have feared that Jacobi’s financial
collapse would have a disastrous effect on his mathematics, but
Bessel reassured him: ‘Fortunately such a talent cannot be
destroyed, but I should have liked him to have the sense of
freedom which money assures.’

The loss of his fortune had no effect whatever on Jacobi’s
mathematics. He never alluded to his reverses but kept on
working as assiduously as ever. In 1842 Jacobi and Bessel
attended the meeting of the Briti^ Association at Manchester,
where the German Jacobi and the Irish Hamilton met in the
fiesh- It was to be one of Jacobi’s greatest glories to continue
the work of Hamilton in dynamics and, in a sense, to complete
what the Irishman had abandoned in favour of a will-o-the”
wisp (which will be followed when we come to it).

At this point in his career Jacobi suddenly attempted to
blossom out into something showier than a mere mathemati-
daii. Not to interrupt the story of his scientific life when we
take it up, we shall dispose here of the illustrious mathemati-
cian’s singular misadventures in politics.

The year following his return from the trip of 1842, Jacobi
had a complete breakdown from overwork. The advancement
of science in the 1840’s in Germany was in the hands of the
benevolent princes and kings of the petty states which were
later to coalesce into the German Empire. Jacobi’s good angel
was the King of Prussia, who seems to have appreciated fiiy
the honour which Jacobi’s researches conferred on the Eung-
dtm. Accordingly, when Jacobi fell ill, the benevolent
urged him to take as long a vacation as he liked in the mild
climate of Italy. After five months at Rome and Naples with
Borchardt (whom we shall meet later in the company of Weier-
stxass) and Dirichlet, Jacobi returned to Berlin in June 1844.
He was now permitted to stay on in Berlin until his health

Z&4t

THE GREAT ALGORIST

should be completely restored but, owing to jealousies, was not
given a professorship in the University, although as a member
of the Academy he was permitted to lecture on anything he
chose. Further, out of his own pocket, practically, the King
granted Jacobi a substantial allowance.

After all this generosity on the part of the King one might
think that Jacobi would have stuck to his mathematics. But on
the utterly imbecile advice of his physician he began meddling
in politics ‘to benefit his nervous system’. If ever a more idiotic
prescription was handed out by a doctor to a patient whose
complaint he could not diagnose it has yet to be exhumed.
Jacobi swallowed the dose. When the democratic upheaval of
1848 began to erupt Jacobi was ripe for office. On the advice of
a friend – who, by the way, happened to be one of the men over
whose head Jacobi had been promoted some twenty years
before – the guileless mathematician stepped into the arena of
politics with all the innocence of an enticingly plump mis-
sionary setting foot on a cannibal island. They got him.

The mildly liberal club to w’hich his slick friend had intro-
duced him ran Jacobi as their candidate for the May election
of 1848. But he never saw the inside of parliament. His elo-
quence before the club comnnced the wiser members that
Jacobi was no candidate for them. Quite properly, it would
seem, they pointed out that Jacobi, the King’s pensioner,
might possibly be the liberal he now professed to be, but that
it was more probable he was a trimmer, a turncoat, and a stool
pigeon for the royalists. Jacobi refuted tlaese base insinuations
in a magnificent speech packed with irrefutable logic – oblivious
of the axiom that logic is the last thing on earth for which a
practical politician has any use. They let him hang himself in
his own noose. He was not elected. Nor was his nervous system
benefited by the uproar over his candidacy ’which rocked the
beer halls of Berlin to their cellars.

Worse was to come. Who can blame the Minister of Educa-
tion for enquiring the following IMay whether Jacobi’s health
had recovered sufficiently for him to return safely to Kdnigs-
berg? Or who can wonder that his allowance from the King was
stopped a few days later? After all even a King may be per-

365

MEN OP MATHEMATICS

mitted a show of petulance when the mouth he tries to feed bites
him. Nevertheless Jacobi’s desperate plight was enough to
escite anybody’s sympathy. Married and practically penniless
he had seven small children to support in addition to his wife.
A Mend in Gotha took in the wife and children, while Jacobi
retired to a dingy hotel room to continue his researches.

He was now (1849) in his forty-fifth year and, except for
Gauss, the most famous mathematician in Europe. Hearing of
his plight, the University of Vienna began angling for him. As
an item of interest here, Littrow, Abel’s Viennese Mend, took
a leading part in the negotiations. At last, when a definite and
generous offer was tendered, Alexander von Humboldt talked
the sulky King round; the allowanee was restored, and Jacobi
was not permitted to rob Germany of her second greatest man.
He remained in Berlin, onw more in favour but definitely out
of politi<^.

The subject, elliptic functions, in which Jacobi did his first
great work, has already been given what may seem like its
fhare of space; for after all it is to-day more or less of a detail in
the vaster theory of functions of a complex variable which, in
its turn, is fading from the ever changing scene as a thing of
living interest. As the theory of elliptic functions will be men-
tioned several times in succeeding chapters we shall attempt a
brief justification of its apparently unmerited prominence.

No mathematician would dispute the claim of the theory of
functions of a complex variable to have been one of the major
fields of nineteenth-century mathematics. One of the reasons
why this theory was of such importance may he repeated here.
Gauss had shown that coinpleas numbers are both necessaiy’ and
sufficient to provide every algebraic equation with a root. Are
any further, more general, kinds of ‘numbers’ possible? How
mi^t such ‘numbers’ arise?

Instead of regarding omvplex numbers as having first pre-
sented themselves in the attempt to solve certain simple
equations, say a;* + 1 s=r o, we may also see their origin in
another problem of elementary algebra, that of factonzation.
To r^lve ^ y* into factors of the first degree we need
nothing more mysterious than the positive and negative

866

THE GREAT ALGORIST

integers: (a3® — y^) = (ss y) (x — y) , But the same problem for
^2 ! y 2 demands ‘imaginaries’: y^ = {ce + yV — 1)

{x — yV — 1), Carrying this up a step in one of many possible
ways open, we might seek to resolve + 2 – into two

factors of the first degree. Are the positives, negatives, and
imaginaxies sufficient? Or must some new kind of ‘number’ be
invented to solve the problem? The latter is the case. It was
found that for the new ‘numbers’ necessary the rules of common
algebra break down in one important particular: it is no longer
true that the order in which ‘numbers’ are rmdtvplied together is
indifierent; that is, for the new numbers it is not true that a X &
is equal to h x fl. More will be said on this when w’e come to
Hamilton. For the moment we note that the elementary alge-
braic problem of factorization quickly leads us into regions
where complex numbers are inadequate.

How far can we go, what are the most general numbers
possible, if we insist that for these numbers all the familiar laws
of common algebra are to hold? It was proved in the latter part
of the nineteenth century that the complex numbers x + iy,
where x,y are real numbers and z = V — 1, are the most
general for which common algebra is true. The real numbers,
we recall, correspond to the distances measured along a fixed
straight line in either direction (positive, negative) from a fixed
point, and the graph of a fimction /(a;), plotted as 2/ = /(a), in
Cartesian geometry, gives us a picture of a function of a real
variable x. The mathematicians of the seventeenth and eigh-
teenth centuries imagined their functions as being of this kind.
But if the common algebra and its extensions into tlie calculus
which they applied to their functions are equally applicable to
complex numbers, which include the real numbers as a very
degenerate case, it was but natural that many of the things the
early analysts found were less than half the whole story
possible. In particular the integral calculus presented many
inexplicable anomalies which were cleared up only when the
field of operations was enlarged to its fullest possible extent and
functions of complex variables were introduced by Gauss and
Cauchy.

The importance of elliptic functions in all this vast and

867

HEX OF MATHEMATICS

fundamental development cannot be over-estimated. Gauss,
Abel, and Jacobi, by their extensive and detailed elaboration
of the theory of elliptic functions, in which complex numbers
appear inevitably, provided a testing ground for the discovery
and improvement of general theorems in the theory of functions
of a complex variable. The two theories seemed to have been
designed by fate to complement and supplement one another –
there is a reason for this, also for the deep eoimexion of elliptic
functions with the Gaussian theory of quadratic forms, which
considerations of space force us to forego. Without the innu-
merable clues for a general theory provided by the special
instances of more inclusive theorems occurring in elliptic func-
tions, the theory of functions of a complex variable would have
developed much more dowly than it did – liouville’s theorem,
the entire subject of multiple periodicity with its impact on
the theory of algebraic functions and their integrals, may be
recalled to mathematical readers. If some of these great monu-
ments of nineteenth-century mathematics are already receding
into the mists of yesterday, we need only remind ourselves that
Picard’s theorem on exceptional values in the neighbourhood
of an essential singularity, one of the most suggestive in current
analysis, was first proved by devices originating in the theory
of elliptic functions. With this partial summary of the reason
why elliptic functions were important in the mathematics of
the nineteenth century we may pass on to Jacobi’s nar dma.l
part in the development of the theory.

The history of elliptic functions is quite involved, and
although of considerable interest to specialists, is not likely to
appeal to the general reader. Accordingly we shall omit the
evidence (letters of G^uss, Abd, Jacobi, Legendre, and othem)
on which the following hare summaiy is based.

First, it is established that Gauss anticipated both Abel and
Jacobi by as much as twenty-seven years in some of their most
striking work. Indeed Gauss says that ‘Abd has followed
exactly the same road that I did in 1798’. That this AUim is just
wiD be admitted by anyone who will study the evidence pub-
lished only after Gauss’ death. Second, it seems to he agreed
that Abel antadpated Jacobi in certain important details, but

868

THE GREAT ALGORIST

that Jacobi made his great start in entire ignorance of his
rival’s work.

A capital property of the elliptic functions is their dovble
periodicity (discovered in 1825 by Abel): if E{x) is an elliptic
function, then there are two distinct numbers, say pgj such
that

E(x -{- Pi) = E(x), and E(x + P 2 ) = B{x)

for all values of the variable a?.

Finally, on the historical side, is the somewhat tragic part
played by Legendre. For forty years he had slaved over elliptic
integrals {not elliptic functions) without noticing what both
Abel and Jacobi saw almost at once, namely that by inverting
his point of view the whole subject would become infinitely
simpler. Elliptic integrals first present themselves in the pro-
blem of finding the length of an arc of an ellipse. To what was
said about inversion in connexion with Abel the following
statement in symbols may be added. This will bring out more
clearly the point which Legendre missed.

If R(t) denotes a polynomial in t, an integral of the type

is called an elliptic integral if R(t) is of either the third or the
fourth degree; if R(t) is of degree higher than the fourth, the
integral is called Abelian (after Abel, some of whose greatest
work concerned such Integrals). If R{t) is of only the second
degree, the integral can be calculated out in terms of elementary
functions. In particular

— ” — dt == sin“^aj,

0 Vi –

(sin“% is read, ‘an angle whose sine is a’). That is, if

r-7^^

J 0 Vl-i*

we consider the upper limiU ts, of the integral, as a function of
the integral itself, namely of y. This irwersion of the problem

369

MEN OF MATHEMATICS

removed most of the dif&eulties ‘which Legendre had grappled
with for forty years. The true theory of these important inte-
grals rushed forth almost of itself after this obstruction had
been removed – like a log-jam going down the river after the
king log has been snaked out.

When Legendre grasped what Abel and Jacobi had done he
encouraged them most cordially, although he realized that their
simpler approach (that of inversion) nullified what was to have
been his own masterpiece of forty years’ labour. For Abel, alas,
Legendre’s praise came too late, but for Jacobi it was an
inspiration to surpass himself. In one of the finest correspon-
dences in the whole of scientific literature the young man in his
early twenties and the veteran in his late seventies strive to
outdo one another in sincere praise and gratitude. The only
jarring note is Legendre’s outspoken disparagement of Gauss,
whom Jacobi xdgorously defends. But as Gauss never con-
descended to publish his researches – he had planned a major
work on elliptic functions when Abel and Jacobi anticipated
him in publication – Legendre can hardly be blamed for holding
a totally mistaken opinion. For lack of space we must omit
extracts from this beautiful correspondence (the letters are
given in full in vol. 1 of Jacobi’s Werke – in French).

The joint creation with Abel of the theory of elliptic functions
was only a small if highly important part of Jacobi’s huge out-
put. Only to enumerate all the fields he enriched in his brief
working life of less than a quarter of a century would take more
space than can be devoted to one man in an account like the
present, so we shall merely mention a few of the other oreat
things he did. ^

Jacobi was the first to apply elliptie functions to the theory
of numbers. This was to become a favourite diversion with some
of the greatest mathematiciaiiLS who followed Jacobi. It is a
curiously recondite subject, where arabesques of ingenious
algebra unexpectedly reveal hitherto unsuspected relations
between the common whole numbers. It was by this means that
Ja(5obi proved the famous assertion of Fermat that every
integer 1,2,3, … is a sum of four integer squares (zero being
counted as an integer) and, moreover, his beautiful analysis

370

THE GREAT ALGORIST

told WiTTi in how many ways any given integer may be expressed
as such a sum.*

For those whose tastes are more practical we may cite
Jacobi’s work in dynamics. In this subject, of fundamental
importance in both applied science and mathematical physics,
Jacobi made the first significant advance beyond Lagrange and
Hamilton. Readers acquainted with quantum mechanics will
recall the important part played in some presentations of that
revolutionary theory by the Hamilton- Jacobi equation. His
work in differential equations began a new era.

In algebra, to mention only one thing of many, Jacobi cast
the theory of determinants into the simple form now familiar
to every student in a second course of school algebra.

To the Newton-Laplace-Lagrange theory of attraction
Jacobi made substantial contributions by his beautiful investi-
gations on the functions which recur repeatedly in that theory
and by applications of elliptic and Abelian functions to the
attraction of ellipsoids.

Of a far higher order of originality is his great discovery in
Abelian functions. Such functions arise in the inversion of an
Abelian integral, in the same way that the elliptic functions
arise from the inversion of an elliptic integral. (The technical
terms were noted earlier in this chapter.) Here he had nothing
to guide him, and for long he wandered lost in a maze that had
no clue. The appropriate inverse functions in the simplest case
are functions of two variables ha\ing Jour periods; in the
general case the functions have n variables and 2n periods; the
elliptic functions correspond to n = 1. This discovery was to
nineteenth-century analysis what Columbus’ discovery of
America was to fifteenth-century geography.

Jacobi did not suffer an early death from overwork, as his
lazier friends predicted that he would, but from smallpox
(18 February 1851) in his forty-seventh year. In taking leave
of this large-minded man we may quote his retort to the great
French mathematical physicist Fourier, who had reproached

• If is odd, the number of ways is 8 times the sum of all the
  divisors of n (1 and n included); if 72 is even, the number of ways is 24^
  times the sum of all the odd divisors of n.

371

MEK OF MATHEMATICS

both Abel and Jacobi for ‘wasting’ their time on elliptic func-
tions while there were still problems in heat-conduction to be ■
solved.

‘It is true’ , Jacobi says, ‘that M. Fourier had the opimon that :
the principal aim of mathematics was public utility and the ‘
explanation of natural phenomena; but a philosopher like him
should have known that the sole end of science is the honour of ‘
the hiunan mind, and that under this title a question about’ ‘
numbers is worth as much as a question about the system, of the ‘;
world.’

If Fourier could re’^dsit the glimpses of the moon he might be
disgusted at what has happened to the analysis he invented for
‘pubic utilty and the explanation of natural phenomena’. So
far as mathematical physics is concerned Fourier analysis to-
day is hut a detaii in the infinitely vaster theory of boundary-
value problems, and it is in the purest of pure mathematics that
the analysis which Fourier invented finds its interest and its
justification. Whether ‘the human mind’ is honoured by these
modem researches may be put up to the experts – provided the
behaviourists have left anything of the human mind to be
honoured.

CHAPTEH XIXETEE!?^-

AN IRISH TRAGEDY
Hamilton

• William eowan Hamilton is by long odds the greatest
  man of science that Ireland has produced. His nationality is
  emphasized because one of the drhing impulses behind Hamil-
  ton’s incessant activity was his avowed desire to put his superb
  genius to such uses as would bring glory to his native land.
  Some have claimed that he was of Scotch descent. Hamilton
  himseK insisted that he was Irish, and it is certainly difficult for
  a Scot to see anything Scotch in Ireland’s greatest and most
  eloquent mathematician.

Hamilton’s father was a solicitor in Dublin, Ireland, where
William, the youngest of three brothers and one sister, was bom
on 3 August 1805.* The father was a first-rate business man
with an ‘exuberant eloquence’, a religious zealot, and last, but
unfortunately not least, a very convivial man, all of which
traits he passed on to his gifted son. Hamilton’s ertraordinary
intellectual brilliance was probably inherited from his mother,
Sarah Hutton, who came of a family well known for its brains.

However, on the father’s side, the swirling clouds of elo-
quence, ‘both of lips and pen’, which made the jolly toper the
life of every party he graced with his reeling presence, con-
densed into something less gaseous in William’s uncle, the
Reverend James Hamilton, curate of the village of Trim (about
twenty miles from Dublin). Uncle James was in fact an inhu-
manly accomplished linguist – Greek, Latin, Hebrew, Sanskrit,
Chaldee, Pali, and heaven knows what other heathen dialects,

• The date on his tombstone is 4 August, 1805. Actually he was
  bom at midnight; hence the confusion in dates. Hamilton, who had
  a passion for accuracy m such trifles, chose 3 August until in later
  life he shifted to 4 August for sentimental reasons.

373

MEX or ilATHEilATICS

came to the tip of Ms tongue as readily as the more civilized
languages of Continental Europe and Ireland. This polyglot
fluency played no inconsiderable part in the early and ex-
tremely extensive zniseducation of the hapless but eager
William, for at the age of three, having already given signs of
genius, he was relieved of his doting mother’s affection and
packed off by his somewhat stupid father to glut himself with
languages under the expert tutelage of the supervoiuble Uncle
James.

Hamilton’s parents had very little to do with his upbringing;
his mother died when he was twelve, his father two years later.
To James Hamilton belongs whatever credit there may be for
having wasted young William’s abilities in the acquisition of
utterly useless languages and turning him out, at the age of
thirteen, as one of the most shocking examples of a linguistic
monstrosiW in history. That Hamilton did not become an
insufferable prig under his misguided parson-uncle’s instruction
testifies to the essential soundness of his Irish common sense.
The education he suffered might well have made a per-
manent ass of even a humorous boy, and Hamilton had no
humour.

The tale of Hamilton’s infantile accomplishments reads like
a bad romance, but it is true; at three he was a superior reader
of English and was considerably advanced in arithmetic; at
four he was a good geographer; at five he read and translated
Latin, Greek, and Hebrew, and loved to recite yards of Dry den,
Collins, ^iilton, and Homer – the last in Greek; at eight he
added a mastery of Italian and French to his coUection and
extemporized fluently in Latin, expressing his unaffected
delight at the beauty of the Irish scene in Latin hexameters
when plain English prose offered too plebeian a vent for Ms
nobly exalted sentiments; and finally, before he was ten he had
laid a firm foundation for Ms extraordinary scholarsMp in
oriental languages by beginning Arabic and Sanskrit.

The tally of Hamilton’s languages is not ye»t complete. When
WBliam was three months under ten years old his uncle reports
that ‘His thirst for the Oriental languages is unabated. TTp> is
now master of most, indeed of all except the minor and com-

IRISH TRAGEDY

paratively provincial ones. The Hebrew, Persian, and Arabic
are about to be confirmed by the superior and intimate acquain-
tance with the Sanskrit, in which he is already a proficient. The
Chaldee and Syriac he is grounded in, also the Hindoostani,
:Malay. Mahratta, Bengali, and others. He is about to commence
the Chinese, but the difficulty of procuring books is very great
It cost me a large sum to supply him from London, but I hope
the money w’as well expended.’ To which we can only throw up
our hands and ejaculate Good God! ^Vliat was the sense of it all?

By thirteen WiUiam was able to brag that he had mastered
one language for each year he had lived. At fourteen he com-
posed a flowery welcome in Persian to the Persian Ambassador,
then visiting Dublin, and had it transmitted to the astonished
potentate. “Wishing to follow up his advantage and slay the
already slain, young Hamilton called on the Ambassador, but
that wily oriental, forewarned by his faithful secretary, ‘much
regretted that on account of a bad headache he was unable to
receive me [Hamilton] personally.’ Perhaps the Ambassador
had not yet recovered from the official banquet, or he may have
read the letter. In translation at least it is pretty awful – just
the sort of thing a boy of fourteen, taking himself with devas-
tating seriousness and acquainted with aU the stickiest and
most bombastic passages of the Persian poets, might imagine a
sophisticated oriental out on a wild Irish spree would relish as
a pick-me-up the morning after. Had young Hamilton really
wished to vdew the Ambassador he should have sent in a salt
herring, not a Persian poem.

Except for his amazing ability, the maturity of his conversa-
tion and his poetical love of nature in all her moods, Hamilton
was like any other healthy boy. He delighted in swimming and
had none of the grind’s interesting if somewhat repulsive pallor.
His disposition was genial and his temper – rather unusually so
for a sturdy Irish boy – invariably even. In later life, however,
Hamilton showed Ins Irish by challenging a detractor – who had
called him a liar – to mortal combat. But the affair was amic-
ably arranged by Hamilton’s second, and Sir WiUiam cannot
be legitimately counted as one of the great mathematical
dueUists. In other respects young Hamilton was not a normal

375

M.M. — VOL. II.

MEX OF MATHEMATICS

boy. The infliction of pain or suffering on beast or man he would
not tolerate. All his life Hamilton loved animals and, what is
regrettably rarer, respected them as equals.

Hamilton’s redemption from senseless devotion to useless
languages began when he was twelve and was completed before
he was fourteen. The humble instrument selected by Providence
to turn Hamilton from the path of error was the American
calculating boy. Zerah Colburn (1804-39), who at the time had
been attending Westminster School in London. Colburn and
Hamilton were brought together in the expectation that the
young Irish genius would be able to penetrate the secret of the
American’s methods, which Colburn himself did not fully
understand fas was seen in the chapter on Fermat). Colburn
was entirely frank in exposing his tricks to Hamilton, w^ho in
his turn improved upon what he had been shown. There was
but little abstruse or remarkable about Colburn’s methods. His
feats were largely a matter of memory. Hamilton’s . acknow-
ledgement of Colburn’s influence occurs in a letter written when
he was seventeen (August 1822) to his cousin Arthur.

By the age of seventeen Hamilton had mastered mathe-
matics through the integral calculus and had acquired enough
mathematical astronomy to be able to calculate eclipses. He
read Newton and Lagrange. All this was his recreation; the
classics were still his serious study, although only a second love.
“SMiat is more important, he had already made ‘’some curious
di scorer i€S% as he wrote to his sister Eliza.

The discoveries to which BLamilton refers are probably the
germs of his first great work, that on systems of rays in optics.
Thus in his seventeenth year Hamilton had already begun his
career of fundamental discove]y\ Before this he had brought
himself to the attention of Dr Brinkley, Professor of Astronomy
at Dublin, by the detection of an error in Laplace’s attempted
proof of the parallelogram of forces.

Hamilton never attended any school before going to the Unh
versity but received all his preliminary training from his uncle
and by private study. His forced devotion to the classics in
preparation for the entrance examinations to Trinity College,
Dublin, did not absorb all of his time, for on 31 May 1823, he

376

AN IRISH TRAGEDY

writes to his cousin Arthur, ‘In Optics I have made a very
curious discovery – at least it seems so to me. . . . ‘

If, as has been supposed- this refers to the •characteristic
function’- which Hamilton will presently describe for us, the
discover}’ marks its author as the equal of any mathematician
in history for genuine precocity. On 7 July 1823 young Hamil-
ton passed, easily first out of 100 candidates, into Trinity
College. His fame had preceded liim. and as was only to be
expected, he quickly became a celebrity: indeed his classical
and mathematical prowess, while he was yet an undergraduate,
excited the curiosity of academic circles in England and Scot-
land as well as in Ireland, and it was even declared by some that
a second Nevlon had arrived. The tale of his undergraduate
triimiphs can be imagined – he carried off practically all the
available prizes and obtained the highest honours in both
classics and mathematics. But more important than all these
triumphs- he completed the first draft of Part I of his epoch-
making memoir on sj’stems of rays. ‘This young man’, Dr
Brinkley remarked, when Hamilton presented his memoir to
the Royal Irish Academy, ‘I do not say zcill be, but fs, the first
mathematician of his age.’

Even his laborious drudgeries to sustain his brilliant acade-
mic record and the hours spent more profitably on research did
not absorb all of young Hamilton’s superabundant energies. At
nineteen he experienced the first of his three serious love affairs.
Being conscious of his own ‘•unworthiness’ – especially as con-
cerned his material prospects – William contented himself with
T;\Titing poems to the young lady, with the usual result: a
solider, more prosaic man married the girl. Early in May 1825
Hamilton learned from his sweetheart’s mother that his love
had married his rival. Some idea of the shock he experienced
caii be inferred from the fact that Hamilton, a deeply religious
man to whom suicide was a deadly sin, was tempted to drown
himself. Fortunately for science he solaced himself with another
poem. All his life Hamilton was a prolific versifier. But his true
poetry, as he told his friend and ardent admirer, William
Wordsworth, was his mathematics. From this no mathemati-
cian will dissent.

377

MEN OF MATHEMATICS

We may dispose here of Hamilton’s lifelong friendships with
some of the shining literary lights of his day – the poets Words-
worth, Southey, and Coleridge, of the so-called Lake School,
Aubrey de Tere, and the didactic novelist Maria Edgeworth, a
lixteratrice after Hamilton’s own pious heart. Wordsworth and
Hamilton first met on the latter’s trip of September 1827 to the
English Lake District. Hax-ing ‘waited on ordsworth at tea\
Hamilton oscDlated back and forth %vith the poet all night, each
desperately tn*ing to see the other home. The following day
Hamilton sent Wordsworth a poem of ninety iron lines which
the poet himself might have warbled in one of his heavier
flights. Naturally Wordsworth did not relish the eager young
mathematician’s unconscious plagiarism, and after damning it
‘with faint praise, proceeded to tell the hopeful author — at great
length “ that ‘the workmanship (what else could be expected
from so young a writer?) is not what it ought to be.’ Two years
later, when Hamilton was already installed as astronomer at
the Dunsink Observ^atory, Wordsworth returned the xusit.
Hamilton’s sister Eliza, on being introduced to the poet, felt
herself ‘involuntarily parodying the first lines of his own poem
Yarrozc Visited:

And this is Wordsworth ! this the man
Of’xJu^m mijfiiHCij cherished
So faithfiilltj a leaking dream.

An image that hath perished !

One great benefit accrued from Wordsworth’s xisit: Hamilton
realized at last that ‘his path must be the path of Science, and
not that of Poetiy^; that he must renounce the hope of habi-
tually cultivating both, and that, therefore, he must brace him-
self up to bid a painful farewell to Poetry’. In short, Hamilton
grasped the ob^ious truth that there was not a spark of poetry
in him, in the literary sense. Nevertheless he continued to ver-
sify all his life. Wordsworth’s opinion of Hamilton’s intellect
was hlglt. In fact he graciously said (in efiect) that only two
men he had ever known gave him a feeling of inferiority,
Coleridge and Hamilton.

Hamilton did not meet Coleridge till 183’2, when the poet had
378

AN IRISH TRAGEDY

piaetically ceased to be anything but a spurious copy of a
mediocre German metaphysician. Nevertheless each formed a
high estimate of the other’s capacity, as Hamilton had for long
been a devoted student of Kant in the original. Indeed philo-
sophical speculation always fascinated Hamilton, and at one
time he declared himself a wholehearted believer – intellec-
tually, but not intestinally – in Berkeley’s de\dtalized idealism.
Another bond between the t’wo was their preoccupation with
the theological side of philosophy (if there is such a side), and
Coleridge favoured Hamilton ‘wdth his half- digested rumina-
tions on the Holy Trinity, by which the devout mathematician
set considerable store.

The close of Hamilton’s undergraduate career at Trinity
College was even more spectacular than its beginning; in fact it
was unique in university annals. Dr Brinkley resigned his pro-
fessorship of astronomy to become Bishop of Cloyne. According
to the usual British custom the vacancy was advertised, and
several distinguished astronomers, including George Biddell
Airy (1801-92), later Astronomer Royal of England, sent in
their credentials. After, some discussion the Governing Board
passed over all the applicants and unanimously elected Hamil-
ton, then (1827) an undergraduate of twenty-two, to the
professorship. Hamilton had not applied. ’Straight was the path
of gold’ for him now, and Hamilton resolved not to disappoint
the hopes of his enthusiastic electors. Since the age of fourteen
he had had a passion for astronomy, and once as a boy he had
pointed out the Obser^’atory on its hill at Dunsink, command-
ing a beautiful ^dew, as the place of all others where he would
like to live were he free to choose. He now, at the age of
twenty-two, had his ambition by the bit; all he had to do was
to ride straight ahead.

He started brilliantly. Although Hamilton was no prac-
tical astronomer, and although his assistant observ’er was
incompetent, these drawbacks were not serious. From its
situation the Dunsink Obsers’atory could never have cut any
important figure in modern astronomy, and Hamilton did
wisely in putting his major efforts on his mathematics. At the
age of twenty-three he published the completion of the “curious

379

MEK OF MATHEMATICS

discoveries’ he had made as a boy of seventeen, Part I of ^
Theory of Systems of Rays, the great classic which does for
optics what Lagrange’s Mecanique analytique does for mechan-
ics and which, in Hamilton’s own hands, was to be extended to
d^maniie*, putting that fundamental science in what is perhaps
its ultimate, pertect form.

The techniques which Hamilton introduced into applied
mathematics in tliLs, his first masterpiece, are to-day indispen-
«^abie in mathematical physics, and it is the aim of many
workers in particular branches of theoretical physics to sum up
tne whole of a theory in a Hamiltonian principle. This magnifi-
cent work is that ‘which caused Jacobi, fourteen years later at
the British Association meeting at Manchester in 1842, to assert
that ‘Hamilton is the Lagrange of your country’-’ – (meaning of
the English-speaking race). As Hamilton himself took great
pains to describe the essence of his new methods in terms com-
prehensible to non-speeialists, we shall quote from his own
abstract presented to the Royal Irish Academy on 23 April

1827. *A Ray, in Optics, is to be considered here as a straight or
      bent or eur-ed line, along w^hich light is propagated; and a
      System of Rays as a collection or aggregate of such lines, con-
      nected by some common bond, some similarity of origin or
      production, in short some optical unity. Thus the rays which
      diverge from a luminous point compose one optical system, and,
      after they have been reflected at a mirror, they compose
      another. To investigate the geometrical relations of the rays of
      a system of which we know (as in these simple cases) the optical
      origin and historj”, to inquire how they are disposed among
      themselves, how they diverge or converge, or are parallel, w’hat
      surfaces or curs’es they touch or cut, and at what angles of
      section, how they can be combined in partial pencils, and how
      each ray in particular can be determined and distinguished
      from every other, is to study that System of Rays. And to
      generalize this study of one srjr’stem so as to become able to pass,
      without change of plan, to the study of other systems, to assign
      general rules and a general method whereby these separate
      optical arrangements may be connected and harmonized to-

380

AX IKISH TRAGEDY

gether, is to form a Theory of Systems of Rays, Finally, to do
this in such a manner as to make available the powers of the
modern mathesis, replacing figures by functions and diagrams
by formulae, is to construct an Algebraic Theory” of such
Systems, or an Application of Algebra to Optics,

“Towards constructing such an application it is natural, or
rather necessary, to employ the method introduced by Des-
cartes for the application of Algebra to Geometry. That great
and philosophical mathematician conceived the possibility, and
employed the plan, of representing or expressing algebraically
the position of any point in space by three co-ordinate numbers
which answer respectively how far the point is in three rectan-
gular directions (such as north, east, and west), from some fixed
point or origin selected or assumed for the purpose; the three
dimensions of space thus receiving their three algebraical
equivalents, their appropriate conceptions and sjTtibols in the
general science of progression [order]. A plane or curved surface
became thus algebraically defined by assigning as iU equation
the relation connecting the three co-ordinates of any point upon
it, and common to all those points: and a line, straight or
curv’ed, was expressed according to the same method, by the
assigning two such relations, correspondent to two surfaces of
w’hich the line might be regarded as the intersection. In this
manner it became possible to conduct general investigations
respecting surfaces and cur^’es, and to discover properties
common to all, through the medium of general investigations
respecting equations between three variable numbers: every
geometrical problem could be at least algebraically expressed,
if not at once resolved, and every improvement or discovery in
Algebra became susceptible of application or interpretation in
Geometrj^. The sciences of Space and Time (to adopt here a view
of Algebra which I have elsewhere ventured to propose) became
intimately intertwined and indissolubly connected with each
other. Henceforth it was almost impossible to improve either
science without impro\ing the other also. The problem of
drawing tangents to cur\”es led to the discovery of Fluxions or
Differentials: those of rectification and quadrature to the inver-
sion of Fluents or Integrals: the investigation of curvatures of

381

3IEN OF MATHEMATICS

surfaces required the Calculus of Partial Differentials: the
isoperimetrical problems resulted in the formation of the
Calculus of Variations. And reciprocally, all these great steps
in Algebraic Science had immediately their applications to
Geometry’, and led to the discovery of new relations between
points or lines or surfaces. But even if the applications of the
method had not been so manifold and important, there would
still have been derivable a high intellectual pleasure from the
contemplation of it as a method.

‘The first important application of this algebraical method of
co-ordinates to the study of optical systems was made by
Plains, a French officer of engineers in Napoleon’s army in
Egypt, and who has acquired celebrity in the history of Phy-
sical Optics as the discoverer of polarization of light by reflec-
tion. Malus presented to the Institute of France, in 1807, a
profound mathematical work which is of the kind above alluded
to, and is entitled Traite cTOptique. The method employed in
that treatise may be thus described:- The direction of a
straight ray of any final optical system being considered as
dependent on the position of some assigned point on the ray,
according to some law which characterizes the particular
sj’Stem and distinguishes it from others; this law may be
algebraically expressed by assigning three expressions for the
three co-ordinates of some other point of the ray, as functions
of the three co-ordinates of the point proposed. Malus accord-
introduces general symbols denoting three such functions
for at least three functions equivalent to these), and proceeds
to draw several important general conclusions, by very compli-
cated yet s\Tmnetric calculations; many of which conclusions,
along with many others, were also obtained afterwards by
myself, when, by a method nearly similar, without knowing
what Malus had done, I began my ovv’n attempt to apply
Algebra to Optics. But my researches soon conducted me to
substitute, fc?r this method of yiaJus, a very different, and (as I
conceive that I have proved) a much more appropriate one, for
the study of optical systems; by which, instead of employing
the three functions above mentioned, or at least their tzoo ratios,
it becomes sufficient to employ one function^ which I call

382

AN IRISH TRAGEDY

characteristic or principal. And thus, whereas he made his de-
ductions by setting out with the tisco equations of a ray, I on the
other hand establish and employ the one equation of a system.

•The function which I have introduced for this purpose, and
made the basis of my method of deduction in mathematical
Optics, had, in another connexion, presented itself to former
writers as expressing the result of a very high and extensive
induction in that science. This known result is usually called the
laxD of least action ^ but sometimes also the principle of least time
[see chapter on Fermat], and includes all that has hitherto been
discovered respecting the rules which determine the forms and
positions of the lines along which light is propagated, and the
changes of direction of those lines produced by reflection or
refraction, ordinary or extraordinary [the latter as in a doubly
refracting crystal, say Iceland spar, in which a single ray is
split into two, both refracted, on entering the crystal]. A
certain quantity which in one physical theory is the action, and
in another the twie, expended by light in going from any first
to any second point, is found to be less than if the light had
gone in any other than its actual path, or at least to have what
is technically called its variation null, the extremities of the
path being unvaried. The mathematical novelty of my method
consists in considering this quantity as a function of the co-
ordinates of these extremities, which varies when they vary,
according to a law which I have called the law of vary mg action;
and in reducing all researches respecting optical systems of rays to
the study of this single function: a reduction which presents
mathematical Optics under an entirely novel view, and one
analogous (as it appears to me) to the aspect under which
Descartes presented the application of Algebra to Geometry.’

Nothing need be added to this account of Hamilton’s, except
possibly the remark that no science, no matter how ably ex-
poimded, is understood as readily as any novel, no matter how
badly written. The whole extract will repay a second reading.

In this great work on systems of rays Hamilton had huilded
better than even he knew. Almost exactly 100 years after the
above abstract was written the methods which Hamilton intro-
duced into optics were found to be just what was required in

383

MEN’ OF ^lATHEMATICS

the wave mechanics associated with the modern quantum
theory and the theory of atomic structure. It may be recalled
that Newton had favoured an emission, or corpuscular, theory
of light, wliile Huygens and his successors up to almost our own
time sought to explain the phenomena of light wholly by means
of a wave theory’. Both points of view^ were united and, in a
purely mathematical sense, reconciled in the modern quantum
theory, w’hich came into being in 1925-6. In 1831?, w’hen he was
twenty-eight, Hamilton realized his ambition of extending the
principles which he had introduced into optics to the whole of
dynamics.

Hamilton’s theory of rays, shortly after its publication when
its author was but twenty-seven, had one of the promptest and
most spectacular successes of any of the classics of mathe-
matics. The theory purported to deal with phenomena of the
actual physical universe as it is obseiv’ed in eveiy’day life and
in scientific laboratories. Unless any such mathematical theory
is capable of predictions which experiments later verify, it is no
better than a concise dictionary of the subject it systematizes,
and it is almost certain to be superseded shortly by a more
imaginative picture which does not reveal its whole meaning at
the first glance. Of the famous predictions which have certified
the value of truly mathematical theories in physical science, we
may recall three: the mathematical discovery by John Couch
Adams (1819-92) and Urhain-Jean- Joseph Levenier (1811-77)
of the planet Neptune, independently and almost simulta-
neously in 1845, from an analysis of the perturbations of the
planet Uranus according to the Newtonian theory of gravita-
tion; the mathematical prediction of wireless waves by James
Clerk Maxwell (1831-79) in 1864, as a consequence of his own
electromagnetic theorj’ of light; and finally, Einstein’s predic-
tion in 1915-10, from his theorj’ of general relativity, of the
deflection of a ray of light in a gravitational field, first con-
firmed by obser\’ations of the solar eclipse on the historic
29 May 1919, and his prediction, also from his theory, that the
spectral lines in light issuing from a massive body would be
shifted by an amount, w^hich Einstein stated, toward the red
end of the spectnim – also confirmed. The last two of these
884

AX IRISH TRAGEDY

instances – Maxwell’s and Einstein’s – are of a different order
from the first: in both, totally unlnioxjcn ay}d unforeseen pheno-
mena were predicted mathematically; that 1% these predictions
were qualitative. Both [Maxwell and Einstein amplified their
qualitative foresight by precise qmntitniive predictions which
precluded any charge of mere guessing when their prophecies
were finally verified experimentally.

Hamilton’s prediction of what is called conical refraction in
optics was of this same qualitative plus quantitative order.
From his theorv’ of systems of rays he predicted mathematically
that a wholly imexpected phenomenon would be found in con-
nexion with the refraction of light in biaxial crystals. ‘VMiile
polishing the Third Supplement to his memoir on rays he sur-
prised himself by a discovery which he thus describes :

“The law of the reflection of light at ordinary mirrors appears
to have been known to Euclid; that of ordinaiy- refraction at a
surface of water, glass, or other uncrystallized medium, was
discovered at a much later date by Snellius; Huygens disco-
vered, and Malus confirmed, the law of extraordinary refraction
produced by uniaxal ciy^stals, such as Iceland spar; and finally,
the law of the extraordinary double refraction at the faces of
biaxal ciy^stals, such as topaz or arragonite, was found in our
own time by Fresnel. But even in these cases of extraordinary
or crystalline refraction, no more than tveo refracted rays had
ever been observed or even suspected to exist, if we except a
theory of Cauchy, that there might possibly be a third ray,
though probabl^’ imperceptible to our senses. Professor Hamil-
ton, however, in investigating by his general method the conse-
quences of the law of Fresnel, was led to conclude that there
ought to be in certain cases, which he assigned, not merely two,
nor three, nor any finite number, but an infinite number, or a
cone of refracted rays within a biaxal crystal, corresponding to
and resulting from a single incident ray; and that in certain
other cases, a single ray witliin such a crystal should give rise
to an infinite number of emergent rays, arranged in a certain
other cone. He was led, therefore, to anticipate from theory
two new laws of light, to which he gave the names of Internal
and External Conical Refraction.^

385

:trEX OF 3IATHEMATI0S

The prediction and its experimental verification by Hum-
phrey Lloyd evoked unbounded admiration for young Hamilton
from those who could appreciate what he had done. Airy, his
former rival for the professorship of astronomy, estimated
Hamilton’s achievement thus: ’Perhaps the most remarkable
prediction that has ever been made is that lately made by
Professor Hamilton.’ Hamilton himself considered this, like any
similar prediction, “a subordinate and secondary result’ com-
pared to the grand object which he had in view, “to introduce
harmony and unity into the contemplations and reasonings of
optics, regarded as a branch of pure science.’

According to some this spectacular success was the high-
water mark in Hamilton’s career; after the great work on optics
and dynamics his tide ebbed. Others, particularly members of
what has been styled the High Church of Quaternions, hold
that Hamilton’s greatest work was still to come – the creation
of what Hamilton himself considered his masterpiece and his
title to immortality, his theory of quaternions. Leading quater-
nions out of the indictment for the moment, we may simply
state that, from his twenty-seventh year till his death at sixty,
two disasters raised havoc with Hamilton’s scientific career,
marriage and alcohol. The second was partly, but not wholly, a
consequence of the unfortunate first.

After a second unhappy love affair, which ended with a
thoughtless remark that meant nothing but which the hyper-
sensitive suitor took to heart, Hamilton married his third fancy,
Helen Maria Bayiey, in the spring of 1833. He was then in his
twenty-eighth year. The bride was the daughter of a country
parson’s widow. Helen was ‘of pleasing ladylike appearance,
and early made a favourable impression upon him [Hamilton]
by her truthful nature and by the religious principles which he
knew her to possess, although to these recommendations was
not added any striking beauty of face or force of intellect.’
Now, any fool can tell the truth, and if truthfulness is all a fool
has to recommend her, whoever commits matrimony with her
vrill get the short end of the indiscretion. In the summer of 1832
Miss Bayiey ’passed through a dangerous illness, , . . , and this
event doubtless drew iiis [the lovelorn Hamilton’s] thoughts
386

AN lEISII TBAGEDY

especially toward her, in the form of anxiety for her recovery,
and, coming at a time [when he had just broken with the girl he
really wanted] when he felt obliged to suppress his former
passion, prepared the way for tenderer and warmer feelings.’
Hamilton in short was properly hooked by an ailing female who
was to become a semi-invalid for the rest of her life and who,
either through incompetence or ill-health, let her husband’s
slovenly ser\’ants run his house as they chose, which at least in
some quarters – especially his study – came to resemble a pig-
sty. Hamilton needed a sympathetic woman with backbone to
keep liim and his domestic affairs in some semblance of order;
instead he got a weakling.

Ten years after his marriage Hamilton tried to pull himself up
short on the slippery trail he realized wdth a brutal shock he was
treading. As a young man, feted and toasted at dinners, he had
rather let himself go, especially as his great gifts for eloquence
and convi\dalit3’ were naturally enough heightened hy a drink
or two. After his marriage, irregular meals or no meals at all, and
his habit of working twelve or fourteen hours at a stretch, were
compensated for by taking nourishment from a bottle.

It is a moot question whether mathematical invent ivetiess is
accelerated or retarded hy moderate indulgence in alcohol, and
until an exhaustive set of controlled experiments is carried out
to settle the matter, the doubt must remain a doubt, preeisety
as in am” other biological research. If, as some maintain, poetic
and mathematical inventiveness are akin, it is hy no means
obvious that reasonable alcoholic indulgence (if there is such a
thing) is destructive of mathematical inventiveness; in fact
numerous well-attested instances would seem to indicate the
contrarj”. In the case of poets, of course, ‘wine and song’ have
often gone together, and in at least one instance – Smnbume –
without the first the second dried up almost completel\”. Mathe-
maticians have frequently remarked on the terrific strain
induced hy prolonged concentrations on a difficult^-, and some
have found the let-down occasioned a drink a decided relief.
But poor Hamilton quickly passed beyond this stage and be-
came careless, not onl}- in the untid}- privacy” of his studj^ but
also in the glaring publicity of a banquet hall. He got drunk at

387

MEX OF MATHEMATICS

a scientific dinner. Realizing what had overtaken him, he
resolved never to touch alcohol again, and for two years he kept
his resolution. Then, during a scientihe meeting at the estate of
Lord Rosse (owner of the largest and most useless telescope
then in existence), his old rival. Airy, jeered at him for drinking
nothing but water. Hamilton gave in, and thereafter took ah he
wanted – which was more than enough. Still, even this handicap
could not put him out of the race, although without it he pro-
bably would have gone farther and have reached a greater
height than he did. However, he got high enough, and moral-
izing may be left to moralists.

Before considering what Hamilton regarded as his master-
piece, we may briefly summarize the principal honours which
came his way. At thirty he held an influential offlce in the
British Association for the Advancement of Science at its
Dublin meeting, and at the same time the Lord-Lieutenant bade
him to “Kneel down, Professor Hamilton’, and then, having
dubbed him on both shoulders with the sword of State, to “Rise
up, Sir William Rowan Hamilton’. This was one of the few
occasions in his life on which Hamilton had nothing whatever
to say. At thirty-two he became President of the Royal Irish
Academy, and at thirty-eight was awarded a Civil List life
pension of £200 a year from the British Government, Sir
Robert Peel, Ireland’s reluctant friend, being then Premier.
Shortly before this Hamilton had made his capital invention —
quaternions.

An honour which pleased him more than any he had ever
received was the last, as he lay on his deathbed: he was elected
the first foreign member of the National Academy of Sciences
of the United States, which was founded during the Chil War.
‘Ihis honour was in recognition of his work in quaternions,
principally , which for some unfathomable reason stirred
American mathematicians of the time (there were only one or
two in existence, Benjamin Peirce of Harvard being the chief)
more profoundly than had any other British mathematics since
Newton’s Principia, The early popularity of quaternions in the
United States is somewhat of a mysten^ Possibly the turgid
eloquence of the Lectures on Quateryiioiis captivated the taste

m

AN IRISH TRAGEDY

of a young and vigorous nation which had yet to outgrow its
morbid addiction to senatorial oratory and Fourth of July
verbal fireworks.

Quaternions has too long a history for the whole story to be
told here. Even Gauss with his anticipation of ISIT was not the
fii’st in the field: Euler preceded him vrith. an isolated result
vrhieh is most simply interpreted in terms of quaternions. The
origin of quaternions may go back even farther than this, for
Augustus de Morgan once half-jokingly offered to trace their
histoiy’ for Hamilton from the ancient Hindus to Queen Vic-
toria. However, we need glance here only at the lion’s share in
the invention and consider briefly what inspired Hamilton.

The British school of algebraists, as v.’ill be seen in the chapter
on Boole, put common algebra on its oto feet during the first
half of the nineteenth century. Anticipating the currently
accepted procedure in developing any branch of mathematics
carefully and rigorously they founded algebra postulationally.
Before this, the various kinds of ‘numbers’ – fractions, nega-
tives, irrationals – which enter mathematics when it is assumed
that all algebraic equations have roots, had been allowed to
function on precisely the same footing as the common positive
integers which were so staled by custom that all mathemati-
cians believed them to be ‘natural’ and in some vague sense
completely understood – they are not, even to-day, as will be
seen when the wDrk of Georg Cantor is discussed. This naive
faith in the self-consistency of a system founded on the blind,
formal juggling of mathematical symbols may have been
sublime but it was also slightly idiotic. The climax of this
credulity was reached in the notorious principle of permanence
of form, which stated in effect that a set of rules which ^^ield
consistent resTiits for one kind of numbers – say the positive
integers – wiH continue to yield consistency when applied to any
other kind – say the imaginaries – even when no interpretation
of the results is e\ident. It does not seem surprising that this
faith in the integrity of meaningless symbols frequently led to
absurdity.

The British school changed all this, although they were
unable to take the final step andpi’orc that their postulates for

3S9

MEN OF 3IATHE3IATICS

common algebra will never lead to a contradiction. That step

taken only in our own generation by the German workers
in the foundations of mathematics. In this connexion it must be
kept in mind that algebra deals only with finite processes ; when
infinite processes enter, as for example in summing an infinite
series, we are thrust out of algebra into another domain. This
is emphubized because the usual elementary text labelled
^41geb^a’ contains a great deal – infinite geometric progressions,
for instance – that is not algebra in the modern meaning of the
word.

The nature of what Hamilton did in his creation of quater-
nions will show up more clearly against the background of a set
of postulates (taken from L. E. Dickson’s Algebras and Their
Ariihmeiics, Chicago, 1923) for common algebra or, as it is
technically called, afield (English writers sometimes use corpus
as the equivalent of the German Korper or French corps).

‘A field F is a system consisting of a set S of elements a, 5,
c, . . . and two operations, called addition and multiplication,
which may be performed upon any two (equal or distinct)
elements a and h of S. taken in that order, to produce uniquely
determined elements « S b and a O boiS. such that postulates
I-V are saiisfied. For simplicity we shall write a -f- b for a © b,
and ah for a C b, and call them the sian and product, respec-
tively, of a and b. ^Moreover, elements of S will be called
elements of F.

T. If a and b are any two elements of F, a -f b and ab
are uniquely determined elements of F, and

b -r a = a >f b, b« = ab.

TI. If are any three elements of F,
ia^h) — c^a-rih-^c), {ah)C = a{bc), a(b -{- c) = ab + ac,

TII. There exist in F two distinct elements, denoted by 0, 1,
such that if // is any element of F, « -f- 0 = a, al = a (w’hence
0 -r fl = a, la — a, by I).

I\ . hatever be the element a of F, there exists in F an
element jc such that a — == 0 (whence a; -f a = 0 by I).

‘V. Whatever be the element a (distinct from 0) of F, there

390

AN IRISH TRAGEDY

exists in jP an element y such that ay = 1 (whence ya = 1, by

I)-’

From these simple postulates the whole of common algebra
follows. A word or two about some of the statements may be
helpful to those who have not seen algebra for years. In II, the
statement {a ~ h) -r c = a (b c), called the associative latv
of addition, says that if a and h are added, and to this sum is
added c, the result is the same as if a and the sum of b and c
are added. Similarly with respect to multiplication, for the
second statement in II. The third statement in II is called the
distributive lav:. In III a ‘zero’ and ‘unity’ are postulated; in IV,
the postulated x gives the negative of a; and the first paren-
thetical remark in V forbids ‘dmsion by zero’. The demands in
Postulate I are called the commutative laws of addition and
multiplication respectively.

Such a set of postulates may be regarded as a distillation of
experience. Centuries of working with numbers and getting
useful results according to the rules of arithmetic – empirically
arrived at – suggested most of the rules embodied in these
precise postulates, but once the suggestions of experience are
understood, the interpretation (here common arithmetic) fur-
nished by experience is deliberately suppressed or forgotten,
and the system defined by the postulates is developed abstractly^
on its owm merits, by common logic plus mathematical tact.

Notice in particular IV, wliich postulates the existence of
negatives. We do not attempt to deduce the existence of nega-
tives from the behaviour of positives. ‘VMien negative numbers
first appeared in experience, as in debits instead of credits, they,
as numbers, were held in the same abhorrence as ‘uimaturar
monstrosities as were later the ‘imaginary’ numbers
V — 2, etc., arising from the formal solution of equations such
as -r 1 = 0, cC- 4- 2 = 0, etc. If the reader will glance back
at what Gauss did for complex numbers he w’Hl appreciate more
fully the complete simplicity of the foUowdng partial statement
of Hamilton’s original way of stripping ‘imaginaries’ of their
silly, purely imaginary mystery. This simple tiling was one of
the steps which led Hamilton to his quaternions, although
strictly it has nothing to do writh them. It is the method and the

391

MEX OF 3IATHEMATICS

point of t’ieu: behind this ingenious recasting of the algebra of
complex numbers TV’hich are of importance for the sequel.

If as usual i denotes – 1, a ‘complex number is a number
of the type a -f hi, where a,b are ‘real numbers’ or, if preferred,
and more generally, elements of the field F defined by the above
postulates. Instead of regarding a 4- bi as one ‘number’,
Hamilton conceived it as an ordered couple of ‘numbers’, and he
designated this couple by T\Titing it {afi). He then proceeded to
impose definitions of sum and product on these couples, as
suggested by the formal rules of combination sublimated from
the experience of algebraists in manipulating complex numbers
as if the laws of common algebra did in fact hold for them.
One advantage of this new way of approaching complex
numbers was this: the definitions for sum and product of
couples were seen to be instances of the general, abstract
definitions of sum and product as in a field. Hence, if the con-
sistency of the system defined by the postulates for a field is
‘proved, the like follows, without further proof, for complex
numbers and the usual rules by which they are combined. It
will be sufficient to state the definitions of sum and product in
Hamilton’s theor\’ of complex numbers considered as couples
{aJi} (c,c?), etc.

The sum of ia,h) and (€,d) is (a -i- b, c -j- d); their product is
(ac — hd, ad -r be). In the last, the minus sign is as in a field;
namely, the element x postulated in IV is denoted by — a. To
the 0, 1 of a field correspond here the couples (0,0), (1,0). With
these definitions it is easily verified that Hamilton’s couples
satisfy aU the stated postulates for a field. But they also accord
with the formal rules for manipulating complex numbers. Thus,
to (clI), (c.d) correspond respectively a -p bi, c H- di, and the
formal \suni’ of these two is (« -f c) -f i{b -f d), to which corre-
sponds the couple (a -h c, 6 -f d). Again, formal multiplication
of a -f hi, c -r id gives (ac — fed) -f i{ad -f- he), to which cor-
responds the couple (ac — bd, ad -f be). If this sort of thing is
new to any reader, it will repay a second inspeetion, as it is an
example of the way in which modem mathematics eliminates
mystery^ So long as there is a shred of my’stery attached to any
eoneept that conc*ept is not mathematical.

zn

AX IRISH TRAGEDY

Ha^ing disposed of complex numbers by couples^ Hamilton
sought to extend his device to ordered triples and quadruples.
Without some idea of what is sought to be accomplished such
an undertaking is of course so vague as to be meaningless.
Hamilton’s object was to invent an algebra which would do for
rotations in space of three dimensions what complex numbers,
or his couples, do for rotations in space of fivo dimensions, both
spaces being Euclidean as in elementarj^ geometry. Now, a
complex number a -f- hi can be thought of as representing a
vector^ that is. a line segment having both length and direction,
as is evident from the diagram, in which the directed segment
(indicated by the arrow) represents the vector OP.

But on attempting to sj^mbolize the behaviour of vectors in
three-dimensional space so as to presen’^e those properties of
vectors which are of use in physics, particularly in the combina-
tion of rotations, Hamilton was held up for years by an unfore-
seen diiBculty whose very nature he for long did not even
suspect. We may glance in passing at one of the clues he
followed. That this led him anywhere ” as he insisted it did – is
all the more remarkable as it is now almost universally regarded
as an absurdity, or at best a metaphysical speculation without
foundation in history or in mathematical experience.

Objecting to the purely abstract, postulational formulation

393

MEN OF 3iathe:matics

of algebra advocated by his British contemporaries, Hamilton
sought to found algebra on something *more reaF, and for this
strictly meaningless enterprise he drew on his knowledge of
Kant’s mistaken notions – exploded by the creation of non-
Euclidean geometrs^ – of space as “a pure form of sensuous
intuition’. Indeed Hamilton, who seems to have been unac-
quainted with non-Euelidean geometiy^, followed Kant in
belie\ing that ‘Time and space are two sources of knowledge
from wiiich various a priori synthetical cognitions can be
derived. Of this, pure mathematics gives a splendid example in
the case of our cognition of space and its various relations. As
they are both pure forms of sensuous intuition, they render
synthetic propositions a priori possible.’ Of course any not
utterly illiterate mathematician to-day knows that Kant was
mistaken in this conception of mathematics, but in the lS40‘s,
when Hamilton was on his way to quaternions, the Kantian
philosophy of mathematics still made sense to those – and they
were nearly all – who had never heard of Lobatchew’skj’. By
what looks like a bad mathematical pun, Hamilton applied the
Kantian doctrine to algebra and drew the remarkable conclu-
sion that, since geometr\’ is the science of space, and since time-
and space are “pure sensuous forms of intuition’, therefore the
rest of mathematics must belong to time, and he wasted much
of his OAvn time in elaborating the bizarre doctrine that algebra
is the scicjice of pure time,

Tliis queer crotchet has attracted many philosophers, and
quite recently it has been exhumed and solemnly dissected bv
owlish metaphysicians seeking the philosopher’s stone in the
gall bladder of mathematics. Just because ‘algebra as the
science of pure time’ is of no earthly mathematical significance,
it will continue to he discussed with animation till time itself
ends. The opinion of a great mathematician on the ‘pure time’
aspect of algebra may be of interest. ‘I cannot myself recognize
the connexion of algebra with the notion of time,’ Cayley con-
fessed: granting that the notion of continuous progression
presents itself and is of importance, I do not see that it is in
any wise the fundamental notion of the science.’

Hamilton’s difficulties in trying to construct an algebra of

394

AN IRISH TRAGEDY

vectors and rotations for three-dimensional space were rooted
in his subconscious conviction that the most important laws of
common algebra must persist in the algebra he was seeking.
How were vectors in three-dimensional space to be multiplied
together?

To sense the difficulty of the problem it is essential to bear in
mind (see Chapter on Gauss) that ordinary complex numbeTS
(2 = V — 1) had been given a simple interpretation in
terms of rotations in a plane, and further that complex numbers
obey all the rules of common algebra, in particular the commuta-
tive law of multiplication: if ..4, B are any complex numbers, then
A A B — B A A, whether A, B are interpreted algebraically,
or in terms of rotations in a plane. It was but human then to
anticipate that the same commutative law would hold for the
generalizations of complex numbers which represent rotations in
space of three dimensions.

HamOton’s great discovery – or invention – was an algebra,
one of the ’natural’ algebras of rotations in space of three
dimensions, in which the commutative law of multiplication
does not hold. In this Hamiltonian algebra of quaternions (as
he called his invention), a multiplication appears in which
A ;< B is not equal to B X ^ but to minus B x A, that is,
A X B = – B X A.

That a consistent, practically useful system of algebra could
be constructed in defiance of the commutative law’ of multipli-
cation was a discovery of the first order, comparable, perhaps,
to the conception of non-Euclidean geometry. Hamilton him-
self was so impressed by the magnitude of what suddenly
dawned on his mind (after fifteen years of fruitless thought) one
day (16 October 1843) when he was out walking with his wife
that he carved the fundamental formulae of the new algebra
in the ^tone of the bridge on which he found himself at the
moment. His great invention showed algebraists the way to
other algebras until to-day, foUo^ving Hamilton’s lead, mathe-
maticians manufacture algebras practically at w’iH by negating
one or more of the postulates. for a field and developing tlit
consequences. Some of these ‘’algebras’ are extremely useful;
the general theories embracing swarms of them include HamH-

395

MEN OF MATHEMATICS

ton’s great invention as a mere detail, although a highly
important one.

In line with Hamilton’s quaternions the numerous brands of
sector analysis favoured by physicists of the past two genera-
tions sprang into being. To-day all of these, including quater-
nions, so far as physical applications are concerned, are being
swept aside by the incomparably simpler and more general
tensor analysis which came into vogue with general relati\ity in

1915. Something vill be said about tills later.

In the meantime it is sufheient to remark that Hamilton’s
deepest tragedy was neither alcohol nor marriage but Ms
obstinate belief that quaternions held the key to the mathe-
matics of the physical universe. History has shown that
Hamilton tragically deceived himself when he insisted “… I
still must assert that this discovery appears to me to be as
important for the middle of the nineteenth centurj’ as the dis-
covery of fluxions [the calculus] was for the close of the seven-
teenth.’ Never was a great mathematician so hopelessly wrong.

The last twenty-two years of Hamilton’s life were devoted
almost exclusively to the elaboration of quaternions, including
their application to dynamics, astronomy, and the wave theory
of light, and his voluminous correspondence. The style of the
overdeveloped Ekments of Qiiaternions, published the year after
Hamilton’s death, shows plainly the effects of the author’s
mode of life. After his death from gout on 2 September 1865 in
the sixty-first year of his age, it was found that Hamilton had
left behind a mass of papers in indescribable confusion and
about sixty huge manuscript books full of mathematics. An
adequate edition of his works is now in progress. The state of
his papers testified to the domestic difficulties under which the
last third of his life had been lived: innumerable dinner plates
with the remains of desiccated, un^dolated chops were found
buried in the mountainous piles of papers, and dishes enough
to supply a large household were dug out from the confusion.
During his last period Hamilton lived as a recluse, ignoring the
meals dhoved at him as he worked, obsessed by the dream that
the last tremendous effort of his magnificent genius would
unmortalize both himself and his beloved Ireland, and stand

396

AN IRISH TRAGEDY

iOreTer unshaken as the greatest mathematical contribution to
science since the Prmcipia of Ne^yton.

His earlj^ work, on which his imperishable glory rests, he
came to regard as a thing of but little moment in the shadow of
what he believed was his masterpiece. To the end he was
humble and devout, and wholly without anxiety for his scien-
tific reputation. T have very long admired Ptolemy’s descrip-
tion of his great astronomical master, Hipparchus, as di/7/p
SlXottovos Kal (l}LAaX7]drj£; a labour-loving and truth-loving
man. Be such my epitaph.’

CHAPTER TWENTY

GE^’IUS AND STUPIDITY
Galois

Abel was done to death by poverty, Galois by stupidity. In all
the histon- of science there is no completer example of the
triumph of crass stupidity over untamable genius than is
afforded by the all too brief life of fivariste Galois. The record
of his misfortunes might well stand as a sinister monument to
all self-assured pedagogues, unscrupulous politicians, and con-
ceited academicians. Galois was no ‘ineffectual angeP, but even
his magnificent powers were shattered before the massed
stupidity aligned against him, and he beat his life out fighting
one unconquerable fool after another.

The first eleven years of Galois* life were happy. His parents
lived in the little \illage of Bourg-la-Reine, just outside Paris,
where fivariste was horn on 25 October 1811. Nicolas-Gabriel
Galois, the father of fivariste. was a relic of the eighteenth
century, cultivated, intellectual, saturated with philosophy, a
passionate hater of royalty and an ardent lover of liberty.
During the Hundred Days after Napoleon’s escape from Elba,
Galois was elected mayor of the village. After Waterloo he
retained his office and served faithfully under the King, hacking
the villagers against the priest and delighting social gatherings
with the old-fashioned rhymes which he composed himself.
These harmless actmties were later to prove the amiable man*s
undoing. From his father, fivariste acquired the trick of
rhyming and a hatred of tyranny and baseness.

Unto the age of twelve Galois had no teacher hut his mother,
Adelaide-Marie Demante. Several of the traits of Galois’
character were inherited from his mother, who came from a
long line of distinguished jurists. Her father appears to have
been somewhat of a Tartar. He gave his daughter a thorough
398

GENIUS AND STUPIDITY

classical and religious education, which she in turn passed on
to her eldest son* not as she had received it, but fused into a
virile stoicism in her o’svn independent mind. She had not
rejected Christianity, nor had she accepted it without question;
she had merely contrasted its teachings mth those of Seneca
and Cicero, reducing all to their basic morality. Her friends
remembered her as a woman of strong character with- a mind
of her own, generous, wdth a marked vein of originality, quiz-
zical, and, at times, inclined to be paradoxical. She died in 1872
at the age of eighty -four. To the last she retained the full vigour
of her mind. She, like her husband, hated tyranny.

There is no record of mathematical talent on either side of
Galois’’ family. His own mathematical genius came on him like
an explosion, probably at early adolescence. As a child he was
affectionate and rather serious, although he entered readily
enough into the gaiety of the recurrent celebrations in his
father s honour, even composing rhymes and dialogues to
entertain the guests. All this changed under the first stings of
petty persecution and stupid misunderstanding, not by his
parents, but by his teachers.

In 1823, at the age of twelve, Galois entered the lycee of
Louis-le- Grand in Paris. It was his first school. The place was a
dismal horror. Barred and grilled, and dominated by a provisor
who was more of a political jailer than a teacher, the place
looked like a prison, and it was. The France of 1823 still remem-
bered the Revolution, It was a time of plots and counterplots,
of riots and rumours of revolution. All this was echoed in the
school. Suspecting the provisor of scheming to bring back the
Jesuits, the students struck, refusing to chant in chapel. With-
out even notif;jTng their parents the pro\isor expelled those
whom he thought most guilty. They found themselves in the
street. Galois was not among them, but it would have been
better for him if he had been.

Till now tyranny had been a mere word to the boy of twelve.
Now he saw it in action, and the experience warped one side of
his character for life. He was shocked into unappeasable rage.
His studies, owing to his mother’s excellent instruction in the
classics, went very well and he won prizes. But he had also

399

gained (something more lasting than any prize, the stubborn
con\iction. right or v^Tong, that neither fear nor the utmost
severity of discipline can extinguish the sense of justice and fair
dealing in young minds experiencing their first unselfish devo-
tion. This his fellow students had taught him by their courage.
Galcis never forgot their example. He was too young not to be
embittered.

The following year marked another crisis in the young boy’s
life. Docile interest in literature and the classics gave way to
boredom; Ins mathematical genius was already stirring. His
teachers ad\dsed that he be demoted, variste’s father objected,
and the boy continued witli his interminable exercises in
rhetoric. Latin, and Greek. His w’ork was reported as mediocre,
his conduct ‘dissipated’, and the teachers had their way.
Galois was demoted. He was forced to lick up the stale leavings
which his genius had rejected. Bored and disgusted he gave his
work perfunctory attention and passed it without effort or
interest. Mathematics was taught more or less as an aside to the
serious business of digesting the classics, and the pupils of
various grades and assorted ages took the elementary mathe-
matical course at the convenience of their other studies.

It was during this year of acute boredom that Galois began
mathematics in the regular school course. The splendid geo-
metry of Legendre came his way. It is said that two years was
the usual time required by even the better mathematicians
among the boys to master Legendre. Galois read the geometry
from cover to cover as easily as other hoys read a pirate yarn.
Tlie book aroused bis enthusiasm; it was no textbook written
by a hack, but a work of art composed by a creative mathe-
matician. A single reading sufficed to reveal the whole structure
of elemcntarj^ geometry in crystal clarity to the fascinated boy.
He had mastered it.

His reaction to algebra is illuminating. It disgusted him, and
for a very good reason when we consider what sort of mind
Galois had. Here was no master like Legendre to inspire him.
The text in algebra was a school book and nothing more.
Galois contemptuously tossed it aside. It lacked, he said, the
creator’s touch that only a creative mathematician can give.

400

GENIUS AND STUPIDITY

Ha’STng made the acquaintance of one great mathematician
through his work, Galois took matters into his own hands.
Ignoring the meticulous pettifogging of his teacher, Galois went
directly for his algebra to the greatest master of the age,
Lagrange. Later he read Abel. The boy of fourteen or fifteen
absorbed masterpieces of algebraical analysis addressed to
mature professional mathematicians – the memoirs on the
numerical solution of equations, the theory of anal3i:icai func-
tions, and the calculus of functions. His class work in mathe-
matics was mediocre: the traditional course was trivial to a
mathematical genius and not necessary for the mastering of real
mathematics.

Galois’ peculiar gift of being able to earn,* on the most diffi-
cult mathematical investigations almost entirely in his head
helped him with neither teachers nor examiners. Their insist-
ence upon details which to him were ob*ious or tri\dal exas-
perated him beyond endurance, and he frequently lost his
temper. Nevertheless he earned off the prize in the general
examination. To the amazement of teachers and students alike
Galois had taken his own kingdom by assault while their backs
were turned.

With this first realization of his tremendous power, Galois’
character underwent a profound change. Knowing his kinship
to the great masters of algebraical analysis he felt an immense
pride and longed to rush on to the front rank to match his
strength with theirs. His family – even his unconventional
mother – found him strange. At school he seems to have
inspired a curious mixture of fear and anger in the minds of his
teachers and fellow students. His teachers were good men and
patient, but they w*ere stupid, and to Galois stupidity was the
unpardonable sin. At the beginning of the year they had
reported him as “very gentle, full of innocence and good quali-
ties, but And they went on to say that “there is something
strange about him.’ No doubt there was. The boy had unusual
brains. A little later they admit that he is not ‘wicked’, but
merely ‘original and queer’, ‘argumentative’, and they complain
that he delights to tease his comrades. All very reprehensible,
no doubt, but they might have used their eyes. The boy had

401

MEX OF MATHEMATICS

discovered mathematics and he was already being driven by
his daemon. By the end of the year of awakening we learn that
‘his queemess has alienated him from all his companions’, and
his teachers obser^^e ‘something secret in his character’. Worse,
they accuse him of ^affecting ambition and originality’. But it
is admitted by some that Galois is good in mathematics. His
rhetoric teachers indulge in a little classical sarcasm; “His
cleverness is now a legend that we cannot credit.’ They rail that
there is only slovenliness and eccentricity in his assigned tasks-
when he deigns to pay any attention to them – and that he goes
out of his way to wear\” his teachers by incessant ‘dissipation’.
The last does not refer to vice, because Galois had no vicious-
ness in him. It is merely a strong w’ord to describe the heinous
inability of a mathematical genius of the first rank to squander
his intellect on the futilities of rhetoric as expounded by
pedants.

One man, to the everlasting credit of his pedagogical insight,
declared that Galois was as able in literary studies as he was in
mathematics. Galois appears to have been touched by this
man’s kindness. He promised to give rhetoric a chance. But his
mathematical de\il was now fully aroused and raging to get
out, and poor Galois fell from grace. In a short time the dissen-
ing teacher joined the majority and made the vote unanimous.
Galois, he sadly admitted, was beyond salvation, ‘conceited
with an insufferable affectation of originality’. But the peda-
gogue redeemed himself by one excellent, exasperated sugges-
tion. Had it been followed, Galois might have lived to eighty.
‘The mathematical madness dominates this boy. I think his
parents had better let him take only mathematics. He is wast-
ing his time here, and all he does is to torment his teachers and
get into trouble.’

At the age of sixteen Galois made a curious mistake. Una-
ware that Abel at the beginning of his career had convinced
himself that he had done the impossible and had solved the
general equation of the fifth degree, Galois repeated the error.
F or a time – a verj- short time, however – he believed that he
had done what cannot be done. This is merely one of several
extraordinary similarities in the careers of Abel and Galois.

402

GENIUS AND STUPIDITY

\Miile Galois at the age of sixteen ’vvas already well started
on his career of fundamental discovery, his mathematical
teacher – Vernier – kept fussing over him like a hen that has
hatched an eaglet and does not know how to keep the unruly
creature’s feet on the good dirt of the barnyard. Vernier
implored Galois to work systematically. The advice was ignored
and Galois, without preparation, took the competitive examina-
tions for entrance to the ficole Polytechnique. This great
school, the mother of French mathematicians, founded during
the French Revolution (some say by Monge), to give chdl and
military engineers the best scientific and mathematical educa-
tion available anywhere in the world, made a double appeal to
the ambitious Galois. At the Pohi:echnique Ms mathematical
talent Y^ould be recognized and encouraged to the utmost. And
his cra\ing for libertj” and freedom of utterance would be grati-
fied: for were not the virile, audacious young Poljiiechnicians,
among them the future leaders of the army, always a thorn in
the side of reactionary schemers who would undo the glorious
work of the Revolution and bring back the corrupt priesthood
and the divine right of kings? The fearless Polyteehnieians, at
least in Galois’ boj^ish eyes, were no race of puling rhetoricians
like the browbeaten nonentities at Louis-le- Grand, but a conse-
crated band of young patriots. Events were presently to prove
liim at least partly right in his estimate.

Galois failed in the examinations. He was not alone in be-
lieving his failure the result of a stupid injustice. The comrades
he had teased unmercifully were stunned. They believed that
Galois had mathematical genius of the highest order and they
suspected his examiners of incompetence in their office. Nearly
a quarter of a century later Terquem, editor of the Nouvelles
Annales de Mathematiques, the mathematical journal devoted
to the interests of candidates for the Polytechnique and Nor-
male schools, reminded Ms readers that the controversy was
not yet dead. Commenting on the failure of Galois and on the
inscrutable decrees of the examiners in another instance. Ter-
quem remarks, candidate of superior intelligence is lost with
an examiner of inferior intelligence. Hie ego barbarus sum quia
non intelUgor illis [Because they don’t understand me, I am a

403

MEX OF MATHEMATICS

barbaxian.] . . . Examinations are mysteries before which I bow.
Like the mysteries of theolog>’, the reason must admit them
with humility, without seeking to understand them.’ As for
Galois, the failure was almost the finishing touch. It drove him
in upon himself and embittered him for life.

In 1828 Galois was seventeen. It was his great year. For the
first time he met a man who had the capacity to understand
his genius, Louis-Paul-Emile Richard (1795-1849), teacher of
advanced mathematics (mathemaiiques speciales) at Louis-le-
Grand. Richard was no conventional pedagogue, but a man of
talent who followed the advanced lectures on geometry at the
Sorbonne in his spare time and kept himself abreast of the
progress of Ihing mathematicians to pass it on to his pupils.
Timid and unambitious on his own account, he threw all his
talent on the side of his pupils. The man who would not go a
step out of his way to advance his own interests coimted no
sacrifice too great where the future of one of his students was
at stake. In his zeal to advance mathematics through the work
of abler men he forgot himself completely, although his scien-
tific friends urged him to wTite, and to his inspired teaching
more than one outstanding French mathematician of the nine-
teenth century has paid grateful tribute: Leverrier, co-disco-
verer with Adams by pure mathematical analysis of the planet
Neptune; Serret, a geometer of repute and author of a classic
on higher algebra in which he gave the first systematic exposi-
tion of Galois’ theory of equations; Hermite, master algebraist
and arithmetician of the first rank; and last, Galois.

Richard recognized instantly what had fallen into his hands –
^the Abel of France’. The original solutions to difficult problems
which Galois handed in were proudly explained to the class,
with just praise for the young author, and Richard shouted
from the housetops that this extraordinary pupil should be
admitted to the Polj’technique without examination. He gave
Galois the first prize and wrote in his term report. This pupil
has a marked superiority above ail his fellow students; he works
only at the most ad-anced parts of mathematics,’ All of which
was the literal truth. Galois at seventeen was making disco-
veries of epochal significance in the theory of equations, dis-

4M

GENIUS AND STUPIDITY

coveries whose consequences are not yet exhausted after more
than a centu^>^ On 1 March 1829, Galois published liis first
paper, on continued fractions. This contains no hint of the
Trreat things he had done, but it ser%’ed to announce him to his
fellow students as no mere scholar but an inventive mathe-
matician.

The leading French mathematician of the time was Cauchy.
In fertility of invention Cauchy has been equalled by but few:
and as we have seen, the mass of his collected works is exceeded
in bulk only by the outputs of Euler and Cayley,* the most
prolific mathematicians of histor\ Mlienever the Academy of
Sciences wished an authoritative opinion ori the merits of a
mathematical work submitted for its consideration it called
upon Cauchy. As a rule he was a prompt and just referee. But
occasionally he lapsed. Unfortunately the occasions of his lapses
were the most important of all. To Cauchy’s carelessness
mathematics is indebted for tw^o of the major disasters in its
historj” ; the neglect of Galois and the shabby treatment of Abel.
For the latter Cauchy w’as only partly to blame, but for
the inexcusable laxity in Galois’ case Cauchy alone is respons-
ible.

Galois had saved the fundamental discoveries he had made
up to the age of seventeen for a memoir to be submitted to the
Academy. Cauchy promised to present this, but he forgot. To
put the finishing touch to his ineptitude he lost the author’s
abstract. That was the last Galois ever heard of Cauchy’s
generous promise. This was only the first of a series of similar
disasters which fanned the thw’^arted boy’s suUen contempt of
academies and academicians into a fierce hate against the whole
of the stupid society’’ in which he w^as condemned to live.

In spite of his demonstrated genius the harassed boy was not
even now left to himself at school. The authorities gave him no
peace to har\^est the rich field of his discoveries, but pestered
him to distraction with petty tasks and goaded him to open
revolt by their everlasting preachings and punishments. Still

• That is, so far as actually published work is concerned up to 1930.
  Euler undoubtedly will surpass Cayley in bulk when the full edition
  of his works is finally printed.

405

MEN OF MATHEMATICS

they could find nothing in him but conceit and an iron deter-
mination to be a mathematician. He already was one, but they
did not know it.

Two further disasters in his eighteenth year put the last”
touches to Galois’ character. He presented himself a second
time for the entrance examinations at the Polytechnique. Men
who were not worthy’ to sharpen his pencils sat in judgement on
him. The result was what might have been anticipated. Galois
failed. It was his last chance: the doors of the Poly technique
were closed forever against iiim.

That examination has become a legend. Galois’ habit of
working almost entirely’ in his head put him at a serious disad-
vantage before a blackboard. Chalk and erasers embarrassed
him – till he found a proper use for one of them. During the oral
part of the examination one of the inquisitors ventured to argue –
a mathematical difficulty with Galois. The man was both
wrong and obstinate. Seeing all his hopes and his whole life as a
mathematician and polyi:echnie champion of democratic liberty
slipping away’ from him, Galois lost all patience. He knew that
he had officially failed. In a fit of rage and despair he hurled the
eraser at his tormentor’s face. It was a hit.

The final touch was the tragic death of Galois’ father. As the
may’or of Bourg-la-Reine the elder Galois was a target for the
clerical intrigues of the times, especially as he had alway’s cham-
pioned the villagers against the priest. After the stormy elec-
tions of 1827 a resourceful young priest organized a scurrilous
campaign against the may’or. Capitalizing the mayor’s well-
known gift for versify’ing, the ingenious priest composed a set
of filthy’ and stupid verses against a member of the may’or’s
family, signed them with Mayor Galois’ name, and circulated
them freely among the citizens. The thoroughly’ decent mayor
developed a persecution mania. During his wife’s absence one
day’ he slipped off to Paris and, in an apartment but a stone’s
throw from the school where his son sat at his studies, com-
mitted suicide. At the funeral serious disorder broke out. Stones
were hurled by’ the enraged citizens; a priest was gashed on the
forehead. Galois saw his father’s coffin lowered into the grave
in the midst of an unseemly’ riot. Thereafter, suspecting every-
40 «

GENIUS AND STUPIDITY

where the injustice which he hated, he could see no good in
anything.

After his second failure at the Polytechnique, Galois returned
to school to prepare for a teaching career. The school now had a
new director, a time-serving, somewhat cowardly stool-pigeon
for the royalists and clerics. This man’s shilly-shally tempor-
izing in the political upheaval which was presently to shake
France to its foundations had a tragic influence on Galois’ last
years.

Still persecuted and maliciously misunderstood by his pre-
ceptors, Galois prepared himself for the final examinations.
The comments of his examiners are interesting. In mathematics
and physics he got ‘very good’. The final oral examination drew
the following comments: ‘This pupil is sometimes obscure in
expressing his ideas, but he is intelligent and shows a remark-
able spirit of research. He has communicated to me some new
results in applied analysis.’ In literature: ‘This is the only
student who has answered me poorly; he knows absolutely
nothing. I was told that this student has an extraordinary
capacity for mathematics. This astonishes me greatly; for, after
his examination, I believed him to have but little intelligence.
He succeeded in hiding such as he had from me. If this pupil is
really what he has seemed to me to be, I seriously doubt
whether he will ever make a good teacher.’ To which Galois,
remembering some of his own good teachers, might have
replied, ‘God forbid.’

In February 1830, at the age of nineteen, Galois was defi-
nitely admitted to university standing. Again his sure know-
ledge of his own transcendent ability was reflected in a wither-
ing contempt for his plodding teachers and he continued to
work in solitude on his own ideas. During this year he composed
three papers in which he broke new ground. These papers con-
tain some of his great work on the theory of algebraic equations.
It was far in advance of anything that had been done, and
Galois had hopefully submitted it all (with further results) in a
memoir to the Academy of Sciences, in competition for the
Grand Prize in Mathematics, This prize was stiU the blue ribbon
m mathematical research; only the foremost mathematicians of

4or

M,M.— VOL. II.

MEN OF MATHEMATICS

the day could sensibly compete. Experts agree that Galois’
memoir Tvas more than worthy of the prize. It was work of the
highest originality. As Galois said with perfect justice, ‘I have
carried out researches which will halt many savants in theirs.’

The manuscript reached the Secretary safely. The Secretary
took it home with him for examination, but died before he had
time to look at it. When his papers were searched after his death
no trace of the manuscript was found, and that was the last
Galois ever heard of it. He can scarcely be blamed for ascribing
his misfortunes to something less uncertain than blind chance.
After Cauchy’s lapse a repetition of the same sort of thing
looked too providential to be a mere accident. ‘Genius’, he said,
Ts condemned by a malicious social organization to an eternal
denial of justice in favour of fawning mediocrity,’ His hatred
grew, and he flung himself into politics on the side of republi-
canism, then a forbidden radicalism.

The first shots of the revolution of 1830 filled Galois with joy.
He tried to lead his fellow students into the fray, but they hung
back, and the temporizing director put them on their honour
not to quit the school. Galois refused to pledge his word, and
the director begged him to stay in till the following day. In his
speech the director displayed a singular lack of tact and a total
absence of common sense. Enraged, Galois tried to escape
during the night, but the wall was too high for him. Thereafter,
all through *the glorious three days’ while the heroic young
Polyteclmicians were out in the streets making history, the
director prudently kept his cjharges under lock and key.
Whichever way the cat should jump the director was prepared
to jump with it. The revolt successfully accomplished, the
astute director very generously placed his pupils at the disposal
of the temporary government. This put the finishing touch to
Galois’ political creed. During the vacation he shocked his
family and boyhood friends with his fierce championship of the
rights of the masses.

The last months of 1830 were as turbulent as is usual after a
tJaorough political stir-up. The dregs sank to the bottom, the
scum rose to the top, and suspended between the two the
moderate element of the population hung in indecision. Galois,

408

GENIUS AND STUPIDITY

back at college, contrasted the time-serving vacillations of the
director and the wishy-washy loyalty of the students with their
exact opposites at the Polytechnique. Unable to endure the
humiliation of inaction longer he wrote a blistering letter to the
Gazette des J^coles in which he let both students and director
have what he thought was their due. The students could have
saved him. But they lacked backbone, and Galois was expelled.
Incensed, Galois wrote a second letter to the Gazette, addressed
to the students. T ask nothing of you for myself’, he wrote: ‘but
speak out for your honour and according to your conscience.’
The letter was unanswered, for the apparent reason that
those to whom Galois appealed had neither honour nor con-
science.

Foot-loose now, Galois announced a private class in higher
algebra, to meet once a week. Here he was at nineteen, a crea-
tive mathematician of the very first rank, peddling lessons to
no takers. The course was to have included ‘a new theory of
imaginaries [what is now known as the theory of “Galois
Imaginaries”, of great importance in algebra and the theory of
numbers]; the theory of the solution of equations by radicals,
and the theory of numbers and elliptic functions treated by
pure algebra’ – all his own work.

Finding no students, Galois temporarily abandoned mathe-
matics and joined the artillery of the National Guard, two of
whose four battalions were composed almost wholly of the
liberal group calling themselves ‘Friends of the People’. He had
not yet given up mathematics entirely. In one last desperate
effort to gain recognition, encouraged by Poisson, he had sent
a memoir on the general solution of equations ~ now called the
‘Galois theory’ – to the Academy of Sciences. Poisson, whose
name is remembered wherever the mathematical theories of
gravitation, electricity, and magnetisna are studied, was the
referee. He submitted a perfunctory report. The memoir, he
said was ‘incomprehensible’, but he did not state how long it
had taken him to reach Ms remarkable conclusion. This was the
last straw. Galois devoted all his energies to revolutionary
politics- ‘If a carcase is needed to stir up the people’, he wrote,
T will donate mine.’

409

MEN OE MATHEMATICS

The ninth of IMay 1831 marked the beginning of the end.
About 200 young republicans held a banquet to protest against
the royal order disbanding the artillery which Galois had
joined. Toasts were drunk to the Revolutions of 1789 and 1793,
to Robespierre, and to the Revolution of 1830. The whole atmo-
sphere of the gathering was revolutionary and defiant. Galois
rose to propose a toast, Ms glass in one hand, Ms open pocket
knife in the other; ‘To Louis Philippe’ – the King. His com-
panions misunderstood the purpose of the toast and whistled
him down. Then they saw the open knife. Interpreting tMs as a
threat against the life of the K ing, they howled their approval,

A Mend of Galois, seeing the great Alexandre Dumas and other
notables passing by the open windows, implored Galois to sit
down, but the uproar continued. Galois was the hero of the
moment, and the artillerists adjourned to the street to celebrate
their exuberance by dancing all night. The following day Galoh
was arrested at Ms mother’s house and thrown into the prison
of Sainte-Pelagie.

A clever lawyer, with the help of Galois’ loyal friends, devised
an ingenious defence, to the effect that Galois had reaUy said;
‘To Louis Philippe, if he turns traitor J* The open knife was easily
explained: Galois had been using it to cut his cMcken. TMs was
the fact. The saving clause in Ms toast, according to Ms friends
who swore they had heard it, was drowned by the wMstling,
and only those close to the speaker caught what was said.
Galois would not claim the saving clause.

During the trial Galois’ demeanour was one of haughty con-
tempt for the court and his accusers. Caring nothing for the
outcome, he launched into an impassioned tirade against all the
forces of political injustice. The judge was a human being with
children of Ms own. He warned the accused that he was not
helping his own case and sharply silenced Mm. The prosecution
quibbled over the point whether the restaurant where the inci-
dent occurred was or was not a ‘public place’ when used for a
semi-private banquet. On tMs nice point of law hung the liberty
of Galois, But it was evident that both court and jury were
moved by the youth of the accused. After only ten minutes’
ddiberation the jury returned a ‘verdict of not guilty. Galois
410

GENIUS AND STUPIDITY

picked up his knife from the evidence table, closed it, slipped
it in his pocket, and left the court-room without a word.

He did not keep his freedom long. In less than a month, on
14 July 1831, he was arrested again, this time as a precau-
tionary measure. The republicans were about to hold a celebra-
tion, and Galois, being a ‘dangerous radical’ in the eyes of the
authorities, was locked up on no charge whatever. The govern-
ment papers of all France played up this brilliant coup of the
police. They now had ‘the dangerous republican, ^Evariste
Galois’, where he could not possibly start a revolution. But they
were hard put to it to find a legal accusation under which he
could be brought to trial. True, he had been armed to the teeth
when arrested, but he had not resisted arrest. Galois was no
fool. Should they accuse him of plotting against the Govern-
ment? Too strong; it wouldn’t go; no jury would convict. Ah!
After two months of incessant thought they succeeded in
trumping up a charge, men arrested Galois had been wearing
his artillery uniform. But the artillery had been disbanded.
Therefore Galois was guilty of illegally wearing a uniform.
This time they convicted him. A friend, arrested with him, got
three months; Galois got six. He was to be incarcerated in
Sainte-Pelagie tiU 29 April 1832. His sister said he looked about
fifty years old at the prospect of the sunless days ahead of him.
Why not? ‘LiCt justice prevail though the heavens fall.’

Discipline in the jail for political prisoners was light, and they
were treated with* reasonable humanity. The majority spent
their waking hours promenading in the courtyard reserved for
their use, or boozing in the canteen – the private graft of the
governor of the prison. Soon Galois, with hist sombre visage,
abstemious habits, and perpetual air of intense concentration,
became the butt of the jovial swilleis. He was concentrating on
his mathematics, but he could not help hearing the taunts
hurlfed at him.

‘What! You drink only water? Quit the Republican Party
and go back to your mathematics.’ – ‘Without wine and women
you 11 never be a man.’ Goaded beyond endurance Galois seized
a bottle of brandy, not knowing or caring what it was, and
drank it down. A decent fellow prisoner took care of him tin he

411

MEN or MATHEMATICS

recovered. His humiliation when he realized what he had done
devastated him.

At last he escaped from what one French writer of the time
calls the foulest sewer in Paris. The cholera epidemic of 1832
caused the solicitous authorities to transfer Galois to a hospital
on 16 March. The ‘important political prisoner’ who had threa-
tened the life of Louis Philippe was too precious to be exposed
to the epidemic.

Galois was put on parole, so he had only too many occasions
to see outsiders. Thus it happened that he experienced his one
and only love affair. In this, as in eversrthing else, he was unfor-
tunate. Some worthless girl (^quelque coquette de bos etage”)
initiated him. Galois took it violently and was disgusted with
love, with himseK, and with his girl. To his devoted friend
Auguste Chevalier he wrote, ‘Your letter, full of apostolic
unction, has brought me a little peace. But how obliterate the
mark of emotions as violent as those which I have expe-
rienced? … On re-reading your letter, I note a phrase in which
you accuse me of being inebriated by the putrefied slime of a
rotten world which has defiled my heart,* my head, and my
hands. . . Inebriation! I am disillusioned of everything, even
love and fame. How can a world which I detest defile me?*
This is dated 25 May 1832. Four days later he was at
liberty. He had planned to go into the country to rest and
meditate.

What happened on 29 May is not definitely known. Extracts
from two letters suggest what is usually accepted as the truth:
Galois had run foul of political enemies immediately after his
release. These ‘patriots’ were always spoiling for a fight, and it
fell to the unfortunate Galois’ lot to accommodate them in an
affair of ‘honour’. In a ‘Letter to All Republicans,’ dated 29
May 1832, Galois writes:

I beg patriots and my friends not to reproach me for
dying otherwise than for my country. I die the victim of an
infamous coquette. It is in a miserahle brawl that my life
is extinguished. OhI why die for so trivial a thing, die for
something so despicable! … Pardon for those who have
killed me, they are of gpod faith.

4X2

GENIUS AND STUPIDITY
In another letter to two unnamed friends :

I have been challenged by two patriots – it was impos-
sible for me to refuse. I beg your pardon for having advised
neither of you. But my opponents had put me on my
honour not to warn smy patriot. Your task is very simple:
prove that I fought in spite of myself, that is to say after
having exhausted every means of accommodation. . . .
Preserv^e my memory since fate has not given me life
enough for my country to know my name. I die your friend

E. Galois.

These were the last words he wrote. All night, before writing
these letters, he had spent the fleeting hours feverishly Ha-ghing
off his scientific last will and testament, writing against time to
glean a few of the great things in his teeming mind before the
death which he foresaw could overtake him. Time after time he
broke off to scribble in the margin ‘I have not time; I have not
time,’ and passed on to the next frantically scrawled outline.
What he wrote in those desperate last hours before the dawn
will keep generations of mathematicians busy for hundreds of
years. He had found, once and for all, the true solution of a
riddle which had tormented mathematicians for centuries:
under what conditions can an equation be solved? But this was
only one thing of many. In this great work, Galois used the
theory of groups (see chapter on Cauchy) with brilliant success.
Galois was indeed one of the great pioneers in this abstract
theory, to-day of fundamental importance in all mathematics.

In addition to this distracted letter Gsdois entrusted his
scientific executor with some of the manuscripts which had
been intended for the Academy of Sciences. Fourteen years
later, in 1’846, Joseph Liouville edited some of the manuscripts
for the Journal de Mathematiques •putts et appliquees* Liouville,
himself a distinguished and original mathematician, and editor
of the great Journal, writes as follows in his introduction:

“The principal work of Evariste Galois has as its object the
conditions of solvability of equations by radicals. The author
lays the foundations of a general theory which he applies in
detail to equations whose degree is a prime number. At the age of

4ia

ilEN OF MATHEMATICS

sixteen, and while a student at the college of Louis-le-Grand . . .
Galois occupied himself with this difficult subject/ Liou\ille
then states that the referees at the Academy had rejected
Galois’ memoirs on account of their obscurity. He continues:
‘An exaggerated desire for conciseness was the cause of tMs
defect which one should strive above all else to avoid when
treating the abstract and mysterious matters of pure Algebra,
Clarity is, indeed, aU the more necessary when one essays to
lead the reader farther from the beaten path and into wilder
territorj% As Descartes said, “When transcendental questions
are under discussion be transeendentally clear.” Too often
Galois neglected this precept; and we can understand how
illu^rious mathematicians may have judged it proper to try,
by the harshness of their sage advice, to turn a beginner, full
of genius but inexperienced, back on the right road. The author
they censured was before them, ardent, active; he could profit
by their advice.

‘But now everything is changed. Galois is no more! Let us not
indulge in useless criticisms; let us leave the defects there and
look at the merits.’ Continuing, liouville tells how he studied
the manuscripts, and singles out one perfect gem for special
mention.

‘My zeal was well rewarded, and I experienced an intense
pleasure at the moment when, having filled in some slight gaps,
I saw the complete correctness of the method by which Galois
proves, in particular, this beautifiil theorem: In order that an
irreducible equation of prime degree be solvable by radicals it is
necessary and sufficient that aU its roofs be rational functions of
any two of them J *

Galois addressed his will to his faithful friend Auguste
Chevalier, to whom the world owes its preservation. ‘My dear
fiiend’, he began, *I have made some new discoveries in analy-
sis/ He then proceeds to outline such as he has time for. They
were epoch-making. He’ concludes: ‘Ask Jacobi or Gauss
publicly to give their opinion, not as to the truth, but as to the
importance of these theorems. Later there wiU be, I hope, some

• The significance of this theorem will be clear if the reader will
  glance tiuongh the extracts from Abel in Chapter 17.

414

GENIUS AND STUPIDITY

people who will find it to their advantage to decipher all this
mess. Je temhrasse avec effusion. E. Galois.’

Confiding Galois! Jacobi was generous; what would Gauss
have said? What did he say of Abel? Whst did he omit to say
of Cauchy, or of Lobatchew^sky? For all his bitter experience
Galois was still a hopeful boy.

At a verj’’ early hour on 13 May 1832, Galois confronted his
adversary on the ‘field of honour’. The duel was with pistols at
twenty-five paces. Galois fell, shot through the intestines. No
surgeon was present. He was left lying where he had fallen. At
nine o’clock a passing peasant took him to the Cochin Hospital.
Galois knew he was about to die. Before the inevitable perito-
nitis set in, and while still in the full possession of his faculties,
he refused the offices of a priest. Perhaps he remembered his
father. His young brother, the only one of his family who had
been warned, arrived in tears. Galois tried to comfort him with
a show of stoicism. ‘Don’t cry’, he said, ‘I need all my courage
to die at twenty.’

Early in the morning of 31 May 1832 Galois died, being then
in the twenty-first year of his age. He was buried in the common
ditch of the South Cemetery, so that to-day there remains no
trace of the grave of ]6variste Galois. His enduring monument
is his collected works. They fill sixty pages.

CHAPTEE TWENTY-ONE

INVARIANT TWINS
Cayley ; Sylvester

• ‘It is difficult to give an idea of the vast extent of modem
  mathematics. The word ‘‘extent” is not the right one: I mean
  extent crowded with beautiful detail – not an extent of mere
  uniformity such as an objectless plain> but of a tract of beautiful
  country seen at first in the distance, but which will bear to be
  rambled through and studied in every detail of hillside and
  valley, stream, rock, wood, and flower. Rut, as for everything
  else, so for a mathematical theory – beauty can be perceived
  but not explained.’

These words from Cayley’s presidential address in 1883 to the
British Association for the Advancement of Science might well
be applied to his own colossal output. For prolific inventiveness
Euler, Cauchy, and Cayley are in a class by themselves, with
Poincare (who died younger than any of the others) a far
second. This applies only to the bulk of these men’s work; its
quality is another matter, to be judged partly by the frequency
with which the ideas originated by these giants recur in mathe-
matical research, partly by mere personal opinion, and partly
by national prejudice.

Cayley’s remarks about the vast extent of modern mathe-
matics suggest that we confine our attention to some of those
features of his own work which introduced distinctly new and
far-reaching ideas. The work on which his greatest fame rests
is in the theory of invariants and what grew naturally out of
that vast theory of which he, brilliantly sustained by his friend
Sylvester, was the originator and unsurpassed developer. The
concept of invariance is of great importance for modem phy-
sics, particularly in the theory of relativity, but this is not its
chief claim to attention. Physical theories are notoriously
416

INVARIANT TWINS

subject to re\usioii and rejection; the theory of invariance as a
permanent addition to pure mathematical thought appears to
rest on firmer ground.

Another of the ideas originated by Cayley, that of the geo-
metry of ‘higher space’ (space of n dimensions) is likewise of
present scientific significance but of incomparably greater
importance as pure mathematics. Similarly for the theory of
matrices, again an invention of Cayley’s. In non-Euclidean
geometry Cayley prepared the way for Klein’s splendid disco-
very that the geometry of Euclid and the non-Euclidean
geometries of Lobatchewsky and Riemann are, aU three, merely
different aspects of a more general kind of geometry which
includes them as special cases. The nature of these contributions
of Cayley’s will be briefly indicated after we have sketched his
life and that of his friend Sylvester.

The lives of Cayley and Sylvester should be written simulta-
neously, if that were possible. Each is a perfect foil to the other,
and the life of each, in large measure, supplies what is lacking
in that of the other. Cayley’s life was serene; Sylvester, as he
him self bitterly remarks, spent much of his spirit and energy
‘fighting the world’. Sylvester’s thought was at times as turbu-
lent as a millrace; Cayley’s was always strong, steady, and
unruffled. Only rarely did Cayley permit himself the printed
expression of anything less severe than a precise matheipatieal
statement – the simile quoted at the beginning of this chapter
is one of the rare exceptions; Sylvester could hardly talk about
mathematics without at once becoming almost orientally
poetic, and his unquenchable enthusiasm frequently caused
hirn to go off half-cocked. Yet these two became close friends
and inspired one another to some of the best work that either
of them did, for example in the theories of invariants and
matrices (described later).

With two such temperaments it is not surprising that the
course of friendship did not always run smoothly. Sylvester was
frequently on the point of exploding; Cayley sat serenely on the
safety valve, confident that his excitable friend would presently
cool down, when he would calmly resume whatever they had
been discussing as if Sylvester had never blown off, while

417

MEN OF MATHEMATICS

Sylvester for ids part ignored his hot-headed indiscretion – till
he got himself all steamed up for another. In many ways this
strangely congenial pair were like a honejnnoon couple, except
that one party to the friendship never lost his temper. Although
Sylvester was Cayley’s senior by seven years, we shall begin
with Cayley, Sylvester’s life breaks naturally into the calm
stream of Cayley’s like a jagged rock in the middle of a deep
river.

Arthur Cayley was bom on 16 August 1821 at Richmond,
Surrey, the second son of his parents, then residing temporarily
in England. On his father’s side Cayley traced his descent back
to the days of the Norman Conquest (1066) and even before, to
a baronial estate in Normandy. The family was a talented one
which, like the Darwin family, should provide much suggestive
material for students of heredity. His mother was Maria
Antonia Doughty, by some said to have been of Russian origin.
Cayley’s father was an English merchant engaged in the Rus-
sian trade; Arthur was bora during one of the periodical visits
of his parents to England.

In 1829, when Arthur was eight, the merchant retired, to live
thenceforth in England. Arthur was sent to a private school at
Blackheath and later, at the of fourteen, to King’s College
School in London. His mathematical genius showed itself very
early. The first manifestations of superior talent were like tliose
of Gauss; young Cayley developed an amazing skill in long
numerical calculations which he undertook for amusement. On
beginning the formal study of mathematics he quickly out-
stripped the rest of the school. Presently he was in a class by
himself, as he was later when he went up to the University, and
his teachers agreed that the boy was a bom mathematician
who should make mathematics his career. In grateful contrast
to Galois’ teachers, Cayley’s recognized his ability from the
beginning and gave him every encouragement. At first the
retired merchant objected strongly to Ms son’s becoming a
mathematiciaii but finally, won over by the Principal of the
schod, gave bis consent, his blessing, and his money. He
decided to send his son to Cambridge,

Cayley bqgan bis timvezsiiy career at the age of seventeen at

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INVARIANT TWINS

Trinity College, Cambridge. Among his fellow students he
passed as ‘a mere mathematician’ with a queer passion for
novel-reading. Cayley was indeed a lifelong devotee of the
somewhat stilted fiction, now considered classical, which
charmed readers of the 1840’s and ’50’s. Scott appears to have
been his favourite, with Jane Austen a close second. Later he
read Thackeray and disliked him; Dickens he could never
bring himself to read. B3rron’s tales in verse excited his admira-
tion, although his somewhat puritanical Victorian taste rebelled
at the best of the lot and he never made the acquaintance of
that diverting scapegrace Don Juan. Shakespeare’s plays, espe-
cially the comedies, were a perpetual delight to him. On the
more solid – or stodgier – side he read and re-read Grote’s
interminable History of Greece and Macaulay’s rhetorical
History of England, Classical Greek, acquired at school, re-
mained a reading-language for him all his life; French he read
and wrote as easily as English, and his knowledge of German
and Italian gave him plenty to read after he had exhausted the
Victorian classics (or they had exhausted him). The enjoyment
of solid fiction was only one of his diversions; others will be
noted as we go.

By the end of his third year at Cambridge Cayley was so far
in front of the rest in mathematics that the head examiner drew
a line under his name, putting the young man in a class by
himself ‘above the first’. In 1842, at the age of twenty-one,
Cayley was senior wrangler in the mathematical tripos, and in
the same year he was placed first in the yet more difficult test
for Snaith’s prize.

Under an excellent plan Cayley was now in line for a fellow-
ship which would enable him to do as he pleased for a few years.
He was elected Fellow of Trinity and assistant tutor for a
period of three years. His appointment might have been
renewed had he cared to take holy orders, but although Cayley
was an orthodox Church of England Christian he could not
quite stomach the thought of becoming a parson to hang on to
his job or to obtain a better one – as many did, without
disturbing either their faith or their conscience.

ffis duties were light almost to the point of non-existence,

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MEN OE MATHEMATICS

He took a few pupils, but not enough to hurt either himself or
his work. Making the best possible use of his liberty he con-
tinued the mathematical researches which he had begim as an
undergraduate. Like Abel, Galois, and many others who have
risen high in mathematics, Cayley went to the masters for his
inspiration. His first work, published in 1841 when he was an
undergraduate of twenty, grew out of his study of Lagrange
and Laplace.

With nothing to do but what he wanted to do after taking
his degree Cayley published eight papers the first year, four the
second, and thirteen the third. These early papers by the young
man who was not yet twenty-five when the last of them
appeared map out much of the work that is to occupy him for
the next fifty years. Already he has begun the study of geo-
metry of n dimensions (which he originated), the theory of
invariants, the enumerative geometry of plane curves, and his
distinctive contributions to the theory of elliptic functions.

During this extremely fruitful period he was no mere grind.
In 1843, when he was twenty-two, and occasionally thereafter
till he left Cambridge at the age of twenty-five, he escaped to
the Continent for delightful vacations of tramping, mountain-
eering, and water-colour sketching. Although he was slight and
frsul in appearance he was tough and wiry, and often after a
Jong night spent in tramping over MQy country, would turn up
as fresh as the dew for breakfast and ready to put in a few hours
at his mathematics. During his first trip he visited Switzerland
and did a lot of moxmtatneering. Thus began another lifelong
passion. His description of the ‘extent of modern mathematics’
is no mere academic exercise by a professor who had never
climbed a mountain or rambled lovingly over a tract of beau-
tiful country, but the accurate simile of a man who had known
nature intimately at first hand.

During the last four months of his first vacation abroad he
became acquainted with northern Italy. There began two
further interests which were to solace him for the rest of his
life: an imderstanding appreciation of architecture and a love
of good painting. He himself delighted in water-colours, in
which he showed marked talent. With his love of good litera-

INVARIANT TWINS

ture, travel, painting, and architecture, and with his deep
understanding of natural beauty, he had plenty to keep him
from degenerating into the ‘mere mathematician’ of conven-
tional literature – written for the most part by people who
may indeed have known some pedantic college professor of
mathematics, but who never in their lives saw a real mathe-
matician in the flesh.

In 1846, when he was twenty-five, Cayley left Cambridge.
No position as a mathematician was open to him unless possibly
he could square his conscience to the formality of ‘holy orders’.
As a mathematician Cayley felt no doubt that it would be
easier to square the circle. Anyhow, he left. The law, which with
the India Civil Service has absorbed much of England’s most
promisiog intellectual capital at one time or another, now
attracted Cayley. It is somewhat astonishing to see how many
of England’s leading barristers and judges in the nineteenth
century were high wranglers in the Cambridge tripos, but it
does not follow, as some have claimed, that a mathematical
training is a good preparation for the law. What seems less
doubtful is that it may be a social imbecility to put a young
man of Cayley’s demonstrated mathematical genius to drawing
up wills, transfers, and leases.

Following the usual custom of those looking toward an
Jfnglish legal career of the more gentlemanly grade (that is,
above the trade of solicitor), Cayley entered Lincoln’s Inn to
prepare himself for the Bar. After three years as a pupil of a
Mr Christie, Cayley was called to the Bar in 1849. He was then
twenty-eight. On being admitted to the Bar, Cayley made a
wise resolve not to let the law run off with bis brains. Deter-
mined not to rot, he rejected more business than he accepted.
For fourteen mortal years he stuck it, making an ample living
and deliberately turning away the opportunity to smother
himself in money and the somewhat blathery sort of renown
that comes to prominent barristers, in order that he might earn
enough, but no more than enougjvto enable him to get on with
Ms work.

BQs patience under the deadening routine of legal
business was exemplary, almost saintly, and his reputation in

L I B R A ^ ^21

MEN OF MATHEMATICS

his hTanch of the profession (conveyancing) rose steadily. It is
even recorded that his name has passed into one of the law
books in connexion with an exemplary piece of legal work he
did* But it is extremely gratifying to record also that Cayley
was no milk-and-water saint but a normal human being who
could, when the occasion called for it, lose his temper. Once he
and his friend Sylvester were animatedly discussing some point
in the theory of invariants in Cayley’s office when the boy
entered and handed Cayley a large batch of legal papers for his
perusal. A glance at what was in his hands brought him down
to earth with a jolt. The prospect of spending days straigh-
tening out some petty muddle to save a few pounds to some
comfortable client’s already plethoric income was too much for
the man with real brains in his head. With an exclamation of
disgust and a contemptuous reference to the ‘wretched rubbish’
in his hands, he hurled the stuff to the floor and went on talking
mathematics. This, apparently, is the only instance on record
when Cayley lost his temper. Cayley got out of the law at the
first opportunity – after fourteen years of it. But during his
period of servitude he had published between 200 and 300
mathematical papers, many of which are now classic.

As Sylvester entered Cayley’s life during the legal phase we
shall introduce him here.

James Joseph – to give him first the name with which he wa|
bom – was the youngest of several brothers and sisters, and was
bom of Jewish parents on 3 September 1814 in London. Very
little is known of his childhood, as Sylvester appears to have
been reticent about his early years. His eldest brother emi-
grated to the United States, where he took the name of Syl-
vester, an example followed by the rest of the family. But why
an orthodox Jew should have decorated himself with a name
favoured by Christian popes hostile to Jews is a mystery.
Possibly that eldest brother had a sense of humour; anyhow,
plain James Joseph, son of Ahraham Joseph, became hence-
forth and for evermore James Joseph Sylvester.

like Cayley’s, Sylvester’s mathematical genius showed itself
early. Between the ages of six and fourteen he attended private
schools. The last five months of his fourteenth year were spent

INVABIANT TWINS

at the University of London, where he studied under De
Morgan. In a paper written in 1840 with the somewhat mystical
title On the Derivation of Coeoaistence, Sylvester says ‘I am in-
debted for this term [recurrents] to Professor De Morgan,
whose pupil I may boast to have been’.

In 1829, at the age of fifteen, Sylvester entered the Royal
Institution at Liverpool, where he stayed less than two years.
At the end of his first year he won the prize in mathematics.
By this time he was so far ahead of his fellow students in
mathematics that he was placed in a special class by himself.
\Thile at the Royal Institution he also won another prize. This
is of particular interest as it establishes the first contact of
Sylvester with the United States of America where some of the
happiest – also some of the most wretched – days of his life were
to be spent. The American brother, by profession an actuary,
had suggested to the Directors of the Lotteries Contractors of
the United States that they submit a difficult problem in
arrangements to young Sylvester. The budding mathemati-
cian’s solution was complete and practically most satisfying to
the Directors, who gave Sylvester a prize of 500 dollars for hia
efforts.

The years at Liverpool were far from happy. Always coura-
geous and open, Sylvester made no bones about his Jewish
faith, but proudly proclaimed it in the face of more than petty
persecution at the hands of the sturdy young barbarians at the
Institution who humorously called themselves Christians. Bat
there is a limit to what one lone peacock can stand from a pack
of dull jays, and Sylvester finally fled to Dublin with only a few
shillings in his pocket. Luckily he was recognized in the street
by a distant relative who took him in, straightened him out, and
paid his way back to Liverpool.

Here we note another curious coincidence: Dublin, or at least
one of its citizens, accorded the religious refugee from liveipocl
decent human treatment on his first visit; on his second, some
eleven years later. Trinity College, Dublin, granted him the
academic degrees (B.A. and M.A.) which his own alma mater,
Cambridge University, had refused him because he could not,
being a Jew, subscribe to that remarkable compost of nonsen-

423

MEN OF MATHEMATICS

sical statements known as the Thirty-Nine Articles prescribed
by the Church of England as the minimum of religious belief
permissible to a rational mind. It may be added here, however,
that when English higher education finally unclutched itself
from the stranglehold of the dead hand of the Church in 1871
Sylvester was promptly given his degrees honoris causa. And it
^ould be remarked that in this as in other difficulties Sylvester
was no meek, long-sufiering martyr. He was full of strength and
courage, both physical and moral, and he knew how to put up
a devil of a fight to get justice for himself – and frequently did.
He was in fact a bom fighter with the untamed courage of a
lion.

In 1831, when he was just over seventeen, Sylvester entered
St John’s College, Cambridge. Owing to severe illnesses his
university career was interrupted, and he did not take the
mathematical tripos till 1837. He was placed second. The man
who beat him was never heard of again as a mathematician.
Not being a Christian, Sylvester was ineligible to compete for
Smith’s prizes.

In the breadth of his intellectual interests Sylvester resembles
Cayley. Physically the two men were no thing alike. Cayley,
though wiry and full of physical endurance as we have seen, was
frail in appearance and shy and retiring in manner. Sylvester,
short and stocky, with a magnificent head set firmly above
broad shoulders, gave the impression of tremendous strength
and vitality, and indeed he had both. One of his students said
he might have posed for the portrait of Hereward the Wake in
Charles Emgsley’s novel of the same name. As to interests out-
side of mathematics, Sylvester was much less restricted .and
far more liberal than Cayley. His knowledge of the Greek and
liatin classics in the originals was broad and exact, and he
tetained his love of them right up to his last illness. Many of his
papers are enlivened by quotations from these classics. The
quotations are always singularly apt and really do illuminate
the matter in band.

The same may be said for his allusions from other literatures.
It might amuse some literary scholar to go through the four
volumes of the collected M^athemaiical Papers and reconstruct

4^

INVARIANT TWINS

Sylvester’s wide range of reading from the credited quotations
and the curious hints thrown out without explicit reference. In
addition to the English and classical literatures he was well
acquainted with the French, German, and Italian in the ori-
ginals. His interest in language and literary form was keen and
penetrating. To him is due most of the graphic terminology of
the theory of invariants. Commenting on his extensive coinage
of new mathematical terms from the mint of Greek and Latin,
Sylvester referred to himself as the ‘mathematical Adam’.

On the literary side it is quite possible that had he not been a
very great mathematician he might have been something a
little better than a merely passable poet. Verse, and the ‘laws*
of its construction, fascinated him all his life. On his own
account he left much verse (some of which has been published),
a sheaf of it in the form of sonnets. The subject-matter of his
verse is sometimes rather apt to raise a smile, but he frequently
showed that he understood what poetry is. Another interest on
the artistic side was music, in which he was an accomplished
amateur. It is said that he once took singing lessons from
Gounod and that he used to entertain working-men’s gather-
ings with his songs. He was prouder of his ‘high C’ than he was
of his invariants.

One of the many marked differences between Cayley and
Sylvester may be noted here: Cayley was an omnivorous reader
of other mathematicians’ work; Sylvester found it intolerably
irksome to attempt to master what others had done. Once, in
later life, he engaged a young man to teach him some thing
about elliptic functions as he wished to apply them to the
theory of numbers (in particular to the theory of partitions,
which deals with the number of ways a given number can be
made up by adding together numbers of a given kind, say all
odd, or some odd and some even). After about the third lesson
Sylvester had abandoned his attempt to learn and was lecturing
to the young man on his own latest discoveries in algebra. But
Cayley seemed to know everything, even about subjects in
which he seldom worked, and his advice as a referee was sought
by authors and editors from ail over Europe. Cayley never for-
got anything he had seen; Sylvester had difficulty in remem-

425

MEN OF MATHEMATICS

bering his own inventions and once even disputed that a certain
theorem of his own could possibly be true. Even comparatively
trivial things that every worldng mathematician knows were
sources of perpetual wonder and dehght to Sylvester. As a
consequence almost any field of mathematics offered an en-
chanting world for discovery to Sylvester, while Cayley glanced
serenely over it all, saw what he wanted, took it, and went on to
something fresh.

In 1838, at the age of twenty-four, Sylvester got his first
regular job, that of Professor of Natural Philosophy (science in
general, physics in particular) at University College, London,
where his old teacher De Morgan was one of his colleagues.
Although he had studied chemistry at Cambridge, and retained
a lifelong interest in it, Sylvester found the teaching of science
thoroughly uncongenial and, after about two years, abandoned
it. In the meantime he had been elected a Fellow of the Royal
Society at the unusually early age of twenty-five. Sylvester’s
mathematical merits were so conspicuous that they could not
escape recognition, but they did not help him into a suitable
position.

At this point in his career Sylvester set out on one of the most
singular misadventures of his life. Depending upon how we look
at it, this mishap is silly, ludicrous, or tragic. Sanguine and filled
with his usual enthusiasm, he crossed the Atlantic to become
Professor of Mathematics at the University of Virginia in
1841 – the year in which Boole published his discovery of
invariants.

Sylvester endured the University only about three months.
The refusal of the University authorities to discipline a young
gentleman who had insulted him caused the professor to resign.
For over a year after this disastrous experience Sylvester tried
vainly to secure a suitable position, soliciting — unsuccessfully –
both Harvard and Columbia Universities. Failing, he returned
to England.

Sylvester’s experiences in America gave binn Ms fill of teach-
ing for the next ten years. On returning to London he became
an energetic actuary for a life insurance company. Such work
for E creative mathematician is poisonous drudgery,

INVARIANT TWINS

Sylvester almost ceased to be a matliematician. However, he
kept alive by taking a few private pupils, one of whom was to
leave a name that is known and fevered in every country of the
world to-day. This was in the early ISoO’s, the ‘potatoes,
prunes, and prisms’ era of female propriety when young women
were not supposed to think of much beyond dabbling in paints
and piety. So it is rather surprising to find that Sylvester’s most
distinguished pupil was a young woman, Florence Nightingale,
the first human being to get some decency and cleanliness into
military hospitals – over the outraged protests of bull-headed
military officialdom. Sylvester at the time was in his late
thirties, Miss Nightingale six years younger than her teacher.
Sylvester escaped from his makeshift ways of earning a living
iD the same year (1854) that Miss Nightingale went out to the
Crimean War.

Before this, however, he had taken another false step that
landed him nowhere. In 1846, at the age of thirty-two, he
entered the Inner Temple (where he coyly refers to himself as
‘a dove nestling among hawks’) to prepare for a legal career,
and in 1850 was called to the Bar. Thus he and Cayley came
together at last.

Cayley was twenty-nine, Sylvester thirty-six at the time;
both were out of the real jobs to which nature had called them.
Lecturing at Oxford thirty-five years later Sylvester paid
grateful tribute to ‘Cayley, who, though younger than myself is
my spiritual progenitor – who first opened my eyes and purged
them of dross so that they could see and accept the higher
mysteries of our common Mathematical faith.’ In 1852, shortly
after their acquaintance began, Sylvester refers to ‘Mr Cayley,
who habitually discomrses pearls and rubies’. Mr Cayley for his
part frequently mentions Mr Sylvester, but always in cold
blood, as it were. Sylvester’s earlier outburst of gratitude in
print occurs in a paper of 1851 where he says, ‘The theorem
above enunciated [it is his relation between the minor deter-
minants of linearly equivalent quadratic forms] was in part
suggested in the course of a conversation with Mr Cayley (to
whom I am indebted for my restoration to the enjoyment of
mathematical life). . ,

427

ilEN OF MATHEMATICS

Perhaps Sylvester overstated the case, but there was a lot ia
what he said. If he did not exactly rise from the dead he at least
got a new pair of lungs: from the hour of his meeting with
Cayley he breathed and lived mathematics to the end of his
days. The two friends used to tramp round the Courts of Lin-
coln’s Inn discussing the theory of invariants which both of
them were creating and later, when Sylvester moved away,
they contmued their mathematical rambles, meeting about
halfway between their respective lodgings. Both were bachelors
at the time.

The theory of algebraic invariants from which the various
extensions of the concept of invariance have grown naturally
originated in an extremely simple observation. As will he noted
in the chapter on Boole, the earliest instance of the idea appears
in Lagrange, from whom it passed into the arithmetical works
of Gauss. But neither of these men noticed that the simple but
remarkable algebraical phenomenon before them was the germ
of a vast theory. Nor does Boole seem to have fully realized
what he had found when he carried on and greatly extended the
work of Lagrange. Except for one slight tiff, Sylvester was
always Just and generous to Boole in the matter of priority, and
Cayley, of course, was always fair.

The simple observation mentioned above can be understood
by anyone who has ever seen a quadratic equation solved, and
is merely this. A necessary and sufficient condition that the
equation -f- c = 0 shall have two equal roots is that

— ac shall he zero. Let us replace the variable x by its value
in terms of y obtained by the transformation y = (prc -f q)l
{tx -r s). Thus a? is to be replaced by the result of solving this for
a?, namely a: = (g — sy)j{Ty — p). This transforms the given
equation into another in t/; say the new equation is Ay^ + 2By
-f C — 0. Carrying out the algebra we find that the new coeffi-
cients At, B, C are expressed in terms of the old a, i», c as follows,

A = as- — 2bsT -f
B = — aqs -f b(qr -j- sp) — cpr,

C = aq- — 2fypq -f- £P“.

From these it is easy to show {by hmte-force reductions, if
428

INVAEIANT TWINS

necessary, although there is a simpler way of reasoning the
result out, without actually calculating A, B, C) that

~ AC = (ps — qrY (&3 — ac).

Now 6® — flc is called the discriminant of the quadratic equation
in 05 ; hence the discriminant of the quadratic in t/ is — AC,
and it has been shown that the discriminant of the transformed
equation is equal to the discriminant of the original equation, times
the factor {ps — gr)® which depends only upon the coefficients
p, g, r, s in the transformation y = (px + q)l(rx + s) by tneans
of which X was expressed in terms of y,

Boole was the first (in 1841) to observe something worth
looking at in this particular trifle. Every algebraic equation has
a discriminant, that is, a certain expression (such as — oc for
the quadratic) which is equal to zero if, and only if, two or more
roots of the equation are equal. Boole first asked, does the dis-
criminant of every equation when its x is replaced by the related
y (as was done for the quadratic) come back unchanged except
for a factor depending only on the coefiBcients of the transfor-
mation? He foxmd that this was true. Next he asked whether
there might not be expressions other than discriminants con-
structed from the coefficients having this same property of
inoariance under transformation. He found two such for the
general equation of the fourth degree. Then another man, the
brilliant young German mathematician, F. M. G. Eisenstein
(1823-52) following up a result of Boole’s, in 1844, discovered
that certain expressions involving both the coefficients and the x
of the original equations exhibit the same sort of invariance: the
original coefficients and the original x pass into the transformed
coefficients and y (as for the quadratic), and the expressions in
question constructed from the originals differ from those con-
structed from the transforms only by a factor which depends
solely on the coefficients of the transformation.

Neither Boole nor Eisenstein had any general method for
finding such invariant expressions. At this point Cayley entered
the field in 1845 with his pathbreaking memoir, On the Theory of
Linear Transformations. At the time he was twenty-four. He
set himself the problem of fi Tiding uniform methods which

429

MEN OF MATHEMATICS

would give him all the invariant expressions of the land
described* To avoid lengthy explanations the problem has been
stated in terms of equations; actually it was attacked otherwise,
but this is of no importance here.

As this question of invariance is fundamental in modem
scientific thought we shall give three further illustrations of
what it means, none of which involves any symbols or algebra.
Imagine any figure consisting of intersecting straight lines and
curves drawn on a sheet of paper. Crumple the paper in any
way you please without tearing it, and try to think what is the
most obvious property of the figure that is the same before and
after crumpling. Do the same for any figure drawn on a sheet
of rubber, stretching but not tearing the rubber in any compli-
cated manner dictated by whim. In this case it is obvious that
sizes of areas and angles, and lengths of lines, have not remained
‘invariant*. By suitably stretching the rubber the straight lines
may be distorted into curves of almost any tortuosity you like,
and at the same time the original curves – or at least some of
them – may be transformed into straight Hues. Yet sometfiing
about the whole figure has remained unchanged; its very sim-
plicity and obviousness might well cause it to be overlooked.
This is the order of the points on any one of the lines of the
figure which mark the places where other lines intersect the
given one. Thus, if moving the pencil along a given line from
A to C, we had to pass over the point B on the Hne before the
figure was distorted, we shall have to pass over B in going from
A to C alter distortion. The order (as described) is an invariant
under the i)artictilar trimsforinaiions which crumpled the sheet
of paper into a crinkly ball, say, or which stretched the sheet
of rubber.’

This illustration may seem trivial, but anyone who has read
a non-mathematical description of the intersections of ‘world-
lines* iihgeneral relativity, and who recalls that an intersection
erf two such lines marks a physical ^pcdnt-event\ will see that
what we have been discussing is of the same stuff as one of our
pictures of the physical universe. The mathematical machinery
powerful enough to handle such complicated ‘txansformations*
and actually to produce the invariants was the creation of

430

INVARIANT TWINS

many workers, including Riemann, Christoffel, Ricci, Levi-
Civita, Lie, and Einstein – all names well known to readers of
popular accounts of relativity; the whole vast programme was
originated by the early workers in the theory of algebraic
invariants, of which Cayley and Sylvester were the true
founders,

, As a second example, imagine a knot to be looped in a string
whose ends are then tied together. Pulling at the knot, and
running it along the string, we distort it into any number of
‘shapes’. What remains ‘invariant’, what is ‘conserved’, under
all these distortions which, in this case, are our transformations?
Ob\dously neither the shape nor the size of the knot is invariant.
But the ‘style’ of the knot itself is invariant; in a sense that
need not he elaborated, it is the same sort of a knot whatever we
do to the string provided we do not untie its ends. Again, in the
older physics, energy was ‘conserved’; the total amount of
energy in the universe was assumed to be an invariant, the same
under ‘all transformations from one form, such as electrical
energy, into others, such as heat and light.

Our third illustration of invariance need be little more than
an allusion to physical science. An observer fixes his ‘position’
in space and time with reference to three mutually perpendi-
cular axes and a standard timepiece. Another observer, moving
relatively to the first, wishes to describe the same physical event
that the first describes. He also has his space-time reference
system; his movement relatively to the first observer can be
expressed as a transformation of his own co-ordinates (or of the
other observer’s). The descriptions given by the two may or
may not differ in mathematical form, according to the parti-
cular kind of transformation concerned. If their descriptions do
differ, the difference is not, obviously, inherent in the physical
event they are both observing, but in their reference systems
aud the transformation. The problem then arises to formulate
only those mathematical expressions of natural phenomena
which shall be independent, mathematically, of any particuldr
reference system and therefore be expressed by all observers in
the same form. This is equivalent to finding the invariants of
the transformation which expresses the most general shift in

431

aiEX OF MATHEMATICS

‘space-time’ of one reference system with, respect to any otha:.
Thus the problem of finding the mathematical expressions for
the intrinsic laws of nature is replaced by an attackable one in
the theorj’- of invariants. More will be said on this when we come
to Riemann.

In 1863 Cambridge University established a new professor-
ship of mathematics (the Sadlerian) and offered the post to
Ca 3 rley, who promptly accepted. The same year, at the age of
forty-two, he married Susan Moline. Although he made less
money as a professor of mathematics than he had at the law,
Cayley did not regret the change. Some years later the affairs
of the University were reorganized and Cayley’s salary was
raised. His duties also were increased from one course of lectures
during one term to two. EEis life was now devoted almost
entirely to mathematical research and university administra-
tion. In the latter his soimd business training, even temper,
impersonal judgement, and legal experience proved invaluable.
He never had a great deal to say, but what he said was usually
accepted as final, for he never gave an opinion without having
reasoned the matter through. His marriage and home life were
happy; he had two children, a son and a daughter. As he
gradually aged his mind remained as vigorous as ever and his
nature became, if anything, gentler. No harsh judgement
uttered in his presence was allowed to pass without a quiet
protest. To younger men and beginners in mathematical careers
he was always generous with his help, encouragement, and
sound advice.

During his professorship the higher education of women was
a hotly contested issue. Cayley threw all his quiet, persuasive
influence on the side of civilization and largely through his
efforts women were at last admitted as students (in their own
nunneries of course) to the monkish seclusion of medieval
Cambridge.

While Cayley was serenely mathematieizing at Cambridge
his friend Sylvester was still fighting the world. Sylvester never
married. In 1854f, at the age of forty, he applied for the profes-
sorship of mathematics at the Royal Military Academy, Wool-
wich. He did not get it- Nor did he get another position for

432

INVARIANT TWINS

whieli he applied at Gresham College, London. His trial lecture
was too good for the governing board. However, the successful
Woolwich candidate died the following year and Sylvester was
appointed. Among his not too generous emoluments was the
right of pasturage on the common. As Sylvester kept neither
horse, cow, nor goat, and did not eat grass himself, it is difficult
to see what particular benefit he got out of this inestimable
boon.

Sylvester held the position at Woolwich for sixteen years, till
he was forcibly retired as ‘superannuated’ in 1870 at the age of
fifty-six. He was still full of vigour but could do nothing against
the hidebound officialdom against him. Much of his great work
was stiU in the future, but his superiors took it for granted that
a man of his age must be through.

Another aspect of his forced retirement roused all his fighting
instincts. To put the matter plainly, the authorities attempted
to swindle Sylvester out of part of the pension which was legiti-
mately his. Sylvester did not take it lying down. To their
chagrin the would-be g3rppers learned that they were not brow-
beating some meek old professor but a man who could give
them a little better than he took. They came through with the
full pension.

While all these disagreeable things were happening in his
material affairs Sylvester had no cause to complain on the
scientific side. Honours frequently came his way, among them
one of those most highly prized by scientific men, foreign corre-
spondent of the French Academy of Sciences. Sylvester was
elected in 1863 to the vacancy in the section of geometry caused
by the death of Steiner.

After his retirement from Woolwich Sylvester lived in Lon-
don, versifying, reading the classics, playing chess, and enjoy-
ing himself generally, but not doing much mathematics. In
1870 he published his pamphlet, The Lems of Verse, by which
he set great store. Then, in 1876, he suddenly came to mathe-
matical life again at the age of sixty-twb. The ‘old ’man was
simply inextinguishable.

The Johns Hopkins University had been founded at Balti-
more in 1875 under the brilliant leadership of President Gilman.

433

MEN OE MATHEMi^TICS

Gilman had been advised to start off with an outstanding
classicist and the best mathematician he could afford as the
nucleus of his faculty. All the rest would follow, he was told,
and it did. Sylvester at last got a job where he might do prac-
tically as he pleased and in which he could do himself justice.
In 1876 he again crossed the Atlantic and took up his professor-
ship at Johns Hopkins. His salary was generous for those days,
five thousand dollars a year. In accepting the call Sylvester
made one curious stipulation; his salary was *to be paid in gold’.
Perhaps he was thinking of Woolwich, which gave him the
equivalent of $2750.00 (plus pasturage), and wished to be sure
that this time he really got what was coming to him, pension or
no pension.

The years from 1876 to 1883 spent at Johns Hopkins were
probably the happiest and most tranquil Sylvester had thus far
known. Although he did not have to ‘fight the world’ any longer
he did not recline on his honours and go to sleep. Forty years
seemed to fall from his shoulders and he became a vigorous
young man again, blazing with enthusiasm and scintillating
with new ideas. He was deeply gratefiil for the opportunity
Johns Hopkins gave him to begin his second mathematical
^ career at the age of sixty-three, and he was not backward in
expressing his gratitude publicly, in his address at the Com-
memoration Day Exercises of 1877.

In this Address he outlined what he hoped to do (he did it) in
his lectures and researches,

*There are thing s called Algebraical Forms. Professor Cayley
calls them Quantics. [Examples: + 2bxy -f cy^, aa^ -f

• -f the numerical coefificients 1,2,1 in the first,

1, 8,3,1 in the second, are binomial coefficients, as in the third
and fourth lines of Pascal’s triangle (Chapter 5); the next in
order would be aj^ -f + to/® -f j/^]. They are not,

properly speaking, GeometriGal Forms, although capable, to
some extent, of being embodied in them, but rather schemes of
proce^, or of operations for forming, for ca-IUng into existence,
as it were, Algebraic quantities*

‘To every such Quantic is associated an infinite variety of
other forms that may be regarded as engendered from and

434

INVARIANT TWINS

floatiQg, like an atmosphere, around it – but infinite as were
these derived existences, these emanations from the parent
form, it is found that they admit of being obtained by composi-
tion, by mixture, so to say, of a certain limited number of
fundamental forms, standard rays, as they might be termed in
the Algebraic Spectrum of the Quantic to which they belong.
And, as it is a leading pursuit of the Physicists of the present
day [1877, and even to-day] to ascertain the fixed lines in the
spectrum of every chemical substance, so it is the aim and
object of a great school of mathematicians to make out the
fundamental derived forms, the Covarianis [that kind of “inva-
riant’ expression, already described, which involves both the
variables and the coefficients of the form or quantic] and
Invariants, as they are called, of these Quantics.’

To mathematical readers it will be evident that Sylvester is
here giving a very beautiful analogy for the fundamental
system and the syzygies for a given form; the non-mathematical
reader may be recommended to re-read the passage to catch the
spirit of the algebra Sylvester is talking about, as the analogy
is really a close one and as fine an example of ‘popularized’
mathematics as one is likely to find in a year’s marching.

In a footnote Sylvester presently remarks ‘I have at present a
class of from eight to ten students attending my lectures on the
Modem Higher Algebra. One of them, a young engineer,
engaged from eight in the morning to six at night in the duties
of his office, with an interval of an hour and a half for his dinner
or lectures, has furnished me with the best proof, and the best
expressed, I have ever seen of what I call [a certain theorem].
. . Sylvester’s enthusiasm ~ he was past sixty – was that of a
prophet inspiring others to see the promised land which he
discovered or was about to discover. Here was teaching at its
best, at the only level, in fact, which justifies advanced teaching
at all.

He had complimentary things to say (m footnotes) about the
country of his adoption: . I believe there is no nation in the
world where ability with character counts for so much, and the
mere possession of wealth (in spite of all that we hear about the
Almighty dollar), for so little as in America. . .

435

MEN OF MATHEMATICS

He also tells how his dormant mathematical instincts were
again aroused to full creative power. ‘But for the persistence of
a student of this University [Johns Hopkins] in urging upon me
his desire to study with me the modern Algebra, I should never
have been led into this investigation. … He stuck with perfect
respectfulness, but with invincible pertinacity, to his point.
He would have the New Algebra (Heaven knows where he had
heard about it, for it is almost unknown on this continent), that
or nothing. I was obliged to yield, and what was the conse-
quence? In trying to throw light on an obscure explanation in
our text-book, my brain took fire. I plunged with requickened
zeal into a subject which I had for years abandoned, and found
food for thoughts which have engaged my attention for a con-
siderable time past, and will probably occupy aU my powers of
contemplation advantageously for several months to come.’

Almost any public speech or longer paper of Sylvester’s con-
tains much that is quotable about mathematics in addition to
technicalities. A refreshing anthology for beginners and even
for seasoned mathematicians could be gathered from the pages
of his collected works. Probably no other mathematician has so
transparently revealed his personality through his writings as
has Sylvester. He liked meeting people and infecting them with
his own contagious enthusiasm for mathematics. Thus he says,
truly in his own case, ‘So long as a man remains a gregarious
and sociable being, he cannot cut himself off from the gratifica-
tion of the instinct of imparting what he is learning, of propa-
gating through others the ideas and impressions seething in his
own brain, without stunting and atroph5Tng his moral nature
and diyiog up the surest sources of his future intellectual
replenishment.’

As a pendant to Cayley’s description of the extent of modem
mathematics, we may hang Sylvester’s beside it. ‘I should be
sorry to suppose that I was to be left for long in sole possession
of so vast a field as is occupied by modem mathematics.
Mathematics is not a book confined within a cover and bound
between brazen clasps, whose contents it needs only patience to
ransack; it is not a mine, whose treasures may take long to
reduce into possession, but which fill only a limited number of

4 ^

INVARIANT TWINS

veins and lodes; it is not a soil, whose fertility can be exhausted
by the yield of successive harvests; it is not a continent or an
ocean, whose area can be mapped out and its contour defined:
it is limitless as that space which it finds too narrow for its
aspirations; its possibilities are as infinite as the worlds which
are forever crowding in and multiplying upon the astronomer’s
gaze; it is as incapable of being restricted within assigned
boundaries or being reduced to definitions of permanent
validity, as the consciousness, the life, which seems to slumber
in each monad, in every atom of matter, in each leaf and bud
and cell, and is forever ready to burst forth into new forms of
vegetable and animal existence.’

In 1878 the American Journal of Mathematics was founded by
Sylvester and placed under his editorship by Johns Hop kins
University. The Journal gave mathematics in the United States
a tremendous urge in the right direction – research. To-day it is
still flourishing mathematically but hard pressed financially.

Two years later occurred one of the classic incidents in
Sylvester’s career. We tell it in the words of Dr Fabian Frank-
lin, Sylvester’s successor in the chair of mathematics at Johns
Hopkins for a few years and later editor of the Baltimore
American, who was an eye (and ear) witness.

‘He [Sylvester] made some excefient translations from
Horace and from German poets, besides writing a number of
pieces of original verse. The tours deforce in the way of rhyming,
which he performed while in Baltimore, were designed to illus-
trate the theories of versification of which he gives illustrations
in his little book called The Laws of Verse. The reading of the
Rosalind poem at the Peabody Institute was the occasion of an
amusing exhibition of absence of mind. The poem consisted of
no less than four hundred lines, all rhyming with the name
Rosalind (the long and short sound of the i both being allowed).
The audience quite filled the hall, and expected to find much
interest or amusement in listening to this unique experiment in
verse. But Professor Sylvester had foxmd it necessary to write
a large number of explanatory footnotes, and he announced
that in order not to interrupt the poem he would read the foot-
notes in a body first. Nearly every footnote suggested some

437

MEN OF MATHEMATICS

additional extempore remark, and the reader was so interested
in each one that he was not in the least aware of the flight of
time, or of the amusement of the audience. ^Vhen he had dis-
patched the last of the notes, he looked up at the clock, and was
horrified to find that he had kept the audience an hour and a
haK before beginning to read the poem they had come to hear.
The astonishment on his face was answered by a burst of good-
humoured laughter from the audience; and then, after begging
all Ms hearers to feel at perfect liberty to leave if they had
engagements, he read the Rosalind poem.*

Doctor Franklin’s estimate of Ms teacher sums the man up
admirably; “Sylvester was quick-tempered and impatient, but
generous, charitable and tender-hearted. He was always ex-
tremely appreciative of the work of others and gave the
warmest recognition to any talent or ability displayed by his
pilpiis. He was capable of flying into a passion on slight provo-
cation, but he did not harbour resentment, and was always glad
to forget the cause of quarrel at the earliest opportunity,*

Before taking up the thread of Cayley’s life where it crossed
Sylvester’s again, we shall let the author of Bosalind describe
how he made one of his most beautiful discoveries, that of what
are called ‘canonical forms’. [This means merely the reduction
of a given ‘quantic* to a ‘standard’ form. For example -f
2h3By -f- cy^ can be expressed as the sum of two squares, say

y2. QQ*h ^ -j- lOcaj^l/2 4- -f-

can be expressed as a sum of three fifth powers, X® -f- y®
4-

‘I discovered and developed the whole theory of canonical
binary forms for odd degrees, and, so far as yet made out, for
even degrees* too, at one sitting, with a decanter of port wine
to sustain nature’s flagging energies, in a back office in Lincoln’s
Inn Fields. The work was done, and well done, but at the usual
cost of racking thought – a brain on fire, and feet feeling, or
feelingless, as if plunged in an ice-pail. Thai night we slept no
tTiore.* Experts agree that the symptoms are unmistakable. But

• This part of the theory was developed many years later by E. K.
  Wakeford (1894-1916), who lost his life in the First World War- ‘Now*
  God be thanked who has matched us with his hour’ (Rupert Brooke).

438

INVARIANT TWINS

it must have been ripe port, to judge by what Sylvester got out
of the decanter.

Cayley and Sylvester came together again professionally
when Cayley accepted an invitation to lecture at Johns Hopkins
for a year in 1881-82. He chose Abelian fimctions, in which
he was researching at the time, as his topic, and the sisty-seven-
vear-old Sylvester faithfully attended every lecture of his
famous friend. Sylvester had still several prolific years ahead of
him, Cayley not quite so many.

We shall now briefly describe three of Ca^dey’s outstanding
contributions to mathematics in addition to his work on the
theory of algebraic invariants. It has already been mentioned
that he invented the theory of matrices, the geometry of space
of n dimensions, and that one of his ideas in geometry threw a
new light (in Klein’s hands) on non-Euclidean geometry. We
shall begin with the last because it is the hardest.

Desargues, Pascal, Poncelet, and others had created projec-
five geometry (see chapters 5, 18) in which the object is to dis-
cover those properties of figures which are invariant under
projection. Measurements – sizes of angles, lengths of lines –
and theorems which depend upon measurement, as for example
the Pythagorean proposition that the square on the longest side
of a right angle is equal to the sum of the squares on the other
two sides, are not projective but metrical, and are not handled
by ordinary projective geometry. It was one of Cayley’s greatest
achievements in geometry to transcend the barrier which,
before he leapt it, had separated projective from metrical pro-
perties of figures. From his higher point of yiew metrical geo-
metry also became projective, and the great power and flexi-
bility of projective methods were shown to be applicable, by
the introduction of ‘imaginary’ elements (for instance points
whose co-ordinates involve v/ — 1) to metrical properties.
Anyone w’ho has done any analytic geometry will recall that
two circles intersect in four points, two of which are always
‘imaginary’. (There are cases of apparent exception, for
example concentric circles, but this is close enough for our
purpose.) The fundamental notions in metrical geometry are the
distance between two points and the angle between two lines-

489 ^

M.1I.–V0L, II.

MEN OF MATHEMATICS

Replacing the concept of distance by another, also involving
‘imaginary’ elements, Cayley pro^dded the means for unifying
Euclidean geometry and the common non-EucIidean geometries
into one comprehensive theory. Without the use of some
algebra it is not feasible to give an intelligible account of how
this may be done; it is sufficient for our purpose to have noted
Cayley’s main advance of uniting projective and metrical
geometry with its cognate unification of the other geometries
just mentioned.

The matter of w -dimensional geometry when Cayley first put
it out was much more mysterious than it seems to us to-day,
accustomed as we are to the special case of four dimensions
(space-time) in relativity. It is still sometimes said that a four-
dimensional geometry is inconceivable to human beings. This
is a superstition which was exploded long ago by Pliicker; it is
easy to put four-dimensional figures on a flat sheet of paper,
and so far as geometry is concerned the whole of a four-dimen-
sional ‘space’ can be easily imagined. Consider first a rather
imeonventional three-dimensional space: all the circles that
may be drawn in a plane. This ‘all’ is a three-dimensional
‘space’ for the simple reason that it takes precisely three
numbers, or three co-ordinates, to individualize any one of the
swarm of circles, namely fsaro to fix the position of the centre
with reference to any arbitrarily given pair of axes, and one to
give the length of the radius.

If the reader now wishes to visualize a four-dimensional space
he may think of straight lines, instead of points, as the element
out of which our common ‘solid’ space is built. Instead of our
fa miliar solid space looking like an agglomeration of infinitely
fine birdshot it now resembles a cosmic haystack of infinitely
thin, infinitely long straight straws. That it is indeed four-
dimensional in straight lines can be seen easOy if we convince
ourselves (as we may do) that precisely Jour numbers are neces-
sary and sufficient to individualize a particular straw in our
haystack. The ‘dimensionaliW’ of a ‘space’ can be anything we
choose to make it, provided we suitably select the elements
(points, circles, lines, etc.) out of which we build it. Of course
if we take points as the elements out of which our space is to be

440

INVARIANT TWINS

constructed, nobody outside of a lunatic asylum has yet suc-
ceeded in visualizing a space of more than three dimensions.

Modem physics is fast teaching some to shed their belief in a
» mysterious ‘absolute space’ over and above the mathematical
‘spaces’ – like Euclid’s, for example – that were constructed by
geometers to correlate their physical experiences. Geometry
to-day is largely a matter of analysis, but the old terminology
of ‘points’, ‘lines’, ‘distances’, and so on, is helpful in suggesting
interesting things to do with our sets of co-ordinates. But it
does not foUow that these particular things are the most useful
that might be done in analysis; it may turn out some day that
all of them are comparative trivialities by more significant
things which we, liideboxmd in outworn traditions, continue to
do merely because we lack imagination.

If there is any mysterious virtue in talking about situations
which arise in analysis as if we were back with Archimedes
drawing diagrams in the dust, it has yet to be revealed. Pictures
after all may be suitable only for very young children ; Lagrange
dispensed entirely with such infantile aids when he composed
his analytical mechanics. Our propensity to ‘geometrize’ our
analysis may only be evidence that we have not yet grown up.
Newton himself, it is known, first got his marvellous results
analytically and reclothed them in the demonstrations of an
Apollonius partly because he knew that the multitude –
mathematicians less gifted than himself – would believe a
theorem true only if it were accompanied by a pretty picture and
a skilled Euclidean demonstration, partly because he himselt still
lingered by preference in the pre-Cartesian twilight of geometry-
The last of Cayley’s great inventions which we have selected
for mention is that of matrices and their algebra in its broad
outline. The subject originated in a memoir of 1858 and grew
directly out of simple observations on the way in which the
transformations (linear) of the theory of algebraic invariants
are combined. Glancing back at what was said on discri mina n t s
and their invariance we note the transformation (the arrow is

liere read ‘is replaced by’) y— Suppose we have two

rx -r 3

such transformations,

£ S

441

MEN OP MATHEMATICS

px ^ q

Pz-^Q
Ez^ s’

the second of wliieh is to be applied to the 02 in the first. We get
{pP~ qE)z (pQ -f qS)

^ (rP – sE)z ^ {rQ ^ sS)‘

Attending only to the coefficients in the three transformations
we write them in square arrays, thus

|p 9 ‘ ii-P Qii I’p-P -r giJ pQ + qS

jr s;:’ ilfi 5 ;i’ jjrP-i-siJ rQ -f sS j’

and see that the result of performing the first two transforma-
tions successively could have been written down by the follow-
ing rule of ‘multiplication’,

Ip 9!! ^ pP «’j ^ ppi’ + qR PQ^ 9 ^]!

\r si i|i 2 jjrP-fsS

where the roa’S of the array on the right are obtained, in an
obvious way, by applying the wxs of the first array on the left
onto the columns of the second. Such arrays (of any number of
rows and columns) are called matrices. Their algebra follows
from a few simple postulates, of which we need cite only the

following. The matrices

are equal (by

definition) when, and only when, a = A^h B^c = C,d = D.
The sum of the two matrices just written is the matrix

la A h + result of multiplying h ^ by w

i;c -r t. a-f n \ ^ [Ic a

(any number) is the matrix The rule for ‘multi-

line ndij

pljdng’, X, (or ‘compounding’) matrices is as exemplified for

\P « i, ii^ above.

11 r sli |jS Sji

A distinctive feature of these rules is that multiplication is
not cemmutativei except for special kmds of matrices. For
example, by the rule we get

X Sil = + Pq + Q.s

||B Sj. jr s!i + Rq+Ss ’

442

INVARIANT TWINS

and the matrix on the right is not equal to that which arises
&om the multiplication

r s B S ‘

All this detail, particularly the last, has been given to illus-
trate a phenomenon of frequent occurrence in the history of
mathematics: the necessary mathematical tools for scientific
applications have often been invented decades before the
science to which the mathematics is the key was imagined. The
bizarre rule of ‘multiplication’ for matrices, by which we get
different results according to the order in which we do the
multiplication (unlike common algebra where a? = ?/ is always
equal toy X x), seems about as far from anything of scientific
or practical use as anything could possibly be. Yet sixty-seven
Tears after Cayley invented it, Heisenberg in 1925 recognized
in the algebra of matrices exactly the tool which he needed for
fais revolutionary work in quantum mechanics.

Cayley continued in creative activity up to the week of his
death, which occurred after a long and painful illness, borne
with resignation and unflinching courage, on 26 January 1895.
To quote the closing sentences of Forsyth’s biography: ‘But he
was more than a mathematician. With a singleness of aim,
which Wordsworth would have chosen for his “Happy War-
rior”, he persevered to the last in his nobly lived ideal. His life
had a significant influence on those who knew him [Forsyth was
a pupil of Cayley and became his successor at Cambridge] : they
admired his character as much as they respected Ms genius: and
they felt that, at his death, a great man had passed from the
world.’

Much of what Cayley did has passed into the main current of
mathematics, and it is probable that much more in his massive
Collected Mathematical Papers (thirteen large quarto volumes of
about 600 pages each, comprising 966 papers) will suggest
profitable forays to adventurous mathematicians for genera-
tions to come. At present the fashion is away from the fields of
Cayley’s greatest interest, and the same may be said for
Sylvester; but mathematics has a habit of returning to its old
problems to sweep them up into more inclusive syntheses.

443

MEN OF MATHEMATICS

In 1833 Henry Jolm Stephen Smith, the hrOliant Irish
specialist in the theoiy of numbers and Saviiian Professor of
Geometry in Oxford UniTersity, died in his scientific prime at
the age of fifty-seven. Oxford invited the aged Sylvester, then
in his seventieth year, to take the vacant chair. Sylvester
accepted, much to the regret of his innumerable friends in
America. But he felt homesick for his native land which had
treated him none too generously; possibly also it gave him a
certain satisfaction to feel that ‘’the stone which the builders
rejected, the same is become the head of the corner’.

The amazing old man arrived in Oxford to take up his duties
with a brand-new mathematical theory (‘Reciprocants’ –
differential invariants) to spring on his advanced students. Any
praise or just recognition always seemed to inspire Sylvester to
outdo himself. Although he had been partly anticipated in his
latest work by the French mathematician Georges Halphen, he
stamped it with his peculiar genius and enlivened it with his
ineffaceable individuality.

The inaugural lecture, delivered on 12 December 1885 at
Oxford when Sylvester was seventy-one, has all the fire and
enthusiasm of his early years, perhaps more, because he now
felt secure and knew that he was recognized at last by that
snobbish world which had fought him. Two extracts will give
some idea of the style of the whole,

*1116 theory I am about to expotmd, or whose birth I am
about to announce, stands to this [“the great theory of Inva-
riants”] in the relation not of a younger sister, but of a brother,
who, though of later birth, on the principle that the masculine
is more worthy than the feminine, or at all events, according to
the regulations of the Salic law, is entitled to take precedence
over his elder sister, and exercise supreme sway over their
united realms.’

Commenting on the unaccountable absencb of a term in a
certain algebraic expression he waxes lyric.

‘Still, in the case before us, this unexpected absence of a
member of the family, whose appearance might have been
looked for, made an impression on my mind, and even went to
the extent of acting on my emotibns,^ I began to thinlr of it as a

444

INVARIANT TWINS

sort of lost PIdad in an Algebraical Constellation, and in the
end, brooding over the subject, my feelings found vent, or
sought relief, in a thymed effusion, a jeu de sottise, which, not
without some apprehension of appearing singular or extrava-
gant, I will venture to rehearse. It will at least serve as an inter-
lude, and give some relief to the strain upon your attention
before I proceed to make my final remarks on the general
theory.

TO A MISSING MEMBER

OF A FAMILY OF TERMS IN AN ALGEBRAICAL FORMULA

Lone and discarded one! divorced by fate,

From thy wished-f or fellows – whither art flown?

Where lingeresi thou in thy bereaved estate.

Like some lost star or buried meteor stone?

Thou mindst me much of that presumptuous one
Who loth, aught less than greatest, to be great,

From Heaven”‘ s immensity fell headlong down
To live forlorn, self-centred, desolate:

Or who, new Heraklid, hard escile bore.

Now buoyed by hope, now stretched on rack of fear.

Till throned Asiraea, wafting to his ear
Words of dim portent through the Atlantic roar.

Bade him ^^the sanctuary of the Muse revere
And strew with flame the dust of Isis” shore,””

Having refreshed ourselves and bathed the tips of our fingers in
the Pierian spring, let us turn hack for a few brief moments to
a light banquet of the reason, and entertain ourselves as a sort
of after-course with some general reflections arising naturally
out of the previous matter of my discourse.’

If the Pierian spring was the old boy’s finger bowl at this
astonishing feast of reason, it is a safe bet that the faithful
decanter of port was never very far from his elbow.

Sylvester’s sense of the kinship of mathematics to the finer
arts found frequent expression in his writings. Thus, in a paper
on Newton’s rule for the discovery of imaginary roots of alge-
braic equations, he asks in a footnote ‘May not Music be

4A&

MEN OF MATHEMATICS

described as the Mathematic of sense, Mathematic as Music of
the reason? Thus the musician feels Mathematic, the mathe-
matician thinks Music – Music the dream, Mathematic the
vrorMng life – each to receive its consummation from the other
when the human intelligence, elevated to its perfect t^’pe, shall
shine forth glorified in some future Mozart-Diiichlet or Beet-
hoven-Gauss — a union aireadj’ not indistinctly foreshadowed
in the genius and labours of a Helmholtz!’

S3dvester loved life, even when he was forced to fight it, and
if ever a man got the best that is in life out of it, he did. He
gloried in the fact that the great mathematicians, except for
what may he classed as avoidable or accidental deaths, have
been long-lived and \igorous of mind to their dying days. In
his presidential address to the British Association in 1869 he
called the honour roll of some of the greatest mathematicians
of the past and gave their ages at death to bear out his thesis
that . there is no study in the world which brings into more
harmonious action all the faculties of the mind than [mathe-
matics], … or, like this, seems to raise them, by successive
steps of initiation, to higher and higher states of conscious
intellectual being. , . . The mathematician lives long and lives
young; the wings of the soul do not early drop off, nor do its
pores become clogged ^vith the earthy particles blown from the
dust\” highways of vulgar life.’

Sylvester was a living example of his own philosophy. But
even he at last began to bow to time. In 1893 – he was then
seventy-nine – his eyesight began to fail, and he became sad and
discouraged because he could no longer lecture with his old
enthusiasm- The following year he asked to he relieved of the
more onerous duties of his professorship, and retired to live,
lonely and dejected, in London or at Tunbridge WeUs. All his
brothers and sisters had long since died, and he had outlived
most of his dearest friends.

But even now he was not through. His mind was stfil vigor-
ous, although he himself felt that the keen edge of his inventive-
ness was dulled for ever. Late in 1896, in the eighty-second year
of his age, he found a new enthusiasm in a field which had
alwaj^ fascinated him, and he blazed up again over the theory

446

INYABIANT TWINS

of compound partitions and Goldbach’s conjecture that every
IS “fclic sum of two jennies*

He had not much longer, mule working at his mathematics
m his London rooms early in March 1897 he suffered a paraMic
stroke which destroyed his power of speech. He died on 15 March
1897, at the age of eighty-three. His life can be summed up in

Ms own words, ‘I really love my subject’.

CHAPTER TWENTY-TWO

MASTER AND PUPIL
Weiersirass; Sonja Kon’alewski

• Young doctors in mathematicsj anxiously seeking positions in
  which their training and talents may have some play, often ask
  whether it is possible for a man to do elementary teaching for
  long and keep alive mathematically. It is. The life of Boole is a
  partial answer; the career of Weierstrass, the prince of analysts,
  ‘the father of modem analysis’, is conclusive.

Before considering Weierstrass in some detail, we place him
chronologically with respect to those of his German contem-
poraries, each of whom, like him, gave at least one vast empire
of mathematics a new outlook during the second half of the
nineteenth centuiy and the first three decades of the twentieth.
The year 1855, which marks the death of Gauss and the break-
ing of the last link with the outstanding mathematicians of the
preceding century, may be taken as a convenient point of refer-
ence. In 1855 Weierstrass (1815-97) was forty; Kronecker
(1S2S-91), thirty-two; Riemann (1826-66), twenty-nine; Dede-
kind (1831-1916), twenty-four; while Cantor (1845-1918) was
a small boy of ten. Thus German mathematics did not lack
recruits to cany on the great tradition of Gauss. Weierstrass
was just gaining recognition; Eronecker was well started; some
of Riemann’s greatest work was already behind him, and
Dedekmd was entering the field (the theory of numbers) in
which he was to gain his greatest fame. Cantor, of course, had
not yet been heard from.

We have juxtaposed these names and dates because four of
the men mentioned, dissimilar and totally unrelated as much of
their finest work was, came together on one of the central
problems of all mathematies, that of irrational numbers:
Weierstrass and Dedekind resumed the discussion of irrationals

m

MASTER AND PUPIL

and continuity practically where Eudoxus had left it in the .
fourth century b.c.; Kronecker, a modern echo of Zeno, made
Weierstrass’ last years miserable by sceptical criticism of the
latter’s revision of Eudoxus; while Cantor, striking out on a new
road of his own, sought to compass the actual infinite itself
which is implicit – according to some – in the very concept of
continuity* Out of the work of Weierstrass and Dedekind deve-
loped the modern epoch of analysis, that of critical logical
precision in analysis (the calculus, the theory of functions of a
complex variable, and the theory of functions of real variables)
in distinction to the looser intuitive methods of some of the
older writers – invaluable as heuristic guides to discovery but
quite worthless from the standpoint of the Pythagorean ideal
of mathematical proof. As has already been noted, Gauss, Abel,
and Cauchy inaugurated the first period of rigour; the move-
ment started by Weierstrass and Dedekind was on a higher
plane, suitable to the more exacting demands of analysis in the
second half of the century, for which the earlier precautions
were inadequate.

One discovery by Weierstrass in particular shocked the intui-
tive school of analysts into a decent regard for caution: he pro-
duced a continuous curve which has no tangent at any point.
Gauss once called mathematics ‘the science of the eye’; it takes
more than a good pair of eyes to ‘see’ the curve which Weier-
strass presented to the advocates of sensual intuition.

Since to every action there is an equal and apposite reaction
it was but natural that all this revamped rigour should engender
its own opposition. Kronecker attacked it vigorously, even
viciously, and quite exasperatingly. He denied that it meant
anything. Although he succeeded in hurting the venerable and
kindly Weierstrass, he made but little impression on his conser-
vative contemporaries and practically none on mathematical
analysis. Kronecker was a generation ahead of his time. Not till
the second decade of the twentietli century did his strictures on
the currently accepted doctrines of continuity and irrational
numbers receive serious consideration. To-day it is true that
not all mathematicians regard Kronecker’s attack as mer^y the
release of his pent-up envy of the more famous Weierstrass

449

MEN OF MATHEMATICS

which some of his contemporaries imagined it to be, and it is
admitted that there may be something ~ not much, perhaps –
in his disturbing objections. “Whether there is or not, Kro-
necker’s attack was partly responsible for the third period of
rigour in modem mathematical reasoning, that which we our-
selves are attempting to enjoy. Weierstrass was not the only
fellow-mathematician whom Kronecker harried; Cantor also
suffered deeply under what he considered his influential col-
league’s malicious persecution. All these men will speak for
themselves in the proper place; here we are only attempting to
indicate that their lives and work were closely interwoven in
at least one comer of the goi^eous pattern.

To complete the picture we must indicate other points of
contact between Weierstrass, Kronecker, and Riemann on one
side and Kronecker and Dedekind on the other. Abel, we recall,
died in 1829, Galois in 1832, and Jacobi in 1851. In the epoch
under discussion one of the outstanding problems in mathe-
matical analysis was the completion of the work of Abel and
Jacobi on multiple periodic functions – elliptic functions,
Abelian functions (see chapters 17, 18). From totally different
points of -view Weierstrass and Riemann accomplished what
was to be done – Weierstrass indeed considered himself in some
degree a successor of Abel; Kronecker opened up new vistas in
elliptic functions but he did not compete with the other two in
the field of Abelian fimctions. Kronecker was primarily an
aiitiurietician and an algebraist; some of his best work went
into the elaboration and extension of the work of Galois in the
theory of equations. Thus Galois found a worthy successor not
too long after his death.

Apart from his forays into the domain of continuity and irra-
tionai numbers. Dedekind’s most original work was in the
higher arithmetic, which he revolutionized and. renovated. In
this Kronecker was his able and sagacious rival, but again their
whole approaches were entirely different and characteristic of
the two men: Dedekind overcame his difficulties in the theory
of algebraic numbers by taking refuge in the infinite (in hfg
theory of ‘ideals’, as will be indicated in the proper place);
Kronecker sought to solve his problems in the finite.

450

MASTER AND PUPIL

Karl Wilhelm Theodor Weierstrass, the eldest son of Wilhelm
Weierstrass (1790-1 8C9) and his wife Theodora Forst, was horn
on 31 October 1815, at Ostenfelde in the district of Miinster,
Germany. The father was then a customs officer in the pay of
the French. It may be recalled that 1815 was the year of
Waterloo; the French were still dominating Europe. That year
also saw the birth of Bismarck, and it is interesting to observe
that whereas the more famous statesman’s life work was shot
to pieces in World War I, if not earlier, the contributions of
his comparatively obscure contemporary to science and the
advancement of chdiization in general are even more highly
esteemed to-day than they were during his lifetime.

The Weierstrass family were devout liberal Catholics all their
lives; the father had been converted from Protestantism, pro-
bably at the time of his marriage. Karl had a brother, Peter
(died in 1904), and two sisters, Klara (1823-96), and Elise
(1826-98) who looked after Ms comfort all their lives. The
mother died in 1826, shortly after Elise’s birth, and the father
married again the following year. Little is known of Karl’s
mother, except that she appears to have regarded her husband
•^th a restrained aversion and to have looked on her marriage
Tvith moderated disgust. The stepmother was a typical German
housewife; her influence on the intellectual development of her
stepchildren was probably ml. The father, on the other hand,
was a practical idealist, and a man of culture who at one time
had been a teacher. The last ten years of his life were spent in
peaceful old age in the house of his famous son in Berlin, where
the two daughters also lived. None of the children ever married,
although poor Peter once showed an inclination toward matri-
mony which was promptly squelched by his father and sisters.

One possible discord in the natural sociability of the children
was the father’s uncompromising righteousness, domineering
authority, and Prussian pigheadedness. He nearly wrecked
Peter’s life with Ms everlasting lecturing and came perilously
close to doing the same by Karl, whom he attempted to force
into an uncongenial career without ascertaining where Ms
hriliiant young son’s abilities lay. Old Weierstrass had the
audacity to preach at his younger son and meddle in his aflairs

451

MEN OF MATHEMATICS

till the ‘boy’ was nearly forty. Lucidly B[ari was made of more
resistant stuff. As we shall see his fight against his father –
although he himself was probably quite unaware that he was
fighting the tyrant – took the not unosual form of making a
mess of the life his father had chosen for him. It was as neat a
defence as he could possibly have devised, and the best of it was
that neither he nor his father ever dreamed what was hap-
pening, although a letter of KarFs when he was sixty shows that
he had at last realized the cause of his early difficulties. Karl at
last got his way, but it was a long, roundabout way, beset with
trials and errors. Only a shaggy man like himself, huge and
rugged of body and mind, could have won through to the end.
Shortly after Karl’s birth the family moved to Western-
kotten, Westphalia, where the father became a customs officer
at the salt works. Westemkotten, like other dismal holes in
which Weierstrass spent the best years of his life, is known in
Germany to-day only because Weierstrass once was condemned
to rot there – only he did not rust; his first published work is
dated as having been written in 1841 (be was then twenty-six)
at Westemkotten. There being no school in the village, Karl
was sent to the adjacent town of Munster whence, at fourteen,
he entered the Catholic Gymnasium at Paderbom. Lake Des-
cartes under somewhat similar conditions, Weierstrass
thoroughly enjoyed his school and made friends of his expert,
civilized instructors. He traversed the set course in considerably
less than the standard time, making a uniformly brilliant record
in all his studies. He left in 1834 at the age of nineteen. Prizes
fell his way with unfailing r^ularity; one year he carried off
seven; he was usually first in Grerman and in two of the three,
Latin, Greek, and mathematics. By a beautiful freak of irony
he never won a prize for calligraphy, although he was destined
to teach penmanship to little boys but recently emancipated
from their mother’s apron strings.

As mathematicians often have a liking for music it is of
interest to note here that Weierstrass, broad as he was, could
not tolerate music in any form. It meant nothing to him and he
did not pretend that it did. “When he had become a success his
solicitous sisters tried to get him to take music lessons to make

452

MASTER AND PUPIL

him more conventional socially, but after a half-hearted lesson
or two he abandoned the distasteful project. Concerts bored
him and grand opera put him to sleep – when his sisters could
drag him out to either.

Like his good father, Karl was not only an idealist but was
also extremely practical – for a time. In addition to capturing
most of the prizes in purely impractical studies he secured a
paying job, at the age of fifteen, as accountant for a prosperous
female merchant in the ham and butter business.

All of these successes had a disastrous effect on Karl’s future.
Old ^yeie^st^ass, like many parents, drew the wrong conclusion
from his son’s triumphs. He ‘reasoned’ as follows. Because the
boy has won a cartload of prizes, therefore he must have a good
mind – this much may be admitted; and because he has kept
himself in pocket money by posting the honoured female butter
and ham merchant’s books efficiently, therefore he will be a
brilliant bookkeeper. Now what is the acme of all bookkeeping?
Obviously a government nest – in the higher branches of course

• in the Prussian civil service. But to prepare for this exalted
  position, a knowledge of the law is desirable in order to pluck
  effectively and to avoid being plucked.

As the grand conclusion of all this logic, paterfamilias Weier-
strass shoved his gifted son, at the age of nineteen, head first into
the University of Bonn to master the chicaneries of commerce
and the quihblings of the law.

Karl had more sense than to attempt either. He devoted his
great bodily strength, his lightning dexterity, and his keen mind
almost exclusively to fencing and the meUow sociability that is
induced by nightly and liberal indulgence in honest German
beer. What a shocking example for ant-eyed Ph.D.’s who
shrink from a spcU of school-teaching lest their dim lights be
dimmed for ever! But to do what Weierstrass did, and get away
with it, one must have at least a tenth of his constitution and
not less than one tenth of 1 per cent of his brains.

Boim found Weierstrass unbeatable, Ifis quick eye, his long
teach, his devilish accuracy, and his lightning speed in fencing
made hlrn an opponent to admire but not to touch. As a matter
of historical fact he never was touched; no jagger scar adorned

453

MEN OF MATHEMATICS

his cheeks, and in all his bouts he never lost a drop of blood.
Whether or not he was ever put under the table in the subse-
quent celebrations of his numerous \dctories is not known. His
discreet biographers are somewhat reticent on this important
point, but to anyone who has ever contemplated one of Weier-
strass’ mathematical masterpieces it is inconceivable that so
strong a head as his could ever have nodded over a half-gallon
stein. His four mis-spent years in the university were perhaps
after all well spent.

His experiences at Bonn did three things of the greatest
moment for Weierstrass: they cured him of his father fixation
without in any way damaging his affection for his deluded
parent; they made him a human being capable of entering fully
into the pathetic hopes and aspirations of human beings less
gifted than himself – his pupils – and thus contributed directly
to his success as probably the greatest mathematical teacher
of all time; and last, the humorous geniality of his boyhood
became a fixed life-habit. So the ‘student years’ were not the
loss his disappointed father and his fluttering sisters – to say
nothing of the panicky Peter – thought they were when Karl
returned, after four ‘empty’ years at Bonn, without a degree,
to the bosom of his wailing family.

There was a terrific row. They lectured him – ‘sick of body
and soul’ as he was, possibly the result of not enough law, too
little mathematics, and too much beer; they sat around and
glowered at him and, worst of all, they began to discuss him as
if he were dead: what was to be done with the corpse? Touching
the law, VTeierstrass had only one brief encounter with it at
Bonn, but it sufficed: he astonished the Dean and his friends by
his acute ‘opposition’ of a candidate for the doctor degree in
law* As for the mathematics at Bonn – it was inconsiderable.
The one gifted man, Julius Pliieker, who might have done
Weierstrass some good was so busy with his manifold duties
that he had no time to spare on individuals and Weierstrass got
nothing out of him.

But like Abel and so many other mathematicians of the first
rank, Weierstrass had gone to the masters in the interludes
between his fencing and drinking: he had been absorbing the

454

MASTER AND PUPIL

Celestial Mechanics of Laplace, thereby laying the foundations
for his lifelong interest in dynamics and systems of simultaneous
diSerential equations. Of course he could get none of this
through the head of his cultured, petty-ofhcial father, and his
obedient brother and his dismayed sisters knew not what the
de\’il he was talking about. The fact alone was sufficient:
brother Karl, the genius of the timorous little family, on whom
such high hopes of bourgeois respectability had been placed,
had come home, after four years of rigid economy on father’s
part, T\dthout a degree.

At last – after weeks – a sensible friend of the family who had
s}Tnpathized with Karl as a boy, and who had an intelligent
amateur’s interest in mathematics, suggested a way out: let
Karl prepare himself at the neighbouring Academy of Minister
for the state teachers’ examination. Young Weierstrass would
not get a Ph.D. out of it, but his job as a teacher would provide
a certain amount of evening leisure in which he could keep alive
mathematically provided he had the right stuff in him. Freely
confessing his ‘sins’ to the authorities, Weierstrass begged the
opportunity’ of making a fresh start. His plea was granted, and
Weierstrass matriculated on 22 May 1839 at Munster to prepare
hims elf for a secondary school teaching career. This was a most
important stepping stone to his later mathematical eminence,
although at the time it looked like a total rout.

Wliat made all the difference to Weierstrass was the presence
at Munster of Christof Gudermann (1798-1852) as Professor of
Mathematics. Gudermann at the time (1839) was an enthusiast
for elliptic functions. We recall that Jacobi had published his
Fundamenta nova in 1829. Although few are now familiar with
Gudermann’s elaborate investigations (published at the instiga-
tion of Crelle in a series of articles in his Journal), he is not to be
dismissed as contemptuously as it is sometimes fashionable to
do merely because he is outmoded. For his time Gudermann
bzd what appears to have been an original idea. The theorj’ of
iiliptic functions can be developed in many’ different ways – too
nany for comfort. At one time some particular way seems the
}est; at another, a slightty different approach is highly adver-
ised for a season and is generally regarded as being more chic*

455

MEN OE MATHEMATICS

Gudennann’s idea was to base everything on the power series
expansion of the functions. (This statement will have to do for
the moment; its meaning will become clear when we describe
one of the leading motivations of the work of Weierstrass.) This
really was a good new idea, and Gudermann slaved over it with
overwhelming German thoroughness for years without, perhaps,
realizing what lay behind his inspiration, and himself never
carried it through. The important thing to note here is that
Weierstrass made the theory of power series – Gudermann’s
inspiration – the nerve of ail his work in analysis. He got the
idea from Gudermann, whose lectures he attended. In later life,
contemplating the scope of the methods he had developed in
analysis, Weierstrass was wont to exclaim, “There is nothing
hut power series!’

At the opening lecture of Gudermann’s course on elliptic
functions (he called them by a different name, but that is of no
importance) there were thirteen auditors. Being in love with his
subject the lectucer quickly left the earth and was presently
soaring practically alone in the aether of pure thought. At the
second lecture only one auditor appeared and Gudermann was
happy. The solitary student was Weierstrass. Thereafter no
incautious third party ventured to profane the holy communion
between the lecturer and his unique disciple. Gudermann and
W’eierstrass were fellow Catholics; they got along splendidly
together.

Weierstrass was duly grateful for the pains Gudermann
lavished on him, and after he had become famous he seized
every opportunity – the more public the better – to proclaim
his gratitude for what Gudermann had done for him. The debt
was not inconsiderable: it is not every professor who can drop
a hint like the one – power series representation of functions as
a point of attack – which iospired Weierstrass. In addition to
the lectures on elliptic functions, Gudermann also gave Weier-
strass private lessons on ‘analytical spherics’ – whatever that
may have been.

In 1841 , at the age of twenty-six, Weierstrass took his exam-
inations for his teacher’s certificate. The examination was in
two sections, written and oral. For the first he was allowed six

45fi

MASTER AND PUPIL

montlis in which to write out essays on three topics acceptable
to ihe examiners. The third question inspired a fine dissertation
on the Socratic method in secondary teaching, a method which
Weierstrass followed with brilliant success when he became the
foremost mathematical teacher of advanced students in the
world.

A teacher – at least in higher mathematics – is judged by his
students. If his students are enthusiastic about his ‘beautifully
dear lectures’, of which they take copious notes, but never do
any original mathematics themselves after getting their
advanced degrees, the teacher is a flat failure as a university
instructor and his proper sphere – if anywhere – is in a secon-
dary school or a small college where the aim is to produce tame
gentlemen but not independent thinkers. Weierstrass’ lectures
were models of perfection. But if they had been nothing more
than finished expositions they would have been pedagogically
worthless. To perfection of form Weierstrass added that intan-
gible something which is called inspiration. He did not rant
about the sublimity of mathematics and he never orated; but
somehow or another he made creative mathematicians out of a
disproportionately large fraction of his students.

The examination which admitted Weierstrass after a year of
probationary teaching to the profession of secondary school
work is one of the most extraordinary of its kind on record. One
of the essays which he submitted must be the most abstruse
production ever accepted in a teacher’s examination. At the
candidate’s request Gudermann had set Weierstrass a real
mathematical problem: to find the power series developments
of the elliptic functions. There was more than this, but the part
mentioned was probably the most interesting,

Gudermann’s report on the work might have changed the
course of Weierstrass’ life had it been listened to, but it made no
practical impression where it might have done good. In a post-
script to the official report Gudermann states that ‘This pro-
blem, which in general would be far too difficult for a young
analyst, was set at the candidate’s express request with the
consent of the commission.’ After the acceptance of his written
work and the successful conclusion of his oral examination^

457

MEX or MATHEMATICS

Weierstrass got a special certificate on his original contribution
to matheiTiatics. Ha\Tiig stated what the candidate had done,
and having pointed out the originality of the attack and the
novelty of some of the results attained, Gudermann declares
that the work evinces a fine mathematical tEilent ‘which, pro-
vided it is not frittered away, will ine\dtabiy contribute to the
advancement of science. For the author’s sake and that of
science it is to be desired that he shall not become a secondary
teacher, but that favourable conditions will make it possible
for him to function in academic instruction. . . . The candidate
hereby enters by birthright into the ranks of the famous
discoverers,’

These remarks, in part underlined by Gudermann, were very
properly stricken from the official report. Weierstrass got his
certificate and that was all. At the age of twenty-six he entered
his trade of secondary teaching which was to absorb nearly
fifteen years of his life, including the decade from thirty to
forty which is usually rated as the most fertile in a scientific
man’s career.

His work was excessive. Only a man with iron determination
and a rugged physique could have done what Weierstrass did.
The nights were his own and he lived a double life. Not that he
became a dull drudge; far from it. Nor did he pose as the \illage
scholar absorbed in mysterious meditations beyond the com-
prehension of ordinary mortals. With quiet satisfaction in his
later years he loved to dwell on the way he had fooled them all;
the gay government officials and the young officers found the
amiable school teacher a thoroughly good fellow and a lively
tavern companion.

But in addition to these boon companions of an occasion^
night out, Weierstrass had another, unknown to his happy-go-
lucky fellows – Abel, with whom he kept many a long rigil. He
himself said that Abel’s works were never veiy far from his
elbow. ‘^Vhen he became the leading analyst in the world and
the greatest mathematical teacher in Europe his first and last
adAuee to his numerous students was ‘Read xAhel!’ For the great
Norwegian he had an unbounded admiration imdimmed by any
shadow of envy, ‘Abel, the lucky fellowl’ he woiQd exclaim: ‘He

458

MASTEE AND PUPIL

has done something everlasting! His ideas will always exercise a
fertilizing influence on our science.’

The same might be said for Weierstrass, and the creative
ideas with which he fertilized mathematics were for the most
part thought out while he was an obscure schoolteacher in
(jismal \Hlages where advanced books were unobtainable, and
at a time of economic stress when the postage on a letter
absorbed a prohibitive part of the teacher’s meagre weekly
wage. Being unable to afford postage, Weierstrass was barred
from scientific correspondence. Perhaps it is as well that he was;
his originality developed unhampered by the fashionable ideas
of the time. The independence of outlook thus acquired charac-
terized his work in later years. In his lectures he aimed to
develop everything from the groimd up in his own way and
made almost no reference to the work of others. This occasion-
ally mystified his auditors as to what was the master’s and
what another’s.

It will be of interest to mathematical readers to note one or
two stages in Weierstrass’ scientific career. After his proba-
tionary year as a teacher at the Gymnasium at Munster,
Weierstrass wrote a memoir on analytic functions in which,
among other things, he arrived independently at Cauchy’s
integral theorem – the so-called fundamental theorem of
analysis. In 1842 he heard of Cauchy’s work but claimed no
priority (as a matter of fact Gauss had anticipated them both
away back in 1811, but as usual had laid his work aside to
ripen). In 1842, at the age of twenty-seven, Weierstrass applied
the methods he had developed to systems of differential equa-
tions – such as those occurring in the Newtonian problem of
three bodies, for example; the treatment was mature and
rigorous. These works were undertaken without thought of
publication merely to prepare the ground on which Weierstrass’
lifework (on Abelian functions) was to be built.

In 1842 Weierstrass was assistant teacher of mathematics
and physics at the Pro-Gymnasium in Deutsch-Kxone, West
Prussia, Presently he was promoted to the dignity of ordinary
teacher. In addition to the subjects mentioned the leading
analyst in Europe also taught German, geography, and writing

459

MEN OF MATHEMATICS

to the little boys under his charge; gymnastics was added in
1845,

In 1848, at the age of thirty-three, Weierstrass was trans-
ferred as ordinary teacher to the Gymnasium at Braunsberg.
This was something of a promotion, but not much. The head of
the school was an excellent man who did what he could to make
things agreeable for Weierstrass although he had only a remote
conception of the intellectual eminence of his colleague. The
school boasted a very small library of carefully selected books
on mathematics and science.

It was in this year that Weierstrass turned aside for a few
weeks from his absorbing mathematics to indulge in a little
delicious mischief. The times were somewhat troubled politi-
cally; the Tiros of liberty had infected the patient German
people and at least a few of the bolder souls were out on the
warpath for democracy. The royalist party in power clamped a
strict censorship on all spoken or printed sentiments not suffi-
ciently laudatory to their regime. Fugitive hymns to liberty
began appearing in the papers. The authorities of course could
tolerate nothing so subversive of law and order as this, and
when Braunsherg suddenly blossomed out with a lush crop of
democratic poets all singing the praises of liberty in the local
paper, as yet uncensored, the flustered government hastily
appointed a local chil servant as censor and went to sleep,
believing that all would he well.

Unfortunately the newly appointed censor had a violent
aversion to aU forms of literature, poetry especially. He simply
could not bring himself to read the stuff! Confining his supervi-
sion to blue-pencilling the dull political prose, he turned over
all the literary efftisions to schoolteacher Weierstrass for cen-
soring. Weierstrass was delighted. Knowing that the official
censor would never glance at any poem, Weierstrass saw to it
that the most inflammatory ones were printed in full right
under the censor’s nose. This went merrily on to the great
delict of the populace till a higher official stepped in and put
an end to the farce. As the censor was the officially responsible
offender, Weierstrass escaped scot-free.

The obscure hamlet of Deutsch-Bfrone has the honour of

480

MASTER AND PUPIL

being the place where Weierstrass (in 1842-43) ihrst broke into
print. German schools publish occasional ‘programmes’ con-
taining papers by members of the staff. Weierstrass contributed
Bemarks on Analytical Factorials. It is not necessary to explain
what these are; the point of interest here is that the subject of
factorials was one which had caused the elder analysts many a
profitless headache. Until Weierstrass attacked the problems
connected with factorials the nub of the matter had been
missed.

Crelie, we recall, wrote extensively on factorials, and we have
seen how interested he was when Abel somewhat rashly in-
formed him that his work contained serious oversights. Crelle
now enters once more, and again in the same fine spirit he
showed Abel.

Weierstrass’ work was not published till 1856, fourteen years
after it had been written, when Crelle printed it in his Journal.
Weierstrass was then famous. Admitting that the rigorous
treatment by Weierstrass clearly exposes the errors of his own
work, Crelle continues as follows: ‘I have never taken the
personal point of view in my work, nor have I striven for fame
and praise, but only for the advancement of truth to the best
of my ability; and it is aU one to me whoever it may be that
comes nearer to the truth – whether it is I or someone else,
provided only a closer approximation to the truth is attained.*
There was nothing neurotic about Crelle. Nor was there abput
Weierstrass.

Whether or not the tiny village of Deutsch-Krone is con-
spicuous on the map of politics and commerce it stands out like
the capital of an empire in the history of mathematics, for it
was there that Weierstrass, without even an apology for a
library and with no scientific connexions whatever, laid the
foundations of his life work – ‘to complete the life work of Abel
and Jacobi growing out of Abel’s Theorem and Jacobi’s dis-
covery of multiple periodic functions of several variables,’

Abel, he observes, cut down in the flower of his youth, had no
opportunity to follow out the consequences of his tremendous
discovery, and Jacobi had failed to see clearly that the true
meaning of his own work was to he sought in Abel’s Theorem^

461

MEN OF MATHEMATICS

‘The consolidation and extension of these gains – the task of
actually exhibiting the functions and working out their pro-
perties – is one of the major problems of mathematics.’ Weier-
strass thus declares his intention of devoting his energies to this
problem as soon as he shall have understood it deeply and have
developed the necessarj’ tools. Later he tells how slowly he
progressed: ‘The fabrication of methods and other difiScuit
problems occupied my time. Thus years slipped away before I
could get at the main problem itself, hampered as I was by an
unfavourable en\lroiLment.’

The whole of Weierstrass’ work in analysis can be regarded
as a grand attack on his main problem. Isolated results, special
developments, and even extensive theories – for example that
of irrational numbers as developed by him – all originated ic
some phase or another of the central problem. He early became
convinced that for a dear understanding of what he was
attempting to do a radical revision of the fundamental concepts
of mathematical analysis was necessary, and from this convic-
tion he passed to another, of more significance to-day perhaps
than the central problem itself: analysis must be founded on the
common whole numbers 1,2,3, . . . The irrationals which give
us the concepts of limits and continuity, from which analysis
springs, must be referred back by irrefrangible reasoning to the
integers; shoddy proofe must be discarded or reworked, gaps
must be filled up, and obscure ‘axioms’ must be dragged out
into the light of critical inquiry till all are understood and all
are stated in comprehensible language in terms of the integers.
This in a sense is the Pythagorean dream of basing all mathe-
matics on the integers, but Weierstrass gave the programme
constructh’-e definiteness and made it work.

Thus originated the nineteenth-century movement known as
the arithmetization of analysis — something quite different from
Kronecker’s arithmetical programme, at which we shall glance
in a later chapter; indeed the two approached were mutually
antagonistic.

In passing it may be pointed out that Weierstrass’ plan for
his Hfe work and his magnificent accomplishment of most of
what he set himself as a young man to do, is a good Olustratloii

462

MASTER AND PUPIL

of the value of the advice Felix Klein once gave a perplexed
student who had asked him the secret of mathematical disco-
very. ‘You must have a problem’, Edein replied. ‘Choose one
definite objective and drive ahead toward it. You may never
reach your goal, but you will find something of interest on the
way.’

From Deutsch-Krone Weierstrass moved to Braimsberg,
where he taught in the Royal Catholic Gymnasium for six
years, beginning in 1848. The school ‘programme’ for 1848-9
contains a paper by Weierstrass which must have astonished
the native: Contributions to the Theory of Abelian Integrals. If
this work had chanced to fall under the eyes of any of the pro-
fessional mathematicians of Germany, Weierstrass would have
been made. But, as his Swedish biographer, Mittag-Lefifier,
dniy remarks, one does not look for epochal papers on pure
mathematics in secondary-school programmes. W^eierstrass
might as well have used his paper to light his pipe.

EDs next effort fared better. The summer vacation of 1853
(Weierstrass was then thirty-eight) was passed in his father’s
house at TVestemkotten. Weierstrass spent the vacation writing
up a memoir on Abelian functions. When it was completed he
sent it to Crelle’s great Journal, It was accepted and appeared
in volume 47 (1854).

This may have been the paper whose composition was respon-
sible for an amusing incident in Weierstrass’ career as a school-
teacher at Braunsberg. Early one morning the director of the
school was startled by a terrific uproar proceeding from the
classroom where Weierstrass was supposed to be holding forth.
On investigation he discovered that Weierstrass had not shown
up. He hurried over to Weierstrass’ dwelling, and on knocking
was bidden to enter. There sat Whierstrass pondering by the
glimmering light of a lamp, the curtains of the room still drawn.
He had worked the whole night through and had not noticed
the approach of dawn. The director called his attention to the
fact that it was broad daylight and told him of the uproar in his
classroom, Weierstrass replied that he was on the trail of an
important discovery which would rouse great interest in the
scientific world and he could not possibly interrupt his work.

463

MEN OF 3IATHEMATICS

The memoir on AbeKan functions published in Crelle’s
Journal in 1854 created a sensation. Here was a masterpiece
from the pen of an unknown schoolmaster in an obscure village
nobody in Berlin had ever heard of. This in itself was suffi-
ciently astonishing. But what surprised those who could appre-
ciate the magnitude of the work even more was the almost
unprecedented fact that the sontarj’ worker had published no
preliminary bulletins announcing his progress from time to
time, but with admirable restraint had held back everything
till the work was completed.

Writing to a friend some ten years later, Weierstrass gives
his modest version of his scientihe reticence: ‘. . . the infinite
emptiness and boredom of those years [as a schoolteacher]
would have been unendurable without the hard work that made
me a recluse – even if I was rated rather a good fellow by the
circle of my friends among the junkers, lawyers, and young
officers of the community. . . . The present offered nothing
worth mentioning, and it was not my custom to speak of the
future.’

Recognition was immediate. At the University of Konigs-
berg, where Jacobi had made his great discoveries in the field
which Weierstrass had now entered with a masterpiece of sur-
passing excellence, Riehelot, himself a worthy successor of
Jacobi in the theory of multiple periodic functions, was Pro-
fessor of Mathematics. His expert eyes saw at once what
Weierstrass had done. He forthwith persuaded his university
to confer the degree of doctor, honoris causa, on Weierstrass
and h ims elf journeyed to Braunsberg to present the diploma.

At the dinner organized by the director of the Gymnasium in
Weierstrass’ honour Riehelot asserted that Ve have all found
our master in Mr Weierstrass’. The Ministry of Education
immediately promoted him and granted him a year’s leave to
prosecute his scientific work. Borchardt, the editor of Crelle’s
Journal at the time, hurried to Braunsberg to congratulate the
greatest analyst in the world, thus starting a warm friendship
’ which lasted till Borchardt’s death a quarter of a century later.

None of this went to Weierstrass’ head. Although he was
deeply moved and profoundly grateful for all the generous

MASTER AND PUPIL

recognition so promptly accorded him, he could not refrain
from easting a backward glance over his career. Years later,
thinking of the happiness of the occasion and of what that
occasion had opened up for him when he was forty years of age,
he remarked sadly that ‘everything in life comes too late.’

Weierstrass did not return to Braunsberg. No really suitable
position being open at the time, the leading German mathema-
ticians did what they could to tide over the emergency and got
Weierstrass appointed Professor of Mathematics at the Royal
Polytechnic School in Berlin. This appointment dated from
1 July 1856; in the autumn of the same year he was made
Assistant Professor (in addition to the other post) at the Uni-
versity of Berlin and was elected to the Berlin Academy.

The excitement of novel working conditions and the strain
of too much lecturing presently brought on a nervous break-
down. Weierstrass had also been overworking at his researches.
In the summer of 1859 he was forced to abandon his course and
take a rest cure. Returning in the autumn he continued his work,
apparently refreshed, but in the following March was suddenly
attacked by spells of vertigo, and he collapsed in the middle of
a lecture.

All the rest of his life he was bothered with the same trouble
off and on, and after resuming his work – as full professor, with
a considerably lightened load – never trusted him self to write
Ms own formulae on the board. His custom was to sit where he
could see the class and the blackboard, and dictate to some
student delegated from the class what was to he written. One
of these ‘mouthpieces’ of the master developed a rash propen-
sity to try to improve on what he had been told to write.
Weierstrass would reach up and rub out the amateur’s efforts
and make him write what he had been told. Occasionally the
battle between the professor and the obstinate student would
go to several rounds, but in the end Weierstrass always won.
He had seen little boys misbehaving before.

As the fame of his work spread over Europe (and later to
America), Weierstrass’ classes began to grow rather unwiddy
and he would sometimes r^ret that the quality of his auditom
lagged far behind their rapidly mounting quantity* Neverthe-

465

MEN OF MATHEMATICS

less he gathered about him an extremely able band of young
mathematicians who were absolutely devoted to him and who
did much to propagate his ideas, for Weierstrass was always
slow about publication, and without the broadcasting of his
lectures which his disciples took upon themselves his influence
on the mathematical thought of the nineteenth century would
have been considerably retarded.

‘VYeierstrass was always accessible to his students and sin-
cerely interested in their problems, whether mathematical or
human. There was nothing of the ‘great man’ complex about
him, and he would as gladly walk home with any of the students

• and there were many – who eared to join him as with the most
  famous of his colleagues, perhaps more gladly when the col-
  league happened to be Kronecker. He was happiest when,
  sitting at a table over a glass of wine with a few of his devoted
  disciples, he became a jolly student again himself and insisted
  on paying the bill for the crowd.

An anecdote (about i^Iittag-LefQer) may suggest that the
Europe of the present century has partly lost something it had
in the 1870’s. The Franco-Prussian war (1870-7’1) had left
France pretty sore at Germany. But it had not befogged the
minds of mathematicians regarding one another’s merits irre-
spective of their nationalities. The like holds for the Napoleonic
wars and the mutual esteem of the French and British mathe-
maticians. In 1873 ll^Iittag-Lefiaer arrived in Paris from Stock-
holm all set and full of enthusiasm to study analysis under
Heimite. ‘You have made a mistake, six’, Hermite told him:
“you should follow Weierstrass’ course at Berlin. He is the
master of all of us.’

Mittag-Leffler took the sound advice of the magnanimous
Frenchman and not so long afterwards made a capital discovery
of his own which is to be found to-day in all books on the theory
of functions. ‘Hermite was a Frenchman and a patriot’, !Mittag-
Leffler remarks; ‘I learned at the same time in what degree he
was also a mathematician.’

The years (1864-97) of Weierstrass’ career at Berlin as Pro-
fessor of Mathematics were full of scientific and human interests
for the man who was acknowledged as the leading anal 5 rst in

466

MASTER AND PUPIL

the world. One phase of these interests demands more thart the
passing reference that might suffice in a purely scientific bio-
graphy of Weierstrass: his friendship with his favourite pupil,
Sonja (or Sophie) Kowalewski.

Madame* Kowalewski’s maiden name was Sonja Corvin-
Eroukowsky; she was born at Moscow, Russia, on 15 January
1850, and died at Stockholm, Sweden, on 10 February 1891, six
years before the death of Weierstrass.

At fifteen Sonja began the study of mathematics. By eigh-
teen she had made such rapid progress that she was ready for
advanced work and was enamoured of the subject. As she came
of an aristocratic and prosperous family, she was enabled to
gratify her ambition for foreign study and matriculated at the
University of Heidelberg.

This highly gifted girl became not only the leading woman
mathematician of modern times, but also made a reputation as
a leader in the movement for the emancipation of women,
particularly as regarded their age-old disabilities in the field of
higher education.

In addition to all this she was a brilliant writer. As a young
girl she hesitated long between mathematics and literature as a
career. After the composition of her most important mathe-
matical work (the prize memoir noted later), she turned to
literature as a relaxation and wrote the reminiscences of her
childhood in Russia in the form of a novel (published first in
Swedish and Danish). Of this work it is reported that “ihe
literary critics of Russia and Scandinavia were unanimous in
declaring that Sonja Kowalewski had equalled the best writers
of Russian literature in style and thought.’ Unfortunately thia
promising start was blocked by her premature death, and only
fragments of other literary works survive. Her one novel was
translated into many languages.

Although Weierstrass never married he was no panicky
bachelor who took to his heels every time he saw a pretty
woman coming. Sonja, according to competent judges who
knew her, was extremely good-looking. We must first teU how
she and Weierstrass met.

Weierstrass used to enjoy his summer vacations in a thor-

467

MEN OE MATHEMATICS

oughly human manner. The Franco-Pmssian war caused him
to forego his usual summer trip in 1870, and he stayed in Berlin,
lecturing on elliptic functions. Owing to the war his class had
dwindled to only twenty instead of the fifty who heard the
lectures two years before. Since the autumn of 1869 Sonja
Kowalewski, then a dazzling young woman of nineteen, h^
been studying elliptic functions imder Leo Konigsberger (bom
1837) at the University of Heidelberg, where she had also
followed the lectures on physics by Kirchhoff and Helmholtz
and had met Bimsen the famous chemist under rather amusing
circumstances – to be related presently, Konigsberger, one of
Weierstrass’ first pupils, was a first-rate publicity agent for his
master, Sonja caught her teacher’s enthusiasm and resolved to
go direct to the master himself for inspiration and enlighten-
ment.

The status of unmarried women students in the 1870’s was
somewhat anomalous. To forestall gossip, Sonja at the age of
eighteen contracted what was to have been a nominal marriage,
left her husband in Russia, and set out for G^ermany. Her one
indiscretion in her dealings with Weierstrass was her neglect to
inform him at the beginning that she was married.

Having decided to learn from the master himself, Sonja took
her courage in her hands and called on Weierstrass in Berlin.
She was twenty, very earnest, very eager, and very determined;
he was fifty-five, vividly grateful for the lift Gudeimann had
given him toward becoming a mathematician by taking him on
as a pupO, and sympathetically understanding of the ambitions
of young people. To hide her trepidation Sonja wore a large and
floppy hat, ‘so that Weierstrass saw nothing of those marv^ellous
•eyes whose eloquence, when she wished it, none could resist.’

Some two or three years later, on a visit to Heidelberg,
Weierstrass learned from Bunsen — a crabbed bachelor – that
Sonja was ‘a dangerous woman’, Weierstrass enjoyed his
friend’s terror hugely, as Bunsen at the time- was unaware that
Sonja had been receiving frequent private lessons from Weier-
strass for over two years.

Poor Bunsen based his estimate of Sonja on bitter personal
experience. He had proclaimed for years that no woman, and

4m

MASTER AND PUPIL

especially no Russian T^^oman, would ever be permitted to pro-
fane the masculine sanctity of his laboratory. One of Sonja’s
Russian girl friends, desiring ardently to study chemistry in
Bunsen’s laboratory, and ha\dng been thrown out herself, pre-
vailed upon Sonja to try her powers of persuasion on the crusty
chemist. Leaving her hat at home, Sonja interviewed Bunsen.
He was only too charmed to accept SonJa^s friend as a student
in his laboratory. After she left he woke up to what she had
done to him. ‘And now that woman has made me eat my own
words,’ he lamented to Weierstrass.

Sonja’s e\ident earnestness on her first visit impressed
Weierstrass favourably and he wrote to Kdnigsberger inquiring
about her mathematical aptitudes. He asked also whether ‘the
lady’s personality offers the necessary guarantees.’ On recehdng
an enthusiastic reply, Weierstrass tried to get the university
senate to admit Sonja to his mathematical lectures. Being
brusquely refused he took care of her himself in his own time.
Every Sunday afternoon was devoted to teaching Sonja at his
house, and once a week W’“eierstrass returned her visit. After
the first few lessons Sonja lost her hat. The lessons began in the
autumn of 1870 and continued with slight interruptions due to
vacations or illnesses till the autumn of 1874. When for any
reason the friends were unable to meet they corresponded.
After Sonja’s death in 1891 Weierstrass burnt all her letters to
him, together with much of his other correspondence and
probably more than one mathematical paper.

The correspondence between Weierstrass and his channing
young friend is warmly human, even when most of a letter is
given over to mathematics. Much of the correspondence was
undoubtedly of considerable scientific importance, but unfor-
tunately Sonja was a very untidy woman when it came to
papers, and most of what she left behind was fragmentary or
in hopeless confusion.

Weierstrass himself was no paragon in this respect. Without
keeping records he loaned his unpublished manuscripts right
and left to students who did not always return what they bor-
rowed. Some even brazenly rehashed parts of their teacher’s
work, spoiled it, and published the results as their own. Al-

469

MEN OF MATHEMATICS

though \yeierstrass complains about this outrageous practice
in letters to Sonja Ms chagrin is not over the petty pilfering of
his ideas but of their bungling in incompetent hands and the
consequent damage to mathematics. Sonja of course never
descended to anything of this sort, but in another respect she
was not entirely blameless. Weierstrass sent her one of his
unpublished works by which he set great store, and that was
the last he ever saw of it. Apparently she lost it, for she dis-
creetly avoids the topic – to judge from his letters – whenever
he brings it up.

To compensate for this lapse Sonja tried her best to get
Weierstrass to exercise a little reasonable caution in regard to
the rest of his unpublished work. It was his custom to carrv
about with him on his frequent travels a large white wooden
box in which he kept all his working notes and the various
versions of papers which he had not yet perfected. His habit
was to rework a theorv^ many times until he found the best, the
“naturaP way in which it should be developed. Consequently he
published slowly and put out a work under his own name only
when he had exhausted the topic from some coherent point of
view. Several of his rough-hewn projects are said to have been
confided to the mysterious box. In 18S0, while Weierstrass was
on a vacation trip, the box was lost in the baggage. It has never
been heard of since.

After taking her degree in absentia from Gottingen in 1874,
Sonja returned to Russia for a rest as she was worn out by
excitement and overwork. Her fame bad preceded her and she
^rested* by plunging into the hectic futilities of a crowded social
season in St Petersburg while Weierstrass, back in Berlin, pulled
wires all over Europe trying to get his favourite pupil a position
worthy of her talents. His fruitless efforts disgusted him with
the narrowness of the orthodox academic mind.

In October 1875 Weierstrass received from Sonja the news
that her father had died. She apparently never replied to his
tender condolences, and for nearly three years she dropped
completely out of his life. In August 1878 he writes to ask
whether she ever received a letter he had written her so long
before that he has forgotten its date, ‘Didn’t you get my letter?

470

MASTER AND PUPIL

Or what can be preventing you from confiding freely in me, your
best friend as you so often called me, as you used to do? This is
a riddle whose solution only you can give me, , .

In the same letter Weierstrass rather pathetically begs her to
contradict the rumour that she has abandoned mathematics:
Tchebychefi, a Russian mathematician, had called on Weier-
strass when he was out, but had told Borchardt that Sonja had
‘gone social’ , as indeed she had. ‘Send your letter to Berlin at
the old address’, he concludes; ‘it will certainly be forwarded to
me.’

Man’s ingratitude to man is a familiar enough theme; Sonja
now demonstrated what a woman can do in that line when she
puts her mind to it. She did not answ^er her old friend’s letter
for two years although she knew he had been unhappy and in
poor health.

The answer when it did come was rather a let-down. Sonja’s
sex had got the better of her ambitions and she had been living
happily with her husband. Her misfortune at the time was to be
the focus for the flattery and unintelligent, sideshow wonder of
a superficially brilliant mob of artists, journalists, and dilettante
litterateurs who gabbled incessantly about her unsurpassable
genius. The shallow praise warmed and excited her. Had she
frequented the society of her intellectual peers she might still
have lived a normal life and have kept her enthusiasm. And she
would not have been tempted to treat the man who had formed
her mind as shabbily as she did.

In October 1878 Sonja’s daughter ‘Foufie’ was born.

The forced quiet after Foufie’s arrival roused the mother’s
dormant mathematical interests once more, and she wrote to
Weierstrass for technical ad\ice. He replied that he must look
up the relevant literature before venturing an opinion. Al-
though she had neglected him, he was still ready with his
ungrudging encouragement. His only regret (in a letter of
October 1880) is that her long silence has deprived him of
the opportunity of helping her. ‘But I don’t like to dwell
so much on the past – so let us keep the future before our
eyes,’

Material tribulations aroused Sonja to the truth. She was a

471

1I.M. — VOL. II,

MEN OF MATHEMATICS

boTn mathematician and could no more keep away from mathe-
matics than a duck can from water. So in October 1880 (she
was then thirty), she wrote begging Weierstrass to advise her
again. Xot waiting for his reply she packed up and left Moscow
for Berlin. His reply, had she received it, might have caused her
to stay where she was. Nevertheless when the distracted Sonja
arrived unexpectedlj’ he devoted a whole day to going over her
difficulties with her. He must have given her some pretty
straight talk, for when she returned to Moscow three months
later she went after her mathematics with such fury that her
gay friends and silly parasites no longer recognized her. At
Weierstrass’ suggestion she attacked the problem of the pro-
pagation of light in a crystalline mediinn.

In 1882 the correspondence takes two new turns, one of
which is of mathematical interest. The other is Weierstrass’
outspoken opinion that Sonja and her husband are unsuited to
one another, especially as the latter has no true appreciation
of her intelleetual merits. The mathematical point refers to
Poincare, then at the beginning of his career. With his sure
instinct for recognizing young talent, Weierstrass hails Poincare
as a coming man and hopes that he will outgrow his propensity
to publish too rapidly and let his researches ripen without
scattering them over too wide a field. To publish an article of
real merit every week – that is impossible’ , he remarks, referring
to Poincare’s deluge of papers.

Sonja’s domestic difficulties presently resolved themselves
through the sudden death of her husband in March 1883, She
was in Paris at the time, he in Moscow. The shock prostrated
her. For four days she shut herself up alone, refused food, lost
consciousness the fifth day, and on the sixth recovered, asked
for paper and pencil, and covered the paper with mathematical
formulae. By autumn she was herself again, attending a scien-
tific congress at Odessa.

Thanks to IMittag-Leffler, Madame Kowalewski at last
obtained a position where she could do herself justice; in the
autumn of 1884} she was lecturing at the University of Stock-
holm, where she was to be appointed (in 1889) as professor for
life. A little later she suffered a rather embarrassing setback

472

MASTEfi. AND PUPIL

when the Italian mathematician Vito Volterra pointed out a
serious mistake in her work on the refraction of light in crystal-
line media. This oversight had escaped Weierstrass, who at the
time was so overwhelmed with official duties that outside of
them he had ‘time only for eating, drinking, and sleeping. . . .
In short’, he says, T am what the doctors call brain- weary.’ He
was now nearly seventy. But as his bodily tils increased his
intellect remained as powerful as ever.

The master’s seventieth birthday was made the occasion for
public honours and a gathering of his disciples and former pupils
from all over Europe. Thereafter he lectured publidy less and
less often, and for ten years received a few of his students at his
own house. ‘When they saw that he was tired out they avoided
mathematics and talked of other things, or listened eagerly
while the companionable old man reminisced of his student
pranks and the dreary years of his isolation from all scientific
friends. His eightieth birthday was celebrated by an even more
impressive jubilee than his seventieth and he became in some
degree a national hero of the German people.

One of the greatest joys Weierstrass experienced in his de-
clining years was the recognition won at last by his favourite
pupil. On Christmas Eve, 1888, Sonja received in person the
Bordin Prize of the French Academy of Sciences for her memoir
On the rotation of a solid body about a fixed poinU

As is the rule in competition for such prizes, the memoir had
been submitted anonymously (the author’s name being in a
sealed envelope bearing on the outside the same motto as that
inscribed on the memoir, the envelope to be opened only if the
competing work won the prize), so there was no opportunity for
jealous rivals to hint at undue influence. In the opinion of the
judges the memoir was of such exceptional merit that they
raised the value of the prize from the previously announced
3,000 francs to 5,000. The monetary value, however, was the
least part of the prize.

Weierstrass was overjoyed- ‘I do not need to tell you’, he
writes, ‘how much your success has gladdened the hearts of
myself and my sisters, also of your friends here. I particularly
experienced a true satisfaction; competent judges have now

F2 478

MEX OF MATHEMATICS

delivered their verdict that my “‘faithful pupil”, my “vreak-
ness” is indeed not a ‘“frivolous humbug”.’

We may leave the friends in their moment of triumph. Two
years later (10 February 1891) Sonja died in Stockholm at the
age of forty-one after a brief attack of influenza which at the
time was epidemic. Weierstrass outlived her six years, dying
peacefully in his eightj^-seeond year on 19 February 1897 at his
home in Berlin after a long illness followed by influenza. His
last wish was that the priest say nothing in his praise at the
funeral but restrict the services to the customary prayers.

Sonja is buried in Stockholm, Weierstrass with his two sisters
in a Catholic cemetery in Berlin. Sonja also was of the Catholic
faith, belonging to the Eastern Church.

We shall now give some intimation of two of the basic ideas
on which Weierstrass foimded his work in analysis. Details or an
exact description are out of the question here, but may be found
in the earlier chapters of any competently written book on the
theory of functions.

A power series is an expression of the form

^0 “T T -r . . . -r -!-•••»

in which the coefficients aQ, 02 . . ., … are constant

numbers and is a variable number; the numbers concerned
may be real or complex.

The sums of 1,2,3, . . . terms of the series, namely a^, 4-

^0 ~r -r . . . are called the partial sums. If for some
particular value of z these partial sums give a sequence of
numbers which converge to a definite limit, the power series is
said to converge to the same limit for that value of z.

All the values of z for which the power series converges to a
limit constitute the domain of convergence of the series; for any
value of the variable z in this domain the series converges; for
other values of z it diverges.

If the series converges for some value of s, its value can be
calculated to any desired degree of approximation, for that
value, by taking a sufficiently large number of terms.

Now, in the majority of mathematical problems which have
applications to science, the ‘answer’ is indicated as the solution

474

MASTER AND PUPIL

in series of a differential equation (or system of such equations),
and this solution is only rarely obtainable as a finite expression
in terms of mathematical functions which have been tabulated
(for instance logarithms, trigonometric functions, elliptic func-
tions, etc.). In such problems it then becomes necessary to do
two things: prove that the series converges, if it does; calculate
its numerical values to the required accuracy.

If the series does not converge it is usually a sign that the
problem has been either incorrectly stated or wrongly solved.
The multitude of functions which present themselves in pure
mathematics are treated in the same way, whether they are
e\’er likely to have scientific applications or not, and finally a
general theory of convergence has been elaborated to cover vast
tracts of all this, so that the individual examination of a parti-
cular series is often referred to more inclusive investigations
already carried out.

Finally, all this (both pure and applied) is extended to power
series in 2, 3, 4, … variables instead of the single variable z
above; for example, in tw’o variables,

a -t^qZ -{• -j- 4- Ciszt) -f- + . , . .

It may be said that without the theory of power series most
of mathematical physics (including much of astronomy and
astro-physics) as we know it would not exist.

Difficulties arising with the concepts of limits, continuity,
and convergence drove Weierstrass to the creation of his theory
of irrational numbers.

Suppose we extract the square root of 2 as we did m school,
carrying the computation to a large number of decimal places.
We get as successive approximations to the required square root

the sequence of numbers 1, 1.4, 1.41, 1.412, With sufficient

labour, proceeding by well-defined steps according to the usual
rule, we could if necessary exhibit the first thousand, or the
first million, of the rational numbers 1, 1.4, … constituting this
sequence of approximations. Examining this sequence we see
that when we have gone far enough we have determined a
perfectly defibnite rational number containing as many decimal
places as we please (say 1,000), and that this rational number
differs from any of the succeeding rational numbers in the

475

MEN OF MATHEMATICS

sequence by a number (decimal), sucb as -000 . . * *000 * . ia
which a correspondingly large number of zeros occur before
another digit (1> 2, … or 9} appears.

This illustrates what is meant by a convergent sequence of
numbers: the raiionals 1, 1-4, . . , constituting the sequence give
us ever closer approximations to the “irrational number’ which
we call the square root of 2, and which we conceive of as having
been defined by the convergent sequence of raiionals, this defini-
tion being in the sense that a method has been indicated (the
usual school one) of calculating any particular member of the
sequence in a finite number of steps*

Although it is impossible actually to exhibit the whole
sequence, as it does not stop at any finite number of terms,
nevertheless we regard the process for constructing any member
of the sequence as a sufficiently clear conception of the whole
sequence as a single definite object which we can reason about.
Doing so, we have a workable method for using the square root
of 2 and s imil arly for any irrational number, in mathematical
analysis.

As has been indicated it is impossible to make this precise m
an account like the present, but e%’en a careful statement might
disclose some of the logical objections glaringly apparent in the
above description – objections which inspired Kronecker and
others to attack Weierstrass’ ‘sequential’ definition of
irrationals.

Nevertheless, right or wrong, Weierstrass and his school
made the theory ucork* The most useful results they obtained
have not yet been questioned, at least on the ground of their
great utility in mathematical analysis and its applications, by
any competent judge in his right mind. This does not mean that
objections cannot he well taken: it merely calls attention to the
fact that in mathematics, as in everything else, this earth is not
yet to be confused with the Kingdom of Heaven, that perfection
is a chimaera, and that, in the words of Crelle, we can only hope
for closer and closer approximations to mathematical truth –
whatever that may be, if anything – precisely as in the Weier-
strassian theory of convergent sequences of ratioiials defining
irrationals.

476

MASTER AND PUPIL

After all, why should mathematicians, who are human beings
like the rest of us, always be so pedantically exact and so in-
humanly perfect? As Weierstrass said, ‘It is true that a mathe-
matician who is not also something of a poet will never be a
perfect mathematician’. That is the answer: a perfect mathe-
matician, by the very fact of his poetic perfection, would be a
mathematical impossibility.

CHAPTER TWENTY-THREE

COMPLETE INDEPENDENCE
Boole

• ‘Oh, we never read anjiJiing the English mathematicians do*’
  This characteristically Continental remark was the reply of a
  distinguished European mathematician when he was asked
  whether he had seen some recent work of one of the leading
  English mathematicians. The ‘‘we’ of his frank superiority in-
  cluded Continental mathematicians in general.

This is not the sort of story that mathematicians like to tell
on themselves, but as it illustrates admirably that characteristic
of British mathematicians – insular originality” – which has been
the chief claim to distinction of the British school, it is an ideal
introduction to the life and work of one of the most insularly
original mathematicians England has produced, George Boole.
The fact is that British mathematicians have often serenely
gone their own way, doing the things that interested them
personally as if they were playing cricket for their own amuse-
ment only, with a self-satisfied disregard for what others,
shouting at the top of their scientific lungs, have assured the
world is of supreme importance. Sometimes, as in the prolonged
idolatry of Newton’s methods, indifference to the leading
fashions of the moment has cost the British school dearly, but
in the long run the take-it-or-leave-it attitude of this school has
added more new fields to mathematics than a slavish imitation
of the Continental masters could ever have done. The theory of
invariance is a case in point; Maxwell’s electrodynamic field
theory is another.

Although the British school has had its share of powerful
developers of work started elsewhere, its greater contribution to
the progress of mathematics has been in the direction of origi-
nality . Boole’s work is a striking illustration of this. When first

478

COMPLETE INDEPENDENCE

put out it was ignored as mathematics ^ except by a few, chiefly
Boole’s own more unorthodox countr^Tnen, who recognized
that here was the germ of something of supreme interest for all
mathematics. To-day the natural development of what Boole
started is rapidly becoming one of the major divisions of pure
mathematics, with scores of workers in practically all countries
extending it to all fields of mathematics where attempts are
being made to consolidate our gains on firmer formdations. As
Bertrand Russell remarked some years ago, pure mathematics
was discovered by George Boole in his work The Laws of Thought
published in 1854. This may be an exaggeration, but it gives a
measure of the importance in ■which mathematical logic and its
ramifications are held to-day. Ofhers before Boole, notably
Leibniz and De Morgan, had dreamed of adding logic itself to
the domain of algebra; Boole did it.

George Boole was not, like some of the other originators in
mathema-tics, born into the lowest economic stratum of society.
Eds fate was much harder. He was born on 2 November 1815 at
Lincoln, England, and was the son of a petty shopkeeper. If we
can credit the pictxire drawn by English writers themselves of
those hearty old days – 1815 was the year of Waterloo – to be
the son of a small tradesman at that time was to be damned by
foreordination.

The whole class to which Boole’s father belonged was treated
with a contempt a trifle more contemptuous than that reserved
for enslaved scullery maids and despised second footmen. The
lower classes’, into whose ranks Boole had been bom, simply
did not exist in the eyes of the ‘upper classes’ – including the
more prosperous wine merchants and moneylenders. It was
taken for granted that a child in Boole’s station should dutifully
and gratefully master the shorter catechism and so live as never
to transgress the strict limits of obedience imposed by that
remarkable testimonial to human conceit and class-conscious
snobbery.

To say that Boole’s early struggles to educate himself into a
station above that to which It had pleased God to him’
were a fair imitation of purgatory is putting it mildly. By an
act of divine providence Boole’s great spirit had been assigned

479

MEN OF MATHEMATICS

to the meanest class; let it stay there then and stew in its own
ambitious juice. Americans may like to recall that Abraham
Lincoln, only six years older than Boole, had his struggle about
the same time. Lincoln was not sneered at but encouraged.

The schools where young gentlemen were taught to knock
one another about in training for their future parts as leaders
in the sweatshop and coal mine systems then coming into vogue
were not for the hkes of George Boole. No; his ‘National School’
was designed chiefly with the end in view of keeping the poor
in their proper, unwashable place.

A wretched smattering of Latin, with perhaps a slight expo-
sure to Greek, was one of the mystical stigmata of a gentleman
in those incomprehensible days of the sooty industrial revolu-
tion. Although few of the boys ever mastered Latin enough to
enable them to read it without a crib, an assumed knowledge of
its grammar was one of the hallmarks of gentility, and its syn-
tax, memorized by rote, was, oddly enough, esteemed as mental
discipline of the highest usefulness in preparation for the
ownership and conservation of property.

Of course no Latin was taught in the school that Boole was
permitted to attend. Making a pathetically mistaken diagnosis
of the abilities which enabled the propertied class to govern
those beneath them in the scale of wealth, Boole decided that
he must learn Latin and Greek if he was ever to get his feet out
of the mire. This was Boole’s mistake. Latin and Greek had
nothing to do with the cause of his difficulties. He did teach
himself Latin with his poor strugg ling father’s sympathetic
encouragement. Although the poverty-stricken tradesman
knew that he himself would never escape he did what he could
to open the door for his son. He knew no Latin. The struggling
boy appealed to another tradesman, a small bookseller and
friend of his father. This good man could only give the boy a
start in the elementary grammar. Thereafter Boole had to go it
alone. Anyone who has watched even a good teacher trying to
get a normal child of eight through Caesar will realize what the
untutored Boole was up against. By the age of twelve he had
mastered enou^ Latin to translate an ode of Horace into
E ngli sh verse. His father, hopefully proud but understanding

460

COMPLETE INDEPENDENCE

nothing of the technical merits of the translation, had it printed
in the local paper. This precipitated a scholarly row, partly
flattering to Boole, partly humiliating.

A classical master denied that a boy of twelve could have
produced such a translation. Little boys of twelve often know
more about some things than their forgetful elders give them
credit for. On the technical side grave defects showed up. Boole
was humiliated and resolved to supply the deficiencies of his
self-instruction. He had also taught himself Greek. Determined
now to do a good job or none he spent the next two years slav-
ing over Latin and Greek, again without help. The effect of all
this drudgerj’ is plainly apparent in the dignity and marked
Latinity of much of Boole’s prose.

Boole got his early mathematical instruction from his father,
who had gone considerably beyond his own meagre schooling by
private study. The father had also tried to interest his son in
another hobby, that of making optical instruments, but Boole,
bent on his own ambition, stuck to it that the classics were the
key to dominant living. After finishing his common schooling he
took a commercial course. This time his diagnosis was better,
but it did not help him greatly. By the age of sixteen he saw
that he must contribute at once to the support of his wretched
parents. School teaching offered the most immediate opportu-
nity of earning steady wages – in Boole’s day ‘ushers’, as assis-
tant teachers were called, were not paid salaries but wages.
There is more than a monetary difference between the two. It
may have been about this time that the immortal Squeers, in
Dickens’ Nicholas Nickleby, was making his great but unappre-
ciated contribution to modem pedagogy at Dotheboys HaH
with his brilliant anticipation of the ‘project’ method. Young
Boole may even have been one of Squeers’ ushers; he taught at
two schools-

Boole spent four more or less happy years teaching in these
elementary schools. The chilly nights, at least, long after the
pupils were safely and mercifully asleep, were his own. He still
was on the wrong track. A third diagnosis of his social unworthi-
ness was similar to his second but a considerable advance over
both his first and second. Lacking anything in the way of

481

MEN OF MATHEMATICS

capital ~ practically every penny the young man earned went
to the support of his parents and the barest necessities of his
own meagre existence – Boole now cast an appraising eye over
the gentlemanly professions. The Army at that time was out of
his reach as he could not afford to purchase a commission. The
Bar made obvious financial and educational demands which he
had no prospect of satisfying. Teaching, of the grade in which
he was then engaged, was not even a reputable trade, let alone
a profession. remained? Only the Church. Boole resolved
to become a clergyman.

In spite of all that has been said for and against God, it must
be admitted even by his severest critics that he has a sense of
humour. Seeing the ridiculousness of George Boole’s ever be-
coming a’clergyman, he skilfully turned the young man’s eager
ambition into less preposterous channels. An unforeseen afBio
tion of greater poverty than any they had yet enjoj^ed com-
pelled Boole’s parents to urge their son to forego all thoughts
of ecclesiastical eminence. But his four years of private prepara-
tion (and rigid privation) for the career he had planned were
not wholly wasted; he had acquired a mastery of French,
German, and Italian, all destined to be of indispensable service
to him on his true road.

At last he found himself. His father’s early instruction now
bore fruit- In his twentieth year Boole opened up a civilized
school of his own. To prepare his pupils properly he had to
teach them some mathematics as it should be taught. His
interest was aroused. Soon the ordinary and execrable text-
books of the day awoke his wonder, then his contempt. Was this
stuff mathematics? Incredible. What did the great masters of
mathematics say? Like Abel and Galois, Boole went directly to
great headquarters for his marching orders. It must be remem-
bered that he had had no mathematical training beyond the
rudiments. To get some idea of his mental capacity we can
imagine the lonely student of twenty mastering, by his own
unaided efforts, the Mecanique celeste of Laplace, one of the
toughest masterpieces ever written for a conscientious student
to assimilate, for the mathematical reasoning in it is full of gaps
and enigmatical declarations that ‘■it is easy to see’, and then
482

COirPLETE INDEPENDENCE

we must think of him making a thorough, understanding study
of the excessively abstract Micanique analytique of Lagrange,
in which there is not a single diagram to illuminate the analysis
from beginning to end. Yet Boole, self-taught, foimd his way
and saw what he was doing. He even got his first contribution to
mathematics out of his unguided efforts. This was a paper on
the calculus of variations.

Another gain that Boole got out of all this lonely study
desers-^es a separate paragraph to itself. He discovered invari-
ants. The significance of this great discovery which Cayley and
Sylvester were to develop in grand fashion has been sufficiently
explained; here we repeat that without the mathematical
theory of invariance (which grew out of the early algebraic
work) the theory of relativity would have been impossible.
Thus at the very threshold of his scientific career Boole noticed
something lying at his feet which Lagrange himself might
easily have seen, picked it up, and found that he had a gem of
the first water. That Boole saw what others had overlooked was
due no doubt to his strong feeling for the symmetry” and beauty
of algebraic relations – when of cotirse they happen to be both
symmetrical and beautiful; they are not always. Others might
have thought his find merely pretty. Boole recognized that it
belonged to a higher order.

Opportunities for mathematical publication in Boole’s day
were inadequate unless an author happened to be a member of
some learned society with a journal or transactions of its own.
Luckily for Boole, The Cambridge Mathematical Journal, under
the able editorship of the Scotch mathematician, D. F. Gregoiy,
was founded in 1837. Boole submitted some of his work. Its
originality and style impressed Gregory favourably, and a cor-
dial mathematical correspondence began a friendship which
lasted out Boole’s life.

It would take us too far afield to discuss here the great con-
tribution which the British school was ma k ing at the time to
the understanding of algebra as algebra, that is, as the abstract
development of the consequences of a set of postulates without
necessarily any interpretation or application to ‘numbers* or
anything else, but it may be mentioned that the modem con-
. 4S3

MEN OF MATHEMATICS

ception of algebra began with the British ‘reformers*, Peacock,
Herschel, De Morgan, Babbage, Gregory, and Boole- MTiat was
a somewhat heretical novelty when Peacock published his
Treatise on Algebra in 1830 is to-day a commonplace in any
competently written school book. Once and for all Peacock
broke away from the superstitition that the ic, i/, z, . , . in such
relations as x -r y ^ y xy = yx, x{y -{- z) = ast/ -f rcz, and
so on, as we find them in elementary algebra, necessarily
‘represent numbers’ ; they do not, and that is one of the most
important things about algebra and the source of its power in
applications. The a;, i/, z, … are merely arbitrary marks, com-
bined according to certain operations, one of which is symbo-
lized as -hj another by x (or simply as xy instead of x x y),
in accordance with postulates laid down at the beginning, like
the specimens x -j- y = y -i- x, etc,, above.

Without this realization that algebra is of itself nothing more
than an abstract system, algebra might still have been stuck
fiast in the arithmetical mud of the eighteenth centxuy, unable
to move forward to its modem and extremely useful variants
under the direction of Hamilton. We need only note here that
this renovation of algebra gave Boole his first opportunity to do
fine work appreciated by his contemporaries. Striking out on
his own initiative he separated the symbols of mathematical
operations from the things upon which they operate and pro-
ceeded to investigate these operations on their own account.
How did they combine? Were they too subject to some sort of
symbolic algebra? He found that they were. His work in this
direction is extremely interesting, but it is overshadowed by
the contribution which is peculiarly his own, the creation of a
isimple, workable system of symbolic or mathematical logic.

To introduce Boole’s splendid invention properly we must
digress slightly and recall a famous row of the first half of the
nineteenth century, which raised a devil of a din in its own day
but which is now almost forgotten except by historians of
pathological philosophy. We mentioned Hamilton a moment
ago. There were two Hamiltons of public fame at this time, one
the Irish mathematician Sir William Rowan Hamilton (1805-
65), the other the Scotch philosopher Sir William Hamilton
484

COMPLETE INDEPENDENCE

(1788-1856). Mathematicians usually refer to the philosopher
as the other Hamilton. After a somewhat unsuccessful career as
a Scotch barrister and candidate for official university positions
the eloquent philosopher finally became Professor of Logic and
Metaphysics in the University of Edinburgh. The mathematical
Hamilton, as we have seen, was one of the outstanding original
mathematicians of the nineteenth century. This is perhaps
unfortunate for the other Hamilton, as the latter had no earthly
use for mathematics, and hasty readers sometimes confuse the
two famous Sir Williams. This causes the other one to turn and
shiver in his grave.

Now, if there is anything more obtuse mathematically than a
thick-headed Scotch metaphysician it is probably a mathema-
tically thicker-headed German metaphysician. To surpass the
ludicrous absiudity of some of the things the Scotch Hamilton
said about mathematics we have to turn to what Hegel said
about astronomy or Lotze about non-Euclidean geometry.
Any depraved reader who wishes to fuddle himself can easily
run down all he needs. It was the metaphysician Hamilton’s
misfortune to have been too dense or too lazy to get more than
the most trivial smattering of elementary mathematics at
school, but ‘omniscience was his foible’, and when he began
lecturing and writing on philosophy, he felt constrained to tell
the world exactly how worthless mathematics is.

Hamilton’s attack on mathematics is probably the most
famous of all the many savage assaults mathematics has sur-
vived, undented. Less than ten years ago lengthy extracts from
Hamilton’s diatribe were vigorously applauded when a pedago-
gical enthusiast retailed them at a largely attended meeting of
America’s National Educational Association. Instead of
applauding, the auditors might have got more out of the
exhibition if they had paused to swallow some of Hamilton’s
philosophy as a sort of compulsory sauce for the proper enjoy-
ment of his mathematical herring. To be fair to him we shall
pass on a few of his hottest shots and let the reader make what
use of them he pleases.

‘Mathematics [Hamilton always used ‘mathematics’ as a
plural, not a smgular, as customary to-day] freeze and parch

485

MEN OF MATHEMATICS

the mind; ‘an excessive study of mathematics absolutely inca- pacitates the mind for those intellectual energies which philo- sophy and life require; ‘mathematics can not conduce to
logical habits at all’; ‘in mathematics dullness is thus elevated
into talent, and talent degraded into incapacity* ; ‘mathematies
may distort, but can never rectify, the mind’.

This is only a handful of the birdshot; we have not room for
the camion balls. The whole attack is most impressive – for a
man who knew far less mathematics than any intelligent child
of ten knows. One last shot deserves special mention, as it
introduces the figure of mathematical importance in the whole
wordy war, De Morgan (180^71), one of the most expert
controversialists who ever lived, a mathematician of vigorous
independence, a great logician who prepared the way for Boole,
the remorselessly good-humoured enemy of all cranks, char-
latans, and humbugs, and finally father of the famous novelist
(Alice foT Shorty etc.). Hamilton remarks, ‘This [a perfectly
nonsensical reason that need not be repeated] is why Mr De
Morgan among other mathematicians so often argues right.
Still, had Mr De Morgan been less of a Mathematician, he might
have been more of a Philosopher; and be it remembered, that
mathematics and dram-drinking tell especially, in the long run.’
Although the esoteric pimctuation is obscure the meaning is
clear enough. But it was not De Morgan who was given to
tippling.

De Morgan, having gained some fame from his pioneering
studies in logic, allowed himself in an absent-minded moment
to be trapped into a controversy with Hamilton over the
latter’s famous principle of ‘the quantification of the predicate.’
There is no need to explain what this mystery is (or was); it is as
dead as a coffin nail, De Morgan had made a real contribution to
the syllogism; Hamilton thought he detected De Morgan’s
diamond in Ms own blue mud; the irate Scottish lawyer-philo-
sopher publicly accused De Morgan of plagiarism – an insanely
imphilosophieal thing to do – and the fight was on. On De
Morgan’s side, at least, the row was a hilarious frolic. De
Morgan never lost his temper; Hamilton had never learned to
keep his.

486

COMPLETE INDEPENDENCE

If this were merely one of the innumerable squabbles over
priority which disfigure scientific history it would not be worth
a passing mention. Its historical importance is that Boole by
now (1848) was a firm friend and warm admirer of De Morgan.
Boole was still teaching school, but he knew many of the leading
British mathematicians personally or by correspondence. He
now came to the aid of his friend – not that the witty De
Morgan needed any mortal’s aid, but because he knew that De
Morgan was right and Hamilton wrong. So, in 1848, Boole
published a slim volume, The Mathematical Analysis of Logic,
his first public contribution to the vast subject which his work
inaugurated and in which he was to win enduring fame for the
boldness and perspicacity of his vision. The pamphlet ~ it was
hardly more than that – excited De Morgan’s warm admiration.
Here was the master, and De Morgan hastened to recognize
him. The booklet was only the promise of greater things to come
six years later, but Boole had definitely broken new, stubborn
ground.

In the meantime, reluctantly turning down his mathematical
friends’ advice that he proceed to Cambridge and take the
orthodox mathematical training there, Boole went on with the
drudgery of elementary teaching, without a complaint, because
his parents were now wholly dependent upon his support. At
last he got an opportunity where his conspicuous abilities as an
investigator and a lecturer could have some play. He was
appointed Professor of hlathematics at the recently opened
Queen’s College at what was then called the city of Cork,
Ireland. This was in 1849.

Needless to say, the brilliant man who had known only
poverty and hard work all his life made excellent use of his
comparative freedom from financial worry and everlasting
grind. His duties would now be considered onerous; Boole found
them light by contrast with the dreary round of elementary
teaching to which he had been accustomed. He produced much
notable miscellaneous mathematical work, hut his ma in effort
went on li cking his masterpiece into shape. In 1854 he pub-
lished it: An Inoestigation of the Laws of Thought, on tohich are
founded the Mathematical Theories of Logic and Probabilities^

487

H,3C-Y0I. n.

Q

MEX OF MATHEMATICS

Boole was thirty-nine when this appeared. It is somewhat
unusual for a mathematician as old as that to produce work of
such profound originality, but the phenomenon is accounted
for when we remember the long, devious path Boole was com-
pelled to follow before he could set his face fairly toward his
goal. (Compare the careers of Boole and Weierstrass.)

A few extracts will give some idea of Boole’s style and the
scope of his work.

‘The design of the following treatise is to investigate the
fundamental laws of those operations of the mind by which
reasoning is performed; to give expression to them in the lan-
guage of a Calculus, and upon this foundation to establish the
science of Logic and construct its method; to make that method
itself the basis of a general method for the application of the
mathematical doctrine of probabilities; and, finally, to collect
from the various elements of truth brought to view in the course
of these inquiries some probable intimations concerning the
nature and constitution of the human mind. . .

‘Shall we then err in regarding that as the true science of
Logic which, lajing down certain elementary laws, confirmed
by the very testimony of the mind, permits us thence to deduce,
by uniform processes, the entire chain of its secondary conse-
quences, and furnishes, for its practical applications, methods
of perfect generality?

‘There exist, indeed, certain general principles founded in the
very nature of language, by which the use of symbols, which are
but the elements of scientific language, is determined. To a
certain extent these elements are arbitrary. Their interpretation
is purely conventional: we are permitted to employ them in
whatever sense we please. But this permission is limited by two
indispensable conditions, – first, that from the sense once con-
\’’entionally established we never, in the same process of reason-
ing, depart; secondly, that the laws by which the process is
conducted he founded exclusively upon the above fixed sense
or meaning of the symbols employed. In accordance with these
principles, any agreement which may be established between
the laws of the symbols of Logic and those of Algebra can but
issue in an agreement of processes. The two pTO\’inces of inter-

4S8

COMPLETE IKDEPENDENCE

pretation remain apart and independent, each, subject to its
own laws and conditions.

‘Now the actual investigations of the following pages exhibit
Logic, in its practical aspect, as a system of processes carried on
by the aid of sjTnbols ha\ing a definite interpretation, and
subject to laws founded upon that interpretation alone. But at
the same time they exhibit those laws as identical in form with
the laws of the general symbols of Algebra, with this single
addition, viz., that the symbols of Logic are further subject to a
special law [x- = x in the algebra of logic, which can be inter-
preted, among other ways, as ‘”the class of all those things
common to a class x and itseK is merely the class a;’*], to which
the symbols of quantity, as such, are not subject.’ (That is, in
common algebra, it is not true that every x is equal to its square,
whereas in the Boolean algebra of logic, this is true.)

This programme is carried out in detail in the book. Boole
reduced logic to an extremely easy and simple type of algebra,
“Reasoning” upon appropriate material becomes in this algebra
a matter of elementary manipulations of formulae far simpler
than most of those handled in a second year of school algebra.
Thus logic itself was brought under the sway of mathematics.

Since Boole’s pioneering work his great invention has been
modified, improved, generalized, and extended in many direc-
tions. To-day symbolic or mathematical logic is indispensable
in any serious attempt to understand the nature of mathematics
and the state of its foundations on which the whole colossal
superstructure rests. The intricacy and delicacy of the diffi-
culties explored by the symbolic reasoning would, it is safe to
say, defy human reason if only the old, pre-Boole methods of
verbal logical arguments were at our disposal. The daring origin-
ality of Boole’s whole project needs no signpost. It is a land-
mark in itself.

Since 1899, when Hilbert published his classic on the founda-
tions of geometry, much attention has been given to the postu-
lationaJ formulation of the several branches of mathematics.
This movement goes back as far as Euclid, but for some strange
reason – possibly because the techniques invented by Des-
cartes, Newton, Leibniz, Euler, Gauss, and others gave mathe-

MEN OF MATHEMATICS

matidans plenty” to do in developing their subject freely and
somcTThat uncritically – the Euclidean method was for long
neglected in ever\’thing but geometry. We have already seen
that the British school applied the method to algebra in the
first half of the nineteenth century. Their successes seem to have
made no verj^ great impression on the work of their contem-
poraries and immediate successors, and it was only with the
work of Hilbert that the postulational method came to be
recognized as the clearest and most rigorous approach to any
mathematical discipline.

To-day this tendency to abstraction, in which the symbols
and rules of operation in a particular subject are emptied of all
meaning and discussed from a purely formal point of view, is
all the rage, rather to the neglect of applications (practical or
mathematical) which some say are the ultimate human justifi-
cation for any scientific activity. Nevertheless the abstract
method does give insights which looser attacks do not, and in
particular the true simplicity of Boole’s algebra of logic is most
easQy seen thus.

Accordingly we shall state the postulates for Boolean algebra
(the algebra of logic) and, having done so, see that they can
indeed be given an interpretation consistent with classical logic.
The following set of postulates is taken from a paper by E. V.
Huntington, in the Transactions of the American Mathematical
Society (vol. 85, 1933, pp. 274-304). The whole paper is easily
understandable by anyone who has had a week of algebra, and
may he found in most large public libraries. As Huntington
points out, this first set of his which we transcribe is not as
elegant some of his others. But as its interpretation in terms
of class inclusion as in formal logic is more immediate than the
like for the others, it is to be preferred here.

The set of postulates is expressed in terms of -EC, + , X , where
K is a class of undefined (wholly arbitrary, without any-
assigned meaning or properties beyond those given in the postu-
lates) elements a, 6, c, … , and a + b and a x b (written also
simply as ab) are the results of two undefined binary operations,
-f , X (‘binary’, because each of -f , x operates on feeo elements
of K). There are ten postulates, I a-VI:

490

COMPLETE INDEPENDENCE

‘I a. If a ctnd b are in the class K, then a bis in the class Kn

‘I b. If a and b are in the class K, then ab is in the class K,

‘II a. There is an element Z such that a + Z = a for ecery
element a, ♦

‘II b. There is an element TJ such that all — a for every
element a.

‘Ill a. a -r 6 = 6 -f a.

‘Ill b. ab = ba.

‘lY a. a -T be = (a -T h) (a c).

‘I\” b. a{b -f c) = a& + ac.

‘V. For every element a there is an element a’ such that a a’
= U and aa’ — Z,

‘\”I. There are at least hjco distinct elements in the class X.’

It ‘Will be readily seen that these postulates are satisfied by
the following interpretation: a, b, c, . are classes; a 4- 5 is the
class of all those things that are in at least one of the dasses,
a, b; ab is the class of all those things that are in both of the
classes a, &; Z is the ‘null class’ – the class that has no members;
r is the ‘universal class’ – the class that contains all the things
in all the classes under discussion. Postulate V then states that
given any class a, there is a class a’ consisting of all those things
which are not in a. Note that VI implies that Z7j Z are not the
same class.

From such a simple and obvious set of statements it seems
rather remarkable that the whole of classical logic can be built
up symbolically by means of the easy algebra generated by the
postulates. From these postulates a theory of what may be
called ‘logical equations’ is developed: problems in logic are
translated into such equations, which are then ‘solved’ by the
devices of the algebra; the solution is then reinterpreted in
terms of the logical data, giving the solution of the original
problem. We shall close this description with the symbolic
equivalent of ‘inclusion’ – also interpretable, when propositions
rather than classes are the elements of as ‘implication’.

‘T^e relation a <h [read, a is included in b] is defined by any
one of the following equations

a + b = b, ab = a, a’ -r b — U, ab’ Z.’

491

MEN OF matheij:atics

To see that these are reasonable, consider for example the
second, ab = a. This states that if a is included in b, then every-
thing that is in both a and b is the whole of a.

From the stated postulates the following theorems on inclu-
sion (with thousands of more complicated ones, if desired) can
be prored. The specimens selected all agree with our intuitive
conception of what ^inclusion’ means.

(1) a < a,

(2) If a <b and b < c, then a < c.

(3) If a <h and b < a, then a = b.

(4i) Z < a {yohere Z is the element tn II a – it is proved to be
the only element satisfying II a).

(5) a < U {xhere U is the element in II b ~ likewise unique),

(6) a < a + b; and if a <y and b <y, then a b <y.

(7) ab < a; and ifx<a and x <b, then x < ab.

(8) Ifx<a and x < a% then ai = Z; andif a <y and a’ < y,
then y = U.

(9) If a < b’ is false^ then there is at least one element as,
distinct from Z, such that x < a and x <h.

It may be of interest to observe that ‘ <’ in arithmetic and
analysis is the symbol for Tess than’. Note that if a, b, c, … are
real numbers, and Z denotes zero, then (2) is satisfied for this
interpretation of and similarly for (4), provided a is posi-
tive; but that (1) is not satisfied, nor is tlie second part of (6) –
as we see from 5 < 10, 7 < 10, but 5 4-7 < 10 is false.

The tremendous power and fluent ease of the method can be
readily appreciated by seeing what it does in any work on
symbolic logic. But, as already emphasized, the importance of
this ‘symbolic reasoning’ is in its applicability to subtle ques-
tions regarding the foundations of all mathematics which, were
it not for this precise method of fixing meanings of ‘words’ or
other ‘symbols’ once for all, would probably be unapproachable
by ordinary mortals.

like nearly all novelties, symbolic logic was neglected for
many years after its invention. As late as 1910 we find eminent
mathematicians scorning it as a ‘philosophical’ curiosity with-
out mathematical significance. The work of ‘VMiitehead and
Russell in Principia Mathemaiica (1910-13) was the first to

492

COMPLETE I^TDEPEXDENCE

convince any considerable body of professional mathematicians
that symbolic logic might be worth their serious attention. One
staunch hater of s^mabolic logic may be mentioned – Cantor,
whose work on the infinite will be noticed in the concluding
chapter. By one of those little ironies which make mathematical
history such amusing reading for the open-minded, symbolic
logic was to play an important part in the drastic criticism of
Cantor’s work that caused its author to lose faith in himself and
his theor\’.

Boole did not long survive the production of his masterpiece.
The year after its publication, still subconsciously striding for
the social respectability that he once thought a knowledge of
Greek could confer, he married Mary Everest, niece of the Pro-
fessor of Greek in Queen’s College. His wife became his devoted
disciple. After her husband’s death, Mary Boole applied some
of the ideas which she had acquired from him to rationalizing
and humanizing the education of young children. In her pamph-
let, Boole’s Psychology, Mary Boole records an interesting
speculation of Boole’s which readers of The Laws of Thought
will recognize as in keeping with the unexpressed but implied
personal philosophy in certain sections. Boole told his wife that
in 1S32, when he was about seventeen, it ‘flashed upon’ hini as
he was walking across a field that besides the knowledge gained
from direct observation, man derives knowledge from some
source undefinable and infusible – which Mary Boole calls ‘the
unconscious’. It will be interesting (in a later chapter) to hear
Poincare expressing a similar opinion regarding the genesis of
mathematical ‘inspirations’ in the ‘subconscious mind’. Any-
how, Boole was inspired, if ever a mortal was, when he wrote
The Laws of Thought.

Boole died, honoured and with a fast-growing fame, on
8 December 1864, in the fiftieth year of his age. His premature
death was due to pneumonia contracted after faithfully keeping
a lecture engagement when he was soaked to the skin. He fully
realized that he had done great work.

CHAPTER TWENTY-FOUR

THE MAX, XOT THE METHOD
Eerm tie

Outstanding unsolved problems demand new methods for
their solution, while powerful new methods beget new problems
to be solved. But, as Poincare observed, it is the man, not the
method, that solves a problem.

Of old problems responsible for new methods in mathematics
that of motion and all it implies for mechanics, terrestrial and
celestial, may be recalled as one of the principal instigators of
the calculus and present attempts to put reasoning about the
infinite on a firm basis. An example of new problems suggested
by powerful new methods is the s^yarm which the tensor
calculus, popularized to geometers by its successes in relativity,
let loose in geometry’. And finally, as an illustration of Poin-
care’s remark, it was Einstein, and not the method of tensors,
that solved the problem of giving a coherent mathematical
account of grayitation. All three theses are sustained in the life
of Charles Hermite, the leading French mathematician of the
second half of the nineteenth century – if we except Hermite’s
pupil Poincare, who belonged partly to our own century.

Charles Hemiite, born at Dieuze, Lorraine, France, on 24
December 1822 could hardly have chosen a more propitious era
for his birth than the third decade of the nineteenth eentuiv’.
His W’as just the rare combination of creative genius and the
ability to master the best in the work of other men which was
demanded in the middle of the century to co-ordinate the
arithmetical creations of Gauss with the discoveries of Abel and
Jacobi in elliptic functions, the striking ady’ances of Jacobi in
Abelian functions, and the y^ast theory of algebraic iny^aiiants
in process of rapid dey^elopment by the English mathematicians
Boole, Cayley, and Sylvester.

494

THE MAN, NOT THE METHOD

Hermite almost lost his life in the French Revolution –
although the last head had fallen nearly a quarter of a century
before he was born. His paternal grandfather was ruined by the
Commune and died in prison; his grandfather’s brother went to
the guillotine. Hermite’ s father escaped owing to his youth.

If Hermite’s mathematical ability was inherited, it probably
came from the side of the father, who had studied engineering.
Finding engineering uncongenial, Hermite senior gave it up,
and after an equally distasteful start in the salt industry, finally
settled down in business as a cloth merchant. This resting place
was no doubt chosen by the rolling stone because he had mar-
ried his employer’s daughter, Madeleine Lallemand, a domi-
neering woman who wore the breeches in her family and ran
everj’thing from the business to her husband. She succeeded in
building both up to a state of solid bourgeois prosperity.
Charles was the sixth of seven children – five sons and two
daughters. He was bom with a deformity of the right leg which
rendered him lame for life – possibly a disguised blessing, as it
effectively barred him from any career even remotely connected
with the army – and he had to get about with a cane. His defor-
mity never affected the uniform sweetness of his disposition.

Hermite’s earliest education was received from his parents.
As the business continued to prosper, the family moved from
Dieuze to Nancy when Hermite was six. Presently the growing
demands of the business absorbed all the time of the parents
and Hermite was sent as a boarder to the hjcee at Nancy. This
school pro^-ing unsatisfactory the prosperous parents decided
to give Charles the best and packed him off to Paris. There he
studied for a short time at the Lycee Henri IV, moving on at
the age of eighteen (1840) to the more famous (or infamous)
Louis-le-Grand – the ‘Alma’ Mater of the wretched Galois – to
prepare for the Polytechnique.

For a while it looked as if Hermite was to repeat the disaster
of his untamable predecessor at Louis-le-Grand. He had the
same dislike for rhetoric and the same indifference to the ele-
mentary mathematics of the classroom. But the competent
lectures on physics fascinated him and won his cordial co-opera-
tion in the bilateral process of acquiring an education. Later on,

495

MEX OF 3IATHEMATICS

Tinpestered by pedants, Hermite became a good classicist and
the master of a beautiful clear prose.

Those who hate examinations will love Hermite. There is
pomethiog in the careers of these two most famous alumni of
Louis-Ie-Grand, Galois and Hermite. which might well cause the
advocates of examinations as a reliable yardstick for arranging
human beings in order of intellectual merit to ask themselves
whether they have used their heads or their feet in arriving at
their conclusions. It was only by the grace of God and the diplo-
matic persistence of the devoted and intelligent Professor
Richard, who had done his unavailing best fifteen years before
to save Galois for science, that Hermite was not tossed out by
stupid examiners to rot on the rubbish heap of failure. ‘While
stni a student at the Hermite, following in the steps of
Galois, supplemented and neglected his elementary lessons by
private reading at the library of Sainte-Genevieve, where he
found and mastered the memoir of Lagrange on the solution of
numerical equations. Sa\’ing up his pennies, he bought the
French translation of the Bisquisiiiones Ariihmeticae of Gauss
and, what is more, mastered it as few before or since have
ma^ered it. B3″ the time he had followed what Gauss had done
Hermite was readjr to go on* Tt was in these two books’, he
loved to ssiy in later life, “that I learned Algebra.’ Euler and
Laplace also instructed him through their works. And yet
Hermite’s performance in examinations was, to sajr the most
fiattering thing possible of it, mediocre. Mathematical nonen-
tities beat him out of sight.

IVIindful of the tragic end of Galois, Richard tried his best to
steer Hermite away from original investigation to the less
exciting though muddier waters of the competitive examina-
tions for entrance to the ficole PoMechnique – the fdthy ditch
in which Galois had drowned himself. Nevertheless the good
Richard could not refrain from telling Hermite’s father that
Charles was ‘a young Lagrange’.

The Nouvelles Annates de Mathematiqnes. a journal devoted
to the interests of students in the higher schools, was founded
in 1842. The first volume contains two papers composed by
Hermite while he was still a student at Louis-le-Grand. The

490

THE MAX, NOT THE METHOD

first is a simple exercise in the anal3i;ic geometry of conic
sections and betrays no originality. The second, which fills only
six and a half pages in Hermite’s collected works, is a horse of
quite a different colour. Its unassuming title is Considerations
on the algebraic solution of the equation of the fifth degree (trans-
lation).

‘It is known’, the modest mathematician of twenty begins,
‘that Lagrange made the algebraic solution of the general equa-
tion of the fifth degree depend on the determination of a root
of a particular equation of the sixth degree, which he calls a

reduced equation [to-day, a “resolvent”] So that, if this

resolvent were decomposable into rational factors of the
second or third degrees, we should have the solution of the
equation of the fifth degree. I shall try to show that such a
decomposition is impossible.’ Hermite not only succeeded in
his attempt – by a beautifully simple argument – but showed
also in doing so that he was an a^ebraist. Mlth but a few slight
changes this short paper will do all that is required.

It may seem strange that a young man capable of genuine
mathematical reasoning of the calibre shown by Hermite in his
paper on the general quintic should find elementary mathe-
matics difficult. But it is not necessary to understand – or even
to have heard of – much of classical mathematics as it has
evolved in the course of its long history in order to be able to
follow or work creatively in the mathematics that has been
developed since 1800 and is stfll of li^dng interest to mathe-
maticians. The geometrical treatment (synthetic) of conic sec-
tions of the Greeks, for instance, need not be mastered to-day
by anyone who wishes to follow modem geometry; nor need
any geometry at all be learned by one whose tastes are alge-
braic or arithmetical. To a lesser degree the same is true for
analysis, where such geometrical language as is used is of the
simplest and is neither necessary nor desirable if up-to-date
proofs are the object. As a last example, descriptive geometry,
of great use to designing engineers, is of practically no use
whatever to a working mathematician. Some quite difficult
subjects that are still mathematically alive require only a school
education in algebra and a clear head for their comprehension.

407

MEN OP MATHEMATICS

Such are the theorj’ of finite groups, the mathematical theory
of the infinite, and parts of the theory of probabilities and the
higher arithmetic. So it is not astonishing that large tracts of
what a candidate is required to know for entrance to a technical
or scientific school, or even for graduation from the same, are
less than worthless for a mathematical career. This accounts
for Hermite’s spectacular success as a budding mathematician
and his narrow escape from complete disaster as an examinee.

Late in 1842, at the age of twenty, Hermite sat for the en-
trance examinations to the ficole Polji:echnique. He passed,
but only as sixty-eighth in order of merit. Already he was a
vastly better mathematician than some of the men who
examined him were, or were ever to become. The humiliating
outcome of this test made an impression on the young master
which all the triumphs of his manhood never effaced.

Hermite stayed only one year at the Polytechnique. It was
not his head that disqualified him but his lame foot which,
according to a ruling of the authorities, imfitted him for any of
the positions open to successful students of the school. Perhaps
it is as well that Hermite was thrown out; he was an ardent
patriot and might easily have been embroiled in one or other
of the political or military” rows so precious to the efferv’escent
French temperament. How’ever, the year was by no means
wasted. Instead of sla\’ing over descriptive geometry, which
he hated, Hermite spent his time on Abelian functions, then
(1842) perhaps the topic of outstanding interest and importance
to the great mathematicians of Europe. He had also made the
acquaintance of Joseph Liomdlle (1809-82), a first-class
mathematician and editor of the Journal des Maihemaiiques,
Liouvilie recognized genius when he saw it. In passing it may
be amusing to recall that Liomdlle inspired William Thomson,
Lord Kel\4n, the famous Scotch physicist, to one of the most
satisfying definitions of a mathematician that has ever been
given, ‘Do you know what a mathematician is?’ Kelvin once
asked a class. He stepped to the board and wrote

e^-dx = V TT,

Putting his finger on what he had written, he turned to the
498

THE MAN, NOT THE METHOD

class. ‘A mathematician is one to whom that is as obvious as that
twice two makes four is to 3 ^ou- liouviHe was a mathematician.’
Young Hermite’s pioneering work in Abelian functions, well
begun before he was twenty-one, was as far beyond Kehnn’s
example in imob\nousness as the example is beyond ‘twice two
makes four.’ Remembering the cordial welcome the aged
Legendre had accorded the revolutionary work of the young
and unknown Jacobi, Liouville guessed that Jacobi would show
a similar generosity to the beginning Hermite. He was not
mistaken.

The first of Hermite’s astonishing letters to Jacobi is dated
from Paris, January” 1843. ‘The study of your [Jacobi’s] memoir
on quadruple periodic functions arising in the theory of Abelian
functions has led me to a theorem, for the di\dsion of the argu-
ments [variables] of these functions, analogous to that which
you gave … to obtain the simplest expression for the roots of
the equations treated by Abel. M. Liouville induced me to
write to you, to submit this work to you; dare I hope, Six, that
you win be pleased to welcome it with all the indulgence it
needs?’ With that he plunges at once into the mathematics.

To recall briefly the bare nature of the problem in question:
the trigonometric functions are functions of one variable with
one period, thus sin (x + 2tt) = sin x, where x is the variable
and 27r is the period; Abel and Jacobi, by ‘inverting’ the elliptic
integrals, had discovered functions of one variable and feoo
periods, say /(a? + p -f 5) = where p, q are the periods (see
Qiapters 12, 18); Jacobi had discovered functions of two
variables and four periods, say

F{x -{■ a b^y + c d) = F{x,y),

where afiyC^d are the periods. A problem early encountered in
trigonometry is to express ^n or sin or generally sin

where n is any given integer, ifi terms of sin x (and

possibly other trigonometric functions of x). The correspond-
ing problem for the functions of two variables and four periods
was that which Hermite attacked. In the trigonometric pro-

499

MEX OF MATHEMATICS

blem vre are finally led to quite simple equations; in Hennite’s
incomparably more difficult problem the upshot is again an
equation (of degree n*), and the unexpected thing about this
equation is that it can be solved algebraically, that is, by
radicals.

Barred from the Polytechnique by his lameness, Hermite
now cast longing eyes on the teaching profession as a haven
where he might earn his li\ing while advancing his beloved
mathematics. The career should have been flung wide open to
him, degree or no degree, but the inexorable rules and regula-
tions made no exceptions. Red tape always hangs the wrong
man, and it nearly strangled Hermite.

Unable to break himself of his *pemicious originality’, Her-
mite continued his researches to the last possible moment when,
at the age of twenty-four, he abandoned the fundamental dis-
coveries he was making to master the trivialities required for
his first degrees (bachelor of letters and science). Two harder
ordeals would normally have followed the first before the
young mathematical genius could be certified as fit to teach, but
fortunately Hermite escaped the last and worst when influential
friends got him appointed to a position where he could mock the
examiners. He passed his examinations (in 1847-8) very badly.
But for the friendliness of two of the inquisitors – Sturm and
Bertrand, both fine mathematicians who recognized a fellow
craftsman when they saw one – Hermite would probably not
have passed at aD. (Hermite married Bertrand’s sister Lomse in
1848.)

By an ironic twist of fate Hermite’s first academic success
was his appointment in 1848 as an examiner for admissions to
the very Pohdechnique which had almost failed to admit him .
A few months later he was appointed quiz master {repeiiteitr) at
the same institution. He was now securely established in a niche
where no examiner could get at him. But to reach this ‘bad
eminence’ he had sacrificed nearly five years of what almost
certainly was his most inventive period to propitiate the
stupidities of the official system,

Ha\’ing finally satisfied or evaded his rapacious examiners,
Hennite settled down to become a great mathematician. EBs

500

THE MAN, NOT THE METHOD

life was peaceful and uneventful. In 1848 to 1850 he substituted
for Libri at the College de France. Six years later, at the early
age of thirty-four, he was elected to the Institut (as a member
of the Academy of Sciences). In spite of his world-wide reputa-
tion as a creative mathematician Hermite was forty-seven
before he obtained a suitable position: he was appointed pro-
fessor in 1869 at the fieole Normale and finally, in 1870, he
became professor at the Sorboime, a position which he held till
his retirement twenty-seven years later. During his tenure of
this influential position he trained a whole generation of distin-
guished French mathematicians, among whom Emile Picard,
Gaston Darboux, Paul Appell, ]£mile Borel, Paul Painleve and
Henri Poincare may be mentioned. But his influence extended
far beyond France, and his classic works helped to educate his
contemporaries in all lands.

A distinguishing feature of Hermite’s beautiful work is closely^
allied to his repugnance to take advantage of his authoritative*
position to re-create all his pupils in his own image: this is the*
unstinted generosity which he invariably displays to his fellow
mathematicians. Probably no other mathematician of modern
times has carried on such a voluminous scientific correspon-
dence with workers all over Europe as Hermite, and the tone of
his letters is always kindly, encouraging, and appreciative.
Many a mathematician of the second half of the nineteenth
century owed his recognition to the publicity which Hermite
gave his first efforts. In this, as in other respects, there is no
finer character than Hermite in the whole history of mathe-
matics. Jacobi was as generous – with the one exception of his
early treatment of Eisenstein – but he had a tendency to sar-
casm (often highly amusing, except possibly to the unhappy
\ictim) which was wholly absent from Hermite’ s general wit.
Such a man deserved the generous reply of Jacobi when the
unknown young mathematician ventured to approach him with
his first great work on Abelian functions. ‘Do not be put out,
Sir’, Jacobi wrote, ‘if some of your discoveries coincide with old
work of my own. As you must begin where I end, there is neces-
sarily a small sphere of contact. In future, if you honour me
with your cjojmnunications, I shall have only to learn.’

501

ilEN OF :MATHEMATrCS

Encouraged by J acobi, Hermite shared with him not onlv the
discoveries in Abelian functions, but also sent him four tremen-
dous letters on the theory of numbers, the first early in 1847,
These letters, the first of which was composed when Hermite
was only twentj^-fom, break new ground (in what respect we
shall indicate presently) and are sufficient alone to establish
Hermite as a creative mathematician of the first rank. The
generality of the problems be attacked and the bold originaiitv
of the methods he demised for their solution assure Hermite’s
remembrance as one of the born arithmeticians of history.

The first letter opens with an apology. ‘Nearly two years have
elapsed without my answering the letter full of goodwill which
you did me the honour to write to me. To-day I shall beg you to
pardon my long negligence and es^press to you all the joy I felt
in seeing myself given a place in the repertory of your works.
[Jacobi has published parts of Hermite’ s letter, with all due
acknowledgement, in some work of his own.] Ha\dng been for
long away from the work, I was greatly touched by such an
attestation of yoxir kindness; allow me, Sir, to believe that it
will not desert me.’ Hermite then says that another research of
Jacobi’s has inspired him to his present efforts.

If the reader will glance at what was said about uniform
functions of a single variable in the chapter on Gauss (a uniform
function takes onli/ one value for each value of the variable),
the following statement of what Jacobi had proved should be
intelligible: a uniform function of only one variable with three
distinct periods is impossible. That uniform functions of one
variable exist having either one period or two periods is proved
by exhibiting the trigonometric functions and the elliptic func-
tions. This theorem of Jacobi’s, Hermite declares, gave him his
own idea for the novel methods which he introduced into the
higher arithmetic. Although these methods are too technical
for description here, the spirit of one of them can be briefly
indicated.

Arithmetic in the sense of Gauss deals with properties of the
rational integers 1,2,3, . . . ; irrationals (like the square root of
2) are excluded. In particular Gauss investigated the integer
solutions of large classes of indeterminate equations in two or

502

THE MAN, NOT THE METHOD

three unkno-wTis, for example as in axr + 2hxy -j-
where a,b,c,m are any given integers and it is required to discuss
all integer solutions aj, y of the equation. The point to be noted
here is that the problem is stated and is to be solved entirely in
the domain of the rational integers, that is, in the realm of
discrete number. To fit analysis, which is adapted to the investi-
gation of continuous number, to such a discrete problem would
seem to be an impossibility, yet this is what Hermite did.
Starting with a discrete formulation, he applied analysis to the
problem, and in the end came out with results in the discrete
domain from which he had started. As analysis is far more
highly developed than any of the discrete techniques invented
for algebra and arithmetic, Hermite’s advance was comparable
to the introduction of modem machinery into a medieval
handicraft.

Hermite had at his disposal much more powerful machinery,
both algebraic and analytic, than any available to Gauss when
he wrote the Disquisitiones Arithmeticae. With Hermite’s own
great invention these more modem tools enabled him to attack
problems which would have baffled Gauss in 1800. At one stride
Hermite caught up with general problems of the type which
Gauss and Eisenstein had discussed, and he at least b^an the
arithmetical study of quadratic forms in any number of un-
knowns. The general nature of the arithmetical ‘theory of
forms’ can be seen from the statement of a special problem.
Instead of the Gaussian equation ax^ -{- 2hoDy -j- = m of

degree two in tmo unknowns {x, y), it is required to discuss the
integer solutions of similar equations of degree n in s imloiowns,
where n, s are any integers, and the degree of each term on the
left of the equation is n (not 2 as in Gauss’ equation). After
stating how he had seen after much thought that Jacobi’s
researches on the periodicity of uniform fimctions depend upon
deeper questions in the theory of quadratic forms, Hermite out-
lines his programmes.

‘But, having once arrived at this point of view, the problems

• vast enough — which I had thought to propose to myself,
  seemed inconsiderable beside the great questions of the general
  theory of forms. In this boundless espanse of researches which

503

MEN OP MATHEMATICS

Monsieur Gauss [Gauss was still living when Hermite wrote this,
hence the polite ‘Monsieur’] has opened up to us, Algebra and
the Theory of Xumbers seem necessarily to be merged in the
same order of analytical concepts, of which our present know-
ledge does not yet permit us to form an accurate idea.’

He then makes a remark which, although not very clear, can
be interpreted as meaning that the key to the subtle connexions
between algebra, the higher arithmetic, and certain parts of the
theory’ of functions will be found in a thorough understanding
of what sort of ‘numbers’ are both necessary and sufficient for
the explicit solution of all types of algebraic equations. Thus,
for — 1 = 0, it is necessary and sufficient to understand
1 ; for a® -}- oa; + & = 0, where afi are any given numbers,
what sort of a ‘number’ x must be invented in order that x may
be expressed explicitly in terms of Gauss of course gave one
kind of answer: any root aj is a complex number. But this is only
a begi n ni ng . Abel proved that if only 2 l finite number of rational
operations and extractions of roots are permitted, then there is
no explicit formula giving x in terms of a,&. We shall return to
this question later; Hermite even at this early date ( 184 j 8 ; he
was then twenty-six) seems’ to have had one of his greatest
discoveries somewhere at the back of his head.

In his attitude toward numbers Hermite was somewhat of a
mystic in the tradition of Pythagoras and Descartes – the
latter’s mathematical creed, as will appear in a moment, was
essentially P 5 i:hagorean. In other matters, too, the gentle
Hermite exhibited a marked leaning toward mysticism. Up to
the age of forty-three he was a tolerant agnostic, like so many
Frenchmen of science of his time. Then, in 1856, he fell suddenly
and dangerously ill. In this debilitated condition he was no
match for even the least persistent evangelist, and the ardent
Cauchy, who had always deplored his brilliant young friend’s
open-mindedness on religious matters, pounced on the prostrate
Hermite and converted him to Homan Catholicism. Thence-
forth Hermite was a devout Catholic, and the practice of his
religion gave him much satisfaction.

Hermite’s number-mysticism is harmless enough and it is one
of those personal things on which argument is futfle. Briefly,
504

THE MAN, NOT THE METHOD

Hermite beKeved that numbers have an existence of their own
above all control by human beings* Mathematicians, he
thought, are permitted now and then to catch glimpses of the
superhuman harmonies regulating this ethereal realm of numer-
ical existence, just as the great geniuses of ethics and morals
have sometimes claimed to have visioned the celestial perfec-
tions of the Kingdom of Heaven.

It is probably right to say that no reputable mathematician
to-day who has paid any attention to what has been done in the
past fifty years (especially the last twenty-five) in attempting
to understand the nature of mathematics and the processes of
mathematical reasoning would agree with the mystical Her-
mite. “^Miether this modem scepticism regarding the other-
worldliness of mathematics is a gain or a loss over Hermite’s
creed must he left to the taste of the reader. Whot is now almost
universally held by competent judges to be the wrong \iew of
■’mathematical existence’ was so admirably expressed by
Descartes in his theory of the eternal triangle that it may be
quoted here as an epitome of Hermite’s mystical beliefs.

‘I imagine a triangle, although perhaps such a figure does not
exist and never has existed anywhere in the world outside my
thought. Nevertheless this figure has a certain nature, or form,
or determinate essence which is immutable or eternal, which I
have not invented and which in no way depends on my Tnind.
This is evident from the fact that I can demonstrate various
properties of this triangle, for example that the sum of its three
interior angles is equal to two right angles, that the greatest
angle is opposite the greatest side, and so forth. \”^Tiether I
desire to or not, I recognize very clearly and convincingly that
these properties are in the triangle although I have never
thought about them before, and even if this is the first time I
have imagined a triangle. Nevertheless no one can say that I
have invented or imagined them.’ Transposed to such simple
‘eternal verities’ aslH-2 = 3, 2 + 2 = 4, Descartes’ everlast-
ing geometry becomes Hennite’s superhuman arithmetic.

One arithmetical investigation of Hennite’s, although rather
technical, may be mentioned here as an example of the pro-
phetic aspect of pure mathematics. Gauss, we recall, introduced

505

MEN OF MATHEMATICS

complex integers fniimbers of the form a — where a, b are
rational integers and i denotes V ~ 1) into the higher arith-
metic in order to give the law of biquadratic reciprocity-* its
simplest expression. Dirichlet and other followers of Gauss then
discussed quadratic forms in which the rational integers
appearing as variables and coefficients are replaced by Gaussian
complex integers. Hermite passed to the general case of this
situation and investigated the representation of integers in
what are to-day called Hermiiian forms. An example of such a
form (for the special case of two complex variables and

their ‘conjugates’ io instead of n variables) is

in which the bar over a letter denoting a complex number indi-
cates the conjugate of that number; namely, x iy is the
complex number, its ‘conjugate’ is a? — iy; and the coefficients
«u. «i 2 > ^22 are such that = ay,, for {i,j) = (1,1), (1,2),

(2,1), (2,2), so that flja ^21 conjugates, and each
Cgg is its own conjugate (so that Cgg are real numbers). It is
easily seen that the entire form is real (free of t) if aU products
are multiplied out, but it is most ‘naturally’ discussed in the
shape given.

When Hermite invented such forms he was interested in
finding what numbers are represented by the forms. Over
seventy years later it was found that the algebra of Hermitian
forms is indispensable in mathematical physics, particularly in
the modem quantum theory. Hermite had no idea that his pure
mathematics would prove valuable in science long after his
death — indeed, like Archimedes, he never seemed to care much
for the scientific applications of mathematics. But the fact that
Hermite’s work has given physics a useful tool is perhaps
another argument favouring the side that believes mathemati-
daxis best justify their abstract existence when left to their own
inscrutable devices.

Leaving aside Hermite’s splendid discoveries in the theory of
algebraic invariants as too technical for discussion here, we
^lall pass on in a moment to two of his most spectacular achieve-
ments in other fields. The high esteem in which Hermite’s

506

THE MAN, NOT THE METHOD

TTork in invariants was held by his contemporaries may, however,
be indicated by Sylvester’s characteristic remark that ‘Cayley,
Hermite, and I constitute an Invariantive Trinity.’ Who was
who in this astounding trinity Sylvester omitted to state; but
perhaps this oversight is immaterial, as each member of such a
trefoil would be capable of transforming himself into himself or
into either of his coinvariantive beings.

The two fields in which Hermite found what are perhaps the
most striking indi\ddual results in all his beautiful work are
those of the general equation of the fifth degree and transcen-
dental numbers. The nature of what he found in the first is
clearly indicated in the introduction to his short note Sur la
resolution de V equation du cinquieme degre (On the Solution of
the [general] Equation of the Fifth Degree; published in the
Comptes rendiis de V Academic des Sciences for 1858, when
Hermite was thirty-six).

‘It is known that the general equation of the fifth degree can
be reduced, by a substitution [on the unknown x] whose coeffi-
cients are determined without using any irrationalities other
than square roots or cube roots, to the form
£c® — a? — a = 0.

[That is, if we can solve this equation for then we can solve the
general equation of the fifth degree.]

‘This remarkable result, due to the English mathematician
Jerrard, is the most important step that has been taken in the
algebraic theory of equations of the fifth degree since Abel
proved that a solution by radicals is impossible. This impossi-
bility shows in fact the necessity for introducing some new
analytic element [some new kind of fxmction] in seeking the
solution, and, on this account, it seems natural to take as an
auxiliary the roots of the very simple equation we have just
mentioned. Nevertheless, in order to legitimize its use rigor-
ously as an essential element in the solution of the general equa-
tion, it remains to see if this simplicity of form actually permits
us to arrive at some idea of the nature of its roots, to grasp
what is peculiar and essential in the mode of existence of these
quantities, of which nothing is known beyond the fact that they
are not expressible by radicals.

507

ilEX OF MATHEilATICS

*Xow it is ven” remarkable that Jerrard’s equation lends it-
self with the greatest ease to this research, and is, in the sense
which we shall explain, susceptible of an actual analytic solu-
tion, For we may indeed conceive the question of the algebraic
solution of equations from a point of view different from that
which for long has been indicated by the solution of equations
of the first four degrees, and to which we are especially com-
mitted.

“Instead of expressing the closely interconnected system of
roots, considered as functions of the coefficients, by a formula
invoking many-valued radicals,* we may seek to obtain the
roots expressed separately by as many distinct uniform [one-
valued] functions of auxiliary variables, as in the case of the
third degree. In this case, where the equation
aj3 _ 3^; = 0

is under discussion, it suffices, as w’e know, to represent the
coefficient a by the sine of an angle, say A, in order that the
roots be isolated as the following well-determined functions

2 sm 2 sm –
3

. A -i- 4t7r

, 2 Sin ■

3 3

[Hermite is here recalling the familiar trigonometric solution’
of the cubic usually discussed in the second course of school
algebra. The ^auxiliary variable’ is A; the ‘uniform functions’
are here sines.]

‘Now it is an entirely similar fact which we have to exhibit
concerning the equation

a?® — a? — a = 0.

Only, instead of sines or cosines, it is the elliptic functions
which it is necessary to introduce. . ,

• For ex^pie, as in the simple quadratic — a = o : the roots are
  3 ! ss 4 – Vflj and x = — V a; the ‘many-valuedness’ of the radical
  involved, here a square root, or irrationality of the second degree,
  appears in the double sign, when we say briefly that the two roots
  are V n. The formula giving the ihree roots of cubic equations involves
  the three-v ahied irrationality *^^1, which has the three values X,
  i( _ 1 4.v”=r8), _ 1 _ -vTTa).

508

THE MAN, NOT THE METHOD

In short order Hermite then proceeds to solve the general
equation of the fifth degree, xising for the purpose elliptic func-
tions (strictly, elliptic modular fimctions, but the distinction is
of no importance here). It is almost impossible to convey to a
non-mathematician the spectacular brilliance of such a feat; to
give a very inadequate simiLe, Hermite found the famous ‘lost
chord’ when no mortal had the slightest suspicion that such an
elusive thing existed anywhere in time and space. Needless to
say his totally unforeseen success created a sensation in the
mathematical world. Better, it inaugurated a new department
of algebra and analysis in which the grand problem is to dis-
cover and investigate those functions in terms of which the
general equation of the nth. degree can be solved explicitly in
finite form. The best result so far obtained is that of Hermite’ s
pupil, Poincar6 (in the ISSO’s), who created the functions gi\Tng
the required solution. These turned out to be a ‘natural’
generalization of the elliptic functions. The characteristic of
those functions that was generalized was periodicity. Further
details would take us too far afield here, but if there is space
we shall recur to this point when we reach Poincare.

Hermite’s other sensational isolated result was that which
established the transcendence (explained in a moment) of the
number denoted in mathematical analysis by the letter c,
namely

where 1! means 1, 2! = 1 x 2, 3! = 1 x 2 x 3, 4! == T X 2
X 3 X 4, and so on; this number is the -base’ of the so-called
‘natural’ system of logarithms, and is approximately
2*718281828 It has been said that it is impossible to con-

ceive of a universe in which e and (the ratio of the circum-
ference of a circle to its diameter) are lacking. However that
may be (as a matter of fact it is .false), it is a fact that e turns up
everywhere in current mathematics, pure and applied. Why
this should be so, at least so far as applied mathematics is con-
cerned, may be inferred from the following fact: e®, considered
as a function of a?, is the only function of aj whose rate of change

509

OF MATHEMATICS

T^ith respect to x is equal to tlie function itself – that is, e® is the
only function -^hich is equal to its derivative.*

The concept of ^transcendence’ is extremely simple, also
extremely important. Any root of an algebraic equation whose
eoeflficients are rational integers (0, . . .) is called an

algebraic number. Thus V — 1, 2 *78 are algebraic numbers,
because they are roots of the respective algebraic equations
a:3 ^ 0, oOx — 139 = 0, in which the coefficients (1, 1 for

the first; 50, – 139 for the second) are rational integers. A
‘number’ which is not algebraic is called transcendental. Other-
wise expressed, a transcendental number is one which satisfies
no algebraic equation with rational integer coefficients.

Xow, given any ‘number’ constructed according to some
definite law, it is a meaningful question to ask whether it is
algebraic or transcendental. Consider, for example, the follow-
ing simply defined number,

To 102 ‘ io« ‘ Yo^i ‘ ‘ =

in which the exponents 2, 6, 24, 120, … are the successive
‘factorials’, namely 2 = 1×2, 6 = lx2x3, 24=1x2x3
X 4, 120 = 1 X2x3x4x5, and the indicated series
continues ‘to infinity”’ according to the same law as that for the

terms given. The next term is

; the sum of the first three

10720

terms is *1 -j- -01 4- *000001, or -llOCOl, sind it can be proved
that the series does actually define some definite number which
is less than -12. Is this number a root of any algebraic equation
with rational integer coefficients? The answer is no, although to
prove this without having been shown how to go about it is a
severe test of high mathematical ability. On the other hand, the
number defined by the infinite series

1 1 1 . 1

^ I i I

10 ^^ 103 10^1 10 ^^

• Strictly, ce’, where a does not depend upon a?, is the most general,
  but the ‘multiplicative constant’ a is tri\dal here.

510

THE MAN, NOT THE METHOD

is algebraic; it is the root of 99900 — 1 = 0 (as may be verified

by the reader who remembers how to sum an infinite convergent
geometrical progression).

The first to prove that certain numbers are transcendental
was Joseph Liouville (the same man who encouraged Hermite
to write to Jacobi) who, in 1844, discovered a very extensive
class of transcendental numbers, of which all those of the form
11 1 1^1

n n® ‘ ^^120 ’

where w is a real number greater than 1 (the example given
above corresponds to n = 10), are among the simplest. But it
is probably a much more difficult problem to prove that a
particular suspect, like e or w, is or is not transcendental than
it is to invent a whole infinite class of transcendentals: the
inventive mathematician dictates – to a certain extent — the
working conditions, while the suspected number is entire
master of the situation, and it is the mathematician in this case,
not the suspect, who takes orders which he only dimly under-
stands. So when Hermite proved in 1873 that e (defined a short
way back) is transcendental, the mathematical world was not
only delighted but astonished at the marvellous ingenuity of
the proof.

Since Heimite’s time many numbers (and classes of numbers)
have been proved transcendental. What is likely to remain a
high-water mark on the shores of this dark sea for some time
may be noted in passing. In 1934 the young Russian mathe-
matician Alexis Gelfond proved that all numbers of the type
where a is neither 0 nor 1 and b is any irraiional algebraic
number, axe transcendental. This disposes of the seventh of
David Hilbert’s list of twenty-three outstanding mathematical
problems which he called to the attention of mathematicians
at the Paris International Congress in 1900. Note that irra-
tional’ is necessary in the statement of Gelfond’s theorem (if
b — film, where n, m are rational integers, then a^, where a is
any algebraic number, is a root of = 0, and it can be

shown that this equation is eqmvalent to one in which all the
coefiScients are rational integers.

511

OF MATHEMATICS

Hermite’s unexpected ^dctory over the obstinate e inspired
mathematicians to hope that ^ Tvould presently be subdued in a
similar manner. For himself, however, Hemiite had had enough
of a good thing. ‘I shall risk nothing”, he wrote to Borchardt, “on
an attempt to prove the transcendence of the number r. If
others undertake this enterprise, no one will be happier than I
at their success, but believe me, my dear friend, it will not fail
to cost them some efforts.” Nine years later (in 1882) Ferdinand
Lindemann of the University of Munich, using methods veiy
similar to those which had sufficed Hermite to dispose of e,
proved that rr is transcendental, thus settling for ever the pro-
blem of ‘squaring the circle’. From what Lindemann proved it
follows that it is impossible with straight-edge and compass
alone to construct a square whose area is equal to that of any
given circle – a problem which had tormented generations of
mathematicians since before the time of Euclid,

As cranks are still tormented by the problem, it may be in
order to state concisely how Lindemann’ s proof settles the
matter. He proved that is not an algebraic number. But any
geometrical problem that is solvable by the aid of straight-edge
and compass alone, when restated in its equivalent algebraic
form, leads to one or more algebraic equations with rational
integer coefficients which can be solved by successive extrac-
tions of square roots. As satisfies no such equation, the circle
cannot be ‘squared’ with the implements named. If other
mechanical apparatus is permitted, it is easy to square the
circle. To all but mild lunatics the problem has been completely
^dead for over half a century. Nor is there any merit at the
present time in computing ir to a large number of decimal
places – more accuracy in this respect is already available than
is ever likely to be of use to the human race if it survives for a
billion to the billionth power years. Instead of trying to do the
impossible, mystics may like to contemplate the following
useful relation between e, tt, — 1 and V — 1 till it becomes
as plain to them as Buddha’s navel is to a blind Hindu
swami,

512

THE 3IAN, XOT THE METHOD

Anyone who can perceive this mystery intuitively will not need
to square the circle.

Since Lindemann settled ct the one outstanding unsolved
problem that attracts amateurs is Fermat’s ‘Last Theorem*.
Here an amateur with real genius undoubtedly has a chance.
Lest this be taken as an invitation to all and simdry to swamp
the editors of mathematical journals with attempted proofs,
recall what happened to Lindemaim when he boldly tackled the
famous theorem. If this does not suggest that more than ordi-
nary’ talent will be required to settle Fermat, nothing can. In
1901 Lindemann published a memoir of seventeen pages pur-
porting to contain the long-sought proof. The vitiating error
being pointed out, Lindemann, undaunted, spent the best part
of the next seven years in attempting to patch the unpatehable,
and in 1907 published sixty-three pages of alleged proof which
were rendered nonsensical by a slip in reasoning near the very
beginning.

Great as were Hermite’s contributions to the technical side of
mathematics, his steadfast adherence to the ideal that science
is beyond nations and above the power of creeds to dominate or
to stultify was perhaps an even more significant gift to civiliza-
tion ia the long ‘view of things as they now appear to a harassed
humanity. “VYe can only look back on his serene beauty of spirit
with a poignant regret that its like is nowhere to be foimd in the
world of science to-day. Even when the arrogant Prussians
were htimiliating Paris in the Franco-Prussian war, Hennite,
patriot though he was, kept his head, and he saw clearly that
the mathematics of ihe enemy’ was mathematics and nothing
else. To-day, even when a man of science does take the civilized
point of view, he is not impersonal about his supposed broad-
mindedness, hut aggressive, as befits a man on the defensive.
To Hennite it was so obvious that knowledge and wisdom are
not the prerogatives of any sect, any creed, or any nation that
he never bothered to put his instinctive sanity into words. In
respect of what Hermite knew by instinct our generation is
two centuries behind him. He died, loved the world over, on
14 January 1901.

CHAPTEB T’^VEXTY-PITE

THE DOUBTER
Kronecker

• Professional mathematicians who could properly be called
  business men are extremely rare. The one who most closely
  approximates to this ideal is Kronecker (1823-91), who did so
  well for himself by the time he was thirty that thereafter he was
  enabled to devote his superb talents to mathematics in consi-
  derably greater comfort than most mathematicians can afford.

The obverse of Kronecker’s career is to be found – according
to a tradition familiar to American mathematicians – in the
exploits of John Pierpont Morgan, founder of the banking
house of Morgan and Company. If there is anything in this tra-
dition, Morgan as a student in Germany showed such extra-
ordinary mathematical ability that his professors tried to
induce him to follow mathematics as bis life work and even
offered him a university position in German^’’ which would have
sent him off to a fl\ing start. Morgan declined and dedicated his
gifts to finance, with results familiar to all. Speculators (in
academic studies, not Wall Street) may amuse themselves by
reconstructing world history on the hj^pothesis that Morgan
had stuck to mathematics.

^Yhat might have happened to Germany had Kronecker not
abandoned finance for mathematics also offers a wide field for
speculation. His business abilities were of a high order; he was
an ardent patriot with an uncanny insight into European diplo-
macy and a shrewd cjTUcism – his admirers called it realism –
regarding the unexpressed sentiments cherished by the great
Powers for one another.

At first a liberal like so many intellectual young Jews,
Ilronecker quickly became a rock-ribbed conservative when he
saw which side his own abundant bread was buttered on – after

514

THE DOUBTER

his financial exploits, and proclaimed himself a loyal supporter
of that callous old truth-doctor Bismarck. The famous episode
of the Ems telegram ‘which, according to some, was the electric
spark that touched off the Franco-Prussian war in 1870, had
Kronecker’s warm approval, and his grasp of the situation was
so firm that before the battle of Weissenburg, when even the
military geniuses of Germany were doubtful as to the outcome
of their bold challenging of France, Kronecker confidently pre-
dicted the success of the entire campaign and was proved right
in detail* At the time, and indeed ah his life, he was on cordial
terms with the leading French mathematicians, and he was
clear-headed enough not to let his political opinions cloud his
just perception of his scientific rivals’ merits. It is perhaps as
well that so realistic a man as Kronecker cast his lot with
mathematics.

Leopold Ejonecker’s life was easy from the day of his birth.
The son of prosperous Jewish parents, he was born on 7
December 1823, at Liegnitz, Prussia. By an unaccountable
oversight Kronecker’s official biographers (Heinrich Weber and
AdoK Kneser) omit all mention of Leopold’s mother, although
he probably had one, and concentrate on the father, who
owned a flourishing mercantile business. The father -was a well-
educated man with an unquenchable thirst for philosophy
which he passed on to Leopold. There was another son, Hugo,
seventeen years yoxmger than Leopold, who became a distin-
guished physiologist and professor at Berne. Leopold’s early
education under a private tutor was supervised by the father;
Hugo’s upbringing later became the loving duty of Leopold.

In the second stage of his education at the preparatory school
for the Gymnasium Leopold -was strongly influenced hy the
co-rector Werner, a man with plnlosophicai and -theological
leanings, who later taught Kronecker when he entered the
Gymnasium. Among other things Kronecker imbibed from
Werner was a liberal draught of Christian theology, for which
he acquired a lifelong enthusiasm. With what looks like his
usual caution, Kronecker did not embrace the Christian faith
till practically on his deathbed when, having seen that it did his
six children no noticeable mischief, he permitted himself to he

515

MEN OF MATHEMATICS

converted from Judaism to evangelical Christianity in his
sixty-eighth year.

Another of Kronecker’s teachers at the G\Tnnasium also
influenced him profoundly and became his lifelong friend,
Ernst Eduard Kummer (1810-93), subsequently professor at
the University of Berlin and one of the most original mathe-
maticians Germany has produced, of whom more will be said
in connexion with Dedekind. These three, Kronecker senior,
Werner, and Kummer, capitalized Leopold’s immense native
abilities, formed his mind, and charted the future course of his
life so cunningly that he could not have departed from it if he
had wished.

Already in this early stage of his education we note an out-
standing feature of Kroneckers genial character, his ability to
get along with people and his instinct for forming lasting
friendships with men who had risen in the world or were to rise,
and who would be useful to him either in business or mathe-
matics. This genius for friendships of the right sort, which is one
of the successful business man’s distinguishing traits, was one
of Kroneeker’s more valuable assets and he never mislaid it.
He was not consciously mercenary, nor was he a snob; he was
merely one of those lucky mortals who is more at ease with the
successful than with the unsuccessful.

Kroneckeris performance at school was uniformly brilliant
and many-sided. In addition to the Greek and Latin classics
which he mastered with ease and fox which he retained a life-
long liking, he shone in Hebrew, philosophy, and mathematics.
His mathematical talent appeared early under the expert
guidance of Kummer, from whom he received special instruc-
tion. Young Kronecker however did not concentrate to any
great extent on mathematics, although it was obvious that his
greatest talent lay in that field, but set himself to acquiring a
broad liberal education commensurate with his manifold
abilities. In addition to his formal studies he took music lessons
and became an accomplished pianist and voc^st. Music, he
declared when he was an old man, is the finest of all the fine
arts, with the possible exception of mathematics, which he
likened to poetiy. These many interests he retained throughout
^ 516

THE DOUBTEK

his life. In none of them was he a mere dabbler: his love of the
classics of antiquity bore tangible fruit in Ms affiliation with
Graeea, a society dedicated to the translation and populariza-
tion of the Greek classics ; his keen appreciation of art made him
an acute critic of painting and sculpture, and his beautiful
house in Berlin became a rendezvous for musicians, among them
Felix Mendelssolm.

Entering the University of Berlin in the spring of 1841,.
Kronecker continued his broad education but began to concen-
trate on mathematics. Berlin at that time boasted Dirichlet
(1S05-59), Jacobi (1804-51) and Steiner (1796-1863) on its
mathematical faculty; Eisenstein (1823-52), the same age as
Kronecker, also was about, and the two became friends.

The influence of Dirichlet on KroneckeFs mathematical
tastes (particularly in the application of analysis to the theory
of numbers) is clear aU through Ms mature writings. Steiner
seems to have made no impression on Mm; Kronecker had no
feeling for geometry. Jacobi gave Mm a taste for elliptic func-
tions wMch he was to cultivate with striking originality and
brilliant success, cMefly in novel applications of magical beauty
to the theory of numbers.

Kronecker’ s xiniversity career was a repetition on a larger
scale of Ms years at school: he attended lectures on the classics
and the sciences and indulged Ms bent for philosophy by pro-
founder studies than any he had as yet undertaken, particularly
in the system of Hegel. The last is emphasized because some
curious and competent reader may be moved to seek the origin
of Kronecker’s mathematical heresies in the abstrusities of
HegePs dialectic – a quest wholly beyond the powers of the
present writer. Nevertheless there is a strange similarity be-
tween some of the weird unorthodoxies of recent doubts con-
cerning the self-consistency of mathematics – doubts for wMch
Kronecker’s ‘revolution’ was partly responsible – and the
subtleties of Hegel’s system. The ideal candidate for such an
undertaking would be a Marxian communist with a sound
training in Polish many-valued logic, though in what incense
tree this rare bird is to be sought God only knows.

Following the usual custom of German students, Kronecker

517

MEN OF MATHEMATICS

did not spend all his time at Berlin but moved about. Part of
his course Tvas pursued at the University of Bonn, where his old
teacher and friend Kummer had taken the chair of mathe-
matics. During Kroneeker’s residence at Bonn the University
authorities were in the midst of a futile war to suppress the
student societies whose cliief object was the fostering of drink-
ing, duelling, and brawling in general. With his customary
astuteness, Kronecker allied himself secretly with the students
and thereby made many friends who were later to prove useful.

Kronecker’s dissertation, accepted by Berlin for his Ph.D. in
1845, was inspired by Kummer s work in the theory of numbers
and dealt with the units in certain algebraic number fields.
Although the problem is one of extreme difficulty when it comes
to actually exhibiting the units, its nature can be understood
from the following rough description of the general problem of
units (for any algebraic number field, not merely for the special
fields which interested Kummer and Kronecker). This sketch
may also serve to make more inteUigible some of the allusions
in the present and subsequent chapters to the work of Kummer,
Kronecker, and Dedekind in the higher arithmetic. The matter
is quite simple but requires several preHminary definitions.

The common whole numbers 1,2,3, , . . are called the (posi-
tive) rational integers. If w is any rational integer, it is the root
of an algebraic equation of the first degree, whose coefficients
are rational integers, namely a? — jn = 0. This, among other
properties of the rational integers, suggested the generalimtion
of the concept of integers to the ‘numbers’ defined as roots of
algebraic equations. Thus if r is a root of the equation

• . – . + = 0 ,

where the a’s are rational integers (positive or negative), and if
further r satisfies no equation of degree less than n, all of
whose coefficients are rational integers and whose leading co-
efficient is 1 (as it is in the above equation, namely the coeffi-
cient of the highest power, a”, of aj in the equation is 1), then r
is called an algebraic integer of degree n. For example, 1 V — 5

is an algebraic integer of degree 2, because it is a root of
a:* — 2aj -f 6 0, and is not a root of any equation of degree

518

THE DOUBTEE

less than 2 “with coefficients of the prescribed kind; in fact
1 — 5 is the root of a; — (1 V — 5) = 0, and the last

coefficient, — (1 ~ V — 5), is not a rational integer.

If in the above definition of an algebraic integer of degree n
suppress the requirement that the leading coefficient be 1,
and say that it can be any rational integer (other than zero,
which is considered an integer), a root of the equation is then
called an algebraic number of degree n. Thus |(1 V — 5) is an
algebraic number of degree 2, but is not an algebraic integer; it
is a root of 2 < 2 )® — 253 -|”

Another concept, that of an algebraic number field of degreen, is
now introduced: if r is an algebraic number of degree n, the
totality of all expressions that can he constructed from r by
repeated additions, subtractions, multiplications, and divisions
(division by zero is not defined and hence is not attempted or
permitted), is called the algebraic number field generated by r,
and may be denoted by JF’[r]. For example, from r we get r + r,
or 2r; from this and r we get 2r/r or 2, 2r — r or r, 2r x r or
2r2, etc. The degree of this F[r] is n.

It can be proved that every member of F[r] is of the form
^ 4- . . . -f where the c’s are rational numbers,

and further every member of F[r] is an algebraic number of
degree not greater than n (in fact the degree is some divisor of
n). Some, but not all, algebraic numbers in jP[r] will be algebraic
integers.

The central problem of the theory of algebraic numbers is to
investigate the laws of arithmetical divisibility of algebraic
integers in an algebraic number field of degree n. To make this
problem definite it is necessary to lay down exactly what is
meant by ^arithmetical divisibility’, and for this we must
understand the like for the rational integers.

We say that one rational integer, m, is divisible by another, d,
if wecanfindarationalinteger,gf,suchthat m = q Xd;d (also g)
is called a divisor of m. For example 6 is a divisor of 12, because
12 = 2 X 6; 5 is not a divdsor of 12 because there does not exist
a rational integer q such that 12 = g x 5,

A (positive) rational pnwie is a rational integer greater than 1

MJi.— TOL. n.

E

519

MEK OP MATHEMATICS

whose only positive di^-isors are 1 and the integer itself. ‘When
we try to extend this definition to algebraic integers we soon
see that we have not found the root of the matter, and we must
seek some property of rational primes which can be carried over
to algebraic integers. This property is the following: if a rational
prime p divides the product a x h of two rational integers, then
(it can be proved that) p di^ddes at least one of the factors a, b
of the product.

Considering the unit, 1, of rational arithmetic, we notice that
1 has the peculiar property that it divides every rational integer;
*— 1 also has the same property, and 1,-1 are the only rational
integers ha\dng this property.

These and other clues suggest something simple that will
work, and we lay down the following definitions as the basis for
a theorj” of arithmetical divisibility” for algebraic integers. We
shall suppose that all the integers considered lie in an algebraic
number field of degree n.

If r,5j< are algebraic integers such that r = s x i, each of s, t
is called a divisor of r.

If j is an algebraic integer which divides every algebraic
integer in the field, j is called a unit (in that field). A given field
may contain an infinity of units, in distinction to the pair 1,
— 1 for the rational field, and this is one of the things that
breeds difficulties.

The next introduces a radical and disturbing distinction
between rational integers and algebraic integers of degree
greater than 1.

An algebraic integer other than a unit whose only divisors are
units and the integer itself, is called irreducible. An irreducible
algebraic integer which has the property that if it divides the
product of two algebraic integers, then it divides at least one of
the factors, is called a prime algebraic integer. Ail primes are
iixeducibles, but not all irreducibles are primes in some alge-
braic number fields, for example in F[V — 5], as will be seen
in a moment. In the common arithmetic of 1,2,3 … the
irreducibles and the primes are the same.

In the chapter on Fermat the fundamental theorem of
(rational) arithmetic was mentioned: a rational int^er is the

520

THE DOUBTER

product of (rational) primes in only one way. From this theorem
springs all the intricate theory of divisibility for rational
integers. Unfortunately the fundamental theorem does not hold
in all algebraic number fields of degree greater than one, and
the result is chaos.

To give an instance (it is the stock example usually exhibited
in text-books on the subject), in the field F[V — 5] we have
6 = 2X3 = (1 + V~S) X (1 – V~5);

each of2, 8, — 3>1 — V — 5 is a prime in this field (as

may be verified with some ingenuity), so that 6, in this field, is
not uniquely decomposable into a product of primes.

It may be stated here that Kronecker overcame this difficulty
by a beautiful method which is too detailed to be explained
untechnically, and that Dedeldnd did likewise by a totally
different method which is much easier to grasp, and which wiO
be noted when we consider his life. Dedekind’s method is the
one in widest use to-day, but this does not imply that Kro-
necker’s is less powerful, nor that it will not come into favour
when more arithmeticians become familiar with it.

In his dissertation of 1845 Kronecker attacked the theory of
the units in certain special fields – those defined by the equa-
tions arising from the algebraic formulation of Gauss’ problem
to divide the circumference of a circle into n equal parts or,
what is the same, to construct a regular polygon of n sides.

We can now close up one part of the account opened by
Fermat. In struggling to prove Fermat’s Xast Theorem’ that
a*” -f == s” is impossible in rational integers x, y, z (none zero)
if n is an integer greater than 2, arithmeticians took what looks
like a natural step and resolved the left-hand side, a?” -f y”,
into its n factors of the first degree (as is done in the usual
second course of school algebra). This led to the exhaustive
investigation of the algebraic number field mentioned above in
connexion with Gauss’ problem — after serious but readily
understandable mistakes had been made.

The problem at first was studded with pitfalls, into which
many a competent mathematician and at least one great one –
Cauchy – tumbled headlong. Cauchy assumed as a matter of

H 2 521

MEK OF MATHEMATICS

course that in the algebraic number field concerned the funda-
mental theorem of arithmetic must hold. After several exciting
but premature communications to the French Academy of
Sciences, he admitted his error. Being restlessly interested in a
large number of other problems at the time, Cauchy turned
aside and failed to make the great discovery which was well
within the capabilities of his prolific genius and left the field to
Kummer. The central difficulty was serious : here was a species
of integers’ – those of the field concerned — which defied the
fundamental theorem of arithmetic; how reduce them to law
and order?

The solution of this problem by the invention of a totally
new kind of ‘number’ appropriate to the situation, which (in
terms of these ‘numbers’) automatically restored the funda-
mental theorem of arithmetic, ranks with the creation of non-
Euclidean geometry as one of the outstanding scientific achieve-
ments of the nineteenth century, and it is well up in the high
mathematical achievements of all history. The creation of the
new ‘numbers’ – so-called ‘ideal numbers’ – was the invention
of Kummer in 1845. These new ‘numbers’ were not constructed
for all algebraic number fields but only for those fields arising
from the division of the circle.

Kummer too had fallen foul of the net which snared Cauchy,
and for a time he believed that be had proved Fermat’s ‘Last
Theorem’. Then Drrichlet, to whom the supposed proof was
submitted for criticism, pointed out by means of an example
that the fundamental theorem of arithmetic, contrary to
Rummer’s tacit assumption, does not hold in the field con-
cerned. This failure of Kummer’s was one of the most fortunate
things that ever happened in mathematics. Like Abel’s initial
mistake in the matter of the general quintic, Rummer’s turned
him into the right track, and he invented his ‘ideal numbers’.

Kummer, Kronecker, and Dedekind, in their invention of the
modem theory of algebraic numbers, by enlarging the scope of
arithmetic ad infinitum and bringing algebraic equations within
the purview of number, did for the higher arithmetic and the
theory of algebraic equations what Gauss, Lobatchewsky,
Johann Bolyai, and Biemann did for geometry in emancipating

522

THE DOUBTEE

it from slavery in Euclid’s too narrow economy. And just as the
inventors of non-Euclidean geometiy” revealed vast and hitherto
unsuspected horizons to geometry and physical science, so the
creators of the theory of algebraic numbers uncovered an
entirely new light, illuminating the whole of arithmetic and
throwing the theories of equations, of systems of algebraic
curves and surfaces, and the very nature of number itself, into
sharp relief against a firm background of shiningly simple
postulates.

The creation of ‘ideals’ – Dedekind’s inspiration from Kum-
mer’s \ision of ‘ideal numbers’ ~ renovated not only arithmetic
but the whole of the algebra which springs from the theory of
algebraic equations and systems of such equations, and it
proved also a reliable clue to the inner significance of the
‘enumerative geometry’ of Pliicker, Cayley and others, which
absorbed so large a fraction of the energies of the geometers of
the nineteenth century who busied themselves with the inter*
sections of nets of curves and surfaces. And last, if Kronecker’s
heres}” against Weierstrassian analysis (noted later) is some day
to become a stale orthodoxy, as all not utterly insane heresies
sooner or later do, these renovations of our familiar integers,
1,2,3, … , on which aU analysis strives to base itself, may ulti-
mately indicate extensions of analysis, and the Pythagorean
speculation may envisage generative properties of ‘number’
that Pythagoras never dreamed of in all his wild philosophy.

Ejonecker entered this beautifully difficult field of algebraic
numbers in 1845 at the age of twenty-two with his famous
dissertation Be Umtatibiis Complecds (On Complex Vniis), The
particular units he discussed were those in algebraic number
fields arising from the Gaussian problem of the division of the
circumference of a circle into n equal arcs. For this work he got
his Ph.D.

The German universities used to have – and may still have –

• One problem in this subject: an algebraic curve may have loops
  on it, or places where the curve crosses its tangents; given the degree
  of the curve, how many such points are there? Or if we cannot
  answer that, what equations connecting the number of these and
  other exceptional points must hold? Similarly for surfaces.

523

MEN OE MATHEMATICS

a laudable custom in connexion mth the taking of a Ph.D. : the
successful candidate was in honour bound to fling a partv ^
usually a prolonged beer bust with all the trimmings – for his
examiners. At such festivities a mock examination consisting
of ridiculous questions and more ridiculous answers was some-
times part of the fun. Kronecker incited practically the whole
facultVs including the Dean, and the memory of that undignified
feast in celebration of his degree was, he declared in later years,
the happiest of his life.

In at least one respect &onecker and his scientific enemy
Weierstrass were much alike: they were both very great gentle-
men, as even those who did not particularly care for either
admitted. But in nearly everything else they were almost
comically different. The climax of Kronecker’s career was his
prolonged mathematical war against Weierstrass, in which
quarter was neither given nor asked. One was a born algebraist,
the other almost made a religion of analysis. Weierstrass was
large and rambling, Kronecker a compact, diminutive man,
not over five feet tall, hut perfectly proportioned and sturdy.
After his student days Weierstrass gave up his fencing;
Kronecker was always an expert gjunnast and swimmer and in
later life a good moimtaineer.

Eye-witnesses of the battles between this curiously mis-
matched pair teU how the big fellow, annoyed by the persistence
of the little fellow, would stand s haking himself iilrf* a good-
natured St Bernard dog trying to rid himself of a determined
fly, only to excite his persecutor to more ingenious attacks, till
Weierstrass, giving up in despair, would amble off, Kronecker
at his heels stiU talking maddeningly. But for all their scientific
differences the two were good friends, and both were great
mathematicians without a particle of the ‘great man’ complex
that too often inflates the shirts of the would-be mighty.

Kronecker was blessed with a rich imcle in the banking busi-
ness, The uncle also controlled extensive farming enterprises.
All this fen into young Kronecker s hands for administration on
the death of the uncle, shortly after the budding mathematician
had taken his degree at the age of twenty-two. The eight years
from 1845 to 1853 were spent in managing the estate and run-

524

THE DOUBTER

ning the business, which. Kjonecker did with great thoroughness
and financial success. To manage the landed property efficiently
he even mastered the principles of agriculture.

In 1848, at the age of twenty-five, the energetic’young busi-
ness man very prudently fell in love with his cousin, Fanny
Prausnitzer, daughter of the defunct wealthy uncle, married
her, and settled down to raise a family. They had six children,
four of whom survived their parents. Kroneeker’s married life
was ideally happy, and he and his wife – a gifted, pleasant
woman — brought up their children with the greatest devotion.
The death of Kronecker’s wife a few months before his own last
illness was the blow which broke him.

During his eight years in business Kronecker produced no
mathematics. But that he did not stagnate mathematically is
shown by his publication in 1853 of a fundamental memoir on
the algebraic solution of equations. All through his activity as a
man of affairs Kronecker had maintained a lively scientific
correspondence with his former master, Kummer, and on
escaping from business in 1853 he visited Paris, where he made
the acquaintance of Hermite and other leading French mathe-
maticians. Thus he did not sever communications with the
scientific world when circumstances forced him into business,
but kept his soul alive by making mathematics rather than
whist, pinochle, or draughts his hobby.

In 1853, when Kronecker’ s memoir on the algebraic solva-
bility of equations (the nature of the problem was discussed in
the chapters on Abel and Galois) was published, the Galois
theory of equations was understood by very few. Kronecker’s
attack was characteristic of much of his finest work. Kronecker
had mastered the Galois theory, indeed he was probably the
only mathematician of the time (the late 1840’s) who had pene-
trated deeply into Galois’ ideas; Liouville had contented him-
self with a sufficient insight into the theory to enable him to
edit some of Galois’ remains intelligently.

A distinguishing feature of Kronecker’s attack was its com-
prehensive thoroughness. In this, as in other investigations in
algebra and the theory of numbers, Ejponecker took the refined
gold of his predecessors, toiled over it like an inspired jeweller,

525

MEX OF MATHEMATICS

added gems of his own, and made from the precious raw
material a flawless work of ark with the unmistakable impress
of his artistic mdi\dduaiity upon it. He delighted in perfect
things; a few of his pages will often exhibit a complete develop-
ment of one isolated result with all its implications immanent
but not loading the unique theme with expressed detail. Conse-
quently even the shortest of his papers has suggested important
developments to his successors, and his longer works are
inexhaustible mines of beautiful things.

Kronecker was what is called an ‘algorist’ in most of his
works. He aimed to make concise, expressive formulae tell the
story and automatically reveal the action from one step to the
next so that, when the climax was reached, it was possible to
glance back over the whole development and see the apparent
inevitability of the conclusion from the premises. Details and
accessory aids were ruthlessly pruned away imtil only the main
trunk of the argument stood forth in naked strength and sim-
plicity. In short, Kronecker was an artist who used mathe-
matical formulae as his medium.

After Kronecker’s works on the Galois theory the subject
passed from the private ownership of a few into the common
property of all algebraists, and Kronecker had wrought so
artistically that the next phase of the theory of equations – the
current postulational formulation of the theory and its exten-
sions – can be traced back to him. His aim in algebra, like that
of Weierstrass in analysis, was to find the ‘natural’ way – a
matter of intuition and taste rather than scientific definition –
to the heart of his problems.

The same artistry and tendency to unification appeared in
another of his most celebrated papers, which occupied only a
couple of pages in his collected works, On the Solution of the
General Equation of the Fifth Degree, first published in 1858,
Hermite, we recall, had given the first solution, by means of
elliptic (modular) functions in the same year, Kronecker attains
Hermite’s solution – or what is practically the same – by
applying the ideas of Galois to the probfem, thereby making the
miracle appear more ‘natural’. In another paper, also short,
over which he has spent most of his time for five years, he

526

THE DOUBTER

returns to the subject in 1861, and seeks the reason why the
general equation of the fifth degree is solvable in the manner in
Tvhich it is, thus taking a step bej’-ond Abel who settled the
question of solvability ‘by radicals’.

Much of Kronecker’s work has a distinct arithmetical tinge,
either of rational arithmetic or of the broader arithmetic of
algebraic numbers. Indeed, if his mathematical activity had
any guiding clue, it may be said to have been his desire, perhaps
subconscious, to ariihmeHze all mathematics, from algebra to
analysis. ‘GJod made the integers’, he said, ‘all the rest is the
work of man.’ Kronecker’s demand that analysis be replaced by
finite arithmetic was the root of his disagreement with Weier-
strass. Universal arithmetization may be too narrow an ideal
for the luxuriance of modern mathematics, but at least it has
the merit of greater clarity than is to be found in some others.

Geometry never seriously attracted Kronecker. The period
of specialization was already well advanced when Kronecker
did most of his work, and it would probably have been impos-
sible for any man to have done the profoundly perfect sort of
work that Kronecker did as an algebraist and in his own
peculiar type of analysis and at the same time have accom-
plished anything of significance in other fields. Specialization is
frequently damned, but it has its virtues.

A distinguishing feature of many of Kronecker’s technical
discoveries was the intimate way in which he wove together the
three strands of his greatest interests ~ the theory of numbers,
the theory of equations, and elliptic functions – into one beau-
tiful pattern in which unforeseen symmetries were revealed as
the design developed and many details were unexpectedly
imaged in others far away. Each of the tools with which he
worked seemed to have been designed by fate for the more
eMcient functioning of the others. Not content to accept this
mysterious unity as a mere mystery, Kronecker sought and
found its underlying structure in Gauss’ theory of binary
quadratic forms, in which the main problem is to investigate
the solutions in integers of indeterminate equations of the
second degree in two unknowns.

Kronecker’s great work in the theory of algebraic numbers

527

HEN OF HATHEilATICS

was not part of this pattern. In another direction he also
departed occasionally from his principal interests when,
according to the fashion of his times, he occupied himself with
the purely mathematical aspects of certain problems (in the
theory of attraction as in Newton’s gravitation) of mathema-
tical physics. His contributions in this field were of mathema-
tical rather than physical interest.

Up till the last decade of his life Kronecker was a free Tnajx
with obligations to no employer. Nevertheless he voluntarily
assumed scientific duties, for which he received no remunera-
tion, when he availed himself of his privilege as a member of the
Berlin Academy to lecture at the University of Berlin. From
1S61 to 1883 he conducted regular courses at the university,
principally on Ms personal researches, after the necessary intro-
ductions. In 1883 Kummer, then at Berlin, retired, and
Kronecker succeeded his old master as ordinary professor. At
this period of his life he travelled extensively and was a frequent
and welcome participant in scientific meetings in Great Britain,
France, and Scandinavia,

Throughout his career as a mathematical lecturer Kjonecker
competed with \Teierstrass and other celebrities whose subjects
were more popular than Ms own. Algebra and the theory of
numbers have never appealed to so wide an audience as have
geometry and analysis, possibly because the connexions of the
latter with physical sdenee are more apparent.

Kronecker took Ms aristocratic isolation good-naturedly and
even with a certain satisfaction. His beautifully clear introduc-
tions deluded Ms auditors into a belief that the subsequent
course of lectures would be easy to follow. This belief evapor-
ated rapidly as the course progressed, until after three sessions
aU but a faithful and obstinate few had silently stolen away –
many of them to listen to Weierstrass. Kronecker rejoiced. A
curtain could now he drawn across the room behind the first
few rows of chairs, he joked, to bring lecturer ^d auditors into
cosier intimacy- The few disciples he retained followed Mm
devotedly, walking home with him to continue the discussions
of the lecture room and frequently affording the crowded side-
walks of Berlin the diverting spectacle of an excited little man

528

THE DOUBTEH

talking with, his whole body – especially his hands – to a spell-
bound group of students blocldng the traffic. His house was
always open to his pupils, for E^roneeker really hked people, and
bis generous hospitality was one of the greatest satisfactions of
Ms life. Several of his students became eminent mathematicians,
but his ‘school’ was the whole world and he made no effort to
acquire an artificially large following.

The last is characteristic of Kjonecker s own most startlingly
independent work. In an atmosphere of confident belief in the
soundness of analysis Kronecker assumed the unpopular role
of the philosophical doubter. Not many of the great mathema-
ticians have taken philosophy seriously; in fact the majority
seem to have regarded philosophical speculations with repug-
nance, and any epistemological doubt affecting the soimdness
of their work has usually been ignored or impatiently brushed
aside.

With Kronecker it was different. The most original part of
his work, in which he was a true pioneer, was a natural out-
growth of his philosophical mclinations. His father, Werner,
Kummer, and his own wide reading in philosophical literature
had influenced him in the direction of a critical outlook on all
human knowledge, and when he contemplated mathematics
from this questioning point of view he did not spare it because
it happened to be the field of his own particular interest, but
infused it with an acid, beneficial scepticism. Although but
little of this found its way into print it annoyed some of his
contemporaries intensely and it has survived. The doubter did
not address himself to the living but, as he said, ‘to those who
shall come after me’. To-day these followers have arrived, and
owing to their united efforts -although they often succeed only
in contradicting one another – we are beginning to get a clearer
insight into the nature and meaning of mathematics.

Weierstrass (Chapter 22) would have constructed mathe-
matical analysis on his conception of irrationals as defined by
infinite sequences of rationals. Kronecker not only disputes
Weierstrass; he would nullify Eudoxus. For him as for Pytha-
goras only the God-given integers 1,2,3, , ‘exist’: all the

rest is a futile attempt of mankind to improve on the Creator.

529

MEN OF MATHEMATICS

Weierstrass on tlie other hand believed that he had at last made
the square root of 2 as comprehensible and as safe to handle as
2 itself; Kronecker denied that the square root of 2 ‘exists’, and
he asserted that it is impossible to reason consistently with or
about the Weierstrassian construction for this root or for any
other irrational. Neither his older colleagues nor the young to
whom Kronecker addressed himself gave his revolutionary idea
a very enthusiastic welcome.

\Yeierstrass himself seems to have felt uneasy: certainly he
was hurt. His strong emotion is released mostly in one tre>
mendous German sentence*^ like a fugue, which it is almost
impossible to preserv^e in English. ‘But the worst of it is’, he
complains, ‘that Kronecker uses his authority to proclaim that
all those who up to new have laboured to establish the theory
of functions are sinners before the Lord. ‘VMien a whimsical
eccentric like Christoffei [the man whose somewhat neglected
work was to become, years after liis death, an important tool in
difierential geometry as it is cultivated to-day in the mathe-
matics of relativity”] says that in twenty or thirty years the
present theory” of functions will be buried and that the whole of
analysis will be referred to the theory of forms, we reply with
a shrug. But when Kronecker delivers himself of the following
verdict which I repeat tcord/or xisord: ‘”If time and strength are
granted me, I myself will show the mathematical world that not
only geometry, but also arithmetic can point the way” to
analysis, and certainly a more rigorous way. If I cannot do it
my”self those who come after me will . . . and they wiD recognize
the incorrectness of all those conclusions with which so-called
analysis works at present” – such a verdict from a man whose
emitjent talent and distinguished performance in mathematical
research I admire as sineereljr and with as much pleasure as all
ids colleagues, is not only humiliating for those whom he adjures
to acknowledge as an error and to forswear the substance of
what has constituted the object of their thought and unremit-
ting labour, hut it is a direct appeal to the yoxmger generation
to desert their present leaders and rally around him as the
disciple of a new system which miist be founded. Truly it is sad.
In a letter to Sonja KowalewsM, 1885 .

530

THE DOUBTER

and it fills me with a bitter grief, to see a man, whose glory is
without flaw, let himself be driven by the well justified feeling
of his own worth to utterances whose injurious effect upon
others he seems not to perceive.

‘But enough of these things, on which I have touched only to
explain to you the reason why I can no longer take the same joy
that I used to take in my teaching, even if my health were to
permit me to continue it a few years longer. But you must not
speak of it ; I should not like others, who do not know me as well
as you, to see in what I say the expression of a sentiment which
is in fact foreign to^me.’

Weierstrass was seventy and in poor health when he wrote
this. Could he have lived till to-day he would have seen his own
great system stiQ flourishing like the proverbial green bay tree.
Kronecker’s doubts have done much to instigate a critical re-
examination of the foundations of all mathematics, but they
have not yet destroyed analysis. They go deeper, and if any-
thing of far-reaching significance is to be replaced by something
firmer but as yet unknown, it seems likely that a good part of
Kronecker’s own work will go too, for the critical attack which
he foresaw has uncovered weaknesses where he suspected
nothing. Time makes fools of us all. Our only comfort is that
greater shall come after us.

Kronecker s “revolution’, as his contemporaries called his sub-
versive assault on analysis, would banish all but the positive
integers from mathematics. GJeometry since Descartes has been
largely an affair of analysis applied to ordered pairs, triples, . . •
of real numbers (the ‘numbers’ which correspond to the dist-
ances measured on a given straight line from a fixed point on the
line); hence it too would come tmder the sway of Kronecker’s
programme. So familiar a concept as that of a negative integer,
— 2 for instance, would not appear in the mathematics Kron-
ecker prophesied, nor would common fractions.

Irrationals, as Weierstrass points out, roused Kroneckeris
special displeasure. To speak of j:- — 2 = 0 having a root would
be meaningless. AH of these dislikes and objections are of course
themselves meaningless unless they can be backed by a definite
programme to replace what is rejected.

531

MEN OF MATHEMATICS

Kxonecker actually did this, at least in outline, and indicated
how the whole of algebra and the theory- of numbers, including
algebraic numbers, can be reconstructed in accordance with his
demand. To get rid of V — 1 , for example, we need only put a
letter for it temporarily, say i, and consider poUmomials con-
taining i and other letters, say . . . Then we manipulate
these polynomials as in elementary’ algebra, treating i like anv
of the other letters, till the last step, when every’ polynomial
containing i is di\ided by P 1 and everything but the re-
mainder obtained from tliis division is discarded. Any’one who
remembers a little elementary’ algebra may- readily’ convince
himself that this leads to all the familiar properties of the
mysteriously misnamed ‘imaginary’’ mmcibers of the text-books.
In a similar manner negatives and fractions and all algebraic
numbers (other than the positive rational integers) are elimi-
nated from mathematics – if desired – and only the blessed
positive integers remain. The inspiration about discarding
V 1 goes back to Cauchy in 1847. This was the germ of
Kronecker s programme.

Those who dislike Kronecker’s ‘revolution’ call it a Putsch,
which is more like a drunken brawl than an orderly’ revolution.
Nevertheless it has led in recent yrears to two constructively
critical movements in the whole of mathematics: the demand
that a construction in a finite number of steps be given or
proved to be possible for any ‘number’ or other mathematical
‘entity’ whose ‘existence’ is indicated, and the banishment
from mathematics of all definitions that cannot be stated expli-
citly in a finite number of words, insistence upon these demands
has already done much to clarify our conception of the nature
of mathematics, but a vast amount remains to be done. As this
work is still in progress we shall defer further consideration of
it until we come to Cantor, when it will be possible to exhibit
examples.

Kronecker’s disagreement with Weierstrass should not leave
an unpleasant impression, as it may^ do if we ignore the rest of
Kronecker’s generous life. Kronecker had no intention of
wounding his kindly old senior; he merely let his tongue run
away with him in the heat of a purely’’ mathematical argument,

582

THE DOUBTEB

and Weierstrass, when he was in good spirits, laughed the whole
attack off, as he should have done, knowing well that just as he
had improved on Eudoxus, so his successors would probably
improve upon him. Possibly if Kronecker had been six or seven
inches taller than he was he would not have felt constrained to
over-emphasize his objections to analysis so vociferously.
Much of the whole wordy dispute sounds suspiciously like the
over-correction of an unjustified inferiority complex.

The reaction of many mathematicians to Kronecker’s ‘revo-
lution’ was summed up by Poincare when he said that Kron-
ecker had been enabled to do so much fine mathematics became
he frequently forgot his own mathematical philosophy. Kike
not a few epigrams this one is just untrue enough to be witty.

Kronecker died of a bronchial illness in Berlin on 29 De-
cember 1891 in his sixty-ninth year.

CHAPTES TWENTY-SIX

ANIMA CANDIDA
Biemann

It has been said of Coleridge that he wrote but Kttle poeti^- of
the highest order of excellence, but that that little should be
bound in gold. The like has been said of Bernhard Riemann, the
mathematical fruits of whose all too brief summer fill only one
octavo volume. It may also be truly said of Riemann that he
touched nothing that he did not in some measure revolutionize.
One of the most original mathematicians of modem times,
Riemann unfortunately inherited a poor constitution, and he
died before he had reaped a tithe of the golden harvests in his
fertile mind. Had he been bom a century later than he was,
medical science could probably have leased him twenty or thirty
more years of life, and mathematics would not now be waiting
for his successor.

Georg Friedrich Bernhard Riemann, the son of a Lutheran
pastor, and the second of six children (two boys, four girls), was
bom in the little village of Breselenz, in Hanover, Germany, on
17 September 1826. His father had fought in the Napoleonic
wars, and on settling down to a less barbarous mode of living
had married Charlotte Ebell, daughter of a court councillor.
H anover in 1826 was not exactly prosperous, and the circum-
stances of an obscure country parson with a wife and six chil-
dren to feed and clothe were far from affluent. It is claimed by
some biographers, apparently with justice, that the frail health
and early deaths of most of the Reimann children were the
result of under-nourishment in their youth and were not due to
poor stamina. The mother also died before her children were
grown.

In spite of poverty the home life was happy, and Riemann
always retained the wannest affection – and homesickness,

584

ANIMA CANDIDA

when he was absent -for all his lovable family. From his earliest
years he was a timid, diffident soul with a horror of speaking in
public or attracting attention to himself. In later life this
chronic shyness proved a very serious handicap and occa-
sioned him much agonized misery till he overcame it by diligent
pr^aration for every public utterance he was likely to make.
The engaging bashfulness of Riemann’s boyhood and early
manhood, which endeared him to all who met him, was in
strange contrast to the ruthless boldness of his matured scien-
tific thought. Supreme in the world of his own creation, he
realized bis transcendent powers and shrank from nobody, real
or imaginary.

■While Riemann was still an infant his father was transferred
to the pastorate of Quickbom. There young Riemann receh^ed
his first instruction, from his father, who appears to have been
an excellent teacher. From the very first lessons Bernhard
showed an unquenchable thirst for learning. His earliest
interests were historical, particularly in the romantic and tragic
history of Poland. As a boy of five Bernhard gave his father no
peace about unhappy Poland, but demanded to be told over
and over again the legend of that heroic country’s gallant (and
at times slightly fatuous) struggles for liberty and, in the late
Woodrow “Wilson’s rich, fruity phrase, ‘self-determinatioii’.

Arithmetic, begun at about six, offered something less har-
rowing for the sensitive young boy to dwell on. His inborn
mathematical genius now asserted itself. Bernhard not only
solved all the problems shoved at him, but invented more
difficult teasers to exasperate his brother and sisters. Already
the creative impulse in mathematics dominated the boy’s
mind. At the age of ten he received mstruction in more ad-
vanced arithmetic and geometry from a professional teacher,
one Schulz, a fairly good pedagogue. Schulz soon found himself
following his pupil, who often had better solutions than he-

At fourteen Riemann went to stay with his grandmother at
Hanover, where he entered his first Gymnasium, in the upper
third class. Here he endured his first overwhelming loneliness.
His shyness made him the butt of his schoolfellows and drove
him in upon his own resources. After a temporary setback his

MEN OF MATHEMATICS

schoolwork was uniformly excellent, but it gave birn no coin-
fort, and his only solace was the joy of buying such inconsider-
able presents as his pocket money would permit, to send home
to his parents and brother and sisters on their birthdays. One
present for his parents he invented and made himself, an
original perpetual calendar, much to the astonishment of his
incredulous schoolfellows. On the death of his grandmother Wo
years later, Riemann was transferred to the Gymnasium at
Liineburg, where he studied till he was prepared, at the age of
nineteen, to enter the University of Gottingen. At Liineburg
Riemann was within walking distance of home. He took fuH
advantage of his opportunities to escape to the warmth of his
own fireside. These years of his secondary education, while his
health was still fair, were the happiest of his life. The tramps
back and forth between the Gymnasium and Quickborn taxed
his strength, but in spite of his mother’s anxiety that he might
wear himself out, Riemann continued to over-exert himself in
order that he might be with his family as often as possible.

While still at the Gymnasium Riemann suffered from the
itch for finality and perfection which was later to slow up his
scientific publication. This defect – if such it was – caused him
great difficulty in his written language exercises and at first
made it doubtful whether he would ‘pass’. But this same trait
was responsible later for the finished form of two of his master-
pieces, one of which even Gauss declared to be perfect. Things
improved when Seyffer, the teacher of Hebrew, took young
Riemann into his own house as a boarder and ironed him out.

The two studied Hebrew together, Riemann frequently giv-
ing more than he took, as the future mathematician at that
time was all set to gratify his father’s wishes and become a
great preacher — as if Riemann, with his tongue-tied bashful-
ness, could ever have thumped hell and damnation or redemp-
tion and paradise out of any pulpit. Riemann himself was
enamoured of the pious prospect, and although he never got as
fax as a probationary sennon, he did employ his mathematical
talents in an attempted demonstration, in the manner of
Spinoza, of the truth of Genesis. Undaunted by his failure
young Riemann persevered in his faith and remained a sincere

536

AXIMA CANDIDA

Christian all his life. As his biographer (Dedekind) states, ‘He
reverently avoided disturbing the faith of others; for him the
main thing in religion was daily self-examination’. By the end
of his GjTimasium course it was plain even to Riemann that
Great Headquarters could have but little use for him as a router
of the devil, but might be able to employ him profitably in the
conquest of nature. Thus once again, as in the cases of Boole
and Kummer, a brand was plucked from the burning, ad
majorem Dei gloriam.

The director of the Gymnasium, Schmalfuss, having ob-
served Riemann’s talent for mathematics, had given the boy
the run of his private library and had excused him from attend-
ing mathematical classes. In this way Riemann discovered his
inborn aptitude for mathematics, but his failure to realize
immediately the extent of liis ability is so characteristic of his
almost pathological modest\’ as to be ludicrous.

Schmalfuss had suggested that Riemann borrow some
mathematical book for private study. Riemann said that would
be nice, provided the book was not too easy, and at the sugges-
tion of Schmalfuss carried off Legendre’s Theorie des Xombres
(Theory of Numbers). This is a mere trifle of 859 large quarto
pages, many of them crabbed with very close reasoning indeed.
Six days later Riemann returned the book. ‘How far did you
read?’ Schmalfuss asked. Without repljdng directly, Riemann
expressed his appreciation of Legendre’s classic. ‘That is cer-
tainly a wonderful book, I have mastered it.’ And in fact he
had. Some time later when he was examined he answered
perfectly, although he had not seen the book for months.

No doubt this is the origin of Riemann’s interest in the riddle
of prime numbers. Legendre has an empirical formula estimat-
ing the approximate number of primes less than any pre-
assigned number; one of Riemann’s profoundest and most
suggestive works (only eight pages long) was to be in the same
general field. In fact ‘Riemann’s hypothesis’, origina t i n g in his
attempt to improve on Legendre, is to-day one of the out-
standing challenges, if not the outstanding challenge, to pure
mathematicians.

To anticipate slightly, we may state here what this hypothe-

537

MEN OF MATHEMATICS

sis is. It occxirs in the famous memoir Ueber die Anzahl der
Primzahlen unter einer gegehenen Grosse (On the number of
prime numbers xmder a given magnitude), printed in the
monthly notices of the Berlin Academy for November 1859,
when Biemann was thirty-three. The problem concerned is to
give a formula which will state how many primes there are less
than any given number n. In attempting to solve this Riemaun
-was driven to an investigation of the infinite series

1111
I + ~ + ~ + 7i-r^ + 9

2® 3^ 4* 5®

in which s is a complex number, say s = w -}- izj (z = V — i),
where u and v are real numbers, so chosen that the series con-
verges, With this proviso the infinite series is a definite function
of s, say ^ (s) (the Greek zeta, is always used to denote this
function, which is called ‘Riemann’s zeta function’); and as s
varies, ^ (s) continuously takes on different values. For what
values of s mill I (s) be zero2 Riemann conjectured that all such
values of s for which u lies between 0 and 1 are of the form J -f
iVi namely, all have their real part equal to
This is the famous hypothesis. IVhoever proves or disproves
it will cover himself with glory and incidentally dispose of many
extremely difficult questions in the theory of prime numbers,
other parts of the higher arithmetic, and in some fields of ana-
lysis. Expert opinion favours the truth of the hypothesis. In
1914 the English mathematician G. H. Hardy proved that cm
infinity of values of s satisfy the hypothesis, but an infinity is
not necessarily all. A decision one way or the other disposing of
Riemann’s conjecture would probably be of greater interest to
mathematicians than a proof or disproof of Fermat’s Last
Theorem, Riemann’s hypothesis is not the sort of problem that
can be attacked by elementary methods. It has already given
rise to an extensive and thorny literature.

Legendre was not the only great mathematician whose
works Riemann absorbed by himself – always with amazing
speed – at the Gymnasium; he became familiar with the cal-
culus and its ramifications through the study of Euler. It is
rather surprising that from such an antiquated start in analysis
538

ANIMA CANDIDA

(Euler’s approach was out of date by the middle 1840’s owing
to the work of Gauss, Abel, and Cauchy), Riemann later became
the acute analyst that he did. But from Euler he may have
picked up something which also has its place in creative mathe-
matical work, an appreciation of symmetrical formulae and
manipulative ingenuity. Although Riemann depended chiefly
on what may be called deep philosophical ideas – those which
get at the heart of a theory – for his greater inspirations, his
work nevertheless is not wholly lacking in the ‘mere ingenuity’
of which Euler was the peerless master and w’hich it is now
quite the fashion to despise. The pursuit of pretty formulae and
neat theorems can no doubt quickly degenerate into a silly vice,
but so also can the quest for austere generalities which are so
very general indeed that they are incapable of application to
any particular. Riemann’ s instinctive mathematical tact pre-
served him from the had taste of either extreme.

In 1S46, at the age of nineteen, Riemann matriculated as a
student of philology and theology at the University of Gottin-
gen. His desire to please his father and possibly help financially
by securing a paying position as quickly as possible dictated the
choice of theology. But he could not keep away from the mathe-
matical lectures of Stem on the theory of equations and on
definite integrals, those of Gauss on the method of least squares,
and Goldschmidt’s on terrestrial magnetism. Confessing all to
his indulgent father, Riemann prayed for permission to alter
his course. His father’s ungrudging consent that Bernhard
follow mathematics as a career made the young man supremely
happy – also profoundly grateful.

After a year at Gottingen, where the instruction was decid-
edly antiquated, Riemann migrated to Berlin to receive from
Jacobi, Dirichlet, Steiner, and Eisenstem his initiation into new
and vital mathematics. From all of these masters he learned
much – advanced mechanics and higher algebra from Jacobi,
the theory of numbers and analysis from Dirichlet, modem geo-
metry from Steiner, wliiie from Eisenstein, three years older
than himself, he learned not only elliptie functions but self-
confidence, for he and the young master had a radical and most
energizing difierence of opinion as to how the theory should be

5ad

MEN OF MATHEMATICS

developed. Eisenstein insisted on beautiful formulae, some’what
in the manner of a modernized Euler; Riemann wanted to intro-
duce the complex variable and derive the entire theory, with a
minimum of calculation, from a few simple, general principles.
Thus, no doubt, originated at least the germs of one of Rie-
mann’s greatest contributions to pure mathematics. As the
origin of Riemann’s work in the theory of functions of a com-
plex variable is of considerable importance in his own history
and in that of modern mathematics, we shall glance at what is
known about it.

Briefly, nothing definite. The definition of an analytic func-
tion of a complex variable, discussed in connexion with Gauss’
anticipation of Cauchy’s fundamental theorem, was essentially
that of Riemann. \Vhen expressed analytically instead of geo-
metricall}” that definition leads to the pair of partial differential
equations* which Riemann took as his point of departure for a
theory of functions of a complex variable. According to Dede-
Idnd, ‘Riemann recognized in these partial differential equa-
tions the essential definition of an [analji:ic] function of a com-
plex variable. Probably these ideas, of the highest importance
for his future career, were worked out by him in the fall vaca-
tion of 1847 [Riemann was then twenty-one] for the first time.’

Another version of the origin of Riemann’ s inspiration is due
to Sylvester, who tells the following story, which is interesting
even if possibly untrue. In 1896, the year before his death,
Sylvester recalls staying a* ‘a hotel on the river at Nirremberg,
where I conversed outside with a Berlin bookseller, bound, like
myself, for Prague. … He told me he was formerly a feUow
pupil of Riemann, at the University, and that, one day, after
receipt of some numbers of the Comptes rendus from Paris, the
latter shut himself up for some weeks, and when he returned to

• H z = X iy, and lo = w in, is an analytic function of 2 ,
  Rieniann’s equations are

du ^ ^ ^ 3d

dx ^ 3t/’ ex’

These equations had been given much earlier by Cauchy, and even
Cauchy was not the first, as D’Alembert had stated the aquations in
the eighteenth century.

540

ANI3IA CANDIDA

the society of his friends said (referring to the newly published
papers of Cauchy), “This is a new mathematic”.’

Riemann spent two years at the University of Berlin. During
the political upheaval of 1848 he served with the loyal student
corps and had one weary spell of sixteen hours’ guard duty
protecting the jittery if sacred person of the king in the royal
palace. In 1849 he returned to Gottingen to complete his mathe-
matical training for the doctorate. His interests were unusually
broad for the pure mathematician he is commonly rated to be,
and in fact he devoted as much of his time to physical science
as he did to mathematics.

From this distance it seems as though Riemann’s real interest
was in mathematical physics, and it is quite possible that had
he been granted twenty or thirty more years of life he would
have become the Xewton or Einstein of the nineteenth century.
His physical ideas were bold in the extreme for his time. Not
till Einstein realized Riemann’s dream of a geometrized (macro-
scopic) physics did the physics which Riemann foreshadowed –
somewhat obscurely, it may be – appear reasonable to physi-
cists. In this direction his only understanding follower till our
own century was the English mathematician William Kingdon
Clifford (1845-79), who also died long before his time.

During his last three semesters at Gottingen Riemann
attended lectures on philosophy and followed the course of
Wilhelm Weber in experimental physics with the greatest
interest. The philosophical and psychological fragments left by
Riemann at his death show that as a philosopliical thinker he
was as original as he was in mathematics and science. “Weber
recognized Riemann’s scientific genius and became his warm
Mend and helpful counsellor. To a far higher degree than the
majority of great mathematicians who have written on phy-
sical science, Riemann had a feeling for what is important – or
likely to be so – in physics, and this feeling is no doubt due to
his work in the laboratory and his contact with men who were
primarily phj^sicists and not mathematicians. The contribu-
tions of even great pmre mathematicians to physicsal science
have usually been characterized by a singular irrelevance so
fax as the universe observed by scientists is concerned. Riemann^

541

MEN OF MATHEMATICS

as a physical mathematician, lisras in the same class as Xewton,
Gauss, and Einstein in his instinct for what is likely to be of
scientific use in mathematics.

As a sequel to his philosophical studies with Johann Friedrich
Herbart (1776-1841), Riemann came to the conclusion in 1850
(he was then twenty-four) that ‘a complete, well-rounded
mathematical theory can be established, which progresses from
the elementary laws for individual points to the processes given
to us in the plenum (“continuously jfilled space”) of reality,
without distinction between gravitation, electricity, magnet-
ism, or thermostatics’. This is probably to be interpreted as
Riemann’s rejection of all ‘action at a distance’ theories in
physical science in favour of field theories. In the latter the
physical properties of the ‘space’ surroxmding a ‘charged
particle’, say, are the object of mathematical investigation.
Riemann at this stage of his career seems to have believed in a
space-filling ‘ether’, a conception now abandoned. But as will
appear from his epochal work on the foundations of geometry,
he later sought the description and correlation of physical
phenomena in the geometry of the ‘space’ of human experience.
This is in the current fashion, which rejects an existent, unob-
servable ether as a cumbersome superfluity.

Fascinated by his work in physics, Riemann let his pure
mathematics slide for a while and in the autumn of 1850 joined
the seminar in mathematical physics which had just been
founded by Weber, Ulrich, Stem, and Listing. Physical experi-
ments in this seminar consumed the time that scholarly prudence
would have reserved for the doctoral dissertation in mathe-
matics, which Riemann did not submit till he was twenty-five.

One of the leaders in the seminar, Johann Benedict Listing
(1808-82), may be noted in passing, as he probably influenced
Riemann’s thought in what was to be (1857) one of his greatest
achievements, the introduction of topological methods into the
theory of functions of a complex variable.

It win be recalled that Gauss had prophesied that analysis
situs would become one of the most important fields of mathe-
matics, and Riemann, by his inventions in the theory of func-
tions, was to give a partial fulfilment of this prophecy. Although

542

ANIMA CANDIDA

topology (now called analysis situs) as first developed bore but
little resemblance to the elaborate theory which to-day absorbs
all the energies of a prolific schools it may be of interest to state
the tri\ial puzzle which apparently started the whole vast and
intricate theory. In Eulers time seven bridges crossed the river
Pregel in Konigsberg, as in the diagram, the shaded bars repre-

senting the bridges. Euler proposed the problem of crossing all
seven bridges without passing twice over any one. The problem
is impossible.

The nature of Riemann’s use of topological methods in the
theory of functions may be disposed of here, although an ade-
quate description is out of the question in untechnical language.
For the meaning of ^uniformity” with respect to a function of a
complex variable we must refer to what was said in the chapter
on Gauss. Now, in the theory of Abelian functions, muliiform
functions present themselves inevitably; an n-vatued function
of s is a function which, except for certain values of z, takes
precisely n distinct values for each value assigned to s, Rlus-
trating multiformity, or many-valuedness, for functions of a real
variable, we note that y, considered as a function of jj, defined
by the equation = ar, is two-valued. Thus, if x =■ 4, we get
y- = 4, and hence 2 ^ = 2or — 2;ifa3is any real number except
zero or ‘infinity’ , y has the two distinct values of V«r and — Vir.
In this simplest possible example y and x are connected by an
algebraic equation, namely — an = 0. Passing at once to the
general situation of which this is a very special case, we might

543

MEN OF MATHEMATICS

discuss the n-valued function y ^rhich is defined, as a function
of X, by the equation

PiW” -i- . . – + P„-t{x)y ^ P^{x) = 0,

in which the P’s are polynomials in x. This equation defines y as
an n-valued function of x. As in the case of y- — ai = 0, there
will be certain values of x for which two or more of these n
values of y are equal. These values of x are the so-called branch
points of the n-valued function defined by the equation.

All this is now extended to functions of complex variables,
and the function w (also its integral) as defined by

P,(z)^- -r -r . . . + + P^{z) = 0,

in which s denotes the complex variable s -f it, where s, t are
real variables and i = V — 1. The n values of w are called the
branches of the function w. Here we must refer (chapter on
Gauss) to what was said about the representation of uniform
functions of z. Let the variable 2 ( = s -f- ii) trace out any path
in its plane, and let the uniform function / (s) be expressed in
the form U -f iV, where U, V are functions of s, t Then, to
every value of z will correspond one, and only one, value for
each of U,V, and, as z traces out its path in the s, i-plane, / (z)
will trace out a corresponding path in the U, F-plane : the path
of / ( 2 ) will be uniquely determined by that of 2 . But if a? is a
multiform (many-valued) function of s, such that precisely n
distinct values of w are determined by each value of 2 (except
at branch points, where several values of w may be equal), then
it is obvious that one re-plane no longer suffices (if n is greater
than 1) to represent the path, the ‘march’ of the function ro. In
the case of a /reo-valued function w, such as that determined by
= Zy two m-planes would be required and, quite generally, for
an n-valued function {n finite or infinite), precisely n such rn-
planes would be required.

The advantages of considering uniform (one-valued) func-
tions instead of n-valued functions (n greater than 1) should be
obvious even to a non-mathematician. What Riemann did was
this: instead of the n. distinct zr-planes, he introduced an n-
sheeted surface, of the sort roughly described in what follows,

544

ANIMA CANDIDA

on which the rmilUform function is uniJoTm, that is, on which
to each ‘place* on the surface corresponds one, and only one,
value of the function represented.

Riemann united, as it were, ail the n planes into a single plane,
and he did this by what may at first look like an inversion of the
representation of the n branches of the n-valued function on n
distinct planes; but a moment’s consideration will show that,
in efiect, he restOTed uniformity. For he superimposed n s-planes
on one another; each of these planes, or sheets, is associated
with a particular branch of the function so that, as long as z
moves in a particular sheet, the corresponding branch of the
function is traversed by td (the n-valued function of z under
discussion), and as passes from one sheet to another, the
branches are changed, one into another, until, on the variable z
ha\’ing traversed all the sheets and having returned to its
initial position, the original branch is restored. The passage of
the variable z from one sheet to another is effected by means of
cuts (which may be thought of as straight-line bridges) joining
branch points; along a given cut providing passage from one
sheet to another, one ‘lip’ of the upper sheet is imagined as
pasted or joined to the opposite lip of the under sheet, and
similarly for the other lip of the upper sheet. Diagrammatically,
in cross-section,

The sheets are not joined along cuts (which may be drawn in
many ways for given branch points) at random, but are so
joined that, as s: traverses its n-sheeted surface, passing from
one sheet to another as a bridge or cut is reached, the analytical
behaviour of the function of s is pictured consistently, parti-
cularly as concerns the interchange of branches consequent on
the variable z, if represented on a plane, having gone completely

545

MEN OF MATHEMATICS

round a branch point. To this circuiting of a branch point on the
single 2 -plane corresponds, on the n-sheeted Riemann surface
the passage from one sheet to another and the resultant inter-
change of the branches of the function.

There are many ways in which the variable may wander
about the n-sheeted Riemami surface, passing from one sheet to
another. To each of these corresponds a particular interchange
of the branches of the function, which may be symibolized bv
writing, one after another, letters denoting the several branches
interchanged. In this way we get the sjmibols of certain
substituiions (as in chapter 15) on n letters; all of these substitu-
tions generate a group which, in some respects, pictures the
nature of the function considered.

Riemann surfaces are not easy to represent pictoriallj”, and
those w’ho use them content themselves with diagrammatical
representations of the connexion of the sheets, in much the
same way that an organic chemist writes a ‘graphical* formula
for a complicated carbon compound which recalls in a schematic
manner the chemical behaviour of the compound but which
does not, and is not meant to, depict the actual spatial arrange-
ment of the atoms in the compound. Riemann made wonderful
advances, particularly in the theory of Abelian functions, by
means of his surfaces and their topology – how shall the cuts be
made so as to render the n-sheeted surface equivalent to a
plane, being one question in this direction. But mathematicians
are like other mortals in their ability to visualize complicated
spatial relationships, namely, a high degree of spatial ”intuition’
is excessively rare.

Early in November, 1851, Riemann submitted his doctoral
dissertation, Grnndlagen fiir eine allgemeine Theorie der Funk-
tionen einer veranderlichcn complexen Grdsse (Foundations for a
general theory of functions of a complex variable), for Gauss*
consideration. This work by the young master of twenty-five
was one of the few modem contributions to mathematics that
roused the enthusiasm of Gauss, then an almost legendary
figure within four years of his death- When Riemann called on
Gauss, after the latter had read the dissertation. Gauss told him
that he himself had planned for years to write a treatise on the

548

ANIMA CANDIDA

same topic. Gauss’ ofEleial report to the Philosophical Faculty
of the University of Gottingen is noteworthy as one of the rare
formal pronouncements in which Gauss let himself go.

“The dissertation submitted by Herr Riemann offers con-
vincing evidence of the author s thorough and penetrating
investigations in those parts of the subject treated in the disser-
tation, of a creative, active, truly mathematical mind, and of a
gloriously fertile originality. The presentation is perspicuous
and concise and, in places, beautiful. The majority of readers
would have preferred a greater clarity of arrangement. The
whole is a substantial, valuable work, which not only satisfies
the standards demanded for doctoral dissertations, but far
exceeds them.’

A month later Riemann passed his final examination, in-
cluding the formality of a public ‘defence’ of his dissertation.
All went ofi successfully, and Riemann began to hope for a posi-
tion in keeping with his talents. ‘I believe I have improved my
prospects with my dissertation’, he wrote to his father; ‘I hope
also to learn to write more quickly and more fluently in time,
especially if I mingle in society and if I get a chance to give
lectures; therefore am I of good courage.’ He also apologizes to
his father for not having gone after a vacant assistantship at the
Gottingen Observatory more energetically, but as he hopes to
be ‘habilitated’ as a Privatdozent the outlook is not as dark as
it might be.

For his HabilitatioJisschrift (probationary essay) Riemann
had planned to submit a memoir on trigonometric series
(Fourier series). But two and a half years were to pass before he
might hang out his sign as an unpaid university instructor
picking up what he could in the way of fees from students not
bound to attend bis lectures. During the autumn of 1852 Rie-
mann profited by Dirichlet’s presence in Gottingen on a vaca-
tion and sought his advice on the embryonic memoir, Rie-
mann’s friends saw to it that the young man met the famous
mathematician from Berlin – second ody to Gauss — socially.

Dirichlet was captivated by Riemann’s modesty and genius*
‘Next morning [after a dinner party] Dirichlet was with me for
two hours,’ Riemann wrote his father. ‘He gave me the notes

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MEN OF MATHEMATICS

I needed for my probationary essay; otherwise I should have
had to spend many hours in the library in laborious research.
He also read over my dissertation with me and was verv
friendly – which I could hardly have expected, considering the
great distance in rank between us. I hope he will remember me
later on.’ During this visit of Dirichlet’s there were excursions
with Weber and others, and Riemann reported to his father
that these human escapes from mathematics did him more
good scientifically than if he had sat all day over his hooks.

From 1853 (Riemann was then twenty-seven) onward he
thought intensively about mathematical physics. By the end
of the year he had completed the probationary” essay, after
many delays due to his growing passion for physical science.

There was still a trial lecture ahead of him before he could be
appointed to the coveted – but unpaid – lectureship. For this
ordeal he had submitted three titles for the faculty to choose
from, hoping and expecting that one of the first two, on which
he had prepared himself, would he selected. But he had mcau-
tiously included as his third offering a topic on which Gauss had
pondered for sixty years or more – the foundations of geometry

• and this he had not prepared. Gauss no doubt was curious to
  see what a Riemann’s ‘gloriously fertile originality’ would make
  of such a profound subject. To Riemann’s consternation Gauss
  designated the third topic as the one on which Riemann should
  prove his mettle as a lecturer before the critical faculty. ‘So I
  am again in a quandary,’ the rash young man confided to his
  father, ‘since I have to work out this one. I have resumed my
  investigation of the connexion between electricity, magnetism,
  light, and gravitation, and I have progressed so far that I
  publish it “without a qualm. I have become more and more
  convinced that Gauss has worked on this subject for years, and
  has talked to some friends (Weber among others) about it. I
  tell you this in confidence, lest I be thought arrogant — I hope
  it is not yet too late for me and that I shall gain recognition as
  an independent investigator.’

The strain of carrying on two extremely difficult investiga-
tions simultaneously, while acting as Weber’s assistant in the
seminar in mathematical physics, combined with the usual

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AXIMA CANDIDA

handicaps of povert\% brought on a temporary breakdown* ‘I
became so absorbed in my investigation of the unity of all
physical laws that when the subject of the trial lecture was
given me, I could not tear myself away from my research.
Then, partly as a result of brooding on it, partly from staying
indoors too much in this vile weather, I fell iU; my old trouble
recurred with great pertinacity and I could not get on with my
work. Only several weeks later, when the weather improved and
I got more social stimulation, I began feeling better. For the
summer I have rented a house in a garden, and since doing so
my health has not bothered me. Having finished two weeks
after Easter a piece of work I could not get out of, I began at
once working on my trial lecture and finished it around Pente-
cost [that is, in about seven weeks]. I had some difficulty in
getting a date for my lecture right away and almost had to
return to Quickborn without having reached my goal. For
Gauss is seriously iU and the physicians fear that his death is
imminent. Being too weak to examine me, he asked me to wait
till August, hoping that he might improve, especially as I
would not lecture anyhow till fall. Then he decided an3rway on
the Friday after Pentecost to set the lecture for the next day at
eleven-thirty. On Saturday I was happily through with every-
thing.’

This is Riemann’s own account of the historic lecture which
was to revolutionize difierential geometry and prepare the way
for the geometrized physics of our own generation. In the same
letter he tells how the work he had been doing around Easter
turned out. ‘VVeber and some of his collaborators ‘had made
very exact measurements of a phenomenon which up till then
had never been investigated, the residual charge in a Leyden
jar [after discharge it is found that the jar is not completely
discharged] … I sent him [one of Weber’s collaborators,
Kohlrausch] my theory of this phenomenon, having worked it
out specially for his purposes. I had found the explanation of
the phenomenon through my general investigations of the con-
nexion between electricity, light, and magnetism. … This
matter was important to me, because it was the first time I
could apply my work to a phenomenon stall unknown, and I

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MEN OE MATHEMATICS

hope that the publication [of it] mil contribute to a favourable
reception of my larger work,’

The reception of Riemann’s probationary lecture (10 June
1854) was as cordial as even he could have wished in the scared
secrecy of Ms modest heart. The lecture had made him sweat
blood to prepare because he had determined to make it intelli-
gible even to those members of the faculty who had but little
knowledge of mathematics. In addition to being one of the
great masterpieces of all mathematics, Riemann’s essay Cher
die Hifpothesen, uoelche der Geometrie zu Grunde liegen (On the
hypotheses which lie at the foundations of geometry), is also a
classic of presentation. Gauss was enthusiastic. ‘Against all
tradition he had selected the tMrd of the three topics submitted
by the candidate, wishing to see how such a difficult subject
would be handled by so young a man. He was surprised beyond
all his expectations, and on returning from the faculty meeting
expressed to Wilhelm Weber Ms Mghest appreciation of the
ideas presented by Riemann, speaking with an enthusiasm
that, for Gauss, was rare.’ ‘\That little can be said here about
tMs masterpiece will be reserved for the conclusion of the
present chapter.

After a rest at home with Ms family in Quickbom, Riemann
returned in September to Gottingen, where he delivered a
hastily prepared lecture (sitting up most of the night to get it
ready on short notice) to a convention of scientists. His topic
was the propagation of electricity in non-conductors. During
the year he continued Ms researches in the mathematical theory
of electricity and prepared a paper on Nobili’s colour rings
because, as he wrote Ms sister Ida: ‘TMs subject is important,
for very exact measurements can be made in connexion with it,
and the laws according to wMch electricity moves can be
tested.’

In the same letter (9 October 1854) he expresses Ms un-
bounded joy at the success of his first academic lecture and his
great satisfaction at the unexpectedly large number of auditors.
Eight students had come to hear him! He had anticipated at the
most two or three. Encouraged by tMs unhoped-for popularity,
Riemann tells Ms father, T have been able to hold my classes

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ANI MA CANDIDA

regularly. My first diffidence and constraint have subsided
more and more, and I get accustomed to think more of the
auditors than of myself, and to read in their expressions whether
I should go on or explain the matter further.’

When Dirichlet succeeded Gauss in 1855, Eiemann’s friends
urged the authorities to appoint Riemann to the security of an
assistant professorship, but the finances of the University” could
not be stretched so far. Nevertheless he was granted the equi-
valent of 200 dollars a year, which was better than the uncer-
tainty of half-a-dozen voluntary students’ fees. His future
worried him, and when presently he lost both his father and his
sister Clara, making it impossible for him to escape for vaca-
tions to Quickbom, Riemann felt poor and miserable indeed.
His three remaining sisters went to live with the other brother,
a postal clerk in Bremen whose salary was princely beside that
of the ‘■economically valueless’ mathematician.

The following year (1856; Riemann was then thirty) the out-
look brightened a little. It was impossible for a creative genius
like Riemann to be downed by despondency so long as he had
the wherewithal to keep body and soul together in order that
he might work. To this period belong part of his characteristi-
cally original work on Abelian functions, his classic on the
hypergeometric series (see chapter on Gauss) and the differ-
ential equations – of great importance in mathematical physics

• suggested by this series. In both of these works Riemann
  struck out on new directions of his own. The generality, the
  irduitivmesSi of his approach was peculiarly his own. His work
  absorbed all his energies and made him happy in spite of
  material worries; possibly, too, the fatal optimism of the con-
  sumptive was already at work in him.

Riemann’ s development of the theory of Abelian functions is
as unlike that of Weierstrass as moonlight is unlike sunlight.
Weierstrass’ attack was methodical, exact in all its details, like
the advance of a perfectly disciplined army imder a generalship
that foresees everything and provides for all contingencies.
Riemann, for his part, looked over the whole field, seeing
everything but the details, which he left to take care of them-
selves, and was content to have grasped the key positions of the

551

MEN OF MATHEMATICS

general topography in his imagination. The method of Weier-
strass was arithmetical, that of Riemann geometricjal and intui-
tive. To say that one is ‘better’ than the other is meaningless;
both cannot be seen from a common point of \iew.

Overwork and lack of reasonable comforts brought on a
nervous breakdown early in his thirty-first year, and Riemann
was forced to spend a few weeks with a friend in the Harz
mountain country, where he was joined by Dedekind. The three
took long tramps together into the mountains and Riemann
soon recovered. Relieved of the strain of having to keep up
academic appearances, Riemann indulged his sense of humour
and kept his companions amused with his spontaneous wit.
They also talked shop together – most mathematicians do when
they get together, just as lawyers or doctors or business men do,
pro\dded they do not have to talk drivel to maintain the social
conventions. One evening after a strenuous hike Riemann
dipped into Brewster’s life of Newton and discovered the letter
to Bentley in which Newton himself asserts the impossibil%
of action at a distance without intervening media. This de-
lighted Riemann and inspired him to an impromptu lecture.
To-day the ‘medium’ which Riemann extolled is not the lumini-
ferous ether, but his own ‘cur\”ed space’, or its reflexion in the
space-time of relativity.

At last, in 1857, at the age of thirty-one, Riemann got his
assistant professorship. EQs salary was the equivalent of about
300 dollars a year, hut as he had had little all his life he missed
less. However, a real disaster presently descended on him: his
brother died and the care of three sisters fell to his lot. It
figured out at exactly seventy-five dollars a year for each of
them. Love on nothing a year in a cottage may be paradise;
existence on next to nothing in a university community is just
plain hell. It was but little different in Riemann’s day. No
wonder he contracted consumption. However, the Lord, who
had so generously given, shortly relieved Riemann of his
youngest sister, IVIarie, so the individual budgets skyrocketed
to 100 dollars a year. If rations had to be watched, affection was
free, and Riemann was more than repaid for his sacrifices by the
self-cpnfidenee inspired in him by his sisters’ devotion and
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AKIMA CANDIDA

encouragement. The Lord may have known that if ever a
stniggling mortal needed encouragement, poor Riemann did;
still, it seems rather an odd way of providing what was required.

In 1S5S Riemann produced his paper on electrodjmamics, of
which he told his sister Ida, ‘My discovery concerning the close
connexion between electricity and light I have dedicated to the
Royal Society [of Gottingen]. From what I have heard, Gauss
had devised another theory regarding this dose connexion,
di^erent from mine, and communicated it to his intimate
friends. However, I am fully convinced that my theory is the
correct one, and that in a few years it win be recognized as such.
As is known, Gauss soon withdrew his memoir and did not
publish it; probably he himself was not satisfied with it.’
Riemann would seem here to have been over-optimistic; Clerk
Maxwell’s electromagnetic theory is the one which to-day holds
the field – in macroscopic phenomena. The present status of
theories of light and the electromagnetic field is too complicated
to be described here; it is sufficient to note that Riemann’s
theory has not survived.

Dirichlet died on 5 May 1859. He had always appreciated
Riemann and had done his best to help the struggling young
man along. This interest of Dirichlet’s and Riemann’s rapidly
mounting reputation caused the government to promote
Riemann to succeed Dirichlet. At thirty-three Riemann thus
became the second successor of Gauss. To ease his domestic
difficulties the authorities let him reside at the Observatory, as
Gauss had done. Recognition of the sincerest kind – praise from
mathematicians who, although older than himself, were in some
degree his rivals – now came in abundance. On a visit to Berlin
he was feted by Borchardt, Kummer, Kronecker, and Weier-
strass. Learned societies, including the Royal Society of
London and the French Academy of Sciences, honoured him
with membership, and in short he got the usual highest distinc-
tions that can come to a man of science. A visit to Paris in 1860
acquainted him with the leading French mathematicians,
particularly Hermite, whose admiration for Riemann was
unbounded. This year, 1860, is memorable in the history of
mathematical physics as that in which Riemann began inten-

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MEN OF MATHEMATICS

geometry. Clifford was no servile copyist but a man with a
brillianth* original mind of Ms own, of whom it may be said, as
Newton said of Cotes, ‘If he had lived we might have known
something.’ The reader who is acquainted with any of the better
available popular accounts of relativistic physics and the wave
theory of electrons wOl recognize several curious adumbrations
of current theories in Clifford’s brief prophecy.

‘Biemann has shown that as there are different Mnds of lines
and surfaces, so there are different kinds of space of three
dimensions; and that we can only find out by experience to
which of these kinds the space in wMch we live belongs. In
particular, the axioms of plane geometry are true within the
limits of experiment on the surface of a sheet of paper, and yet
we know that the sheet is really covered with a number of smafi
ridges and furrows, upon wMch (the total curvature being not
zero) these axioms are not true. Similarly, he says, although the
axioms of solid geometry are true within the limits of experi-
ment for finite portions of our space, yet we have no reason to
conclude that they are true for very small portions; and if any
help can be got thereby for the explanation of physical pheno-
mena, we may have reason to conclude that they are not true
for very small portions of space.

‘I wish here to indicate a manner in which these speculations
may be applied to the investigation of physical phenomena. I
hold in fact

(1) That small portions of space are in fact of a nature analo-
. gous to little hills on a surface which is on the average flat;

namely, that the ordinary laws of geometry are not valid in
them.

(2) That tMs property of being curved or distorted is con-
tinually being passed on from one portion of space to another
after the manner of a wave.

(3) That tMs variation of the curvature of space is what
really happens in that phenomenon wMch we call the motion of
maMet, whether ponderable or ethereal.

(4) That in the physical world nothing else takes place hut
this variation, subject (possibly) to the law of continuity.

am endeavouring in a g^eral way to explain the laws of

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ANIMA CANDIDA

double refraction on this hypothesis, hut have not yet arrived
at any results sufficiently decisive to he communicated,’

Riemann also believed that his new geometry woxiid prove of
scientific importance, as is shown hy the conclusion of his
memoir (Clifford’s translation):

‘Either therefore the reality which underlies space must form
a discrete manifold, or we must seek the ground of its metric
relations outside it, in binding forces which act upon it.

‘The answer to these questions can only be got by starting
from the conception of phenomena which has hitherto been
justified by experience, and which Newton assumed as a foun-
dation, and by making in this conception the successive changes
required by facts which it cannot explain.’ And he goes on to
say that researches like his own, starting from general notions,
‘can be useful in preventing this work from becoming hampered
by too narrow views, and progress of knowledge of the inter-
dependence of things from being checked by traditional
prejudices.

•This leads us into the domain of another science, that of
physics, into which the object of this work does not allow us to
go to-day.’

Riemann’s work of 1854 put geometry in a new light. The
geometry he visions is non-Euclidean, not in the sense of
Lobatchewsky and Johann Bolyai, nor in that of Riemann’s
own elaboration of the hypothesis of the obtuse angle (as
explained in chapter 16), but in a more comprehensive sense
depending on the conception of measurement. To isolate
measure-TelatioTis as the nerve of Riemann’ s theory is to do it an
injustice; the theory contains much more than a workable
philosophy of metrics, but this is one of its main features. No
paraphrase of Riemann’s concise memoir can bring out all that
is in it; nevertheless, we shall attempt to describe some of his
basic ideas, and we shall select three: the concept of a manifold,
the definition of distance, and the notion of curcature of a
manifold.

A manifold is a class of objects (at least in common mathe-
matics) which is such that any member of the class can be
completely specified by assigning to it certain numbers, in a

esfisir

MEN OF MATHEMATICS

defiLnite order, corresponding to ‘numberable’ properties of the
elements, the assignment in the given order corresponding to a
preassigned ordering of the ‘numberable’ properties. Granted
that this may be even less comprehensible than Riemann’s
definition, it is nevertheless a ‘working basis from which to
start, and all that it amoiints to in plain mathematics is this:
a manifold is a set of ordered ‘n-tuples’ of numbers ,

where the parentheses, (), indicate that the numbers
a: 1,332, . . are to be written in the order given. Two such
K-tuples, . . . , and … , 2/J are equal when, and

only when, corresponding numbers in them are respectively
equal, namely, when, and only when, — 2^2 = »

= 2/n-

If precisely n numbers occur in each of these ordered n-tuples
in the manifold, the manifold is said to be of n dimensions. Thus
we are back again talking co-ordinates with Descartes. If each
of the numbers in . . . , is a positive, zero, or negative

integer, or if it is an element of any countable set (a set whose
elements may he counted ofi 1,2,3, . . . ), and if the like holds for
every n-tuple in the set, the manifold is said to be discrete. If
the numbers x^,x»i … , a3„, may take on values continuously
(as in the motion of a point along a line), the manifold is
continuous.

This working definition has ignored – deliberately – the ques-
tion of whether the set of ordered n-tuples is ‘the manifold’ or
whether something “represented by’ these is ‘the manifold’.
Thus, when we say (x,y) are the co-ordinates of a point in a
plane, we do not ask what ‘a point in a plane’ is, but proceed to
work with these ordered couples of numbers {x,y) where x,y run
through all real numbers independently. On the other hand it
may sometimes be advantageous to fix our attention on what
such a symbol as {x,y) represents. Thus if a? is the age in seconds
of a man and y his height in centimetres, we may be interested
in the man (or the class of all men) rather than in his co-ordinates
miih which alone the mathematics of our enquiry is concerned. In
this same order of ideas, geometry is no longer concerned with
what ‘space’ ‘is’ – whether ‘is’ means anything or not in relation
to ‘space’. Space, for a modem mathematician, is merely a

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ANIMA CANDIDA

number-manifold of the kind described above, and this con-
ception of space grew out of Riemann’s ‘manifolds’.

Passing on to measurement, Riemann states that ‘Measure-
ment consists in a superposition of the magnitudes to be com-
pared. If this is lacking, magnitudes can be compared only
when one is part of another, and then only the more or less, but
not the how much, can be decided.’ It may be said in passing
that a consistent and useful theory of measurement is at present
an urgent desideratum in theoretical physics, particularly in all
questions where quanta and relativity are of importance.

Descending once more from philosophical generalities to less
mystical mathematics, Riemann proceeded to lay down a defi-
nition of distance^ extracted from his concept of measurement,
which has proved to be extremely fruitful in both physics and
mathematics. The Pythagorean proposition

that a = 6® + or a = where a is the length of the

longest side of a right-angled triangle and b,c are the lengths of
the other two sides, is the fundamental formula for the measure-
ment of distances in a plane. How shall this be extended to a
curved surfaced To straight hnes on the plane correspond geode-
sies (see chapter 14) on the surface; but on a sphere, for
example, the Pythagorean proposition is not true for a right-
angled tria n gle formed by geodesics. Riemann generalized the
Pythagorean formula to any manifold as follows:

Let (aii,aj 2 , . . . , a?„}, {x^ + -f ajg’, . . . , -f a?/) be the

co-ordinates of two ‘points’ in the manifold which are ‘infinite-
simally near’ one another. For our present purpose the meaning
of ‘infinitesimally near’ is that powers higher than the second of

• . • ? which measure the ‘separation’ of the two

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MEN OF MATHEMATICS

points in the manifold, can he neglected. For simplicity we shall
state the definition when n = 4 – giving the distance between
two neighbouring points in a space of four dimensions: the
distance is the square root of

• -f gl^iSSi + gi^PHiOS^’

4” gz^z’^z^ 4“ g^ 4 p^Z^\

4 –

in which the ten coefficients … ,^34 are functions of
For a particular choice of the g’s, one ‘space’ is
defined. Thus we might ha%^e = 1, = 1, = 1, ===

— 1, and all the other g’s zero; or we might consider a space in
which all the g’s except ^44 and ^34 were zero, and so on. A space
considered in relativity is of this general kind in which all the
g’s except gii,g22>^3 33^44 zero, and these are certain simple

expressions invohdng

In the case of an n-dimensional space the distance between
neighbouring points is defined in a similar manner; the general
expression contains J72(w -{- 1) terms. The generalized Pytha-
gorean formula for the distance between neighbouring points
being given, it is a solvable problem in the integral calculus to
find the distance between any two points of the space, A space
whose metric (system of measurement) is defined by a formula
of the type described is called Riemannian.

Curvature, as conceived by Riemann (and before him hy
Gauss; see chapter on the latter) is another generalization from
common experience. A straight line has zero curvature; the
^measure’ of the amount by which a curved line departs from
straightness may be the same for every point of the curve (as it
is for a circle), or it may vary from point to point of the curve,
when it becomes necessary again to express the ‘amount of
curvature’ through the use of infimtesimals. For curv^ed sur-
faces, the curvature is measured similarly by the amount ci
departure from a plane, which has zero curvature. This may be
generalized and made a little more precise as follows. For sim-
plicity we state first the situation for a two-dimensional space,

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ANIMA CANDIDA

namely for a surface as we ordinarily imagine surfaces. It is
possible from the formula

expressing (as before) the square of the distance between neigh-
bouring points on a given surface (determined when the func-
tions iivgiztizz given), to calculate the measure of curvature
of any point of the surface wholly in terms of the given functions
giiigi2sg22′ ^ ordinary language, to speak of the ‘curva-
ture’ of a space of more than two dimensions is to make a
meaningless noise. Nevertheless Riemann, generalizing Gauss,
proceeded in the same mathematical way to build up an expres-
sion involving all the g’s in the general case of an n-dimensional
space, which is of the same kind mathematically as the Gaussian
expression for the curvature of a surface, and this generalized
expression is what he called the measure of curvature of the
space. It is possible to exhibit visual representations of a curved
space of more than two dimensions, but such aids to perception
are about as useful as a pair of broken crutches to a man with
no feet, for they add nothing to the understanding and they are
mathematically useless.

Why did Riemann do all this and what has come out of it?
Not attempting to answer the first, except to suggest that
Riemann did what he did because his daemon drove him, we
may briefly enmnerate some of the gains that have accrued
from Riemann’ s revolution in geometrical thought. First, it put
the creation of ‘spaces’ and ‘geometries’ in unlimited number
for specific purposes – use in dynamics, or in pure geometry, or
in physical science – within the capabilities of professional geo-
meters, and it baled together huge masses of important geo-
metrical theorems into compact bundles that could he handled
easily as wholes. Second, it clarified our conception of space, at
least so far as mathematicians deal in ‘space’, and stripped that
mystic nonentity Space of its last shred of mystery- Biemann’s
achievement has taught mathematicians to disbelieve in any
geometry, or in any space, as a necessary mode of human per-
ception. It was the last nail in the coffin of absolute space, and
the first in that of the ‘absolutes’ of nineteenth-centuiy physics,

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MEN OF MATHEMATICS

Finally, the curvature whicb Riemann defined, tlie processes
wMch he de\ised for the investigation of quadratic diterential
fonns {those giving the forniula for the square of the distaiicE’
hetween neighhouring points in a space of any numher of ■
diiiiensions), and his recognition of tne fact that the cirrvatiiie.
is an invariant (in the technical sense explained in, pre\ioiis
chapters), all found their physical interpretations in the theow
of reiatimty. “V^Tiether the latter is in its final form or not is
beside the point; since relathdty our outlook on physical science
is not what it was before. Without the work of Riemann this
revolution in scientific thought would have been impossible-
unless some later man had created the concepts and the mathe-
matical methods that Riemann created.

CHAPTER TWENTY-SEVEX

ARITHMETIC THE SECOND
Kummer ; Dedekind

It is a curious fact that although arithmetic – the theory of
numbers – has been the fertile mother of more profound pro-
blems and powerful methods than any other discipline of
mathematics, it is usually regarded as standing rather to one
side of the main progress as a more or less cold-blooded spec-
tator of the flashier achievements of geometry and anabasis,
particularly in their services to physical science, and compara-
tively few of the great mathematicians of the past 2,000 years
have expended their more serioTis efforts on the advancement
of the science of ‘pure number’.

Many causes have determined this strange neglect of what,
after all, is mathematics par excellence. Among these we need
note only the following: arithmetic at present is on a higher
plane of intrinsic difficulty than the other great fields of mathe-
matics; the immediate applications of the theory of numbers
to science are few and not readily perceptible to the ordinary
run of creative mathematicians, although some of the greatest
have felt that the proper mathematics of nature will be found
ultimately in the behaviour of the common whole numbers;
and, finally, it is only human for mathematicians – at least for
some, even the great – to court reputation and popularity in
their own generation by reaping the easier harvests of a spec-
tacular success in analysis, geometry, or applied mathematics.
Even Gauss succumbed, to his keen regret in middle life.

Modem arithmetic – after Gauss – began with Kummer. The
origin of Kummer’s theory in his attempt to prove Fermat’s
Last Theorem has already been noted (Chapter 25). Something
of the man’s long life may be told before we pass to Dedeldnd.
Kummer was a typical German of the old school with all the

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MEN or MATHEMATICS

blunt simplicity, good nature, and racy humour, ^hidi
characterized that fast-vanishing species at its best. Museum
specimens, aged in the wood, could be foimd behind the bar in
any San Francisco German beer garden a generation ago.

Although Ernst Eduard Kummer (29 January 1810-14 Slav
1893) was bom only five years before the deflation of Napoleo^
the glorious Emperor of the French played an important if
unwitting part in Hummer’s life. The son of a physician of
Sorau (then in the principality of Brandenburg), Geimanv,
Kummer at the age of three lost his father: the lousy remnant
of Napoleon’s Grand Army, filtering back through Germany to
France, brought with it the characteristically Russian gift of
typhus, which it shared freely with the well- washed Germans.
The overworked physician caught the disease, died of it, and
left Ernst and an elder brother to the care of his widow. Young
Kummer grew up in cramping poverty, but Ms struggling
mother contrived somehow or another to see her sons through
the local Gymnasium. The arrogance and exactions of the
Napoleonic French, no less than the memory of his father,
which the mother kept alive, made young Kummer an ex-
tremely practical patriot, and it was with real gusto that he
devoted much of his superb scientific talent in later life to
traimug German army officers in ballistics at the war college of
Berlin, Many of his students gave good accounts of themselves
in the Franco-Prussian War.

At the age of eighteen (in 1828) Kummer was sent by his
mother to the University of Halle to study theology and other-
wise fit himself for a career in the church. Owing to his poverty
Kummer did not reside at the University, but tramped back
and forth every day from Sorau to Halle with his food and
books in a knapsack on his back. Regarding his theological
studies Kummer makes the interesting observation that it is
more or less a matter of accident or environment whether a
mind with a gift for abstract speculation turns to philosophy or
to mathematics. The accident in his own case was the presence
at Halle of Hemrich Ferdinand Scherk (1798-1885) as professor
of mathematics. Scherk was rather old-fashioned, but he had
an enthusiasm for algebra and the theory of numbers which he

564

ARITHMETIC THE SECOND

imparted to young Kummer. Under Scherk’s guidance Kummer
soon abandoned his moral and theological studies in favour of
mathematics. Echoing Descartes^ Kummer said he preferred
mathematics to philosophy because ‘mere errors and false views
cannot enter mathematics.’ Had Kummer lived till to-day he
might have modified his statement, for he was a broadminded
man, and the present philosophical tendencies in mathematics
are sometimes curiously reminiscent of medieval theology. In
his third year at the University Kummer solved a prize problem
in mathematics and was awarded his Ph.D. degree (10 Sep-
tember 1831) at the age of twenty-one. No university position
being open at the time, Kummer began his career as a teacher
in his old Gymnasium.

In 1832 he’ moved to Liegnitz, where he taught for ten years
in the Gymnasium. It was there that he started Kronecker off
on his revolutionary career. Fortunately Kummer was not so
hard up as Weierstrass under similar circumstances and was
able to afford postage for scientific correspondence. The eminent
mathematicians (including Jacobi) with whom Kummer shared
his mathematical discoveries saw to it that the young genius of
a school teacher was lifted into a more suitable position at the
earliest opportunity, and in 1842 Kummer was appointed
Professor of Mathematics at the University of Breslau. He
taught there till 1855, when the death of Gauss caused exten-
sive revisions in the mathematical map of Europe.

It had been assumed that Dirichlet was contented at Berlin,
then the mathematical capital of the world. But when Gauss
died, Dirichlet could not resist the temptation of succeeding the
Prince of Mathematicians and his own former master as pro-
fessor at Gottingen. Even to-day the glory of being a ‘successor
of Gauss’ has an almost irresistible attraction for mathemati-
cians who might easily earn more money in other positions, and
until quite recently GSttingen could choose whom it would.
The high esteem in which Kummer was held by his fellow
mathematicians can be judged by the fact that he was the
unanimous choice to succeed Dirichlet at Berlin. Since the age
of twenty-nine he had been a corresponding member of the
Royal Berlin Academy. He now (1855) succeeded IMriehlet in

565

MEN OF MATHEMATICS

botli the University and the Academy, and was also appointed
professor at the Berlin War College.

Knmmer was one of those rarest of all scientific geniuses who
are first class in the most abstract mathematics, the applica-
tions of mathematics to practical affairs, including war, which
is the most unblushingly practical of all human idiocies, and
finally in the ability to do experimental phj-sies of a high degree
of excellence. His finest work was in the theory of numbers
where his profound originality led him to inventions of the very
first order of importance, but in other fields – analysis, geome-
try, and applied physics – he also did outstanding work.
Although Kummer’s advance in the higher arithmetic was of
the pioneering sort that justifies comparing him with the
creators of non-Euelidean geometry, we somehow get the
impression, on reviewing his life of eighty-three years, that
splendid as bis achievement was, he did not accomplish all that
he must have had in him. Possibly his lack of personal ambition
ion instance is given presently), his easy-going geniality, and
his broad sense of humour prevented him from winding himself
in an attempt to beat the record.

The nature of what Kummer did in the theory of numbers
has been described in the chapter on Kronecker: he restored the
fundamental theorem of arithmetic to those algebraic number fields
which arise in the attempt to prove Fermat* s Last Theorem and in
the Gaussian theory of cyclotomy, and he effected this restoration
by the creation of an entirely new species of numherSy his so-called
Hdeal numbers*. He also carried on the work of Gauss on the law
of biquadratic reciprocity and sought the laws of reciprocity for
degrees higher than the fourth.

As has already been mentioned in preceding chapters, Rum-
mer’s ‘ideal numbers’ are now largely displaced by Dedekind’s
‘ideals’, which will be described when we come to them, so it is
not necessary to attempt here the almost impossible feat of
explaining in untechrdcal language what Kummer’s ‘numbers’
are. But what he accomplished by means of them can be stated
with sufficient accuracy for an account like the present: Rum-
mer ^proved that ^ = 2®’, where p is a prime, is impossible
in integers all different from zoro, for a whole very exten-

666

ARITHMETIC THE SECOND

sive class of primes p. He did not succeed ia proving Fermat’s
theorem for all primes; certam slippery ‘exceptional primes’
eluded Kummer’s net – and still do. Nevertheless the step ahead
which he took so far surpassed everything that all his prede-
cessors had done that Kummer became famous almost in spite
of him self. He was awarded a prize for which he had not
competed.

The report in full of the French Academy of Sciences on the
competitition for its ‘Grand Prize’ in 1857 ran as follows.
‘Report on the competition for the grand prize in mathematical
sciences. Already set in the competition for 1853 and prorogued
to 1856. The committee, having found no work which seemed to
it worthy of the prize among those submitted to it in competi-
tion, proposed to the Academy to award it to M. Kummer,
for his beautiful researches on complex numbers composed of
roots of unity* and integers. The Academy adopted this
proposal.’

Kummer’s earliest work on Fermat’s Last Theorem is dated
October 1835. This was followed by further papers in 1844-47,
the last of which was entitled Proof of Fermafs Theorem on the
ImpossiMliiy o/ai^ -f for an Infinite^ Number of Primes

p. He continued to add improvements to his theory, including
its application to the laws of higher reciprocity, till 1874, when
he was sixty-four years old.

Although these highly abstract researches were the field of
his greatest interest, and although he said of himself, *To

• If atp -f yp = s?’, then ajp = sp — yp, and resohing sp — yo into its
  p factors of the first degree, we

asP = (s-y) (=-r®y) . . . (s-rp-^y),

in which r is a ‘pth root of unity’ (other than 1), namely tp — 1 = 0,
with r not equal to 1. The algebraic integers in the field of degree p
generated by r are those which Kummer introduced into the study of
Fermat’s equation, and which led him to the invention of his ‘ideal
numbers’ to rt^tore unique factorization in the field – an int^er in
such a field is not uniquely the product of primes in the field for aU
primes p.

f The ‘infinite’ in Kummer’s title is still (1936) unjustified; ‘many’
should be put for ‘infinite*.

567

MEN OF 3IATHEMATICS

describe my personal scientific attitude more exactly, I may
conveniently designate it as theoretical . . . ; I have particularlv
striven for that mathematical knowledge which finds its proper
sphere in mathematics without reference to applications,’
Kummer was no narrow specialist. Somewhat like Gauss, he
appeared to take equal pleasure in both pure and applied
science. Gauss indeed, through his works, was Kummer’ s real
teacher, and the apt pupil proved his mettle by extending his
master’s work on the h\^ergeometric series, adding to what
Gauss had done substantial developments which to-day are of
great use in the theory of those differential equations which
recur most frequently in mathematical physics.

Again, the magnificent work of Hamilton on systems of rays
(in optics) inspired Kummer to one of his own most beautiful
inventions, that of the surface of the fourth degree which is
known by his name and which plays a fundamental part in the
geometry of EucKdean space when that space is four-dimen-
sional (instead of three-dimensional, as we ordinarily imagine
it), as happens when straight lines instead of points are taken
as the irreducible elements out of which the space is con-
structed. This surface (and its generalizations to higher spaces)
occupied the centre of the stage in a whole department of nine-
teenth-century geometry; it was found (by Cayley) to be repre-
sentable (parametrically – see the chapter on Gauss) by means
of the quadruple periodic functions to which Jacobi and Her-
inite devoted some of their best efforts.

Quite recently (since 1934) it has been observed by Sir Arthur
Eddington that Kummer’s surface is a sort of cousin to Dirac’s
wave equation in quantum mechanics (both have the same
finite group; Rummer’s surface is the wave surface in space of
four dimensions).

To complete the circle, Kummer was led back by his study of
systems of rays to physics, and he made important contribu-
tions to the theory of atmospheric refraction. In his work at the
War College he astonished the scientific world by proving him-
self a first-rate experimenter in his work on ballistics. With
characteristic humour Kummer excused himself for this bad
fall from mathematical grace; *When Z attack a problem expen-
ses

ARITHMETIC THE SECOND

mentally,’ lie told a young friend, ‘it is a proof that the problem
is mathematically impregnable.’

Remembering his own struggles to get an education and his
mother’s sacrifices, Kummer was not only a father to his stu-
dents but something of a brother to their parents. Thousands of
grateful young men who had been helped on their way by
Kummer at the University of Berlin or the War College
remembered him all their lives as a great teacher and a great
friend. Once a needy young mathematician about to come up
for his doctor’s examination was stricken with smallpox and
had to return to his home in Posen near the Russian border. No
word came from him, but it was known that he was desperately
poor. When Kummer heard that the young man was probably
unable to afford proper care, he sought out a friend of the
student, gave him the requisite money and sent him off to Posen
to see that what was necessary was done. In his teaching
Kummer was famous for his homely similes and philosophical
asides. Thus, to drive home the importance of a particular
factor in a certain expression, he observed that ‘K you neglect
this factor you will be like a man who in eating a plum swallows
the stone and spits out the pulp.’

The last nine years of Kummer s life were spent in complete
retirement. ‘Nothing wiU be found in my posthumous papers,’
he said, thinking of the mass of work which Gauss left to be
edited after his death. Surrounded by his family (nine children
survived him), Kummer gave up mathematics for good when
he retired, and except for occasional trips to the scenes of his
boyhood lived in the strictest seclusion. He died after a j^ort
attack of influenza on 14 May 1893, aged eighty-three.

Rummer’s successor in arithmetic was Julius Wilhelm
Richard Dedekind (he dropped the first two names when he
grew up), one of the greatest mathematicians and one of the
most original Germany – or any other country – has produced.
Like Kummer, Dedekind had a long life (6 October 1831-12
February 1916), and he remained mathematically active to
within a short time of his death. When he died m 1916 Dade-
kind had been a mathematical classic for well over a generation.
As Edmimd Landau (himself a friend and follower of Dedekind

569

MEN OF MATHEMATICS

in some of his work) said in his commemorative address to the
Royal Society of Gottingen in 1917 ; “Richard Dedekind was not
only a great mathematician, but one of the wholly great in the
history of mathematics, now and in the past, the last hero of a
great epoch, the last pupil of Gauss, for four decades himself a
classic, from whose works not only we, but our teachers and the
teachers of our teachers, have drawn.’

Richard Dedekind, the youngest of the four children of Julius
Levin Ulrich Dedekind, a professor of law, was bom in Bruns-
wick, the natal place of Gauss.* From the age of seven to sixteen
Richard studied at the Gymnasium in his home town. He gave
no early evidence of unmistakable mathematical genius; in fact
his first loves were physics and chemistry, and he looked upon
mathematics as the handmaiden – or scullery slut – of the
sciences. But he did not wander long in darkness. By the age of
seventeen he had smelt numerous rats in the alleged reasoning
of physics and had turned to mathematics for less objectionable
logic. In 1848 he entered the Caroline College – the same institu-
tion that gave the youthful Gauss an opportunity for self-
instruction in mathematics. At the college Dedekind mastered
the elements of analytic geometry, ‘advanced’ algebra, the
calculus, and ‘higher’ mechanics. Thus he was well prepared to
begin serious work when he entered the University of Gottingen
in 1850 at the age of nineteen. His principal instmctors were
Moritz Abraham Stem (1807-94), who wrote extensively on the
theory of numbers, Gauss, and Whhelm ‘VVeher the physicist.

• No adequate biography of Dedekind has yet appeared. A life was
  to have been included in the third volume of his collected works
  (1932), but was not, owing to the death of the editor in chief (Robert
  Fricke), The account here is based on Landau’s commemorative
  address. Note that, following the good old Teutonic custom of some
  German biographers, Landau omits all mention of Dedekind’s
  mother. This no doubt is in accordance with the theory of the ‘three
  K’s’ propounded by the late Kaiser of GJermany and heartily
  endorsed by Adolf Hitler: ‘A woman’s whole duty is comprised in the
  three big ICs – Kissing, Koofcing [kooking is spelt with a K in
  Germank and Kids,’ Still, one would like to know at least the maiden
  name of a great man’s mother.

570

ARITHMETIC THE SECOND

From tliese three men Dedekind got a thorough grounding in
the calculus, the elements of the higher arithmetic, least
squares, higher geodesy, and experimental phj^sics.

In later life Dedekind regretted that the mathematical in-
struction available during his student years at Gottingen, while
adequate for the rather low requirements for a state teacher’s
certificate, was inconsiderable as a preparation for a mathe-
matical career. Subjects of living interest were not touched
upon, and Dedekind had to spend two years of hard labour
after taking his degree to get up by himself elliptic functions,
modem geometry, higher algebra, and mathematical physics –
all of which at the time were being brilliantly expounded at
Berlin by Jacobi, Steiner, and Dirichlet. In 1852 Dedekind got
his doctor’s degree (at the age of twenty-one) from Gauss for a
short dissertation on Eulerian integrals. There is no need to
explain what this was: the dissertation was a useful, indepen-
dent piece of work, but it betrayed no such genius as is e\1dent
on every page of Dedeldnd’s later works. Gauss’ verdict on the
dissertation will be of interest: ‘The memoir prepared by Herr
Dedekind is concerned with a research in the integral calculus,
which is by no means commonplace. The author evinces not
only a very good knowledge of the relevant field, but also such
an independence as augurs favourably for his future achieve-
ment. As a test essay for admission to the examination I find
the memoir completely satisfying.’ Gauss evidently saw more
in the dissertation than some later critics have detected;
possibly his close contact with the young author enabled him
to read between the lines. However, the report, even as it
stands, is more or less the usual perfunctory politeness custo-
mary in accepting a passable dissertation, and we do not know
whether Gauss really foresaw Dedeldnd’s penetrating oiigm-
ality.

In 1854 Dedekind was appointed lecturer {Pmaidozent) at
Gottingen, a position which he held for four years. On the death
of Gauss in 1855 Dirichlet moved from Berlin to Gottingen.
For the remaining three years of his stay at Gottingen, Dede-
kind attended Dirichlet’s most important lectures. Later he was
to edit Dirichlet’s famous treatise on the theory of numbers and

671

MEN OF M..AuTHEMATICS

add to it the epoch-making ‘Eleventh Supplement’ containing
an outline of his own theory of algebraic numbers. He also
became a friend of the great Hiemann, then beginning his
career. Dedekind’s university lectures were for the most part
elementary, but in 1857-8 he gave a course (to two students.
Selling and Auwers) on the Galois theory of equations. This was
probably the first time that the Galois theory had appeared
formally in a university course. Dedekind was one of the first
to appreciate the fimdamental importance of the concept of a
group in algebra and arithmetic. In this early work Dedekind
already exhibited two of the leading characteristics of his later
thought, abstractness and generality. Instead of regarding a
finite group from the standpoint offered by its representation
in terms of substitutions (see chapters on Galois and Cauchy),
Dedekmd defined groups hy means of their postulates (substan-
tially as described in Chapter 15) and sought to derive their
properties from this distillation of their essence. This is in the
modem manner: abstractness and therefore generality. The
second characteristic, generality, is, as just implied, a conse-
quence of the first.

At the age of twenty-six Dedekind was appointed (in 1857)
ordinary professor at the Zurich polytechnic, where he stayed
five years, returning in 1862 to Brunswick as professor at the
technical high school. There he stuck for half a century. The
most important task for Dedekind’s official biographer – pro-
vided one is unearthed – Aviil be to explain (not explain away)
the singular fact that Dedekind occupied a relatively obscure
position for fiity years while men who were not fit to lace his
dioes filled important and influential university chairs. To say
that Dedekind preferred obscurity U one explanation. Those
who believe it should leave the stock market severely alone, fox
as surely as God made little lambs they will be fleeced.

Till his death (1916) in his eighty-fifth year Dedekind re-
mained fresh of mind and robust of body. He never married, but
lived with his sister Julie, remembered as a novelist, tfll her
death in 1914, His otlier sister, IMathiJde, died in 1860; his
brother became a distinguished jurist.

Such are the bare facts of any importance in Dedekind’s

572

ARITHMETIC THE SECOND

material career. He lived so long that although some of liis
^ork (his theory of irrational numbers, described presently)
had been familiar to all students of analysis for a generation
before his death, he himself had become almost a legend and
many classed him Tdth the shadowy dead. Twelve years before
his death, Teubner’s Calendar for Mathematicians listed Dede-
kind as ha\ing died on 4 September 1899, much to Dedekind’s
amusement. The day, 4 September, might possibly prove to be
correct, he wrote to the editor, but the year certainly was
wrong. ‘According to my owm memorandum I passed this day
in perfect health and enjoyed a very stimulating conversation
on ’isystem and theory”’ with my luncheon guest and honoured
friend Georg Cantor of HaUe.’

Dedekind’s mathematical activity impinged almost wholly
on the domain of number in its widest sense. We have space for
only two of his greatest achievements and we shall describe
first his fundamental contribution, that of the ‘Dedekind cut”,
to the theory of irrational numbers and hence to the founda-
tions of analysis. This being of the very first importance we may
recall briefly the nature of the matter. If a, b are common whole
numbers, the fraction a,& is called a rational number; if no
whole numbers m, n exist such that a certain ‘number’ A is
expressible as min, then A’ is called an irrational number. Thiis
V 2 , Vs, Vs are irrational numbers. K an irrational number be
expressed in the decimal notation the digits following the
decimal point exhibit no regularities – there is no ‘period’ which
repeats, as in the decimal representations of a rational number,
say 13/11, = 1T8181S . . . , where the ‘18’ repeats indefinitely.
How then, if the representation is entirely lawless, are decimals
equivalent to irrationals to be defined, let alone manipulated?
Have we even any clear conception of what an irrational
number is? Eudoxus thought he had, and Dedekind’s definition
of equality between numbers, rational or irrational, is identical
with that of Eudoxus (see Chapter 2).

If two rational numbers are equal, it is no doubt obvious that
their square roots are equal. Thus 2×3 and 8 are equal; so
also then are V 2 x 3 and V 6- But it is not obvious that V 2 X
Vs = V 2 X 3, and hence that V2 X V3 V6. The un^

573

MEX OE MATHEMATICS

ob\dousness of this simple assumed equality, V2 x Vs = ye,
taken for granted in school arithmetic, is evident if ‘^e visualize
what the equality implies: the ‘lawless’ square roots of 2, 3 , q
are to be extracted, the first two of these are then to be multi-
plied together, and the result is to come out equal to the third.
As not one of these three roots can be extracted exactly, no
matter to how many decimal places the computation is carried,
it is clear that the verification by multiplication as just de-
scribed will never be complete. The whole human race toiling
incessantly through all its existence could never prove in this
way that V2 X Vs = Vd. Closer and closer approximations
to equality would be attained as time went on, but finalitv
would continue to recede. To make these concepts of ‘approxi-
mation’ and ‘equality’ precise, or to replace our first crude
conceptions of irrationals by sharper descriptions which will
obviate the difficulties indicated, was the task Dedekind set
himself in the early 1870’s – his work on Continuity and
Irrational Numbers was published in 1872.

The heart of Dedekind’s theory of irrational numbers is his
concept of the ‘cut’ or ‘section’ (Schniuy. a cut separates aU
rational numbers into too classes, so that each number in the
first class is less than each number in the second class ; every such
cut which does not ‘correspond’ to a rational number ‘defines’
an irrational number. This bald statement needs elaboration,
particularly as even an accurate exposition conceals certain
subtle difficulties rooted in the theory of the mathematical
infinite, which will reappear when we consider the life of
Dedekind’s friend Cantor.

Assume that some rule has been prescribed which separates
all rational numbers into txo dasses, say an ‘upper’ class and a
‘lower’ class, such that each number in the l&wer class is less than
every number in the upper class. (Such an assumption would
not pass unchallenged to-day by all schools of mathematical
philosophy. However, for the moment, it may be regarded as
unobjectionable.) On this assumption one of three mutually
exclusive situations is possible.

(A) There may be a number in the lower class which is grecdei
than every other number in that class.

574

ARITHMETIC THE SECOND

(B) There may be a number in the tipper class which is less
than every other number in that class.

(C) y either of the numbers (greatest in [A], least in [B])
described in (A), (B) may exist.

The possibility which leads to irrational numbers is (C), For
if (C) holds, the assumed rule ‘defines’ a definite break or ‘cut’
in the set of all rational numbers. The upper and lower classes
strive, as it were, to meet. But in order for the classes to meet
the cut must be filled with some ‘number’, and, by (C), no such
filling is possible.

Here we appeal to intuition. All the distances measured from
any fixed point along a given straight line •correspond’ to
‘numbers’ which ‘measure’ the distances. If the cut is to be left
unfilled, we must picture the straight line, which we may con-
ceive of as having been traced out by the continuous motion of
a point, as now having an unbridgeable gap in it. This violates
our intuitive notions, so we say, by definition, that each cut
does define a number. The number thus defined is not rational,
namely it is irrational. To provide a manageable scheme for
operating with the irrationals thus defined by cuts (of the kind
[C] ) we now consider the lower class of raiionals in (C) as being
equivalent to the irrational which the cut defines.

One example will suffice. The irrational square root of 2 is
defined by the cut whose upper class contains all the positive
rational numbers whose squares are greater than 2, and whose
lower class contains all other rational numbers.

If the somewhat elusive concept of cuts is distasteful two
remedies may be suggested: devise a definition of irrationals
which is less mystical than DedeMnd’s and fully as usable;
follow Kronecker and, denying that irrational numbers exist,
reconstruct mathematics without them. In the present state of
mathematics some theory of irrationals is convenient. But,
from the verj^ nature of an irrational number, it would seem to
be necessary to understand the mathematical infinite
thoroughly before an adequate theory of irrationals is
possible. The appeal to infinite classes is ob^dous m Dede-
kind’s definition of a cut. Such classes lead to serious logical
difficulties.

575

MEN or MATHEMATICS

It depends upon the indi\idual mathematician’s level of
sophistication ^whether he regards these difficulties as relevant
or of no consequence for the consistent development of mathe-
matics. The courageous analyst goes boldly ahead, piling one
Babel on top of another and trusting that no outraged god of
reason mil confound him and all his works, while the critical
logician, peering cynically at the foundations of his brother’s
imposing skyscraper, makes a rapid mental calculation pre-
dicting the date of collapse. In the meantime all are busy and
all seem to be enjoying themselves. But one conclusion appears
to be inescapable: without a consistent theory of the mathe-
matical infinite there is no theory of irrationals; without a
theory of irrationals there is no mathematical analysis in any
form even remotely resembling what we now have; and finally,
without analysis the major part of mathematics – including
geometry and most of applied mathematics – as it now exists
would cease to exist.

The most important task confronting mathematicians would
therefore seem to he the construction of a satisfactory theory of
the infinite. Cantor attempted this, with what success will be
seen later. As for the Dedekind theory of irrationals, its author
seems to have had some qualms, for he hesitated over two years
before venturing to publish it. If the reader wiU glance back at
Eudoxus’ definition of ‘same ratio’ (Chapter 2) he will see that
^infinite difficulties’ occur there too, specifically in the phrase
‘any whatever equimultiples’. Nevertheless some progress has
been made since Eudoxus wrote; we are at least beginning to
understand the nature of our difficulties.

The other outstanding contribution which DedeMnd made
to the concept of ‘number’ was in the direction of algebraic
numbers. For the nature of the fundamental problem con-
cerned we must refer to what was said in the chapter on
Kronecker concerning algebraic number fields and the resolu-
tion of algebraic integers into their prime factors. The crux of
the matter is that in seme such fields resolution into prime
factors is not unique as it is in common arithmetic; Dedekind
restored this highly desirable uniqueness by the invention of
what he called ideals. An ideal is not a number, but an infinite

576

asith:iietic the second

class of numbers, so again Dedekind overcame his difficulties by
taking refuge in che infinite.

The concept of an ideal is not hard to grasp, although there
is one twist – the more inclusive class divides the less inclusive, as
will be explained in a moment – which shocks common sense.
However, common sense was made to be shocked; had we
nothing less dentable than shock-proof common sense we
should be a race of mongoloid imbeciles. An ideal must do at
least two things: it must leave common (rational) arithmetic
substantially as it is, and it must force the recalcitrant alge-
braic integers to obey that fundamental law of arithmetic –
unique decomposition into primes – which they defy.

The point about a more inclusive class di\ndmg a less inclu-
sive refers to the following phenomenon (and its generalization,
as stated presently). Consider the fact that 2 divides 4 – anth >
metically, that is, without remainder. Instead of this ob\dous
fact, which leads nowhere if followed into algebraic number
fields, we replace 2 by the class of all its integer multiples, . . . ,
_ 8^ — _ 4, — 2, 0, 2, 4, 6, 8, … As a matter of convenience

we denote this class by (2). In the same way (4) denotes the
class of all integer multiples of 4. Some of the numbers in (4)
aie . . . , – 16, -12, -8,-4, 0 , 8, 12, 16, … It is now
obvious that (2) is the more inclusive class; in fact (2) contains
aR the numbers in (4) and in addition (to mention only two) — 6
and 6. The fact that (2) contains (4) is symbolized by writing
(2)1(4). It can be seen quite easily that if m, n are any common
whole numbers then (m) | (n) when, and only when, m divides n.

This might suggest that the notion of common arithmetical
divisibility be replaced by that of class inclusion as just
described. But this replacement would be futile if it f^ed to
preserve the characteristic properties of arithmetical divisi-
bility. That it does so preserve them can be seen in detail, but
one instance must suffice. If m divides n, and n divides I, then
m divides I – for example, 12 divides 24 and 24 di\ides 72, and
12 does in ffict divide 72. Transferred to classes, as above, this
becomes: if (m) j (») and (n) | (Z), then (m) j (Z) or, in English, if
the (»i) contains the class (n), and if the class (n) contams
the class (Q, then the class (m) contains the class (1) – which

577

ilEN OF -MATHEMATICS

obviously is true. The upshot is that the replacement of nuift.
bers by their corresponding classes does what is required wiicn
we add the dejanition of ‘multiplication’ : (m) x (n) is defined to
be the class (mn); (2) x (6) = (12). Notice that the last is a
definition; it is not meant to follow from the meanings of (ijtj
and (n).

Dedekind’s ideals for algebraic numbers are a generalizaticm
of what precedes. Following his usual custom Dedekind gave
an abstract definition, that is, a definition based upon essentia
properties rather than one contingent upon some particular
mode of representing, or picturing, the thing defined.

Consider the set (or class) of all algebraic integers in a given
algebraic number field. In this all-inclusive set will be subsets.
A subset is called an ideal if it has the two following properties,

A. The sum and difference of any two integers in the subset
are also in the subset.

B. If any integer in the subset be multiplied by any integei
in the all-inclusive set, the resulting integer is in the subset.

An ideal is thus an infinite class of integers. It will be seen
readily that (m), (n), … , previously defined, are ideals accord-
ing to A, B. As before, if one ideal contains another, the fiisl
is said to divide the second.

It can be proved that every ideal is a class of integers all d
which are of the form

where … » are Ju&ed integers of the field of degree*

concerned, and each of ajj, … , may be any integer what-
ever in the field. This being so, it is convenient to symbolize ai
ideal by exhibiting only the fixed integers, Uj, Ogj • • • > ^

this is done by writing (a^, » a^) as the symbol of th(

ideal. The order in which a … , are written in the
symbol is immaterial.

‘Multiplication’ of ideals must now be defined: the product di
the two ideals (fli, . . . , (hi, … > h^) is the ideal whoa
symbol is (flihi, . . . , • • • > ^ which all possibit*

products, etc., obtained by multiplying an integer in the!
first symbol by an integer in the second occur. For example, the

5T8

ARITHMETIC THE SECOND

product of (fli, and &.>) is {a-pi, ajbi, aj)^. It is
alTS-ays possible to reduce any such product-symbol (for a field
of degree t?) to a symbol containing at most n integers.

One final short remark completes the synopsis of the story.
An ideal whose symbol contains hut one integer, such as (Gj), is
called 2 l – principal ideal. Using as before the notation {a-^ ‘ (b^)
to signify that (a^) contains (6^), we can see without difficulty
that (ai) i (hi) u:hen, and only when^ the integer a-^ divides the
integer by As before, then, the concept of arithmetical di^dsi-
bility is here – for algebraic integers – completely equivalent to
that of class inclusion. A prim^ ideal is one which is not ‘divi-
sible by’ – included in ~ any ideal except the all-inclusive ideal
which consists of all the algebraic integers in the given field.
Algebraic integers being now replaced by their corresponding
principal ideals, it is proved that a given ideal is a product of
prime ideals in one way only, precisely as in the ‘fundamental
theorem of arithmetic’ a rational integer is the product of
primes in one way only. By the above equivalence of arith-
metical divisibility for algebraic integers and class inclusion,
the fundamental theorem of arithmetic has been restored to
integers in algebraic number fields.

Anyone who will ponder a little on the foregoing bare outline
of Dedekind’s creation will see that what he did demanded
penetrating insight and a mind gifted far above the ordinary
good mathematical mind in the power of abstraction. Dedekmd
was a mathematician after Gauss’ own heart: ^At nostro guidem
judicio hujusmodi veritates ex notionibus potius quam ex noia-
tionibus hauriri debeanf (But in our opinion such truths
[arithmetical] should be derived from notions rather than from
notations). Dedekmd always relied on his head rather than on
an ingenious symbolism and expert manipulations of formulae
to get him forward. If ever a man put notions into mathematics,
Dedekmd did, and the wisdom of his preference for creative
ideas over sterile symbols is now apparent although it may not
have been during his lifetime. The longer mathematics lives the
more abstract – and therefore, possibly, also the more practical

• it becomes.

CHAPTER TWE^’TY-EIGHT

THE LAST UNIVERSALIST
Poincare

Ix the History of his Life and Times the astrologer William
Lilly (1602-81) records an amusing – if incredible – aeeount of
the meeting between John Napier (1550-1617), of Merchiston,
the inventor of logarithms, and Henry Briggs (1561-1631) of
Gresham College, London, who computed the first table of
common logarithms. One John IVIarr, ‘an excellent mathemati-
cian and geometrician’, had gone ‘into Scotland before Mr
Briggs, purposely to be there when these two so learned persons
should meet. ]Vlr Briggs appoints a certain day when to meet m
Edinburgh; but failing thereof, the lord Napier was doubtful he
would not come. It happened one day as John IVIarr and the
lord Napier were speaking of Mr Briggs; “Ah John (said Mer-
chiston), IVir Briggs will not now come.” At the very moment
one knocks at the gate; John Marr hastens down, and it proved
hlr Briggs to his great contentment. He brings Mr Briggs up
into my lord’s chamber, where almost one quarter of an km
tjoas spent, each beholding other with admiration, before one
zvord was spohe.^

Recalling this legend Sylvester tells how he himself went
after Briggs’ world record for flabbergasted admiration when,
in 1885, he called on the author of numerous astonishingly
mature and marv^ellously original papers on a new branch of
analysis which had been swamping the editors of mathematical
ioumals since the early 1880’s.

‘I quite entered into Briggs’ feelings at his inter\dew with
Napier’, Sylvester confesses, ‘when I recently paid a visit to
Poincare [1854r-1912] in his airy perch in the Rue Gay-Lussac,
… In the presence of that mighty reser^^oir of pent-up intel-
lectual force my tongue at first refused its office, and it was not

580

THE LAST UNIVEKSALIST

until I had taken some time (it may be two or three minutes) to
peruse and absorb as it were the idea of his external youthful
lineaments that I found myself in a condition to speak,’

Elsewhere Sylvester records his bewilderment when, after
having toiled up the three flights of narrow stairs leading to
Poincare’s ‘airy perch’, he paused, mopping his magnificent
bald head, in astonishment at beholding a mere boy, *so blond,
so young’, as the author of the deluge of papers which had
heralded the advent of a successor to Cauchy.

A second anecdote may give some idea of the respect in
which Poincare’s w’ork is held by those in a position to appre-
ciate its scope. Asked by some patriotic British brass hat in the
rabidly nationalistic days of World War I ~ when it was
obligatory on all academic patriots to exalt tbeir aesthetic allies
and debase their boorish enemies – who was the greatest man
France had produced in modem times, Bertrand Russell
answered instantly, ‘Poincare.’ ‘What! That man?’ his unin-
formed interlocutor exclaimed, believing Russell meant Ray-
mond Poincare, President of the French Republic. ‘Oh,’ Russell
explained when he imderstood the other’s dismay, ‘I was
fhmldng of Raymond’s cousin, Henri Poincare.’

Poincare was the last man to take practically all mathe-
matics, both pure and applied, as his province. It is generally
believed that it would be impossible for any human being
starting to-day to xmderstand comprehensively, much less do
creative work of high quality in more than two of the four main
divisions of mathematics – arithmetic, algebra, geometry,
analysis, to say nothing of astronomy and mathematacal
physics. However, even in the 1880’s, when Poincare’s great
career opened, it was commonly thought that Gauss was the
last of the mathematical universalists, so it may not prove
impossible for some future Poincare once more to cover the
entire field.

As mathematics evolves it both expands and contracts,
somewhat like one of Lemaitre’s models of the universe. At
present the phase is one of explosive expansion, and it is quite
impossible for any man to familiarize himself with the entire
inchoate mass of mathematics that has been dumped on the

581

MEN OF MATHEMATICS

world since the year 1900. But already in certain important
sectors a most welcome tendency towards contraction is plainly
apparent. This is so, for example, in algebra, where the whole-
sale introduction of postulational methods is making the sub-
ject at once more abstract, more general, and less disconnected.
Unexpected similarities – in some instances amounting to
disguised identity – are being disclosed by the modem attack,
and it is conceivable that the next generation of algebraists will
not need to know much that is now considered valuable, as
many of these particular, difScult things will have been sub-
sumed under simpler general principles of wider scope. Some-
thing of this sort happened in classical mathematical physics
when relativity put the complicated mathematics of the ether
on the shelf. ^

Another example of this contraction in the midst of expan-
sion is the rapidly growing use of the tensor calculus in prefer-
ence to that of numerous special brands of vector analysis.
Such generalizations and condensations are often hard for older
men to grasp at first and frequently have a severe struggle to
sur\dve, but in the end it is usually realized that general
methods are essentially simpler and easier to handle than
miscellaneous collections of ingenious tricks devised for special
problems. When mathematicians assert that such a thing as
the tensor calculus is easy ~ at least in comparison with some
of the algorithms that preceded it – they are not trying to
appear superior or mysterious but are stating a valuable truth
which any student can verify for himself. This quality of inclu-
sive generality was a distinguishing trait of Poincare’s vast
output.

If abstractness and generality have obvious advantages of
the kind indicated, it is also true that they sometimes have
serious drawbacks for those who must be interested in details.
Of what immediate use is it to a working physicist to know that
a particular differential equation occurring in his work is solv-
able, because some pure mathematician has proved that it is,
when neither he nor the mathematician can perform the Her-
culean laboiur demanded by a numerical solution capable of
application to specific problems?

582

THE LAST UKIVERSALIST

To take an example from a field in vrliich Poincare did some
of his most original work, consider a homogeneous, incompres-
sible fluid mass held together by the gra\dtation of its particles
aPxd rotating about an axis. Under what conditions will the
motion be stable and what will be the possible shapes of such a
stably rotating fluid? MacLatirin, Jacobi, and others proved
that certain ellipsoids will be stable; Poincare, using more
intuitive, ‘less arithmeticar methods than his predecessors,
once thought he had determined the criteria for the stability of
a pear-shaped body. But he had made a slip. His methods were
not adapted to numerical computation and later workers,
including G. H. Darwin, son of the famous Charles, undeterred
by the horrific jungles of algebra and arithmetic that must be
cleared out of the way before a definite conclusion can be
reached, undertook a decisive solution.*

The man interested in the evolution of binary stars is more
comfortable if the findings of the mathematicians are presented
to him in a form to which he can apply a calculating machine.
And since E^necker’s fiat of ‘no construction, no existence’,
some pure mathematicians themselves have been less enthu-
siastic than they were in Poincare’s day for existence theorems
which are not constructive. Poincare’s scorn for the kind of
detail that users of mathematics demand and must have before
they can get on with their work was one of the most important
contributory causes to bis universality. Another was his extra-
ordinarily comprehensive grasp of all the machinery of the
theory of functions of a complex variable. In this he had no
equal. And it may be noted that Poincare turned his universa-
lity to magnificent use in disclosing hitherto unsuspected con-
nexions between distant branches of mathematics, for example
between (continuous) groups and linear algebra.

♦ This famous question of the ‘piriform body’, of considerable
importance in cosmogony, was apparently settled in 1905 by liapou-
noff, whose conclusion was confirmed in 1915 by Sir James Jeans;
they found that the motion is unstable. Few have had the courage to
check the calculations. After 1915 Leon Lichtenstein, a fellow-
countryman of liapounoff , made a general attack on the problem of
rotating fluid masses. The problem seems to be unlucky; both L’s
had violent deaths.

H-M.— von. u K 583

MEN OF MATHEMATICS

One more characteristic of Poincare’s outlook must be recalled
for completeness before \ve go on to his life: few mathematicians
have had the breadth of philosophical vision that Poincare badj
and none is his superior in the gift of clear exposition. Probably
he had always been deeply interested in the philosophical
implications of science and mathematics, but it was only in
1902, when his greatness as a technical mathematician was
established beyond all cavil, that he turned as a side-interest to
what may be called the popular appeal of mathematics and let
hims elf go in a sincere enthusiasm to share with non-profes-
sionals the meaning and human importance of his subject.
Here his liking for the general in preference to the particular
aided him in telling intelligent outsiders what is of more than
technical importance in mathematics without talking down to
his audience. Twenty or thirty years ago workmen and shop-
girls could be seen in the parks and cafes of Paris avidly^ reading
one pr other of Poincare’s popular masterpieces in its cheap
print and shabby paper cover. The same works in a richer
format could also be foimd – well thumbed and evidently read-
on the tables of the professedly cultured. These hooks were
translated into English, German, Spanish, Hungarian, Swedish,
and Japanese. Poincare spoke the universal languages of
mathematics and science to all in accents which they recog-
nized. His style, peculiarly his own, loses much by translation,
Por the literary excellence of his popular writings Poincar^
was accorded the highest honour a French writer can get, mem-
bership in the literary section of the Institut. It has been some-
what spitefully said by envious novelists that Poincar^ achieved
this distinction, unique for a man of science, because one of the
functions of the (literary) Academy is the constant compilation
of a definitive dictionary of the French language, and the
universal Poincaxd was obviously the man to help out the poets
and grammarians in their struggle to tell the world what auto-
morphic functions are. Imi)artial opinion, based on a study of
Poincare’s writings, agrees that the mathematician deserved no
less than he got.

Closely allied to his interest in the philosophy of mathematies
was Poincare’s preoccupation with the psychology of math©-

584

THE LAST UNITEESALIST

matical creation. How do mathematicians make their disco-
veries? Poincare will tell us later his own observatiotts on this
mystery in one of the most interesting narratives of personal
discovery that was ever written. The upshot seems to be that
mathematical discoveries more or less make themselves after
a long spell of hard labour on the part of the mathematician.
As in literature – according to Dante Gabriel Rossetti – ‘a
certain amount of fundamental brainwork’ is necessary before
a poem can mature, so in mathematics there is no discovery
irithout preliminary drudgery, but this is by no means the
whole story. All ‘explanations’ of creativeness that fail to
provide a recipe whereby a gifted human being can create are
open to suspicion. Poincare’s excursion into practical psycho-
logy, like some others in the same direction, failed to bring back
the Gk>lden Fleece, but it did at least suggest that such a thing
is not wholly mythical and may some day be found when
human beings grow intelligent enough to understand their own
bodies.

Poincare’s intellectual heredity on both sides was good. We
shall not go farther back than bis paternal grandfather. During
the Napoleonic campaign of 1814 this grandfather, at the early
age of twenty, was attached to the military hospital at Saint-
Quentin. On settling in 1817 at Rouen he married and had two
sons: Leon Poincare, bom in 1828, who became a first-rate
physician and a member of a medical faculty; and Antoine, who
rose to the inspector-generalship of the department of roads
and bridges. Leon’s son Henri, bom on 29 April 1854, at Nancy,
Lorraine, became the leading mathematician of the early
twentieth century; one of Antoine’s two sons, Rajnnond, went
in for law and rose to the presidency of the French Republic
dyrrin gr World War I; Antoine’s other son became director of
secondary education. A great-unde who had followed Napoleon
into Russia disappeared and was never heard of after the
Moscow fiasco.

From this distinguished list it might be thought that H^iri
would have exhibited some administrative ability, but he did
not, except in bis early childhood when he freely invented
political games for his sister and young friends to pia-y* la these

MEN OF MATHEMATICS

games he was always fair and scrupulously just, seeing that
each of his playmates got his or her full share of office-holding.
This perhaps is conclusive evidence that ‘the child is father to
the man’ and that Poincare was constitutionally incapable of
imderstanding the simplest principle of administration, which
his cousin Raymond applied intuitively.

Poincare’s biography was written in great detail by his fellow
countryman Gaston Darboux (1842-1917), one of the leading
geometers of modem times, in 1913 (the year following Poin-
care’s death). Something may have escaped the present writer,
but it seems that Darboux, after ha\Tng stated that Poincare’s
mother ‘coming from a family in the Meuse district whose
[ the mother’s] parents lived in Arrancy , was a very good person,
very active and very intelligent’, blandly omits to mention her
maiden name. Can it he possible that the French took over the
doctrine of ‘the three big K’s’ – noted in connexion with Dede-
Idnd – from their late instructors after the kultural drives of
Germany into France in 18T0 and 1914? However, it can he
deduced” from an anecdote told later by Darboux that the
famOy name tnay have been Lannois. We learn that the mother
devoted her entire attention to the education of her two
yoimg children, Henri and his younger sister (name not men-
tioned). The sister was to become the wife of fimile Boutroux
and the mother of a mathematician (who died young).

Owing partly to his mother’s constant care, Poincare’s mental
development as a child was extremely rapid. He learned to talk
very early, but also very badly at first because he thought more
rapidly than he could get the words out. From infancy his
motor co-ordination was poor. When he learned to write it was
discovered that he was ambidextrous and that he could write
or draw as badly with his left hand as with his right. Poincari
never outgrew this physical awkwardness. As an item of some
interest in this connexion it may be recalled that when Poincari
was acknowledged as the foremost mathematician and leading
popularizer of science of his time he submitted to the Binet
tests and made such a disgraceful showing that, had he been
judged as a child instead of as the famous mathematician he
was, he would have been rated – by the tests – as an imbecile.

586

THE LAST UNIVEESALIST

At the age of five Henri suffered a bad setback from diph-
theria which left him for nine months with a paralyzed larjmx.
This misfortune made him for long delicate and timid, but it
also turned him back on his own resources as he was forced to
shun the rougher games of children his own age.

His principal diversion was reading, where his unusual talents
first showed up. A book once read – at incredible speed – became
a permanent possession, and he could always state the page and
line where a particular thing occurred. He retained this power-
ful memory all his life. This rare faculty, which Poincare shared
with Euler who had it in a lesser degree, might be called visual
or spatial memory. In temporal memory – the ability to recall
with uncanny precision a sequence of events long passed — he
was also unusually strong. Yet he unblushingly describes his
memory as ‘badh His poor eyesight perhaps contributed to a
third peculiarity of his memory. The majority of mathemati-
cians appear to remember theorems and formulae mostly by
eye; with Poincare it was almost wholly by ear. Unable to see
the board distinctly when he became a student of advanced
mathematics, he sat back and listened, following and remem-
bering perfectly without taking notes – an easy feat for him,
but one incomprehensible to most mathematicians. Yet he
must have had a vivid memory of the ‘inner eye’ as well, for
much of his work, like a good deal of Riemann’s, was of the
kind that goes with facile space-intuition and acute \dsualiza-
tion. His inability to use his fingers skilfully of course handi-
capped him in laboratory exercises, which seems a pity, as some
of his own work in mathematical physics might have been closer
to reality had he mastered the art of experiment- Had Poincare
been as strong in practical science as he was in theoretical he
might have made a fourth with the incomparable three,
Archimedes, Newton, and Gauss.

Not many of the great mathematicians have been the absent-
minded dreamers that popular fancy likes to picture them.
Poincare was one of the exceptions, and then only in compara-
tive trifles, such as carrying off hotel linen in his baggage. But
many persons who are anything but absent-minded do the
same, and some of the most alert mortals living have even been

587

MEN OF MATHEMATICS

knoTini to slip restaurant silver into their pockets and get away
with it.

One phase of Poincare’s absent-mindedness resembles some-
thing quite different. Thus (Darboux does not tell the story, but
it should be told, as it illustrates a certain brusqueness of Poiu-
care’s later years), when a distinguished mathematician had
come all the way from Finland to Paris to confer with Poincare
on scientific matters, Poincare did not leave his study to greet
his caller when the maid notified him, but continued to pace
back and forth – as was his custom when mathematicizing – for
three solid hours. All this time the diffident caller sat quietly in
the adjoining room, barred from the master only by flimsy
portieres. At last the drapes parted and Poincare’s buffalo head
was thrust for an instant into the room. ‘Fows me derange
beancoup^ (You are disturbing me greatly), the head exploded,
and disappeared. The caller departed without an interview,
which was exactly what the ‘absent-minded’ professor wanted.

Poincare’s elementary school career was brilliant, although
he did not at first show any marked interest in mathematics.
His earliest passion was for natural history, and all his life he
remained a great lover of animals. The first time he tried out a
rifle he accidentally shot a bird at which he had not aimed. This
mishap affected him so deeply that thereafter nothing (except
compulsory military drill) could induce him to touch firearma.
At the age of nine he showed the first promise of what was to be
one of his major successes. The teacher of French composition
declared that a short exercise, original in both form and sub-
stance, which young Poincare had handed in, was ‘a little
masterpiece’, and kept it as one of his treasures. But he
also advised his pupil to be more conventional – stupider

• if he wished to make a good impression on the schod
  examiners.

Being out of the more boisterous games of his schoolfellows,
Poincare invented his own. He also became an indefatigable
dancer. As all Ms lessons came to him as easily as breathing he
spent most of his time on amusements and helping his mothcf
about the house. Even at this early stage of Ms career Poincai^
exhibited some of the more suspicious features of Ms matosEe

dSd

THE LAST UNIVETtSALIST

‘absentmindedness’ : he frequently forgot his meals and almost
ne%’er remembered whether or not he had breakfasted. Perhaps
he did not care to stuff himself as most boys do.

The passion for mathematics seized hiin at adolescence or
shortly before (when he was about fifteen). From the first he
exhibited a lifelong peculiarity: his mathematics was done in
his head as he paced restlessly about, and was committed to
paper only when all had been thought through. Talking or other
noise never disturbed him while he was working. In later life he
wrote his mathematical memoirs at one dash without looking
back to see what he had written and limiting himself to but a
very few erasures as he wrote. Cayley also composed in this way,
and probably Euler, too. Some of Poincare’s work shows the
marks of hasty composition, and he said himself that he never
finished a paper without regretting either its form or its sub-
stance. More than one man who has written well has felt the
same. Poincare’s flair for classical studies, in which he excelled
at school, taught him the importance of both form and
substance.

The Franco-Prussian war broke over France in 1870 when
Poincare was sixteen. Although he was too young and too
frail for active service, Poincare nevertheless got his full
share of the horrors, for Nancy, where he lived, was sub-
merged by the full tide of the invasion, and the young boy
accompanied his physician-father on his rounds of the ambu’-
lances. Later he went with his mother and sister, under
terrible difficulties, to Arrancy to see what had happened to his
maternal grandparents, in whose spacious country garden
the happiest days of his childhood had been spent during the
long school vacations. Arrancy lay near the battlefield of Samt-
Privat. To reach the town the three had to pass Mn glacial cold’
through burned and deserted villages. At last they reached their
destination, only to find that the house had been thoroughly
pillaged, *not only of things of value but of things of no value’,
and in addition had been defiled in the bestial manner made
familiar to the French by the 1914 sequel to 1870. The grand-
parents had been left nothing; their evening meal on the day
they viewed the great purging was supplied by a poor woman

589

MEN OF MATHEMATICS

had refused to abandon the ruins of her cottage and who
insisted on sharing her meagre supper with them,

Poincare never forgot this, nor did he ever forget the long
occupation of Nancy by the enemy. It was during the war that
he mastered German. Unable to get any French news, and eager
to learn what the Germans had to say of France and for them-
selves, Poincare taught himself the language. What he had seen
and what he learned from the official accounts of the invaders
themselves made him a flaming patriot for life but, like Her-
mite, he never confused the mathematics of his country’s
enemies with their more practical activities. Cousin Raymond,
on the other hand, could never say anything about les Alle-
mands (the Germans) without an accompanying scream of hate.
In the bookkeeping of heU which balances the hate of one
patriot against that of another, Poincard may be checked off
against Kummer, Hermite against Gauss, thus producing that
perfect zero implied in the scriptural contract *an eye for an eye
and a tooth for a tooth’.

Following the usual French custom Poincare took the
examinations for his first degrees (bachelor of letters, and of
science) before specializing. These he passed in 1871 at the age
of seventeen – after almost failing in mathematics! He had
arrived late and flustered at the examination and had fallen
down on the extremely simple proof of the formula giving the
sum of a convergent geometrical progression. But his fame had
preceded him. *Any student other than Poincare would have
been plucked’, the head examiner declared.

He next prepared for the entrance examinations to the School
of Forestry, where he astonished his companions by capturing
the first prize in mathematics without having bothered to take
any lecture notes. His classmates had previously tested him out,
believing him to be a trifler, by delegating a fourth-year student
to quiz him on a mathematical difficulty which had seemed
pairticularly tough. Without apparent thought, Poincare gave
the solution immediately and walked off, leaving his crestfallen
baiters asking ‘How does he do it?’ Others were to ask the same
question all through Poincare’s career. He never seemed to
think when a mathematical difficulty was submitted to him by
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THE LAST UNIYEESALIST

his colleagues: ‘The reply came like an arrow’.

At the end of this year he passed first into the ficole Poly-
technique. Several legends of his unique examination survive.
One tells how a certain examiner, forewarned that young Poin-
care was a mathematical genius, suspended the examination for
three-quarters of an hour in order to devise ‘a “nice” question’ –
a refined torture. But Poincare got the better of him and the
inquisitor ‘congratulated the examinee warmly, telling hiin he
had won the highest grade’. Poincare’s experiences with his
tormentors would seem to indicate that French mathematical
examiners have learned something since they ruined Galois and
came within an ace of doing the like by Hermite.

At the Polytechnique Poincar^ was distinguished for his
brilliance in mathematics, his superb incompetence in ail
physical exercises, including gymnastics and military drill, and
his utter inability to make drawings that resembled anything in
heaven or earth. The last was more than a joke; his score of zero
in the entrance examination in drawing had almost kept him
out of the school. This had greatly embarrassed his examiners:
‘ . a zero is eliminatory. In everything else [but drawing] he is
absolutely without an equal. If he is admitted, it will be as
first; but can he be admitted?’ As Poincare was admitted the
good examiners probably put a decimal point before the zero
and placed a 1 after it.

In spite of his ineptitude for physical exercises Poincar^ was
extremely popular with his classmates. At the end of the year
they organized a public exhibition of his artistic masterpieces,
carefully labelling them in Greek, ‘this is a horse’, and so on –
not always accurately. But Poincare’s inability to draw also
had its serious side when he came to geometry, and he lost first
place, passing out of the school second in rank.

On leaving the Polytechnique in 1875 at the age of twenty-
one Poincare entered the School of Mines with the intention of
becoming an engineer. His technical studies, although faithfully
carried out, left him some leisure to do mathematics, and he
showed what was in him by attacking a general problem in
differential equations. Three years later he presented a thesis,
on the same subject, but concerning a more dfflcult and yet

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MEN or MATHEMATICS

more general question, to the Faculty of Sciences at Paris for
the degree of doctor of mathematical sciences, ‘At the first
glance’, says Darhoux, who had been asked to examine the
work, it was clear to me that the thesis was out of the ordinary and amply merited acceptance. Certainly it contained results enough to supply material for several good theses. But, I must not be afraid to say, if an accurate idea of the way Poincare worked is wanted, many points called for corrections or expla- nations. Poincare was an intuitionist. Having once arrived at the summit he never retraced his steps. He was satisfied to have crashed through the difficulties and left to others the pains of mapping the royal roads destined to lead more easily to the
end. He willingly enough made the corrections and tidying-up
which seemed to me necessary. But he explained to me when I
asked him to do it that he had many other ideas in his head; he
was already occupied with some of the great problems whose
solution he was to give us.’

Thus young Poincare, like Gauss, was overwhelmed by the
host of ideas which besieged his mind but, unlike Gauss, his
motto was not ‘Few, but ripe’. It is an open question whether a
creative scientist who hoards the fruits of his labour so long that
some of them go stale does more for the advancement of science
than the more impetuous man who scatters broadcast every-
thing he gathers, green or ripe, to fall where it may to ripen or
rot as wind and weather take it. Some believe one way, some
another. As a decision is beyond the reach of objective criteria
everyone is entitled to his own purely subjective opinion.

Poincare was not destined to become a mining engineer, but
during his apprenticeship he showed that he had at least the
courage of a real engineer. After a mine explosion and fire which
had claimed sixteen victuns he went down at once with tlie
rescue crew. But the calling was uncongenial and he welcomed
the opportunity to become a professional mathematician which
his thesis and other early work opened up to him. His first
academic appointment was at Caen on 1 December 1879, as

• ‘There is no royal road to geometry*, as Menaechmus is said to
  have told Alexander the Great when the latter wished to conqaer
  geometry in a hurry.

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THE LAST UNIYEESALIST

Professor of Mathematical Analysis. Two years later he was
promoted (at the age of twenty-seven) to the University of
Paris where, in 1886, he was again promoted, taking charge of
the course in mechanics and experimental physics (the last
seems rather strange, in view of Poincare’s exploits as a student
in the laboratory). Except for trips to scientific congresses in
Europe and a visit to the United States in 1904 as an in\dted
lecturer at the St Lnuis Exposition, Poincare spent the rest of
his life in Paris as the ruler of French mathematics.

Poincare’s creative period opened with the thesis of 1878 and
closed with bis death in 1912 – when he was at the apex of his
powers. Into this comparatively brief span of thirty-four years
he crowded a mass of work that is sheerly incredible when we
consider the difficulty of most of it. His record is nearly 500
papers on new mathematics, many of them extensive memoirs,
and more than thirty books covering practically all branches of
mathematical physics, theoretical physics, and theoretical
astronomy as they existed m his day. This leaves out of account
his classics on the philosophy of science and his popular essays.
To give an adequate idea of this immense labour one would
have to be a second Poincar^, so we shall presently select two
or three of his most celebrated works for brief description,
apologizing here once for all for the necessary inadequacy.

Poincare’s first succsesses were in the theory of differential
equations, to which he applied ail the resources of the analysis
of which he was absolute master. This early choice for a major
effort already indicates Poincare’s leaning toward the applica-
tions of mathematics, for differential equations have attracted
swarms of workers since the time of Newton chiefly because
they are of great importance in the exploration of the physical
universe, ‘Pure’ mathematicians sometimes like to imagine that
all their activities are dictated by their own tastes and that the
applications of science suggest nothing of interest to them.
Nevertheless some of the purest of the pure drudge away their
lives over differential equations that first appeared in the
translation of physical situations into mathematical symbolism,
and it is precisely these practically suggested equations which
ajre the heart of the theory. A particular equation suggested by

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MEN OF MATHEMATICS

science may be generalized by tbe mathematicians and then be
turned back to the scientists (frequently without a solution in
any form that they can use) to be applied to new physical pro-
blems, but first and last the motive is scientific. Fourier summed
up this thesis in a famous passage which irritates one type of
mathematician, but which Poincare endorsed and followed in
much of his work.

‘The profound study of nature’, Fourier declared, ‘is the most
fecund source of mathematical discoveries. Not only does this
study, by offering a definite goal to research, have the advan-
tage of excluding vague questions and futile calculations, but
it is also a sure means of moulding analysis itself and discover-
ing those elements in it which it is essential to know and which
science ought always to conserve. These fundamental elements
are those which recur in all natural phenomena,’ To which some
might retort; No doubt, but what about arithmetic in the sense
of Gauss? However, Poincare followed Fourier’s advice
whether he believed in it or not – even his researches in the
theory of numbers were more or less remotely inspired by others
closer to the mathematics of physical science.

The investigations on differential equations led out in 1880,
when Poincare was twenty-six, to one of his most brilliant dis-
coveries, a generalization of the elliptic functions (and of some
others). The nature of a (uniform) periodic function of a single
variable has frequently been described in preceding chapters,
but to bring out what Poincare did, we may repeat the essen-
tials. The trigonometric function sin z has the period 27r, namely,
sin (s -{- 27 t) = sin z; that is, when the variable 3 is increased by
277 , the sine function of z returns to its i n i ti al value. For an
elliptic function, say JS(z), there are two distinct periods, say pi
andpa, such that E(z + pj = JS(z), E[z + Pa) = ^(2)* Poincare
found MisX periodicity is merely a special case of a more general
property: the value of certain functions is restored when the
variable is replaced by any one of a denuTnerable infinity of
linear fractional transformations of itself, and all these trans-
formations form a group. A few symbols will clarify this state-
ment.

THE LAST UNIVERSALIST

az b

Let z be replaced by — Then, for a denumerable infinity

of sets of values of a,b,c,d, there are uniform functions of 2 , say
T{z) is one of them, such that

F

= m-

Further, if and are any two of the sets of

values of a.b,c,d, and if 2 be replaced first by and then,

_ , _ , , , Qo2 -f- 6*>

m this, 2 be replaced hy — = — ; ,

CgS -p dg

only do we have

givmg, say.

Ci2 -p di

Az 4- B

Cz 4- D

, then not

but also

_ n.),

\CjZ 4- di/ \C22 4- dj

( Az 4“ T-,/

^ — = F(z).

[Cz -hJDj ^ ^

Further the set of all the substitutions

az 4“ h
cz– d

(the arrow is read ^is replaced by’) which leave the value of
F(z) unchanged as just explained/om a group: the result of the
successive performance of two substitutions in the set,

OjZ 4- hi fljsS + ^2

Z —> i — V’ ^ ^ , jt *

c^z 4- di CjjZ 4- d*

is in the set; there is an ‘identity substitution’ in the set, namely
z^z (here a = 1, 5 = 0, c = 0, d = 1); and finally each substi-
tution has a unique ‘inverse’ — that is, for each substitution in
the set there is a single other one which, if applied to the first,
will produce the identity substitution. In summary, using the
terminology of previous chapters, we see that jP(z) is afttnciicn
Tjchich is invariant under an infinite group of linear fractional
transformations. Note that the infinity of substitutions is a

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MEN OE MATHEMATICS

denumerable infinity , as first stated: the substitutions can be
counted off 1,2,3, … , and are not as numerous as the points ‘ “
on a line. Poincare actually constructed such functions and
developed their most important properties in a series of
papers in the ISSO’s. Such functions are. called automorphic.

Only two remarks need he made here to indicate what Poin-
care achieved by this “wonderful creation. First, his theory
includes that of the elliptic functions as a detail. Second, as the
distinguished French mathematician Georges Humbert said,
Poincare found two memorable propositions which ‘gave him
the keys of the algebraic cosmos’ :

Two automorphic functions* invariant under the same group
are connected by an algebraic equation;

Conversely, the co-ordinates of a point on any algebraic
curve can be expressed in terms of automorphic functions, and
hence by uniform functions of a single parameter (variable).

An algebraic curve is one whose equation is of the type
F(x,y) = 0, where P(x,y) is a polynomial in x and y. As a simple
example, the equation of the circle whose centre is at the origin

• (0,0) – and whose radius is u, is According to the

second of Poincare’s ‘keys’, it must be possible to express x,y as
automorphic functions of a single parameter, say f. It is; for if
X = a cos t and y == a sin f, then, squaring and adding, we get
rid of t (since cos® t -f sin® t = 1), and find — a®. But the

trigonometric functions cos t, sin t are special cases of elliptic
functions, which in turn are special cases of automorphic
functions-

The creation of this vast theory of automorphic functions was
but one of many astonishing things in analysis which Poinear^
did before he was thirty. Nor was all his time devoted to analy-
sis; the theory of numbers, parts of algebra, and mathematical
astronomy also shared his attention. In the first he recast the
Gaussian theory of binary quadratic forms (see chapter oa

• Poincar^ called some of his functions ‘Fuchsian’, after the
  Grerman mathematician Lazarus Fuchs (1833-1902) one of the
  creators of the modem theory of differential equations, for reasons
  that need not be gone into here. Others he called ‘Kleinian’ after
  Felix Klem – in ironic acknowledgement of disputed priority.

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Gauss) in a geometrical shape which appeals particularly to
those who, like Pomcar6, prefer the intuitive approach. This of
course was not ail that he did in the higher arithmetic, but
limitations of space forbid further details.

Work of this calibre did not pass unappreciated. At the
unusually early age of thirty-two (in 1887) Poincar6 was elected
to the Academy. His proposer said some pretty strong things, but
most mathematicians will subscribe to their truth; ‘[Poincare’s]
work is above ordinary praise and reminds us inevitably of
what Jacobi wrote of Abel ~ that he had settled questions
which, before him, were unimagined. It must indeed be recog-
nized that we are witnessing a revolution in Mathematics com-
parable in every way to that which manifested itself, half a
century ago, by the accession of elliptic functions.’

To leave Poincare’s work in pure mathematics here is like
rising from a banquet table after having just sat down, but we
must turn to another side of his universality.

Since the time of Newton and his immediate successors astro-
nomy has generously supplied mathematicians with more pro-
blems than they can solve. Until the late nineteenth century the
weapons used by mathematicians in their attack on astronomy
were practically aU immediate improvements of those invented
by Newton himself, Euler, Lagrange, and Laplace. But all
through the nineteenth century, particularly since Cauchy’s
development of the theory of functions of a complex variable
and the investigations of himself and others on the convergence
of infinite series, a huge arsenal of untried weapons had been
accumulating from the labours of pure mathematicians. To
Poincare, to whom analysis came as naturally as thinking, this
vast pile of unused mathematics seemed the most natural thing
in the world to use in a new offensive on the outstanding pro-
blems of celestial mechanics and planetary evolution. He picked
and chose what he liked out of the heap, improved it, invented
new weapons of his own, and assaulted theoretical astronomy
in a grand fashion it had not been assaulted in for a century. He
jnodemized the attack; indeed his campaign was so extremely
modem to the majority of experts in celestial mechanics that
even to-day, forty years or more after Poincare opened his

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3IEN OF MATHEMATICS

oSeosive, few have mastered his weapons and some, unable to
bend Ms bow, insinuate that it is worthless in a practical attach.
Nevertheless Poincare is not without forceful champions whose
conquests would have been impossible to tlie men of the pre-
Poincare era.

Poincare’s first (1889) great success in mathematical astro-
nomy grew out of an unsuccessful attack on ‘the problem of n
bodies.’ For == 2 the problem was completely solved by
Newton; the famous ‘problem of three bodies’ {n = 3) will be
noticed later: when n exceeds 3 some of the reductions applic-
able to the ease n = 3 can be carried over.

According to the Newtonian law of gravitation two particles
of masses jk, at a distance D apart attract one another with

a force proportional to — — ■ Imagine n material particles

distributed in any manner in space; the masses, initial motions’
and the mutual distances of all the particles are assumed known
at a given instant. If they attract one another according to the
Newtonian law, what will be their positions and motions (velo-
cities) after any stated lapse of time? For the purposes of mathe-
matical astronomy the stars in a cluster, or in a galaxy, or in a
cluster of galaxies, may he thought of as material particles
attracting one another according to the Newtonian law* The
‘problem of n bodies’ thus amounts – in one of its applications –
to asking what will be the aspect of the heavens a year from
now, or a billion years hence, it being assumed that we have
suSicient observational data to describe the general configura-
tion now. The problem of course is tremendously complicated
by radiation ~ the masses of the stars do not remain constant
for of years; but a complete, calculable solution of the

problem of n bodies in its Newtonian form would probably give
results of an accuracy sufidcient for all human purposes — the
human race will likely be extinct long before radiation can
introduce observable inaccuKwnes.

This was substantially the problem proposed for the prize
offered by Oscar II of Sweden in 1887. Poincard did not
solve the problem, but in 1889 he was awarded the prize any-
how by a iury consisting of Weierstrass, Hermite, and IVIittag-

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LefiSer for his general discussion of the differential equations of
dynamics and an attack on the problem of three bodies. The
last is usually considered the most important case of the n*body
problem, as the Earth, Moon, and Sun furnish an instance of the
case 71 = 3, In his report to Mittag-Leffler, Weierstrass TOOte,
‘You may tell your Sovereign that this work cannot indeed be
considered as furnishing the complete solution of the question
proposed, but that it is nevertheless of such importance that its
publication will inaugurate a new era in the history of Celestial
Mechanics, The end which His Majesty had in view in opening
the competition may therefore be considered as having been
attained.’ Not to be outdone by the King of Sweden, the
French Government followed up the prize by making Poincare
a Knight of the Legion of Honour – a much less expensive
acknowledgement of the young mathematician’s genius than
the King’s 2,500 crowns and gold medal.

As we have mentioned the problem of three bodies we may
now report one item from its fairly recent history; since the
time of Euler it has been considered one of the most difficult
problems in the whole range of mathematics. Stated mathe-
matically, the problem boils down to sohdng a system of nine
simultaneous differential equations (aH linear, each of the
second order). Lagrange succeeded in reducing this system to a
simpler. As in the majority of physical problems, the solution
is not to be expected in finite terms; if a solution eodsts ai all it
will he given by infinite series. The solution will ‘exist’ if these
series satisfy the equations (formally) and moreover converge
for certain values of the variables. The central difficulty is to
prove the convergence. Up till 1905 various special solutions
had been found, but the existence of anything that could be
called general had not been proved.

In 1906 and 1909 a considerable advance came from a rather
unexpected quarter — Finland, a country which sophisticated
Europeans even to-day consider barely civilized, especially for
its queer custom of paying its debts, and which few Americans
thought advanced beyond the Stone Age till Paavo Nurmi ran
the legs off the United States. Excepting only the rare case
when all three bodies collide simultaneously, Karl Frithiof

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MEN OF MATHEMATICS

Sundman of Helsingfors, utilizing analytical methods due to the
Italian Levi-Civita and the French Painleve, and making an
ingenious transformation of his own, proved the existence of a
solution in the sense described above. Sundman’s soiution is
not adapted to numerical computation, nor does it give much
information regarding the actual motion, but that is not the
point of interest here; a problem which had not been known to
be solvable was proved to be so. Many had struggled desperately
to prove this much; when the proof was forthcoming, some,
humanly enough, hastened to point out that Sundman had
done nothing much because he had not solved some problem
other than the one he had. This kind of criticism is as common
in mathematics as it is in literature and art, showing once more
that mathematicians are as human as anybody.

Poincare’s most original work in mathematical astronomy
was summed up in his great treatise Les mithodes nouveUes de la
m^canique celeste (New Methods of Celestial Mechanics; three
volumes, 1892, 1893, 1899). This was followed hy another three-
volume work in 1905-10 of a more immediately practical
nature, Legons de mecanique ciieste^ and a little later hy the
publication of his course of lectures Sur les figures d’iquilibrc
tftt3iemassfi:yZwidc(OntheFiguresofEquilibrium of aFluid Mass),
and a historical-critical book Sur les hypotMses cosmogonigues
(On Cosmological Hypotheses).

Of the first of these works Daihoux (seconded by many
others) declares that it did indeed start a new era in celestial
mechanics and that it is comparable to the Mecanique cilesteot
Laplace and the earlier work of D’Alembert on the precession
of the equinoxes. ‘Following the road in analytical mechanics
opened up by Lagrange,’ Darboux says, ‘ . Jacobi had estab-
lished a theory which appeared to be one of the most complete
in dynamics. For fifty years we lived on the theorems of the
illustrious German mathematician, applying them and studjii^
them from all angles, but without adding anything essenriaL
It was Poinear6 who first shattered these rigid frames in which
the theory seemed to be encased and contrived for it vistas and
new windows on the external world. He introduced or used, in
the study of dynamical problems, different notions; the first,

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THE LAST UNIVERSALIST

which had been given before and which, moreover, is applicable
not solely to mechanics, is that of varicdional equaticms, namely,
linear differential equations that determine solutions of a pro-
blem infinitely near to a given solution; the second, that of
integral invariants^ which belong entirely to him and play a
capital part in these researches. Further fundamental notions
were added to these, notably those concerning so-called
“periodic” solutions, for which the bodies whose motion is
studied return after a certain time to their initial positions and
original relative velocities.’

The last started a whole department of mathematics, the
investigation of periodic orbits: given a system of planets, or of
stars, say, with a complete specification of the initial positions
and relative velocities of all members of the sj’stem at a stated
epoch, it is required to determine under what conditions the
system will return to its initial state at some later epoch, and
hence continue to repeat the cycle of its motions indefinitely.
For example, is the solar system of this recurrent t}^, or if not,
would it be were it isolated and not subject to perturbations by
external bodies? Needless to say the general problem has not
yet been solved completely.

Much of Poincare’s work in his astronomical researches was
qualitative rather than quantitative, as befitted an intuitionist,
and this characteristic led him, as it had Riemann, to the study
of analysis situs. On this he published six famous memoirs
which revolutionized the subject as it existed in his day. The
work on analysis situs in its turn was freely applied to the
mathematics of astronomy.

We have already alluded to Poincare’s work on the problem
of rotating fluid bodies – of obvious importance in cosmogony,
one brand of which assumes that the planets were once suffi-
ciently like such bodies to be treated as if they actually were
without patent absurdity. Whether they were or not is of no
importance for the mathematics of the situation, which is of
interest in itself. A few extracts from Poincare’s own summary
will indicate more dearly than any paraphrase the nature of
what he mathematicized about in this difficult subject-

*Let us imagine a [rotating] fluid body contracting by csool-

001

MEN OF MATHEMATICS

ing, but slowly enough to remain homogeneous and fox the
rotation to be the same in all its parts.

‘At first very approximately a sphere, the figure of this mass
will become an ellipsoid of revolution which will flatten more
and more, then, at a certain moment, it will be transformed into
an ellipsoid with three imequal axes. Later, the figure will cease
to be an ellipsoid and will become pear-shaped until at last the
mass, hollowing out more and more at its “waist”, will separate
into two distinct and unequal bodies.

‘The preceding hypothesis certainly cannot he applied to the
solar system. Some astronomers have thought that it might be
true for certain double stars and that double stars of the type
of Beta Lyrae might present transitional forms analogous to
those we have spoken of.’

He then goes on to suggest an application to Saturn’s rings,
and he claims to have proved that the rings can be stable only
if their density exceeds 1/16 that of Saturn. It may be remarked
that these questions were not considered as fully settled as late
as 1935. In particular a more drastic mathematical attack on
poor old Saturn seemed to show that he had not been completely
vanquished by the great mathematicians, including Clerk
Maxwell, who have been firing away at him off and on for the
past seventy years.

Once more we must leave the banquet having barely tasted
anything and pass on to Poincare’s voluminous work in mathe-
matical physics. Here his luck was not so good. To have cashed
in on his magnificent talents he should have been bom thirty
years later or have lived twenty years longer. He had the mis-
fortune to be in his prime just when physics had reached one of
its recurrent periods of senility, and he was so thoroughly
saturated with nineteenth-century theories when physics began
to recover its youth – after Planck, in 1900, and Einstein, in
1905, had performed the difficult and delicate operation of
endowing the decrepit roui with its first pair of new glands –
that he had barely time to digest the miracle before his death
in 1912. All his mature life Poincard seemed to absorb know-
ledge through his pores without a conscious effort. Like Cayley,
he was not only a prolific creator but also a profoundly erudite

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scholar. His range was probably wider than even Cayley^s, for
Cayley never professed to be able to understand everything that
was going on in applied mathematics. This unique erudition
may have been a disadvantage when it came to a question of
li^-ing science as opposed to classical.

Everything that boiled up in the melting pots of physics was
grasped instantly as it appeared by Poincare and made the
topic of several purely mathematical investigations. ^Vhen
wireless telegraphy was invented he seized on the new thing and
worked out its mathematics. While others were either ignoring
Einstein’s early work on the (special) theory of relativity or
passing it by as a mere curiosity, Poincare was already busy
with its mathematics, and he was the hrst scientific man of high
standing to tell the world what had arrived and urge it to watch
Einstein as probably the most significant phenomenon of the
new era which he foresaw but could not himself usher in. It was
the same with Planck’s early form of the quantum theor\
Opinions differ, of course; but at this distance it is beginning to
look as if mathematical physics did for Poincare what Ceres did
for Gauss; and although Poincare accomplished enough in
mathematical physics to make half a dozen great reputations,
it was not the trade to which he had been bom and science
would have got more out of him if he had stuck to pure mathe-
matics – his astronomical work was nothing else. But science
got enough, and a man of Poincare’s genius is entitled to his
hobbies.

We pass on now to the last phase of Poincare’s universality
for which we have space: his interest in the rationale of mathe-
matical creation. In 1902 and 1904 the Swiss mathematical
periodical L* Enseignement Mathematique undertook an enquiry
into the working habits of mathematicians. Questionnaires
were issued to a number of mathematicians, of whom over a
hundred replied. The answers to the questions and an analysis
of general trends were published in final form in 1912.* Anyone
wishing to look into the ‘psychology’ of mathematicians will

*Enquite de *L*Enseignement ^lathematique’ sur la mcthode de
tracail des maUiSmatieiens, Available either in the periodical or in
book form (8 -j- 137 pp.) from Gauthier-Yillais, Paris.

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MEK OP MATHEMATICS

find much of interest in this unique work and many confirma-
tions of the views at which Poincare had arrived independently
before he saw the residts of the questionnaire. A few points of
general interest may he noted before we quote from Poincare.

The early interest in mathematics of those who were to be-
come great mathematicians has been frequently exemplified in
preceding chapters. To the question ‘At what period . . . and
under what circumstances did mathematics seize you?’ ninety-
three replies to the first part were received; thirty-five said
before the age of ten; forty-three said eleven to fifteen; eleven
said sixteen to eighteen; three said nineteen to twenty; and the
lone laggard said twenty-six.

Again, anyone with mathematical friends will have noticed
that some of them like to work early in the morning (I know one
very distinguished mathematician who begins his day’s work
at the inhuman hour of five a.m-), while others do nothing till
after dark. The replies on this point indicated a curious trend –
possibly significant, although there are numerous exceptions:
mathematicians of the northern races prefer to work at night,
while the Latins favour the morning. Among night-workers
prolonged concentration often brings on insomnia as they grow
older and they change – reluctantly – to the morning. Felix
Klein, who worked day and night as a young man, once indi-
cated a possible way out of this difficulty. One of his American
students complained that he could not sleep for thinking of his
mathematics. ‘Can’t sleep, eh?’ Klein snorted. ‘What’s chloral
for?’ However, this remedy is not to be recommended indiscri-
minately; it probably had something to do with Klein’s orm
tragic breakdown.

Probably the most significant of the replies were those
received on the topic of inspiration versus drudgery as the
source of mathematical discoveries. The conclusion is that
‘Mathematical discoveries, small or great . • . are never bom of
spontaneous generation. They always presuppose a soil seeded
with preliminary knowledge and well prepared by labour, both
conscious and subconsdoxis.’

Those who, like Thomas Alva Edison, have declared that
genius is 99 per cent perspiration and only 1 per cent inspira-

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tioiif are not contradicted by those ‘W’ho would reverse the
figtires. Both are right; one man remembers the dradgerj’’ while
another forgets it all in the thrill of apparently sudden discoveiy,
but both, when they analyse their impressions, admit that
without drudgery and a flash of ‘inspiration” discoveries are not
made. If drudgery alone sufficed, how is it that many gluttons
for hard work who seem to know everything about some branch
of science, while excellent critics and commentators, never
themselves make even a small discovery? On the other hand,
those who believe in ‘inspiration’ as the sole factor in discovery
or invention – scientific or literary – may find it instructive to
look at an early draft of any of Shelley’s ‘completely sponta-
neous’ poems (so far as these have been preserved and repro-
duced), or the successive versions of any of the greater novels
that Balzac inflicted on his maddened printer.

Poincare stated his views on mathematical discovery in an
essay first published in 1908 and reproduced in his Science et
2Iethode* The genesis of mathematical discovery, he says, is a
problem which should interest psychologists intensely, for it is
the activity in which the human mind seems to borrow least
from the external world, and by understanding the process of
mathematical thinking we may hope to reach what is most
essential in the human mind.

How does it happen, Pomcar6 asks, that there are persons
who do not understand mathematics? ‘This should surprise us,
or rather it would surprise us if we were not so accustomed to
it.’ If mathematics is based only on the rules of logic, such as all
normal minds accept, and which only a lunatic would deny
(according to Poincare), how is it that so many are mathemati-
cally impermeable? To which it may be answered that no
exhaustive set of experiments substantiating mathematical
incompetence as the normal human mode has yet been pub-
lished. ‘And further’, he asks, ‘how is error possible in mathe-
matics?’ Ask Alexander Pope; ‘To err is human’, whidi is as
unsatisfactory a solution as any other. The chemistry of the
digestive system may have something to do with it, but Poin-
care prefers a more subtle explanation – one which could not be
tested by feeding the ‘vile body’ hasheesh and alcohol.

605

ilEN OF MATHEMATICS

‘Tlie answer seems to me evident’, lie declares. Logic lias very
little to do with discovery or invention, and memory plays
tricks. Memory however is not so important as it might be. His
own memory, he says without a blush, is had: ‘Why then does
it not desert me in a difi&cult piece of mathematical reasoning
where most chess players [whose ‘memories’ he assumes to be
excellent] would be lost? Evidently because it is guided by the
general course of the reasoning. A mathematical proof is not a
mere juxtaposition of syllogisms; it is syllogisms arranged in a
certain order ^ and the order is more important than the elements
themselves.’ If he has the ‘intuition’ of this order, memory is at
a discount, for each syllogism will take its place automatically
in the sequence.

Mathematical creation, howev^er, does not consist merely in
making new combinations of things already known; ‘anyone
could do that, but the combinations thus made would be infinite
in number and most of them entirely devoid of interest. To
create consists precisely in avoiding useless combinations and
in making those which are useful and which constitute only a
small minority. Invention is discernment, selection.’ But has
not all this been said thousands of times before? WTiat artist
does not know that selection – an intangible – is one of the
secrets of success? We are exactly where we were before the
investigation began.

To conclude this part of Poincare’s observations it may he
pointed out that much of what he says is based on an assump-
tion which may indeed be true but for which there is not a
particle of scientific evidence. To put it bluntly he assumes that
the majority of human beings are mathematical imbeciles.
Granting him this, we need not even then accept his purely
romantic theories. They belong to inspirational literature and
not to science. Passing to something less controversial, we shall
now quote the famous passage in which Poincai’e describes how
one of his own greatest ‘inspirations’ came to him. It is meant
to substantiate his theory of mathematical creation. Whether
it does or not may be left to the reader.

He first points out that the technical terms need not be
understood in order to follow his narrative; ‘What is of interest

606

THE LAST UNIVEKSALIST
to the psychologist is not the theorem but the circumstances/

For fifteen days I struggled to prove that no functions
analogous to those I have since called Fiichsian functions
could exist; I Tvas then very ignorant. Every day I sat
down at my work table where I spent an hour or two; I
tried a great number of combinations and arrived at no
result. One evening, contrary to my custom, I took black
coffee; I could not go to sleep; ideas swarmed up in clouds;
I sensed them clashing until, to put it so, a pair would hook
together to form a stable combination. By morning I had
established the existence of a class of Fuehsian functions,
those derived from the hypergeometric series. I had only to
write up the results, which took me a few hours.

Next I wished to represent these functions by the quo-
tient of two series; this idea was perfectly conscious and
thought out; analogy with elliptic functions guided me. I
asked myself what must be the properties of these series if
they existed, and without diiTiculty I constructed the series
which I called thetafuchsian.

I then left Caen, where I was living at the time, to parti-
cipate in a geological trip sponsored by the School of
Mines, The exigencies of travel made me forget my mathe-
matical labours; reaching Coutances we took a bus for
some excursion or another. The instant I put my foot on
the step the idea came to me, apparently with nothing
whatever in my previous thoughts having prepared me for
it, that the transformations which I had used to define
Fuehsian functions were identical with those of non-
Euclidean geometry, I did not make the verification; I
should not have had the time, because once in the bus I
resumed an interrupted conversation; but I felt an instant
and complete certainty. On returning to Caen, I verified
the result at my leisure to satisfy my conscience.

I then undertook the study of certain arithmetical ques-
tions without much apparent success and without suspect-
ing that such matters could have the slightest connexion
with my previous studies. Disgusted at my lack of success,
I went to spend a few days at the seaside and thought of
something else. One day, while walking along the cliffs,
the idea came to me, again with the same characteristics
of brevity, suddenness, and immediate certainty, that the

6U7

MEN OF MATHEMATICS

transformations of indefinite ternary quadratic forms were
identical with those of non-Euclidean geometry.

On returning to Caen, I reflected on this result and
deduced its consequences; the example of quadratic forms
showed me that there were Fuchsian groups other than
those corresponding to the hypergeometric series; I saw
that I could apply to them the theory of thetafuchsian
functions, and hence that there existed thetafuchsian
functions other than those derived from the hjq^ergeo-
metric series, the only ones I had known up tiU then.
Naturally I set myself the task of constructing all these
functions, I conducted a systematic siege and, one after
another, carried all the outworks; there was however one
which still held out and whose fall would bring about that
of the whole position. But all my efforts served only to
make me better acquainted with the difficulty, which in
itself was something. AU this work was perfectly conscious.

At this point I left for Mont-Valerien, where I was to
discharge my military service. I had therefore very
different preoccupations. One day, while crossing the
boulevard, the solution of the difficulty which had stopped
me appeared to me all of a sudden. I did not seek to go into
it immediately, and it was only after my service that I
resumed the question. I had all the elements, and had only
to assemble and order them. So I wrote out my definitive
memoir at one stroke and with no difficulty.

Many other examples of this sort of thing could be given horn
his own work, he says, and from that of other mathematicians
as reported in UEnsei^riement MathimaMque. From his experi-
ences he believes that this semblance of ‘sudden illuminatian
[is] a manifest sign of previous long subconscious work’, and be
proceeds to elaborate his theory of the subconscious mind and
its part in mathematical creation. Conscious work is necessary
as a sort of trigger to fire off the accumulated dynamite whidb
the subconscious has been excreting – he does not put it so, bat
what he says amounts to the same. But what is gained in the
way of rational explanation if, following Poincar6, we foist off
on the ‘subconscious mind’, or the ‘subliminal self’, the very
activities which it is our object to understand? Instead of ear
dowing this mysterious agent with a hypothetical tact enahlii^
608

THE LAST UNIVERSALIST

it to discriminate between the ‘exceedingly numerous’ possible
combinations presented (how, Poincare does not say) for its
inspection, and calmly saying that the ‘subconscious’ rejects a|l
but the ‘useful’ combinations because it has a feeling for sym-
metry and beauty, sounds suspiciously like solving the initial
problem by gi^dng it a more impressive name. Perhaps this is
exactly what Poincare intended, for he once defined mathe-
matics as the art of giving the same name to difierent things; so
here he may be rounding out the symmetry of his view by giv-
ing different names to the same thing. It seems strange that a
man who could have been satisfied with such a ‘psychology’ of
mathematical invention was the complete sceptic in religious
matters that Poincare was. After Poincare’s brilliant lapse into
psychology sceptics may well despair of ever disbelieving
anything.

During the first decade of the twentieth century Poincare’s
fame increased rapidly and he came to he looked upon, espe-
cially in France, as an oracle on ail things mathematical. His
pronoxmcements on all manner of questions, from politics to
ethics, were usually direct and brief, and were accepted as final
by the majority. As almost invariably happens after a greaft
man’s extinction, Poincare’s dazzling reputation during Ms
lifetime passed through a period of partial eclipse in the decade
following his death. But his intuition for what was likely to be
of interest to a later generation is always justifying it^lf. To
take but one instance of many, Poincare was a vigorous oppo-
nent of the theory that all mathematics can be rewritten in
terms of the most elementary notions of classical logic; some-
thiog more than logic, he believed, makes mathematics what it
is. Although he did not go quite so far as the current intuitionist
school, he seems to have believed, as that school does, that at
least some mathCTaatical notions precede logic, and if one is to
be derived from the other it is logic which must come out of
mathematics, not the other way about. Whether this is to be
the ultimate creed remains to be seen, but at present it appears
as if the theory which Poincare assailed with all the irony at Ms
command is not the final one, whatever may be its merits*
Except for a distressing illness during his last four years

509

MEN OF MATHEMATICS

Poincare’s busy life was tranquil and happy. Honours were
showered upon him by all the leading societies of the world, and
in 1906, at the age of fifty-two, he achieved the highest distinc-
tion possible to a French scientist, the Presidency of the
x\cademy of Sciences. None of all this inflated liis ego, for
Poincare was truly humble and unaffectedly simple. Ke knew
of course that he was without a dose rival in the days of his
maturity, but he could also say without a trace of affectation
that he knew nothing compared to what is to be known. He
was happily married and had a son and three daughters in
whom he took much pleasure, especially when they were chil-
dren. His wife was a great-granddaughter of fitienne-GeoScoy
Saint-Hilaire, remembered as the antagonist of that pugnacious
comparative anatomist Cuvier. One of Poincare’s passions was
symphonic music.

At the International Mathematical Congress of 1908, held at
Rome, Poincare was prevented by illness from reading his
stimulating (if premature) address on The Future of Maike-
matical Physics, His trouble was hypertrophy of the prostate,
which was relieved by the Italian surgeons, and it was thought
that he was permanently cured. On his return to Paris he
resumed Ms work as energetically as ever. But in 1911 he began
to have presentiments that he might not live long, and on
9 December wrote asking the editor of a mathematical journal
whether he would accept an unfinished memoir – contrary to
the usual custom – on a problem wMch Poincjare considered of
the highest importance: . at my age, I may not be able to

solve it, and the results obtained, susceptible of putting re-
searchers on a new and unexpected path, seem to me too full of
promise, in spite of the deceptions they have caused me, that I
should resign myself to sacrificing them. . . . ’ He had spent the
better part of two fruitless years trying to overcome his
difficulties.

A proof of the theorem which he conjectured would have
enabled him to make a striking advance in the problem of three
bodies; in particular it would have permitted him to prove the
existence of an infinity of periodic solutions in cases more
general than those Mtherto considered. The desired proof was

610

THE LAST UKIVEESALIST

given shortly after the publication of Poincare’s ‘unfinished
symphony’ liy a young American mathematician, George
David Biikhoi (1884^1944).

In the spring of 1912 Poincare fell ill again and underwent a
second operation on 9 July. The operation was successful, but
on 17 July he died veiy suddenly from an embolism while
dressing. He was in the fifty-ninth year of his age and at the
height of his powers – ‘the living brain of the rational sciences*,
in the words of Painlev^.

CHAPTER TWENTY-NINE

PARADISE LOST?
Cantor

The controversial topic of Mengenlehre (theory of sets, or
classes, particularly of infinite sets) created in 1874-95 by
Georg Cantor (1845-1918) may well be taken, out of its chrono-
logical order, as the conclusion of the whole stor}^ This topic
typifies for mathematics the general collapse of those principles
which the prescient seers of the nineteenth century, foreseeing
everything but the grand debacle, believed to be fundamentaliy
sound in aU things from physical science to democratic govern-
ment.

If ‘collapse’ is perhaps too strong to describe the transition
the world is doing its best to enjoy, it is nevertheless true that
the evolution of scientific ideas is now proceeding so vertigi-
nously that evolution is barely distinguishable from revolution.

Without the errors of the past as a deep-seated focus of dis-
turbance the present upheaval in physical science would per-
haps not have happened; but to credit our predecessors with ail
the inspiration for what our own generation is doing, is to give
them more than their due. This point is worth a moment’s
consideration, as some may be tempted to say that the corre-
sponding ‘revolution’ in mathematical thinking, whose begin-
nings are now plainly apparent, is merely an echo of Zeno and
other doubters of ancient Greece*

The difficulties of Pythagoras over the square root of 2 and
the paradoxes of Zeno on continuity (or ‘infinite divisibility’)
are – so far as we know – tdie origins of our present mathe-
matical schism. Mathematicians to-day who pay any attention
to the philosophy (or foxindations) of their subject are split into
at least two factions, apparently beyond present hope of recon-
ciliation, over the validity of the reasoning used in mathemati-

PARADISE LOST?

cal analysis, and this disagreement can be traced back through
the centuries to the Middle Ages and thence to ancient Greece-
All sides have had their representatives in all ages of mathe-
matical thought, whether that thought was disguised in provo-
cative paradoxes, as with Zeno, or in logical subtleties, as with
some of the most exasperating logicians of the Middle Ages.
The root of these differences is commonly accepted by mathe-
maticians as being a matter of temperament; any attempt to
convert an analyst like Weierstrass to the scepticism of a
doubter like Kronecker is bound to be as futile as trying to
convert a Christian fundamentalist to rabid atheism.

A few dated quotations from leaders in the dispute may serve
as a stimulant – or sedative, according to taste ~ for our enthu-
siasm over the singular intellectual career of Georg Cantor,
whose ‘positive theory of the infinite* precipitated, in our own
generation, the fiercest frog-mouse battle (as Einstein once
called it) in history over the validity of traditional mathema-
tical reasoning.

In 1831 Gauss expressed his ‘honor of the actual infinite’ as
follows. ‘I protest against the use of infinite magnitude as some-
thing completed, which is never permissible in mathematics.
Infinity is merely a way of speaking, the true meaning being a
limit which certain ratios approach indefinitely close, while
others are permitted to increase without restriction.’

Thus, if X denotes a real number, the fraction Ijx diminishes
as X increases, and we can find a value of x such that l[x differs
from zero by any preassigned amount (other than zero) which
may be as small as we please, and as x continues to increase, the
difference remains less than this preassigned amount; the limit
of l/a, ‘as X tends to infinity,’ is zero. The symbol of infinity is
00 ; the assertion l/oo = 0 is nonsensical for two reasons: ‘divi-
sion by infinity’ is an operation which is undefined, and hence
has no meaning; the second reason was stated by Gauss.
Similarly 1/0 = oo is meaningless.

Cantor agrees and disagrees with Gauss. Writing in 1886 on
the problem of the actual (what Gauss called completed) mfi-
uite, Cantor says that ‘in spite of the essential difference
between the concepts of the potential and the actual “infinite’*,

618

MEN or MATHEMATICS

the former meaning a variable finite magnitude increasiiig
beyond all finite limits (like ar in l/a: above), while the latter is a
fixed^ constant magnitude lying beyond all finite magnitudes, it
happens only too often that they are confused.’

Cantor goes on to state that misuse of the infinite in mathe-
matics had justly inspired a horror of the infinite among careful
mathematicians of his day, precisely as it did in Gauss. Never-
theless he maintains that the resulting ‘uncritical rejection of
the legitimate actual infinite is no less a violation of the nature
of things [whatever that may be – it does not appear to have
been revealed to mankind as a whole], wliich must be taken as
they are’ – however that may be. Cantor thus definitely aligns
himself with the great theologians of the Middle Ages, of whom
he was a deep student and an ardent admirer.

Absolute certainties and complete solutions of age-old pro-
blems always go down better if well salted before swallowing.
Here is what Bertrand Russell had to say in 1901 about
Cantor’s Promethean attack on the infinite.

‘Zeno was concerned with three problems. . . . These are the
problem of the infinitesimal, the infinite, and continuity. . . .
From his day to our own, the finest intellects of each generation
in turn attacked these problems, but achieved, broadly speak-
ing, nothing. . . . Weierstrass, Dedeldnd, and Cantor . . . have
completely solved them. Their solutions . . . are so clear as to
leave no longer the slightest doubt of difficulty. This achieve-
ment is probably the greatest of which the age can boast. . . .
The problem of the infinitesimal was solved by Weierstrass, the
solution of the other two was begun by Dedekind and definitely
accomplished by Cantor.’*

The enthusiasm of this passage warms us even to-day,
although we know that Russell in the second edition (1924) of
his and A. N. Whitehead’s Prindpia Mathematica admitted
that ail was not well with the Dedekind ‘cut’ (see Chapter 27),
which is the spinal cord of analysis. Nor is it well to-day. More
is done for or against a particular creed in science or mathe-

• Quoted from R. E. Moritz’ Memorabilia McUhcmaiica, 1914. The
  original source is not accessible to me.

614

PARADISE LOST?

matics in a decade than was accomplished in a century of anti-
quity”, the hliddle Ages, or the late renaissance. More good
minds attack an outstanding scientific or mathematical problem
to-day than ever before, and finality has become the private
property of fimdamentalists. Not one of the finalities in
Russell’s remarks of 1901 has survived. A quarter of a century
ago those who were unable to see the great light which the
prophets assured them was blazing overhead like the noonday
sun in a midnight sky were called merely stupid. To-day for
every- competent expert on the side of the prophets there is an
equally competent and opposite expert against them. If there
is stupidity anywhere it is so evenly distributed that it has
ceased to be a mark of distinction. We are entering a new era,
one of doubt and decent humility.

On the doubtful side about the same time (1905) we find
Poincare. T have spoken … of our need to return continually
to the first principles of our science, and of the advantages of
this for the study of the human mind. This need has inspired
two enterprises which have assumed a very prominent place in
the most recent development of mathematics. The first is
Cantorism. . . • Cantor introduced into science a new way of
considering the mathematical infinite . • . but it has come about
that we have encountered certain paradoxes, certain apparent
contradictions that would have delighted Zeno the Eleatic and
the school of Megara. So each must seek the remedy. I for my
part – and I am not alone – think that the important thing is
never to introduce entities not completely definable in a finite
number of words. Whatever be the cure adopted, we may pro-
mise ourselves the joy of the physician called in to treat a
beautiful pathologic case.’

A few years later Poincare’s interest in pathology for its own
sake had abated somewhat. At the International Mathematical
Congress of 1908 at Rome, the satiated physician delivered
himself of this prognosis: ^Later generations will regard Mengen-
lehre as a disease from which one has recovered.’

It was Cantor’s greatest merit to have discovered in spite of
himself and against his own wishes in the matter that the ‘body
mathematic’ is profoundly diseased and that the sickness with

C15

3LM.— TOl.n.

MEX OF MATHEMATICS

which Zeno infected it has not yet been alleviated. His disturb-
ing discovery is a curious echo of his own intellectual life. We
shall first glance at the facts of his material existence, not of
much interest in themselves, perhaps, but singularly illumina-
tive in their later aspects of his theory.

Of pure Jewish descent on both sides, Georg Ferdinand
Ludwig Philipp Cantor was the first child of the prosperous
merchant Georg Waldemar Cantor and his artistic wife Maria
Bohm. The father was bom in Copenhagen, Denmark, but
migrated as a young man to St Petersburg, Russia, where the
mathematician Georg Cantor was bom on 3 March 1845. Pul-
monary disease caused the father to move in 1856 to Frankfurt,
Germany, where he lived in comfortable retirement till his
death in 1863. From this curious medley of nationalities it is
possible for several fatherlands to claim Cantor as their son.
Cantor himself favoured Germany, but it cannot be said that
Germany favoured him very cordially.

Georg had a brother Constantin, who became a German army
oiBficer (what a career for a Jew!), and a sister, Sophie Nobiling,
The brother was a fine pianist; the sister an accomplished
designer. Georg’s pent-up artistic nature found its turbulent
outlet in mathematics and philosophy, both classical and
scholastic. The marked artistic temperaments of the children
were inherited from their mother, whose grandfather was a
musical conductor, one of whose brothers, living in Vienna,
taught the celebrated violinist Joachim. A brother of Maria
Cantor was a musician, and one of her nieces a painter. If it is
true, as claimed by the psychological proponents of drab medio-
crity, that normality and phlegmatic stability are equivalent,
all this artistic brilliance in his family may have been the root
of Cantor’s instability.

The family were Christians, the father having been converted
to Protestantism; the mother was bom a Roman Catholic. Like
his arch-enemy Kxonecker, Cantor favoured the Protestant side
and acquired a singular taste for the endless hairsplitting of
medieval theology. Had he not become a mathematician it is
quite possible that he would have left Ms mark on pMLosophy
or theology. As an item of interest that may be noted in. this

616

PARADISE LOST?

connexion, Cantor’s theory of the infinite was eagerly pounced
on by the Jesuits, whose keen logical minds detected in the
mathematical imagery beyond their theological comprehension
indubitable proofs of the existence of God and the self-consis-
tency of the Holy Trinity with its three-in-one, one-in-three,
co-equal and co-etemal. Mathematics has strutted to some
pretty queer tunes in the past 2,500 years, hut this takes the
cake. It is only fair to say that Cantor, who had a sharp wit and
a sharper tongue when he was angered, ridiculed the pretentious
absurdity of such ‘proofs’, devout Christian and expert
theologian though he himself was.

Cantor’s school career was like that of most highly gifted
mathematicians – an early recognition (before the age of fifteen)
of his greatest talent and an absorbing interest in mathematical
studies. His first instruction was under a private tutor, followed
by a course in an elementary school in St Petersburg. ‘\Yhen the
family moved to Germany, Cantor first attended private schools
at Frankfurt and the Darmstadt non-classical school, entering
the Wiesbaden Gymnasium in 1860 at the age of fifteen.

Georg was determined to become a mathematician, hut his
practical father, recognizing the boy’s mathematical ability,
obstinately tried to force him into engineering as a more pro-
mising bread-and-butter profession. On the occasion of Cantor’s
confirmation in 1860 his father wrote to him expressing the high
hopes he and all Georg’s numerous aimts, uncles, and cousins in
Germany, Denmark, and Russia had placed on the gifted boy:
‘They expect from you nothing less than that you become a
Theodor Schaeffer and later, perhaps, if Gk)d so wills, a shinin g
star in the engineermg firmament.’ When will parents recognize
the presumptuous stupidity of trying to make a cart horse out
of a bom racer?

The pious appeal to God which was intended to blackjack the
sensitive, religious boy of fifteen into submission in 1860 would
to-day (thank God!) rebound like a tennis ball from the harder
heads of our own younger generation. But it hit Cantor pretty
hard. In fact it knocked him out cold. Losing his father
devotedly and being of a deeply religious nature, young Cantor
could not see that the old man was merely rationalizing his own

617

L 2

MEN OF MATHEMATICS

greed for money. Thus began the first waxping of Georg Cantor’s
acutely sensitive mind. Instead of rebelling, as a gifted hoy
to-day might do with some hope of success, Georg submitted
till it became apparent even to the obstinate father that he was
wrecking his son’s disposition. But in the process of trying to
please his father against the promptings of his own instincts
Georg Cantor sowed the seeds of the self-distrust which was to
make him an easy victim for ICronecker’s vicious attack in later
life and cause him to doubt the value of his work. Had Cantor
been brought up as an independent human being he would
never have acquired the timid deference to men of established
reputation which made his life wretched.

The father gave in when the mischief was already done. On
Georg’s completion of his school course with distinction at the
age of seventeen, he was permitted by ‘dear papa’ to seek a
university career in mathematics. ‘My dear papa!’ Georg writes
in his boyish gratitude : ‘You can realize for yourself how greatly
your letter delighted me. The letter fixes my future. . . , Now I
am happy when I see that it will not displease you if I follow my
feelings in the choice. I hope you will live to find joy in me, dear
father; since my soul, my whole being, lives in my vocatbn;
what a man desires to do, and that to which an inner compulsion
drives him, that will he accomplish!’ Papa no doubt deserves a
vote of thanks, even if Georg’s gratitude is a shade too servile
for a modem taste.

Cantor began his university studies at Zurich in 1862, but
migrated to the University of Berlin the following year, on the
death of his father. At Berlin he specialized in mathematics,
philosophy, and physics. The first two divided his interests
about equally; for physics he never had any sure feeling. In
mathematics his instructors were Kummer, Weierstrass, and
his future enemy Kronecker. Following the usual German
custom, Cantor spent a short time at another university, and
was in residence for one semester of 1866 at Gottingen.

With Kummer and ICronecker at BerMn the mathematical
atmosphere was highly charged with arithmetic. Cantor made a
profound study of the Disquisitiones Arithmeiicae of Gauss and
wrote his dissertation, accepted for the Ph,D. degree in 1887,

618

PARADISE LOST?

on a difficult point which Gauss had left aside concerning the
solution in integers y, z of the indeteiminate equation

ax- -r by^ -l cs® = 0,

where a, c are any given integers. This was a hue piece of
work, but it is safe to say that no mathematician who read it
anticipated that the conser\^ative author of twenty-two was to
become one of the most radical originators in the history of
mathematics. Talent no doubt is plain enough in this first
attempts but genius – no. There is not a single hint of the great
originator in this severely classical dissertation.

The like may be said for all of Cantor’s earliest work pub-
lished before he was twenty-nine. It was excellent, but might
have been done by any brilliant man who had thoroughly
absorbed, as Cantor had, the doctrine of rigorous proof from
Gauss and Weierstrass. Cantor’s first love was the Gaussian
theory of numbers, to which he was attracted by the hard,
sharp, clear perfection of the proofs. From this, under the influ-
ence of the Weierstrassians, he presently branched off into
rigorous analysis, particularly in the theory of trigonometric
series (Fourier series).

The subtle difficulties of this theory (where questions of con-
vergence of infinite series are less easily approachable than in
the theory of power series) seem to have inspired Cantor to go
deeper for the foundations of analysis than any of his contem-
poraries had cared to look, and he was led to his grand attack
on the mathematics and philosophy of the infinite itself, which
is at the bottom of all questions concerning continuity, limits,
and convergence. Just before he was thirty, Cantor published
his fijpst revolutionary paper (in Crelle’s Journal) on the theory
of infinite sets. This will be described presently. The unex-
pected and paradoxical result concerning the set of all algebraic
numbers which Cantor established in this paper and the com-
plete novelty of the methods employed immediately marked the
young author as a creative mathematician of extraordinary
originalit3\ Whether all agreed that the new methods were
sound or not is beside the point; it was universally admitted
that a man bad arrived with something fundamentally new in

619

MEN OF MATHEMATICS

mathematics. He should have been given an influential position
at once.

Cantor’s material career was that of any of the less eminent
German professors of mathematics. He never achieved hig
ambition of a professorship at Berlin, possibly the highest Ger-
man distinction during the period of Cantor’s greatest and most
original productivity (1874-84, age twenty-nine to thirty-nine).
All his active professional career was spent at the University of
Halle, a distinctly third-rate institution, where he was
appointed Privatdozeni (a lecturer who lives by what fees he can
collect from his students) in 1869 at the age of twenty-four. In
1872 he was made assistant professor and in 1879 – before the
criticism of his work had begun to assume the complexion of a
malicious personal attack on himself – he was appointed full
professor. His earliest teaching experience was in a girl’s school
in Berlin. For tlois curiously inappropriate task he had qualified
himsfilf by listening to dreary lectures on pedagogy by an unin-
spired mathematical mediocrity before securing his state licence
to teach children. More social waste.

Rightly or wrongly, Cantor blamed Kronecker for his failure
to obtain the coveted position at Berlin, The aggressive clan-
nishness of Jews has often been remarked, sometimes as an
argument against employing them in academic work, but it has
not been so generally observed that there is no more vicious
academic hatred than that of one Jew for another when they
disagree on purely scientific matters or when one is jealous or
afraid of another. Gentiles either laugh these hatreds off or go
at them in an efficient, underhand way which often enables
them to accomplish their spiteful ends under the guise of sincere
friendship. When two intellectual Jews fall out they disagree all
over, throw reserve to the dogs, and do everything in their
power to cut one another’s throats or stab one another in the
back. Perhaps after all this is a more decent way of fighting – if
men must fight – than the sanctimonious hypocrisy of the
other. The object of any war is to destroy the enemy, and being
sentimental or chivalrous about the unpleasant business is the
mark of an incompetent fighter. Kronecker was one of the most
competent warriors in the history of scientific controversy;

620

PAEABISE LOST?

Cantor, one of the least competent. Kronecker won. But, as will
appear later, Kxonecker’s bitter animosit 3 ” towards Cantor was
not wholly personal but at least partly scientific and dis-
interested.

The year 1874 which saw the appearance of Cantor’s first
revolutionary paper on the theory of sets was also that of his
marriage, at the age of twenty-nine, to Vally Guttmann, Two
sons and four daughters were bom of this marriage. None of the
children inherited their father’s mathematical ability.

On their honeymoon at Interlaken the young couple saw a lot
of Dedekind, perhaps the one first-rate mathematician of the
time who made a serious and sympathetic attempt to under-
stand Cantor’s subversive doctrine.

HimseK somewhat of a persona non grata to the leading Ger-
man overlords of mathematics in the last quarter of the nine-
teenth century, the profoundly original Dedekind was in a posi-
tion to sympathize with the scientifically disreputable Cantor.
It is sometimes imagined by outsiders that originality is always
assured of a cordial welcome in science. The history of mathe-
matics contradicts this happy fantasy: the way of the trans-
gressor in a well-established science is likely to be as hard as it
is in any other field of human conservatism, even when the
transgressor is admitted to have found something valuable by
overstepping the narrow bounds of bigoted orthodoxy.

Both Dedekind and Cantor got what they might have
expected had they paused to consider before striking out in new
directions. Dedekind spent his entire working life in mediocre
positions; the claim – now that Dedekind’s work is recognized
as one of the most important contributions to mathematics that
Germany has ever made – that Dedekind preferred to stay in
obscure holes while men who were in no sense his intellectual
superiors shone like tin plates in the glory of public and aca-
demic esteem, strikes observers who are themselves ‘Aryans’
but not Germans as highly diluted eyewash.

The ideal of German scholarship in the nineteenth century
was the lofty one of a thoroughly co-ordinated ‘safety first’, and
perhaps rightly it showed an extreme Gaussian caution towards
radical originality – the new thing might conceivably be not

821 ‘

MEN or MATHEMATICS

quite Tight. After all an honestly edited encyclopaedia is in
general a more reliable source of information about the soar-
ing habits of skylarks than a poem, say Shelley’s, on the same
topic.

In such an atmosphere of cloying alleged fact, Cantor’s theory
of the infini te – one of the most disturbingly original contribu-
tions to mathematics in the past 2,500 years – felt about as
much freedom as a skylark trying to soar up through an atmo-
sphere of cold glue. Even if the theory was totally wrong – and
there are some who believe it cannot be salvaged in any shape
resembling the thing Cantor thought he had launched – it
deserved something better than the brickbats which were
hurled at it chiefly because it was new and unbaptized in the
holy name of orthodox mathematics.

The pathbreaking paper of 1874 undertook to establish a
totally unexpected and highly paradoxical property of the set
of dU algebraic numbers. Although such numbers have been
frequently described in preceding chapters, we shall state once
more what they are, in order to bring out clearly the nature of
the astounding fact which Cantor proved – in saying ‘proved’
we deliberately ignore for the present all doubts as to the
soundness of the reasoning used by Cantor.

If r satisfies an algebraic equation of degree n with rational
integer {common whole number) coefficients, and if r satisfies
no such equation of degree less than n, then r is an algebraic
number of degree w-

This can be generalized. For it is easy to prove that any root
of an equation of the type

-f + . • . + = 0,

in which the c’s are any given algebraic numbers (as defined
above), is itself an algebraic number. For example, according to
this theorem, all roots of

(1 – 8 V~l)a3 – (2 + sVlT) X + ■^^90’ = 0

are algebraic numbers, since the coefficients are. (The first co-
622

PARADISE LOST?

eiScient satisfies a- — 2ai 4 – 10 = 0, the second, x- — 4z
— 421 = 0, the third, — 90 = 0, of the respective degrees
2,2,3;.

Imagine (if j’ou can) the set of all algebraic numbers. Among
these will be all the positive rational integers 1, 2, 3, … , since
any one of them, say n, satisfies an algebraic equation, as — n =

0. in which the coefficients (1, and — n) are rational integers.
   But in addition to these the set of all algebraic numbers will
   include all roots of all quadratic equations with rational integer
   coefficients, and all roots of all cubic equations with rational
   integer coefficients, and so on, indefinitely. Is it not iniuiivoely
   evident that the set of all algebraic numbers will contain infi-
   niiely more members than its subset of the rational integers
1. 2, 3, … ? It might indeed be so, but it happens to be false.

Cantor proved that the set of all rational integers 1, 2, 8, …

contains preciseh^ as many members as the ^infinitely more
inclusive’ set of all algebraic numbers.

A proof of this paradoxical statement cannot be given here,
but the kind of device – that of ‘one-to-one correspondence*’ –
upon which the proof is based can easOy be made intelligible.
This should induce in the philosophical mind an understanding
of what a cardinal number is. Before describing this simple but
somewhat elusive concept it will be helpful to glance at an
expression of opinion on this and other definitions of Cantor’s
theory which emphasizes a distinction between the attitudes of
some mathematicians and many philosophers toward all
questions regarding ‘niimber’ or ‘magnitude’.

‘A mathematician never defines magnitudes in themselves,
as a philosopher would be tempted to do; he defines their
equality, their sum, and their product, and these definitions
determine, or rather constitute, all the mathematical properties
of magnitudes. In a yet more abstract and more formal manner
he lays dawn symbols and at the same time prescribes the rules
according to which they must be combined; these rules suffice
to characterize these symbols and to give them a mathematical
value. Briefly, he creates mathematical entities by means of
arbitrary conventions, in the same way that the several chess-
men are defined by the conventions which govern their moves

623

MEN OE MATHEMATICS

and the relations between them.’* Not ail schools of mathe-
matical thought would subscribe to these opinions, but they
suggest at least one ‘philosophy’ responsible for the following
definition of cardinal numbers.

Note that the initial stage in the definition is the description
of ‘same cardinal number’, in the spirit of Couturat’s opening
remarks; ‘cardinal number’ then arises phoenix-like from the
ashes of its ‘sameness’. It is ail a matter of relations between
concepts not explicitly defined.

Two sets are said to have the same cardinal number when
all the things in the sets can be paired off one-to-one. After
the pairing there are to be no unpaired things in either
set.

Some examples will clarify this esoteric definition. It is one
of those trivially obvious and fecund nothings which are so
profound that they are overlooked for thousands of years. The
sets (a?, ?/, 2 ;), (fl, 6, c) have the same cardinal number (we shall not
commit the blunder of saying ‘Of course! Each contains three
letters’) because we can pair off the things x,y,zm. the first set
with those, a, 6, c in the second as follows, x with a, y with h,
z with c, and having done so, find that none remain unpaired in
either set. Obviously there are other ways for effecting the
pairing. Again, in a Christian community practising technical
monogamy, if twenty married couples sit down together to
dioner, the set of husbands will have the same cardinal number
as the set of wives.

As another instance of this ‘obvious’ sameness, we recall

• L. Couturat, De Vinflni madiemaiigue, Paris, 1896, p. 49. ‘With
  the caution that much of this work is now hoi)eIessly out of date, it
  can be recommended for its clarity to the general reader. An account
  of the elements of Cantorism by a leading Polish expert which is
  within the comprehension of anyone with a grade-school education
  and a taste for abstract reasoning is the Legons sur les nomhres trans^
  finis f by Wadaw Sierpinski, Paris, 1928. The preface by Borel
  suppKes the necessary danger signal. The above extract from
  Couturat is of some historical interest in connexion with Hilbert’s
  programme. It anticipates by thirty years Hilbert’s statement of his
  formalist creed.

624

PAEADISE LOST?

Galileo’s example of the set of all squares of positive integers
and the set of all positive integers:

12 92 02 4.2 «2

1 ,233)4<9… jTl}…

The ‘paradoxical’ distinction between this and the preceding
examples is apparent. If all the wives retire to the drawing
room, leaving their spouses to sip port and tell stories, there will
be precisely twenty human beings sitting at the table, just half
as many as there were before. But if all the squares desert the
natural numbers, there are just as many left as there were
before. Dislike it or not as we may (we should not, if we are
rational animals), the crude miracle stares us in the face that a
part of a set may hate the same cardinal number as the entire set
If anyone dislikes the ‘pairing’ definition of ‘same cardinal
number’, he may be challenged to produce a comelier. Intuition
(male, female, or mathematical) has been greatly overrated.
Intuition is the root of all superstition.

Notice at this stage that a difficulty of the first magnitude
has been glossed. What isaset^ora class? ‘That’, in the words of
Hamlet, ‘is the question’. We shall return to it, but we shall not
answer it. Whoever succeeds in answering that innocent ques-
tion to the entire satisfaction of Cantor’s critics will quite likely
dispose of the more serious objections against his ingenious
theory of the infinite and at the same time establish mathe-
matical analysis on a non-emotional basis. To see that the
difficulty is not trivial, try to imagine the set of all positive
rational integers, I, 2, 3, … , and ask yourself whether, with
Cantor, you can hold this totality – which is a ‘class’ – in your
mind as a definite object of thought, as easily apprehended as
the class x, y, z of three letters. Cantor requires us to do just
this thing in order to reach the iransflnite numbers which he
created.

Proceeding now to the definition of ‘cardinal number’, we
introduce a convenient technical term: two sets or classes
whose members can be paired oft one-to-one (as in the examples
given previously) are said to be similar. Haw many things are
there in the set (or class) x, y, zl Obviously three. But what is

325

MEN OF MATHEMATICS

Hhree’? An answer is contained in tlie following definition:
‘The number of things in a given class is the class of all classes
that are similar to the given class.’

This definition gains nothing from attempted explanation:
it must be grasped as it is. It was proposed in 1879 by Gottlob
Frege, and again (independently) by Bertrand Russell in 1901.
One advantage which it has over other definitions of ‘cardinal
number of a class’ is its applicability to both finite and infinite
classes. Those who believe the definition too mystical for mathe-
matics can avoid it by following Couturat’s advice and not
attempting to define ‘cardinal number’. However, that way also
leads to difficulties.

Cantor’s spectacular result that the class of all algebraic
numbers is similar (in the technical sense defined above) to its
sub-class of all the positive rational integers was but the first
of many wholly unexpected properties of infinite classes.
Granting for the moment that his reasoning in reaching these
properties is sound, or, if not unobjectionable in the form in
which Cantor left it, that it can be made rigorous, we must
admit its power.

Consider for example the ‘existence’ of transcendental
numbers. In an earlier chapter we saw what a tremendous eSort
it cost Hermite to prove the transcendence of a particular
number of this kind. Even to-day there is no general method
known whereby the transcendence of any number which we
suspect is transcendental can be proved; each new type
requires the invention of special and ingenious methods. It is
suspected, for example, that the number (it is a constant,
although it looks as if it might be a variable from its definition)
which is defined as the limit of

1+ – + – + log ri

2 3 n

as n tends to infinity, is transcendental, but we cannot prove
that it is. What is required is to show that this constant is not
a root of any algebraic equation with rational integer co-
efficients.

All this suggests the question ‘How many transcendental
626

PARADISE LOST?

numbers are there? Are they more numerous than the integers,
or the rationals. or the algebraic numbers as a whole, or are they
less numerous? Since (by Cantor’s theorem) the integers, the
rationals, and all algebraic numbers are equally numerous, the
question amounts to this: can the transcendental numbers be
counted off 1, 2, 3, …? Is the class of all transcendental
numbers similar to the class of all positive rational integers?
The answer is no; the transcendentals are infinitely more
rnmerous than the integers.

Here we begin to get into the controversial aspects of the
theory of sets. The conclusion just stated was like a challenge
to a man of Kxonecker’s temperament. Discussing Lindemann’s
proof that tt is transcendental (see Chapter 24), Kronecker
asked, ‘Of what use is your beautiful investigation regarding ;r?
^Yhy study such problems, since irrational [and hence trans-
cendental] numbers do not exist?’ We can imagine the effect
on such a scepticism of Cantor’s proof that the transcendentals
are infinitely more numerous than the integers 1, 2, 3, . • ,
which, according to Kronecker, are the noblest work of God
and the only numbers that do ‘exist’.

Even a summary of Cantor’s proof is out of the question here,
but something of the kind of reasoning he used can be seen
from the following simple considerations. If a class is similar
(in the above technical sense) to the class of ail positive
rational integers, the class is said to be denumerable. The thing *,
in a denumerable class can be counted off 1, 2, 3, – . . ; the
t hing s in a non-denum erable class can not be counted off
1, 2, 3, , . . ; there will be more things in a non-denumerabie
class than in a denumerable class. Do non-denumerabie classes
exist? Cantor proved that they do. In fact the class of all points
on any line-segment, no matter how small the segment is
(provided it is more than a single point), is non-denumerabie.

From this we see a hint of why the transcendentals are non-
denumerabie. In the chapter on Gauss we saw that any root of
any algebraic equation is representable by a point on the plane
of Cartesian geometry. All these roots constitute the set of ail
algebraic numbers, which Cantor proved to be denumerable.
But if the points on a mere line-segment are non-denmnerable,

627

MEN OP MATHEMATICS

it follows that all the points on the Cartesian plane are like^
non-denumerahle. The algebraic numbers are spotted over the
plane like stars against a black sky; the dense blackness is the
firmament of the transcendentals.

The most remarkable thing about Cantor’s proof is that it
provides no means whereby a single one of the transcendentals
can be constructed. To Kronecker any such proof was sheer
nonsense. Much milder instances of ‘existence proofs’ roused
his wrath. One of these in particular is of interest as it prophe-
sied Brouwer’s objection to the full use of classical (Aristo-
telian) logic in reasoning about infinite sets.

A polynomial -f- + Z, in which the coeiB-

cients a,b, I are rational numbers is said to be inedimbh if
it cannot he factored into a product of two polynomials both of
which have rational number coefficients. Now, it is a meaningful
statement to most human beings to assert, as Aristotle would,
that a given polynomial either is irreducible oris not irreducible.

Not so for Kronecker. Until some definite process, capable
of being carried out in a finite number of non-tentative steps, is
provided whereby we can settle the reducibility of any given
polynomial, we have no logical right, according to Kronecker,
to use the concept of irreducibility in our mathematical proofs.
To do otherwise, according to him, is to court inconsistencies in
our conclusions and, at best, the use of ‘irreducibility’ without
the process described can give us only a Scotch verdict of ‘not
proven’. All such non-comtructioe reasoning is – according to
Kronecker – illegitimate.

As Cantor’s reasoning in Ms theory of infinite classes is
largely non-constructive, Kronecker regarded it as a dangerous
type of mathematical insanity. Seeing mathematics headed for
the madhouse under Cantor’s leadersMp, and being passion-
ately devoted to what he considered the truth of mathematics,
Kronecker attacked ‘the positive theory of infinity’ and its
hypersensitive author vigorously and viciously with every
weapon that came to his hand, and the tragic outcome was that
not the theory of sets went to the asylum, but Cantor. Kron-
ecker’s attack broke the creator of the theory.

In the spring of 1884, in his fortieth year. Cantor experienced

628

PARADISE LOST?

the first of those complete breakdowns which were to recur with
varying intensity throughout the rest of his long life and drive
him from society to the shelter of a mental clinic. His explosive
temper aggravated his difficulty. Profound fits of depression
humbled himself in his own eyes and he came to doubt the
soundness of bis work. During one lucid interval he begged the
authorities at Halle to transfer him from his professordiip of
mathematics to a chair of philosophy. Some of his best work on
the positive theory of the infinite was done in the intervals
between one attack and the next. On recovering from a seizure
he noticed that his mind became extraordinarily dear.

Rronecker perhaps has been blamed too severely for Cantor’s
tragedy; his attack was but one of many contributing causes.
Lack of recognition embittered the man who believed he had
taken the first – and last – steps toward a rational theory of the
infinite and he brooded himself into melancholia and irration-
ality. Kronecker, however, does appear to have been largely
responsible for Cantor’s failure to obtain the position he craved
in Berlin. It is usually considered not quite sporting for one
scientist to deliver a savage attack on the work of a contem-
porary to his students. The disagreement can be handled objec-
tively in scientific papers. Kronecker laid himself out in 1891
to criticize Cantor’s work to his students at Berlin, and it
became obvious that there was no room for both under one
roof. As Kronecker was already in possession, Cantor resigned
himself to staying out in the cold.

However, he was not without some comfort. The sympathetic
I^Iittag-Leffler not only published some of Cantor’s work in his
journi (Acta Mathematica) but comforted Cantor in his fight
against Kronecker. In one year alone Mittag-Leffler received no
less than fifty-two letters from the suffering Cantor. Of those
who believed in Cantor’s theories, the genial Hennite was one
of the most enthusiastic. His cordial acceptance of the new
doctrine warmed Cantor’s modest heart; *The pmises which
Hennite pours out to me in this letter … on the subject of the
theory of sets are so high in my eyes, so unmerited, that I
should not care to publish them lest I incur the reproach of
being dazzled by them.’

629

MEN OF MATHEMATICS

With the opening of the new century Cantor’s work gradually
came to be accepted as a fundamental contribution to all
mathematics and particularly to the foundations of analysis*
But unfortunately for the theory itself the paradoxes and anti-
nomies which still infect it began to appear simultaneously.
These may in the end be the greatest contribution which
Cantor’s theory is destined to make to mathematics » for their
unsuspected existence in the very rudiments of logical and
mathematical reasoning about the infinite was the direct in-
spiration of the present critical movement in all deductive
reasoning. Out of this we hope to derive a mathematics which
is both richer and ‘truer’ ~ freer from inconsistency – than the
mathematics of the pre-Cantor era.

Cantor’s most striking results were obtained in the theory of
non-denumerable sets, the simplest example of which is the set
of all points on a line-segment. Only one of the simplest of his
conclusions can be stated here. Contrary to what intuition
would predict, two unequal line-segments contain the same
number of points. Remembering that two sets contain the same
number of things if, and only if, the things in them can be
paired off one-to-one, we easily see the reasonableness of

Cantor’s conclusion* Place the unequal segments AB, CD as in
the figure. The line OPQ cuts CD in the point P, and AB in
Q; P and Q are thus paired off. As OPQ rotates about O, the

aso

PARADISE LOST?

point P traverses CD, while Q simultaneously traverses AB,
and each point of CD has one, and only one, ‘paired’’ point of
AB,

An even more unexpected result can be proved. Any line-
segment, no matter how small, contains as many points as an
infini te straight line. Further, the segment contains as many
points as there are in an entire plane, or in the whole of three-
dimensional space, or in the whole of space of n dimensions
(where n is any integer greater than zero) or, finally, in a space
of a denumerably infinite number of dimensions.

In all this we have not yet attempted to define a class or a set»
Possibly (as Russell held in 1912) it is not necessary to do so in
order to have a clear conception of Cantor’s theory or for that
theory to be consistent with itself – which is enough to demand
of any mathematical theory. Nevertheless present disputes
seem to require that some clear, self-consistent definition be
given. The following used to be thought satisfactor\

A set is characterized by three qualities: it contains all things
to which a certain definite property (say redness, or volume, or
taste) belongs; no thing not having this property belongs to the
set; each thing in the set is recognizable as the same thing and
as different from all other things in the set – briefly, each thing
in the set has a permanently recognizable individuality. The
set itself is to be grasped as a whole. This definition may be too
drastic for use. Consider, for example, what happens to Cantor’s
set of all transcendental numbers under the third demand.

At this point we may glance back over the whole history of
mathematics – or as much of it as is revealed by the treatises of
the master mathematicians in their purely technical works –
and note two modes of expression which recur constantly in
nearly all mathematical exposition. The reader perhaps has
been irritated by the repetitious use of phrases such as ‘we can
find a whole number greater than 2% or Ve can choose a number
less than n and greater than n ~ 2.’ The choice of such phrase-
ology is not merely stereotyped pedantry. There is a reason for
its use, and careful writers mean exactly what they say when
they assert that ‘we can find, etc’. They mean that they can do
what they say.

631

MEN OF MATHEMATICS

In sharp distinction to this is the other phrase which is reiter-
ated over and over again in mathematical writing; ‘There
exists.’ For example, some would say ‘there exists a whole
number greater than 2’, or ‘there exists a number less than n
and greater than n — 2.’ The use of such phraseology definitely
commits its user to the creed which Kronecker held to be
untenable, unless^ of course, the ‘existence’ is proved by a con-
struction. The existence is not proved for the sets (as defined
above) which appear in Cantor’s theory.

These two ways of speaking divide mathematicians into two
types: the ‘we can’ men believe (possibly subconsciously) that
mathematics is a purely human invention; the ‘there exists’
men believe that mathematics has an extra-human ‘existence’
of its own, and that ‘we’ merely come upon the ‘eternal truths’
of mathematics in our journey through life, in much the same
way that a man taking a walk in a city comes across a number
of streets with whose planning he had nothing whatever to do.

Theologians are ‘exist’ men; cautious sceptics for the most
part *we’ men. ‘There exist an infinity of even numbers, or of
primes’, say the advocates of extra-human ‘existence’ ; ‘produce
them’, say Kronecker and the ‘we’ men.

That the distinction is not trivial can be seen from a famous
instance of it in the New Testament. Christ asserted that the
Father ‘exists’ ; Philip demanded ‘Show us the Father and it
suflficeth us.’ Cantor’s theory is almost wholly on the ‘existence’
side. Is it possible that Cantor’s passion for theology deter-
mined his allegiance? If so, we shall have to explain why
Kronecker, also a connoisseur of Christian theology, was the
rabid Sve’ man that he was. As in all such questions ammunition
for either side can be filched from any pocket.

A striking and important instance of the ‘existence’ way of
looking at the theory of sets is afforded by what is known as
Zennelo’s postulate (stated in 1904). ‘For every set M whose
elements are sets P (that is, M is a set of sets, or a class of
ctoses), the sets P being non-empty and non-overlapping (no
two contain things in common), there exists at least one set
which contains precisely one element from each of the sets P
which constitute M.’ Comparison of this with the previous^
632

PARADISE DOST?

stated definition of a set (or class) will show that the ‘we^ men
would not consider the postulate self-e\ddeiit if the set M con-
sisted, say, of an infinity of non-overlapping line segments. Yet
the postulate seems reasonable enough. Attempts to prove it
have failed. It is of considerable importance in all questions
relating to continuity,

A word as to how this postulate came to he introduced into
mathematics will suggest another of the unsolved problems of
Cantor’s theory. A set of distinct, countable things, like ail the
bricks in a certain wall, can easily be ordered; we need only
count them off 1, 2, 3, … in any of dozens of different ways
that win suggest themselves. But how would we go about
ordering all the points on a straight line? They cannot be
counted off 1, 2, 3, … The task appears hopeless when we
consider that between any two points of the line ‘we can find’,
or ‘there exists’ another point of the line. If every time we
counted two adjacent bricks another sprang into being between
them in the wall our counting would become slightly confused.
Nevertheless the points on a straight line do appear to have
some sort of order; we can say whether one point is to the right
or the left of another, and so on. Attempts to order the points
of a line have not succeeded. Zermelo proposed his postulate as
a means for making the attempt easier, but it itself is not
universally accepted as a reasonable assumption or as one
which it is safe to use.

Cantor’s theory contains a great deal more about the actual
infinite and the ‘arithmetic’ of transfinite (infinite) numbers
than what has been indicated here. But as the theory is still in
the controversial stage, we may leave it with the statement of
a last riddle. Does there ‘exist*, or can we ‘construct’, an infinite
set whidi is not similar (technical sense of one-to-one matching)
either to the set of all the positive rational integers or to the set
of all points of a line? The answer is unknown.

Cantor died in a mental hospital in Halle on 6 January 191S
at the age of seventy-three. Honours and recognition were his
at the last, and even the old bitterness against lironecker was
forgotten. It was no doubt a satisfaction to Cantor to recall that
he and Kjonecker had become at least superficially reconciled

MEN OE MATHEMATICS

some years before Kronecker’s death in 1891* Could Cantor
have lived till to-day he might have taken a Just pride in the
movement toward more rigorous th i nkin g in all mathematics
for which his own efforts to found analysis (and the infinite) on
a sound basis were largely responsible.

Looking back over the long struggle to make the concepts of
real number^ continuity^ limit, and infinity precise and consis-
tently usable in mathematics, we see that Zeno and Eudoxus
were not so far in time from Weierstrass, Dedekind, and Cantor
as the twenty-four or twenty-five centuries which separate
modem Germany from ancient Greece might seem to imply.
There is no doubt that we have a clearer conception of the
‘ nature of the difficulties involved than our predecessors had,
because we see the same unsolved problems cropping up in new
guises and in fields the ancients never dreamed of, but to say
that we have disposed of those hoary old difficulties is a gross
mis-statement of fact. Nevertheless the net score records a
greater gain than any which our predecessors could rightfully
claim. We are going deeper than they ever imagined necessary,
and we are discovering that some of the ‘laws’ – for instance
those of Aristotelian logic – which they accepted in theii
reasoning are better replaced by others – pure conventions –
in our attempts to correlate our experiences. As has already
been said, Cantor’s revolutionary work gave our present acti-
\Tty its initial impulse. But it was soon discovered – twenty-one
years before Cantor’s death – that his revolution was either too
revolutionary or not revolutionary enough. The latter now
appears to be the case.

The first shot in the counter-revolution was fired in 1897 by
the Italian mathematician Burali-Forti who produced a flagrant
contradiction by reasoning of the type used by Cantor in his
theory of infinite sets- This particular paradox was only the
first of several, and as it would require lengthy explanations to
make it intelligible, we shall state instead Russell’s of 1908.

We have already mentioned Frege, who gave the ‘class of all
classes similar to a given class’ definition of the cardinal number
of the given class. Frege had spent years trying to put the
mathematics of numbers on a sound logical basis. Ris life woifc

684

PARADISE LOST?

is Ms Grundgeselze der Aritkmetik (The Fundamental La-ws oi
Arithmetic), of wMch the first volume was published in 1893,
the second in 1903. In tMs work the concept of sets is used.
There is also a considerable use of more or less sarcastic invec-
tive against previous writers on the foundations of arithmetic
for their manifest blunders and manifold stupidities. The
second volume closes with the following acknowledgement.

A scientist can hardly encounter anything more unde-
sirable than to have the foundation collapse just as the
work is finished. I was put in this position by a letter from
Mr Bertrand Bussell when the work was almost through
the press.

Russell had sent Frege his ingenious paradox of ‘the set of all
sets which are not members of themselves.’ Is this set a member
of itseK? Either answer can be puzzled out with a little thought
to be wrong. Yet Frege had freely used ‘sets of all sets’.

Many ways were proposed for evading or eliminating the con-
tradictions wMch began exploding like a barrage in and over
the Frege-Dedekind-Cantor theory of the real numbers, con-
tinuity, and the infinite. Frege, Cantor, and Dedekind quit the
field, beaten and disheartened. RusseD proposed his ‘vicious
circle principle’ as a remedy: ‘Whatever involves all of a collec-
tion must not be one of the collection’; later he put forth his
‘axiom of reducibility’, wMch, as it is now practically aban-
doned, need not be described. For a time these restoratives
were brilliantly effective (except in the opinion of the Germssx
mathematicians, who never swallowed them). Gradually, as the
critical examination of all mathematical reasoning gained head-
way, physic was thrown to the dogs and a concerted effort was
begun to find out what really ailed the patient in his irrational
and real number system before administering further nostrums*
The present effort to understand our difficulties originated in
the work of David Hilbert (1862-1943) of Gottingen in 1899 and
in that of L. E, J. Brouwer (1881- ) of Amsterdam in 1912.

Both of these men and their numerous followers have the com-
mon purpose of putting mathematical reasoning on a sound
basis, although in several respects t he i r methods and phiio-

635

MEN OF MATHEMATICS

sopMes are ‘violently opposed. It seems unlikely that both can
be as ‘w^holiy right as each appears to believe he is.

Hilbert returned to Greece for the beginning of his philosophy
of mathematics. Resuming the Pythagorean programme of a
rigidly and fully stated set of postulates from which a mathe-
matical argument must proceed by strict deductive reasoning,
Hilbert made the programme of the postulational development
of mathematics more precise than it had been ‘vvith the Greeks,
and in 1899 issued the first edition of his classic on the founda-
tions of geometry. One demand which Hilbert made* and ‘Prhich
the Greeks do not seem to have thought of, ‘was that the
proposed postulates for geometry shall be proved to be self-
consistent (free of internal, concealed contradictions). To
produce such a proof for geometry it is sho’wn that any contra-
diction in the geometry developed from the postulates would
imply a contradiction in arithmetic. The problem is thus
shoved back to pro-ving the consistency of arithmetic, and there
it remains to-day.

Thus -we are back once more asking the sphinx to tell us what
a number is. Both Dedekind and Frege fled to the infinite –
Dedekind ‘with his infinite classes defining irrationals, Frege
-with his class of aU classes similar to a given class defining a
cardinal number – to interpret the numbers that puzzled
Pythagoras. Hilbert, too, would seek the answer in the infinite
which, he believes, is necessary for an understanding of the
finite. He is quite emphatic in his belief that Cantorism ‘will
ultimately be redeemed from the purgatory in which it now
tosses. ‘This [Cantor’s theory] seems to me the most admirable
fruit of the mathematical mind and indeed one of the hipest
achievements of man’s intellectual processes.’ But he admits
that the paradoxes of Burali-Forti, Russell, and others are not
resolved. However, his faith surmounts all doubts: ‘No one
shah expel us from the paradise which Cantor has created for
us.’

But at this moment of exaltation Brouwer appears with
something that looks suspiciously like a flaming sword in his
strong right hand. The chase is on: Dedekind, in the role of
Adam, and Cantor disguised as Eve at his side, are already

PARADISE LOST?

2yemg the gate apprehensively under the stem regard of the
uncompromising Dutchman. The postulational method for
securing freedom from contradiction proposed by Hilbert ‘will,
says, Brouwer, accomplish its end – produce no contradictions,
but ‘nothing of mathematical value will be attained in this
manner; a false theory which is not stopped by a contradiction
is none the less false, just as a criminal policy unchecked by a
reprimanding court is none the less criminaL’

The root of Brouwer’s objection to the ‘criminal pokey’ of his
opponents is something new – at least in mathematics. He
objects to an unrestricted use of Aristotelian logic, particularly
in dealing with infinite sets, and he maintains that such logic
is bound to produce contradictions when applied to sets which
cannot be definitely constructed in Kronecker’s sense (a rule of
procedure must be given whereby the things in the set can be
produced). The law of ‘excluded middle’ (a thing must have a
certain property or must not have that property, as for example
in the assertion that a number is prime or is not prime) is
legitimately usable only when applied to finite sets. Aristotle
devised his logic as a body of working rules for finite sets,
basing his method on human experience ot finite sets, and there
is no reason whatever for supposing that a logic which is ade-
quate for the finite will continue to produce consistent (not
contradictory) results when applied to the infinite. This seems
reasonable enough when we recall that the very definition of an
infinite set emphasizes that a part of an infinite set may contain
precisely as many things as the whole set (as we have illustrated
many times), a situation which never happens for a finite set
when ‘part’ means some, but not all (as it does in the definition
of an infinite set).

Here we have what some consider the root of the trouble in
Cantor’s theory of the actual infinite. For the definition of a set
(as stated some time hack), by which all things having a certain
quality are ‘united’ to form a ‘set’ (or ‘class’), is not suitable as
a basis for the theory of sets, in that the definition either is not
constructive (in Kronecker’s sense) or assumes a constructibility
which no mortal can produce. Brouwer claims that the use of
the law of excluded middle in such a situation is at best merely

637

MEN OE MATHEMATICS

a iieuristie guide to propositions which may be true, hut which
are not necessarily so, even when they have been deduced by a
rigid application of Aristotelian logic, and he says that
numerous false theories (including Cantor’s) have been erected
on this rotten foundation during the past half century.

Such a revolution in the rudiments of mathematical thinkiiifir
does not go unchallenged. Brouwer’s radical move to the left is
speeded by an outraged roar from the reactionary right, ‘What
Weyl and Brouwer are doing [Brouwer is the leader, Weyi his
companion in revolt] is mainly following in the steps of Kro-
necker’, according to Hilbert, the champion of the siatiis quo.
‘They are trying to establish mathematics by jettisoning every-
thing which does not suit them and setting up an embargo. The
eSect is to dismember our science and to run the risk of losing
part of our most valuable possessions. Weyl and Brouwer
condemn the general notions of irrational numbers, of functions

• even of such functions as occur in the theory of numbers –
  Cantor’s transdnite numbers, etc., the theorem that an infinite
  set of positive integers has a least, and even the “law of ex-
  cluded middle”, as for example the assertion: Either there is
  only a finite number of primes or there are infinitely many.
  These are examples of [to them] forbidden theorems and modes
  of reasoning. I believe that impotent as Kronecker was to
  abolish irrational numbers (Weyl and Brouwer do permit us to
  retain a torso), no less impotent will their efforts prove to-day.
  No! Brouwer’s programme is not a revolution, but merely the
  repetition of a futile coup de rmin with old methods, but which
  was then imdertaken with greater verve, yet failed utterly.
  To-day the State [mathematics] is thoroughly armed and
  strengthened through the labours of Frege, Dedeldnd, and
  Cantor. The efforts of Brouwer and Weyl are foredoomed to
  futility,’

To which the other side replies by a shrug of the shoulders
and goes ahead with its great and fundamentally new task of re-
establishing mathematics (particularly the foundations of
analysis) on a firmer basis than any laid down by the men of
the past 2,500 years from Pythagoras to Weierstrass.

What will mathematics be like a generation hence when – we

638

FABABISE LOST?

tope these difficulties will ‘ have been cleared up? Only a
prophet or the seventh son of a prophet sticks Ms head into the
noose of prediction. Bnt if there is any continuity at all in the
evolution of ‘ mathematics – and the majority of dispassionate
observers believe that there is – we shall find that the mathe-
matics wMch is to come will be broader, firmer, and richer in
content than that which we or our predecessors have known.

‘ Already the controversies of the past third of a century have
added new fields – including totally new logics – to the vast
domain of mathematics, and the new is being rapidly consol-
dated and co-ordinated with the old. If we may rasMy venture
a prediction, what is to come wfii be fresher, younger in every
respect, and closer to human thought and human needs – freer
of appeal for its Justification to extra-human ‘existences’ – than
what is now being vigorously refasMoned. The spirit of mathe-
matics is eternal youth. As Cantor said, ‘The essence of mathe-
matics resides in its freedom’; the present ‘revolution’ is but
another assertion of that freedom.

INDEX

Abel, Niels Henrik, 1, 179, 182, 245,
251, 252, 285, 296, 297, 299, 324,
337-358, 360, 361, 366, 368 fi., 372.
398, 401, 402, 414, 415, 420, 449,
450, 464, 458, 461. 4S2, 494, 499,
504, 522, 525, 526, 539, 597
Abel, Anne Marie Simonsen, 337
Abelian integral, 369, 371, 463
Adams, John Coucli, 384, 404
Airy, G. B., 216, 379, 386, 388
Alexander, J. W., 294
Alexander the Great, 592
Algebraic forms, 434, 506
Algebraic integers, 518 ff., 666, 676 ff.
Algebraic numbers, 610 5., 518, 519,
522, 623, 527, 632, 672, 576, 678, 619,
622, 623, 626 ff.

Algebraic number field, 518 ff., 666, 676,
579

Algorithm, 152, 582
Ampere, A. M., 349
Analysis situs, 239, 294, 543, 601
Antoinette, Marie, 181, 186
ApoUonius, 6, 28, 84, 348, 441
Appell, Paul, 501

Arago, r. J, B., 151, 164, 207, 211, 224
Archimedes, 5, 6, 19, 20, 29 2., 63, 111,
124, 130, 131, 160, 167, 176, 239, 241,
242, 262. 260, 263, 264, 279 2., 441,
506, 587
Archytas,26

Aristotle, 20, 26, 84, 263, 306, 628, 634,
637,638

Arithmetical theory of forms, 391
Amauid, A., 90, 139, 141
Assodatiye, assodatiyity, 306, 308, 391
Ausonius, 42
Austen, Jane, 419

Arioms, 21, 336, 336, 365, 462, 556, 635
Ayscou^, Rev. Wm, 98, 99

Babbage, Charles, 484

Baehet de M&irxao, 77

BaiQet, A., 41

Ball, ■W.W.R.. 251

BaJaac, Honore de, 606

Baipow, L, 103, 104, 116, 116, 128

Bartels, Johann Martin, 243 fi.

Bauer, Heinrich, 360
Beethoven, L, van, 446
Bexm, Eriedrich, 240

640

Berkeley, Bishop, 379
EemouHis, 124, 137, 143-50, 156, 157,
170

Berthollet, Claude-Louis, Count, 200,
206 ff., 211,213,214, 300
Bertrand, J. L. F., 500
Bessel, Friedrich Wilhelm, 269, 272,
275, 364

Biot, J. B., 198, 199

Birkhog, George David, 611

Bismarck, 0. E. L., Prince von, 451, 515

Blake, William, 9

Bliss, G. A., 145, 146

Boeckh, P. A., 361

Bohr, N., 19

Bolyai, Johann, 253, 522, 667
Bolyai, Wolfgang, 241, 253, 266
Boole, George, 128, 131, 134, 233, 389,
428, 429, 448, 478-93, 494, 537
Boole, Mary Everest, 493
Borchardt, C. W., 364, 464, 471, 612,
553

Borel, Emile, 501, 624
Boundary values, 200, 372
Boutroux, Emile, 586
Bonvellcs, Charles, 91
Brahe, Tycho, 118
Branches, branch points, 644 g.
Brewster, Sir David, 121, 295, 552
Brianchon, C. J., 237
Briggs, Henry, 580
Brinkley, John, 376, 377, 379
Brochard, Jeanne, 38
Brooke, Rupert, 438
Brouwer, L. E. J., 19, 24, 305, 628,
635 fi.

Bruno, Giordano, 49
Bunsen, B, W., 408, 469
Borali-Forti, C., 634, 636
Burnet, John, 25

Byron, Geoige Gordon, Lord, 2S2, 317,
419

Calculus of variations, 124, 146, 169,
170, 296, 382,483
Calculus, tensor, 234, 280, 494, 5S2
Campanella, Tammaso, 49
Cantor, Georg F. L. P., 19, 25, 3S9,
448 f 493, 532,673, 574, 676, 612-39
Cantor, Georg Waldemar, 616
Cantor, Maria Bohm, 616

INDEX

Cantor, iloritz, 17
Cantor, Vally Guttmaun, 621
CarcaTi, 75
Cardan, H., 355

Camot, Ijazare-Nicolas-Margueiite, 1^27,
312,313

Catherine the Great, 148, 153, 162
Cauchy, Aloise de Bure, 314
Cauchy, Augustin-Louis, 165, 179, 180,
182, 184, 245, 275, 285, 296-322,
34S fi., 367, 385, 405, 408, 413, 415,
416, 449, 459, 504, 521 S., 532, 539 ff.,
572, 5S1, 597

Cauchy, Loms-Franjoia, 298
Cauchy, jJarieOIaddeine Deseatre, 298
Causahty, 336
Cavalieri, B-, 128

Cayley, Arthui, 1, 2, 233 2,, 297, 309,

394, 405, 416-47, 4S3, 494, 507, 523,
568,539,602, 603

Cayley, llaiia Antonia Doughty, 413
Ckanute (French Ambassador}, 53 2.
Charactenstie, 383
Charles 1, 100
Ohaadet, Father, 38, 39, 64
Chevalier, Auguste, 412, 414
Christine, Queen of Sweden, 52 2., 89
Chnsto2^ E. B., 280, 430, 530
Ciccaro, 59, 399

Cass, 625 2., 631, 633, 634, 033, 637
Clifford, Wm K., 323, 541, 555 ff.
Colburn, Zerah, 71, 376
Coleridge, Samuel Taylor, 378, 379, 534
Columbus, Christopher, 371
Comhination, rule of, 306, 307
Commutative, 391, 395, 442
Compleac number, variable, 256, 256,
272 ff., 286, 293, 311, 366 2., 391, 392,

395, 449, 504, 506, 540, 542 2., 546,
597

Complex: units, 523

Condorcet, N. C. de, 163, 166, 197, 205,
206

Congruence, 247 2., 257, 25S, 277, 321
Conjugates, 6(^6
Conou, 21

Convergence, 165, 244, 300, 314, 474 2.,
538,590,597,599,619
Copernicus, Nicolas, 49, 323, 336
Corneille, Kerre, 82
Corpus, 390
Cotes, R., 556
CoutDrat, Xu, 131, 624, 626
Co variant, 435

CreUe, August Leopold, 345 2., 350,
351, 358, 455, 461, 463, 464, 470, ol9
CJromwell, Oliver, 100
Curvature, 2S9, 200, 557, 560 2.

Cuts, 545, 546, 573 2., 614
Cyelotomy, 566

D’Alembert, Jean le Rond, 1C2, IGG n.,
174 2., 189, 203, 540, COO
Darboux, Gaston, 501, 5S6, 5S8, £02,
60<J

Darwin, Charles, 16, 150, 5S3
Darwin, G. H., 583
De Bagne, Cardinal, 46, 47
De Bemlle, Cardinal, 46, 47
Dedekind, Julie, 572
Dfrdekind. Richard, 19, 25, 245, 261,
27S, 44S 2., 516, 518, 521 2., 537. 54*),
552, 554, 555, 563-79, 5S6, 614, 021,
024,2. 63S

Delambre, J. B. J., 168
De Morgan, A-, 159, 3S9, 423, *426, 479,
48-4,480,487
Denumerable class, 627
Denumerable infinity, 594 2.

Do Pastoret, M., 196
Desargues, G„ 80, 83, 85, 20Cr, 228, 234,
312,439

Descartes, Bene, 5, 6, 14, 15, 20, 33, 37-
59, 60, 61, 63, 67, 63, 79, 80, b6, 87.
89, 100, 120, 131, 132, MO, 152, 16C»,
231, 233, 255, 268, 291, 2D2, 3sl, 414,
452, 4S9, 504, 505, 531, 665
Determinants, 371, 427
Dickens, Charles, 419, *481
Dickson, lu E., 287, 390
Diderot, Denis, lob
Diophantes, 7C, 77, 165
Dir.ic, P. A. M., 19. 5GS
Dirichlet, P. G. Lejeune, 259, 20»i, 34;\
S64. 446, 506, 517, 522. 539. 547, c48,
651, 553, 565. 571

Discrete, 12, 13, 21, 24, 127, 177, SxO,
o(>3

Discriminant, 429, 441
Distance, 559, 5d0
Distributive law, 291
Divergent series, 314, 474
Dc sitheua, 31

Dually, principle of, 229, 235, 237, 235
Dmnas, Alexandre, 410

Eddington, Sir Arthur, 56S
Edgeworth, JIaria, 378

641

INDEX

Edison, Thomas A., 604
Einstein, Albert, 3, 6, 15, 19, 150, 168,
234, 280, 335, 351, 384, 385, 431, 494,
541, 542, 602, €03, 613
Eisonateiii, F. M. G.. 74, 260, 278, 429,
501, 603, 617, 639, 640
EUjah, 75

Eli^beth, Princess, 43, 51, 53, 54, 140
Elliptic functions, see Functions, dlip-
tio

Elliptic integrals, 349, 334, 356, 369,
371, 499

Eratosthenes, 31, 32
Essenbeck, Nees von, 263
Euclid, 6, 14, 19, 20, 28, 81, 82, 167, 192,
245, 292, 329 S., 335, 336, 345, 3S5,
417, 489, 512

Eudoxus, 19, 25 fi., 449, 529, 533, 573,
576, 634

Euler, Albert, 162
Euler, CatbLarina Gsell, 158
Euler, Leonard, 74, 124, 144, 145, 147
£f., 161-66, 170, 174, 191, 244 fi., 248,
259, 285, 268, 289, 297, 298, 304, 312.
33S, 339, 360, 361, 389, 405, 416, 489,
408, 638 ff., 543, 587, 5S9, 597, 599
Euler, Marguerite Brucker, 155
Euler, Paul, 155
Euler, Salome Abigail Gsell, 163
Extrema, 335

Faotoriaas,347, 461, 610
Factorization, 366, 628
Factorization, unique, 521, 667, 576
Factors, prime, 576
Ferdinand II, 41

Ferdinand, Duke of BrunsYTick, 245,
253, 254, 264, 266 fi,, 270, 272
Fermat, dement-Samuel, 62
Fermat, Dominique, 61
Fermat, Pierre, 6, 6, 38, ^)-78, 79, 90,
91, 93, 94, 96, 128, 129, 145, 146, 162,
165, 176, 177, 269 ff., 277, 287, 311,
341, 370, 376, 383, 613, 520 SE., 638,
563, 666, 667
Field, 390
Flamsteed, 118
Fleming, Admiral, 53
Foncenex, D. le, 168
Formalism, 314
Forsyth, A. B., 443

Fourier, Jean-Baptiste-doseph, 113,
192, 194, 195, 200-25, 349, 371, 372,
594,619

im

Fractions, continued, 405
Franldin, Fabian, 437, 438
Frederick the Great, 153, 161, 167,
173 ff., 181, 267

Frege, Gottlob, 026, 634 ff., 638
Fresnel, A. J., 385
Fricke, Robert, 570
Fuchs, Lazarus, 696
Functions, Abelian, 371, 439, 450, 459,
463, 464, 498, 499, 501, 502, 543, 548
551

Functions, automorphic, 5S4, 696
Functions, elliptic, 138, 220, 250, 251,
278, 285, 2S6, 354, 366, 360, 361, 363,
366 ff., 409, 420, 425, 450, 455 ff., 468,
494, 602, 508, 609, 517, 526, 527, 630,
594, 697, 607

Functions, multiple periodic, 450, 461,
464

Galileo, 16, 20, 27, 38, 44, 46, 49, 86, 91,
97, 100,132, 140,319,625
Galois, Adelaide-Marie Demante, 398
Galois, Evariste, 1, 179, 180, 182, 296,
342, 398-415, 418, 420, 460, 482, 495,
496, 525, 526, 672, 591
Galois, Nicolas-Gabriel, 398
Gallon, Francis, 150, 221, 354, 355
Gauss, Carl Friedrich, 1, 2, 20, 29, 69,
72, 74, 78, 114, 116, 130, 131, 159,
165, 176, 177, 182, 203, 239-95, 296,
297, 311, 312, 314, 327, 338, 343 fiE.,
351, 359, 360, 363, 364, 366 ff., 370,
389, 391, 395, 414, 415, 418, 428,446,
448, 449, 459, 439, 494, 496, 602
521, 522, 627, 536, 639, 640, 642, 1544,
546 2., 653, 661, 663, 565, 566, 668
579, 581, 687, 590, 592, 694, 697, 603,
613, 614, 618, 619, 627
Gauss, Dorothea Benz, 240, 241, 246
Gauss, Gerhard Diederich, 239, 242
Gauss, Joharme Osthof, 266
Gauss, Minna Waldeck, 266
Gelfond, Alesls, 511
Gelon, 30

Geodesic, 332, 333, 335, 336, 650
Geometry, analytical, 4, 5, 33, 60,

62, 60, 61, 68, 100, 134, 145, 161, 160,
168, 218, 219, 231, 232, 255, 272, 439,
497,670

Geometry, descriptive, 200, 202, 203,
227, 498

Geometry, dxSerential, 289, 530
Geometry, enumeratiT^ 623

INDEX

Greometry, foundations of, 2S5, 54S, 550,
554, 655

G«onaetrv, n-dimensional, 417, 420,
439 fi/

Geometry, non-Euclidiaii,4, 5, 116, 133,
203, 2^, 245, 252. 253, 263, 2S6, 323,
327, 329 ff., 394, 395, 417, 439, 440,
4S3, 522, 523, 554 ff., 557, 559 fi., 5r36,
fc07, COS

Geometrr, pxojectiTe, 84, 228, 233,
235 ff.) 439, 440
Germain, Sopbie, 277, 2S6 ff.

Gibbtan, Edward, 2S2
Gilbert, Win, 3S
Gilman, Arthur, 433
Goethe, J. W. von, 281:

Goldbach, C., 447
Gotmod, Charles Francois, 423
Grassmann, Hermann, 134
Gregory, D. F., 4S3, 4t4
Gregory, James, 136
Grote, Geo., 419

Groups, 5, 73, ISO, 294, 2QS, 305 fi.. 413,
49S, 548, 568, 672. 683, 595, 608
Groups, abstract, 309
Groups, continuD’ts, 294, 310, 5&S
Gudermann, Ckristof, 455 5., 46S

Hachette, J. X. P., 340, 352
Hadamard, Jacques. 5&, 233
Halley, Edmund, 114, 116 tT., Ic7
Haiphen, Greorges, 444
Hamilton, Eliza, 378
Hamilton, Rev. James, 370, 371
Hamilton, Sarah Hutton. STS
Hamilton, Sir William, 4S4 f .

Hamilton, William Bnowan, 15, 20, SO.
134, 145, 146, 169, 2S5, 230, 200, 1*6-^,
367 , 373—9 / , 4S4, 4So, 56S
Hansteen, Christoph, 36u
Hardy, G. H., 53&

Harvey, Wm, 38

Hegel,” G. W. F.. 262, 2G3, 265, 4S5, 6iT
Heiberg, J. L., 32
Heisenberg, W., 19, 443
Helmholtz, H. von, 446, 4GS
Henry, C., ^

Eera^tns, 12

Herbart, Johann Friedrich, 542
.Hamite, Charles, 2, 179, 299. 337, 302,
371, 404, 466, 494-513, 525, 526, 253.
568, 590,591,595,626,629
Hocmitiaa forma, 506
Herschel, Sir William, 262, 4S4

Hertz, Heinrich, 16
Hieron 11, 30, 35

Hilbert, David, 67, 262, 4S9, 490, 511,
624, 635 3.

Hipparchus, 1 IS, 397
Hitler, Adol^ 570

Kolmhogt Bint ASchael, 338 £f., 345,
345, 349, 357, SCI
Hooke, Robert, 115 3.

Horace, 352
KumLert, Georges, 566
Humboldt, Alexander von, 22j, 26fi,
268, 2S4, 366
Hume, David, 159
HTiutingt-on, E. V., 4S0
jELuTgens, Christiaan, SI, USy lUi 3.,
140,146, 384, SS5

Ideals, 450, 530, 57G 5,

Ideal, prime, 579

Ideniicai operation, identity, SOd,
Imaginarics, 5, 3i4, 3G7, S*?, 291, 4’^,
439, 532, 338, 544

Invariance, invariants, 5, 55. 18.’, 233, 31i, 410, 41, 420, 422, 423, 427
435, 441, 444, 4S3, 404, 506, 5U7, 5i>2,
695, COl

L’/edodhles, 520, 568, 628

Jacobi, Carl Gustav Jacob, 16, 21, 56,
152, 231, 252, 2S5, 296, 352, 353, 355,
356, 359-72, 3S0, 414, 415, 4!:-J, 455,
461, 464, 494, 499, 601 fi., oil, 517,
539, 5B5, 568, 571, 5h3, 507, tfl;
Jacobi, 11 H., 359
James II, 100, 120
Jansen, Corc’-lius, 86, S7
Jeans, Sir James H-, 16, 21, i:4^ 5*^3
Js&eys, George, 120
Jtrraid, G. E.,6»}7,505
J-.ats Christ, 632
Joachim, Joseph, Gib
Jourdain, P, E. E., 205

Kent, L, 194, 26H. 379, 394
Kelvir, D^rd {William Thcinivi- it
21T,32l,4%4&ti
Eempis, Thu-imas a, SCid
Kepler, J., 20, lOi?, 114, ilS,

Kicraley, Chatlts. 424
Klpilng, Rudyard, 35C
Kirchhofi, G- R.,46b
KJein, FelLv, 235, 311, 417, 4fi:v 46^*?,
506,604

648

INDEX

KiwRcr, Adolf, 515
Konigsbcrgor, L., 468, 409
Kowalewsfed, iSonja (Sophie), 286, 448-
77, 530

Kroneoker, Leopold, 19, 179, 257,
201, 321, 448 ft., 462, 466, 47C, 51-1-
33, 653, 666, 60G, 675, 570, 583, 613,
616, 018, 620, 621, 627 ff., G32 f[., 637,
638

Kuoimor, Enist Eduard, 201, 278, 516,
518, 622, 525, 628, 529, 637, 553, 663-
70, 690, 618

Lacroix, S. “F., 349

Ijagrange, Joscph-Iiouis, 3, 8, 67, 124,
146, 164, 161, 162, 107-87, 190, 194,
196, 108, 202, 210, 216, 244 ff., 259,
271, 288, 280, 296, 297, 300 C., 304,
312, 338, 341, 360, 361, 371, 378, 380,
401, 420, 428, 441, 483, 496, 497, 697,
690, 600

Lagcange, Maiic-Thorose Gros, 167
liamarck, J. B. A. E., 198
Lamb, Horace, 14
Landau, Edmund, 669, 670
La]>laoo, Biorro*Simon, Marquis, 112,
113, 120, 170, 177, 178, 185, 186, 188-
<)0, 200, 204, 216, 220, 244, 246, 231,
263 IL, 270 IL, 284, 206, 300 ff., 314,
340, 361, 371, 376, 420, 465, 482, 400,
597, 600

Laviuaicr, A. L., 181 ff., 185, 200
Leawl squares, theory of, 240, 284, 286,
280,517

Lofschetz, S., 294

ItOgeudra, Adrion-Marie, 178, 190, 210,
246, 248, 249, 259, 284, 285, 290, 304,
312, 343, 348, 349, 352 ff., 366, 368
400,409,637,538

Leibniz, Gottfried Wilhelm, 6, 13, 15,
17, 33, OO, 04, 73, 82, 84, 95, 106, 111,
122 IL, 127-42, 145 H., 152, 153, 176,
184, 244, 253, 479, 489
Lcmaitre, Eather, 681
Lemonnier, P. 0., 186
U-veriier, U. J. J.. 318, 384, 404
Levi-Civita, T., 280, *130, 600
Liap<)unoli, A., 683
Liln’i, G., 353
Lichtenstein, Leon, 683
Lhs S„430
Lilly, Wm, 580
limit, 474, 634
Lincoln, Abraham, 479

644

Lindcmann, F., 345, 612, 613, 627
Linus (of Li5go), 110
liouville, Joseph, 368, 413, 414, 49*
499, 511, 525
Listing, J. B., 642
Littrow, J. J. von, 324, 351, 366
Lloyd, Humphrey, 386
Lobatclicwsky, Nikolas Ivaiiovitc
323-36, 394, 416, 417, 622, 557
Lobalchewsky, Praskovia Ivanovna
323

Locke, John, 121

l 4 )gio, symbolic, 4, 6, 13, 131, 134, 47,’
486 ff.

long, Claire de, 01
Long, Louise de, 62
Lotze,R. 1L,486
Lucas (of liege), 116

MacLaurin, C., 583
Macaulay, Thos. B., 282, 419
Magnitude, 623

Malus, Etierme-Louis, 304,305, 332,
Manifold, 201, 557 ff.

Mapping, conformal, 289, 292, 293
MarcoUus, 20, 35, 36
Marie, Abb6, 173, 178
Marr, John, 580
Matrices, 417, 439, 441 ff.

Miuirico, Prince of Orange, 41
Maxwell, Jamtss Clerk, 293, 334,
478, 553, 002
Monacchmus, 593
Mendelssohn, Felix, 617
Mercator, N., 115,130,292 V

Mer6, Gombaud Antoine, Chevalier ‘
94 ff.

Mersenne, P., 39, 48, 67, 82, 87
IMilton, Jolin, 38
Minkowsld, H., 6

Mittag.Lefflcr, G. M., 347, 366, 463,’
472, 698, 590, 629
Modulus, 247 ff., 257, 268, 277, 321
Mongo, Gaspai-d, 187, 200-25, 227,
312,313,316,403
Monge, Jacques, 201
Moiiogonicity, 273 ff.

Montagu, Charles, 121
More, L. T., 00
Morgan, John I’ierpoint, 61 i
Moritz, R. E., 614
Mozart, W. A. 0., 446
JMidtiformity, 643

Napier, John, 680

INDEX

Napoleon, Bonaparte 135, 167, 184, 186,
190 207 g., 221 fi., 226, 267, 26S,

270 g., 276, 297, 301, 302, 304, 313,
382, 398, 466, 564, 585
Newton, Hajanah Ayscongh, 97
jCfewton, Isaac, 3, 6, 13, 17 ff., 27, 29,
30, 33, 33, 60, 63 g., 69, 79, 97-126,

■ 127 g., 136, 137, 140, 141, 145, 147,
161, 152, 155. 160, 163, 168, 169, 171,
174, ISl, 1S3, 1S4, 186, ISS, 190, 191,
197, 217, 234, 239, 241, 242, 244 g.,
252, 260, 261, 263, 264, 272, 278.g.,
2S8, 317, 338. 339, 371, 376, 3S4, 388,
G 397,441,445,478,489,528,541,542,
GL 552, 556, 557, 587, 503, 597, 598
Gk-iiglitingale, Florence, 427
G ron-dennmerable classes, 627
C Normal, 290

r dumber, cardinal, 623 g., 634, 636
Numbers, ideal, 522, 523, 5G6
Numbers, irrational, 22, 25, 28, 257,
L 443. 462, 475, 476, 602, 511, 529, 573

G. g., 627, 636, 638

.. ’umbers, ncgatrre, 391, 531, 532
G lumbers, prime, 70 g., 519, g. 537, 538,
G 560,567,632,637,638
C lumbers, transcendental, 507, 509 g.,
626 g., 631

I Numbers, transfinite, 624, 625, 633, 6^
E Nurmi, Paavo, 599
S

H ‘fibers, H- W. iL, 261, 270, 285, 288

H, ‘Idenbuig, H., 117
Hfc*rder, 308, 633

H»3car n. King of Sweden, 598
E

5 “^ainleTe, Paul, 501, 600, 611
Parameter, 292, 596
Parametric representation, 290 g., 568
p Parmenides, 24
^Partitions, theory of, 425, 447
H^ascal, Antoinette Begone, 79
p^ ascal, Blaise, 1, 8, 38, 61, 79-96, 124,
E 129, 135, 139, 2<30, 228, 233, 237, 312,
H. 434,439
J(if Tscal, fitierme, 79
‘jg^iscal, Gflberte (Madame P4rierJ, 61,
79, 81, 82, 86, 88, 89
]^iscal, Jacqueline, 79, SO, 82, 86 g.

Tff ul, Jean, se& Richter, J. P. P.

.^acock, G., 484
eel. Sir Robert, 388
sirce, Benjamin, 3S3
»’Pepys, Samuel, 121

Periodicity, 218 g., 251, 368, 369. 4y9>.

502, 503, 509, 573, 594, 601, Cly
Permanence of form, 3S9
Permutation, 306, 307, 310
Peter the Great, 140, 153, 157, 1 78
Pfag, Johann Friediich, 253, 2c -j
tPheidias, 30
Phidias, 354
‘Philip, Apostle, 632
Piazzi, Gniscppe, 262
Picard, fcile, 501
Picard, Jean, 3CS
Planck, M., 602, 603
Plato, 3, 16, 20 21, 26, 27, 33, 2v3
Plucker, J., 440, 454, 523
Plutarch, 29, 35

Poincare, Henri, 8, 17, 172, 291, 416,
472, 493, 494, 501, 509, 533, 5S*:>-6n,
615

Poincare, Raymond, 531, 556, 5*.}<J
Poinsot, L., 304
Poisson, S. B., 349, 409
Poacelet, Jean-Victor, 205, 226-235,
312,439

Pope, Alexander, 605
Postulate, 21, 23, 306, 307, 309. 329 g.,
333, 335, 336, 389 g.. 395, 483, 490 g.,
523, 626, 672, 632, 633, 636
Power series, 466, 467, 474, 476, 6! 9
Probability, mathematical theorv ‘if, 5,
61, 79, 90,93, 129, 145, 146, 149, 169,
188,193, 194,349,487,498
Problem of » bodies, 598, 599
Progression, 390, 394, 590
Pseudo-sphere, 335, 336
Ptolemy, 118, 192, 397
Pythagoras, 16, 20 g., 27, 292, 439, 44l\
504, 523, 529, 569, 560, 612, 636, 638

Quadratic forms, 176, 368, 427, 503,
506, 527, 554, 596,

Qnantics, 434, 435, 438
Quantum theory, 68, 96, 115, 354, S-Jtv
559, 568, 603*

Quatemions, 285, 286, 386, 3SS g.,
394 g.

Quintic, 339 g., 344, 345, 34S, 360, 4frT,
507,509,522,526

Radicals, 409, 413, 414, 500, 607, C<»S,
526

Ramanigan Srinivasa, 152, 360
Ratio, anharmonic or cross, 234, 235
Ratios, 28, 3S2, 576
Rays, systems of, 3S0 g., 668

645