Lectures on ten British mathematicians of the nineteenth century

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MATHEMATICAL MONOGRAPHS

EDITED BY

MANSFIELD MERRIMAN and ROBERT S. WOODWARD

No. 17

LECTURES ON

TEN BRITISH MATHEMATICIANS

of the Nineteenth Century

BY

ALEXANDER MACFARLANE,

Late President of the International Association for Promoting
the Study of Quaternions

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No. 17 Copyrighted, 1016,

BY

HELEN S. MACFARLANE

PREFACE

During the years 1 901- 1904 Dr. Alexander Macfarlane
delivered, at Lehigh University, lectures on twenty-five British
mathematicians of the nineteenth century. The manuscripts
of twenty of these lectures have been found to be almost ready
for the printer, although some marginal notes by the author
indicate that he had certain additions in view. The editors
have felt free to disregard such notes, and they here present
ten lectures on ten pure mathematicians in essentially the same
form as delivered. In a future volume it is hoped to issue
lectures on ten mathematicians whose main work was in physics
and astronomy.

These lectures were given to audiences composed of students,
instructors and townspeople, and each occupied less than an
hour in delivery. It should hence not be expected that a lecture
can fully treat of all the activities of a mathematician, much
less give critical analyses of his work and careful estimates of
his influence. It is felt by the editors, however, that the lectures
will prove interesting and inspiring to a wide circle of readers
who have no acquaintance at first hand with the works of the
men who are discussed, while they cannot fail to be of special
interest to older readers who have such acquaintance.

It should be borne in mind that expressions such as " now,"
" recently," " ten years ago," etc., belong to the year when a
lecture was delivered. On the first page of each lecture will
be found the date of its delivery.

For six of the portraits given in the frontispiece the
editors are indebted to the kindness of Dr. David Eugene
Smith, of Teachers College, Columbia University.

3

4 PREFACE

Alexander Macf arlane was born April 21, 185 1, at Blairgowrie,
Scotland. From 187 1 to 1884 he was a student, instructor and
examiner in physics at the University of Edinburgh, from 1885
to 1894 professor of physics in the University of Texas, and from
1895 to 1908 lecturer in electrical engineering and mathematical
physics in Lehigh University. He was the author of papers on
algebra of logic, vector analysis and quaternions, and of Mono-
graph No. 8 of this series. He was twice secretary of the sec-
tion of physics of the American Association for the Advancement
of Science, and twice vice-president of the section of mathematics
and astronomy. He was one of the founders of the International
Association for Promoting the Study of Quaternions, and its
president at the time of his death, which occured at Chatham,
Ontario, August 28, 1913. His personal acquaintance with
British mathematicians of the nineteenth century imparts to
many of these lectures a personal touch which greatly adds
to their general interest.

Alexander Macfarlane
From a photograph of 189

CONTENTS

Portraits of Mathematicians Frontispiece

George Peacock (1791-1858) page 7

A Lecture del. vered Apri 12, 1901.

Augustus De Morgan (1806-1871) 19

A Lecture delivered April 13, 1901.

Sir William Rowan Hamilton (1805-1865) 34

A Lecture delivered April 16, 1901.

George Boole (1815-1864) 50

A Lecture de ivered April 19, 1901.

Arthur Cayley (1821-1895) 64

A Lecture delivered April 20, 1901.

William Kingdon Clifford (1845-1879) 78

A Lecture delivered April 23, 1901.

Henry John Stephen Smith (1826-1883) 92

A Lecture delivered March 15, 1902.

James Joseph Sylvester (1814-1897) 107

A Lecture delivered March 21, 1902.

Thomas Penyngton Kirkman (1806-1895 122

A Lecture delivered April 20, 1903.

Isaac Todhunter (1820-1884) 134

A Lecture delivered April 13, 1904.

Index 147

5

TEN BRITISH MATHEMATICIANS

GEORGE PEACOCK*
(1791-1858)

George Peacock was born on April 9, 1791, at Denton
in the north of England, 14 miles from Richmond in Yorkshire.
His father, the Rev. Thomas Peacock, was a clergyman of the
Church of England, incumbent and for 50 years curate of the
parish of Denton, where he also kept a school. In early life
Peacock did not show any precocity of genius, and was more
remarkable for daring feats of climbing than for any special
attachment to study. He received his elementary education
from his father, and at 17 years of age, was sent to Richmond,
to a school taught by a graduate of Cambridge University to
receive instruction preparatory to entering that University.
At this school he distinguished himself greatly both in classics
and in the rather elementary mathematics then required for
entrance at Cambridge. In 1809 he became a student of Trinity
College, Cambridge.

Here it may be well to give a brief account of that Uni-
versity, as it was the alma mater of four out of the six mathe-
maticians discussed in this course of lectures. f

At that time the University of Cambridge consisted of seven-
teen colleges, each of which had an independent endowment,
buildings, master, fellows and scholars. The endowments, gen-
erally in the shape of lands, have come down from ancient times;
for example, Trinity College was founded by Henry VIII in

* This Lecture was delivered April 12, 1901— Editors.

t Dr. Maefarlane's first course included the first six lectures given in this
volume. — Editors.

7

8 TEN BRITISH MATHEMATICIANS

1546, and at the beginning of the 19th century it consisted of a
master, 60 fellows and 72 scholars. Each college was provided
with residence halls, a dining hall, and a chapel. Each college
had its own staff of instructors called tutors or lecturers, and
the function of the University apart from the colleges was
mainly to examine for degrees. Examinations for degrees con-
sisted of a pass examination and an honors examination, the
latter called a tripos. Thus, the mathematical tripos meant
the examinations of candidates for the degree of Bachelor of
Arts who had made a special study of mathematics. The
examination was spread over a week, and those who obtained
honors were divided into three classes, the highest class being
called wranglers, and the highest man among the wranglers,
senior wrangler. In more recent times this examination de-
veloped into what De Morgan called a " great writing race;"
the questions being of the nature of short problems. A candidate
put himself under the training of a coach, that is, a mathema-
tician who made it a business to study the kind of problems
likely to be set, and to train men to solve and write out the
solution of as many as possible per hour. As a consequence
the lectures of the University professors and the instruction of
the college tutors were neglected, and nothing was studied ex-
cept what would pay in the tripos examination. Modifications
have been introduced to counteract these evils, and the con-
ditions have been so changed that there are now no senior
wranglers. The tripos examination used to be followed almost
immediately by another examination in higher mathematics to
determine the award of two prizes named the Smith's prizes.
" Senior wrangler " was considered the greatest academic dis-
tinction in England.

In 18 1 2 Peacock took the rank of second wrangler, and the
second Smith's prize, the senior wrangler being John Herschel.
Two years later he became a candidate for a fellowship in his
college and won it immediately, partly by means of his exten-
sive and accurate knowledge of the classics. A fellowship then
meant about £200 a year, tenable for seven years provided
the Fellow did not marry meanwhile, and capable of being

GEORGE PEACOCK 9

extended after the seven years provided the Fellow took clerical
Orders. The limitation to seven years, although the Fellow
devoted himself exclusively to science, cut short and prevented
by anticipation the career of many a laborer for the advance-
ment of science. Sir Isaac Newton was a Fellow of Trinity
College, and iis limited terms nearly deprived the world of the
Principia.

The year after taking a Fellowship, Peacock was appointed
a tutor and lecturer of his college, which position he continued
to hold for many years. At that time the state of mathematical
learning at Cambridge was discreditable. How could that be?
you may ask; was not Newton a professor of mathematics in
that University? did he not write the Principia in Trinity
College? had his influence died outso soon? The true reason
was he was worshipped too much as an authority; the Univer-
sity had settled down to the study of Newton instead of Nature,
and they had followed him in one grand mistake — the ignoring
of the differential notation in the calculus. Students of the
differential calculus are more or less familiar with the controversy
which raged over the respective claims of Newton and Leibnitz
to the invention of the calculus; rather over the question whether
Leibnitz was an independent inventor, or appropriated the
fundamental ideas from Newton's writings and correspondence,
merely giving them a new clothing in the form of the differential
notation. Anyhow, Newton's countrymen adopted the latter
alternative; they clung to the fluxional notation of Newton;
and following Newton, they ignored the notation of Leibniz
and everything written in that notation. The Newtonian
notation is as follows: If y denotes a fluent, then y denotes its
fluxion, and y the fluxion of J\ if y itself be considered a fluxion,
then y' denotes its fluent, and y" the fluent of y' and so on;
a differential is denoted by o. • In the notation of Leibnitz y

is written -r, y is written ~, y' is ( ydx, and so on. The
ax ax1 J

result of this Chauvinism on the part of the British mathema-
ticians of the eighteenth century was that the developments of
the calculus were made by the contemporary mathematicians

10 TEN BRITISH MATHEMATICIANS

of the Continent, namely, the Bernoullis, Euler, Clairault,
Delambre, Lagrange, Laplace, Legendre. At the beginning
of the 19th century, there was only one mathematician in Great
Britain (namely Ivory, a Scotsman) who was familiar with the
achievements of the Continental mathematicians. Cambridge
University in particular was wholly given over not merely to the
use of the fluxional notation but to ignoring the differential
notation. The celebrated saying of Jacobi was then literally
true, although it had ceased to be true when he gave it utterance.
He visited Cambridge about 1842. When dining as a guest at
the high table of one of the colleges he was asked who in his
opinion was the greatest of the living mathematicians of England ;
his reply was " There is none."

Peacock, in common with many other students of his own
standing, was profoundly impressed with the need of reform,
and while still an undergraduate formed a league with Babbage
and Herschel to adopt measures to bring it about. In 18 15 they
formed what they called the Analytical Society, the object of
which was stated to be to advocate the d'ism of the Continent
versus the dot-age of the University. Evidently the members
of the new society were armed with wit as well as mathematics.
Of these three reformers, Babbage afterwards became celebrated
as the inventor of an analytical engine, which could not only
perform the ordinary processes of arithmetic, but, when set
with the proper data, could tabulate the values of any function
and print the results. A part of the machine was constructed,
but the inventor and the Government (which was supplying the
funds) quarrelled, in consequence of which the complete machine
exists only in the form of drawings. These are now in the
possession of the British Government, and a scientific commis-
sion appointed to examine them has reported that the engine
could be constructed. The third reformer — Herschel — was a
son of Sir William Herschel, the astronomer who discovered
Uranus, and afterwards as Sir John Herschel became famous
as an astronomer and scientific philosopher.

The first movement on the part of the Analytical Society
was to translate from the French the smaller work of Lacroix on

GEORGE PEACOCK 11

the differential and integral calculus; it was published in 1816.
At that time the best manuals, as well as the greatest works on
mathematics, existed in the French language. Peacock followed
up the translation with a volume containing a copious Collection
of Examples of the Application of the Differential and Integral
Calculus, which was published in 1820. The sale of both books
was rapid, and contributed materially to further the object of
the Society. Then high wranglers of one year became the
examiners of the mathematical tripos three or four years after-
wards. Peacock was appointed an examiner in 1817, and he did
not fail to make use of the position as a powerful lever to advance
the cause of reform. In his questions set for the examination
the differential notation was for the first time officially employed
in Cambridge. The innovation did not escape censure, but he
wrote to a friend as follows: " I assure you that I shall never
cease to exert myself to the utmost in the cause of reform, and
that I will never decline any office which may increase my
power to effect it. I am nearly certain of being nominated to
the office of Moderator in the year 1818-1819, and as I am an
examiner in virtue of my office, for the next year I shall pursue
a course even more decided than hitherto, since I shall feel that
men have been prepared for the change, and will then be
enabled to have acquired a better system by the publication of
improved elementary books. I have considerable influence as
a lecturer, and I will not neglect it. It is by silent perseverance
only, that we can hope to reduce the many-headed monster of
prejudice and make the University answer her character as the
loving mother of good learning and science." These few sen-
tences give an insight into the character of Peacock: he was
an ardent reformer and a few years brought success to the cause
of the Analytical Society.

Another reform at which Peacock labored was the teaching
of algebra. In 1830 he published a Treatise on Algebra which
had for its object the placing of algebra on a true scientific
basis, adequate for the development which it had received at
the hands of the Continental mathematicians. As to the state
of the science of algebra in Great Britain, it may be judged

12 TEN BRITISH MATHEMATICIANS

of by the following facts. Baron Maseres, a Fellow of Clare
College, Cambridge, and William Frend, a second wrangler,
had both written books protesting against the use of the nega-
tive quantity. Frend published his Principles of Algebra in 1 796,
and the preface reads as follows: " The ideas of number are the
clearest and most distinct of the human mind; the acts of the
mind upon them are equally simple and clear. There cannot
be confusion in them, unless numbers too great for the com-
prehension of the learner are employed, or some arts are used
which are not justifiable. The first error in teaching the first
principles of algebra is obvious on perusing a few pages only
of the first part of Maclaurin's Algebra. Numbers are there
divided into two sorts, positive and negative; and an attempt
ismade to explain the nature of negative numbers by allusion
to book debts and other arts. Now when a person cannot
explain the principles of a science without reference to a meta-
phor, the probability is, that he has never thought accurately
upon the subject. A number may be greater or less than another
number; it may be added to, taken from, multiplied into, or
divided by, another number; but in other respects it is very
intractable; though the whole world should be destroyed, one
will be one, and three will be three, and no art whatever can
change their nature. You may put a mark before one, which
it will obey; it submits to be taken away from a number greater
than itself, but to attempt to take it away from a number less
than itself is ridiculous. Yet this is attempted by algebraists
who talk of a number less than nothing ; of multiplying a negative
number into a negative number and thus producing a positive
number; of a number being imaginary. Hence they talk of
two roots_to every equation of the second order, and the learner
is to try which will succeed in a given equation; they talk of
solving an equation which requires two impossible roots to make
it soluble; they can find out some impossible numbers which
being multiplied together produce unity. This is all jargon,
at which common sense recoils; but from its having been once
adopted, like many other figments, it finds the most strenuous
supporters among those who love to take things upon trust and

GEORGE PEACOCK 13

hate the colour of a serious thought." So far, Frend. Peacock
knew that Argand, Francais and Warren had given what
seemed to be an explanation not only of the negative quantity
but of the imaginary, and his object was to reform the teaching
of algebra so as to give it a true scientific basis.

At that time every part of exact science was languishing in
Great Britain. Here is the description given by Sir John
Herschel: "The end of the 18th and the beginning of the 19th
century were remarkable for the small amount of scientific move-
ment going on in Great Britain, especially in its more exact
departments. Mathematics were at the last gasp, and Astronomy
nearly so — I mean in those members of its frame which depend
upon precise measurement and systematic calculation. The
chilling torpor of routine had begun to spread itself over all
those branches of Science which wanted the excitement of
experimental research." To elevate astronomical science the
Astronomical Society of London was founded, and our three
reformers Peacock, Babbage and Herschel were prime movers
in the undertaking. Peacock was one of the most zealous
promoters of an astronomical observatory at Cambridge, and
one of the founders of the Philosophical Society of Cambridge.

The year 1831 saw the beginning of one of the greatest
scientific organizations of modern times. That year the British
Association for the Advancement of Science (prototype of the
American, French and Australasian Associations) held its first
meeting in the ancient city of York. Its objects were stated
to be: first, to give a stronger impulse and a more systematic
direction to scientific enquiry; second, to promote the inter-
course of those who cultivate science in different parts of the
British Empire with one another and with foreign philosophers;
third, to obtain a more general attention to the objects of
science, and the removal of any disadvantages of a public kind
which impede its progress. One of the first resolutions adopted
was to procure reports on the state and progress of particular
sciences, to be drawn up from time to time by competent per-
sons for the information of the annual meetings, and the first
to be placed on the list was a report on the progress of mathe-

14 TEN BRITISH MATHEMATICIANS

matical science. Dr. Whewell, the mathematician and phil-
osopher, was a Vice-president of the meeting : he was instructed
to select the reporter. He first asked Sir W. R. Hamilton, who
declined; he then asked Peacock, who accepted. Peacock had
his report ready for the third meeting of the Association, which
was held in Cambridge in 1833; although limited to Algebra,
Trigonometry, and the Arithmetic of Sines, it is one of the best
of the long series of valuable reports which have been prepared
for and printed by the Association.

In 1837 he was appointed Lowndean professor of astronomy
in the University of Cambridge, the chair afterwards occupied
by Adams, the co-discoverer of Neptune, and now occupied by
Sir Robert Ball, celebrated for his Theory of Screws. In 1839
he was appointed Dean of Ely, the diocese of Cambridge. While
holding this position he wrote a text book on algebra in two
volumes, the one called Arithmetical Algebra, and the other
Symbolical Algebra. Another object of reform was the stat-
utes of the University; he worked hard at it and was made
a member of a commission appointed by the Government for
the purpose; but he died on November 8, 1858, in the 68th year
of his age. His last public act was to attend a meeting of the
Commission.

Peacock's main contribution to mathematical analysis is his
attempt to place algebra on a strictly logical basis. He founded
what has been called the philological or symbolical school of
mathematicians; to which Gregory, De Morgan and Boole
belonged. His answer to Maseres and Frend was that the
science of algebra consisted of two parts — arithmetical algebra
and symbolical algebra — and that they erred in restricting the
science to the arithmetical part. His view of arithmetical
algebra is as follows: " In arithmetical algebra we consider
symbols as representing numbers, and the operations to which
they are submitted as included in the same definitions as in
common arithmetic; the signs + and — denote the operations
of addition and subtraction in their ordinary meaning only,
and those operations are considered as impossible in all cases
where the symbols subjected to them possess values which

GEORGE PEACOCK 15

would render them so in case they were replaced by digital
numbers; thus in expressions such as a +b we must suppose
a and b to be quantities of the same kind; in others, like a — b,
we must suppose a greater than b and therefore homogeneous

with it; in products and quotients, like ab and - we must suppose

the multiplier and divisor to be abstract numbers; all results
whatsoever, including negative quantities, which are not strictly
deducible as legitimate conclusions from the definitions of the
several operations must be rejected as impossible, or as foreign
to the science."

Peacock's principle may be stated thus: the elementary
symbol of arithmetical algebra denotes a digital, i.e., an integer
number; and every combination of elementary symbols must
reduce to a digital number, otherwise it is impossible or foreign
to the science. If a and b are numbers, then a+b is always
a number; but a — b is a number only when b is less than a.
Again, under the same conditions, ab is always a number, but

- is really a number only when b is an exact divisor of a.
b

Hence we are reduced to the following dilemma : Either - must

b

be held to be an impossible expression in general, or else the

meaning of the fundamental symbol of algebra must be extended

so as to include rational fractions. If the former horn of the

dilemma is chosen, arithmetical algebra becomes a mere shadow;

if the latter horn is chosen, the operations of algebra cannot

be defined on the supposition that the elementary symbol is an

integer number. Peacock attempts to get out of the difficulty

by supposing that a symbol which is used as a multiplier is

always an integer number, but that a symbol in the place of the

multiplicand may be a fraction. For instance, in ab, a can denote

only an integer number, but b may denote a rational fraction.

Now there is no more fundamental principle in arithmetical

algebra than that ab = ba; which would be illegitimate on

Peacock's principle.

One of the earliest English writers on arithmetic is Robert

16 TEN BRITISH MATHEMATICIANS

Record, who dedicated his work to King Edward the Sixth.
The author gives his treatise the form of a dialogue between
master and scholar. The scholar battles long over this diffi-
culty,— that multiplying a thing could make it less. The master
attempts to explain the anomaly by reference to proportion;
that the product due to a fraction bears the same proportion to
the thing multiplied that the fraction bears to unity. But the
scholar is not satisfied and the master goes on to say: "If I
multiply by more than one, the thing is increased; if I take it
but once, it is not changed, and if I take it less than once, it
cannot be so much as it was before. Then seeing that a fraction
is less than one, if I multiply by a fraction, it follows that I do
take it less than once." Whereupon the scholar replies, " Sir,
I do thank you much for this reason, — and I trust that I do per-
ceive the thing."

The fact is that even in arithmetic the two processes of
multiplication and division are generalized into a common mul-
tiplication ; and the difficulty consists in passing from the origi-
nal idea of multiplication to the generalized idea of a tensor,
which idea includes compressing the magnitude as well as
stretching it. Let m denote an integer number; the next step

is to gain the idea of the reciprocal of m, not as — but simply as

m

/m. When m and /n are compounded we get the idea of a
rational fraction; for in general m/n will not reduce to a number
nor to the reciprocal of a number.

Suppose, however, that we pass over this objection; how
does Peacock lay the foundation for general algebra? He calls
it symbolical algebra, and he passes from arithmetical algebra
to symbolical algebra in the following manner: "Symbolical
algebra adopts the rules of arithmetical algebra but removes
altogether their restrictions; thus symbolical subtraction dif-
fers from the same operation in arithmetical algebra in being
possible for all relations of value of the symbols or expressions
employed. All the results of arithmetical algebra which are
deduced by the application of its rules, and which are general
in form though particular in value, are results likewise of

GEORGE PEACOCK 17

symbolical algebra where they are general in value as well as in
form; thus the product of am and an which is am+n when m
and n are whole numbers and therefore general in form though
particular in value, will be their product likewise when m and
n are general in value as well as in form; the series for (a+b)n
determined by the principles of arithmetical algebra when n
is any whole number, if it be exhibited in a general form, without
reference to a final term, may be shown upon the same principle
to the equivalent series for (a+b)n when n is general both in
form and value."

The principle here indicated by means of examples was
named by Peacock the " principle of the permanence of equiva-
lent forms," and at page 59 of the Symbolical Algebra it is thus
enunciated: " Whatever algebraical forms are equivalent when-
the symbols are general in form, but specific in value, will be
equivalent likewise when the symbols are general in value as
well as in form."

For example, let a, b, c, d denote any integer numbers, but
subject to the restrictions that b is less than a, and d less than
c; it may then be shown arithmetically that

(a — b)(c — d) =ac+bd—ad—bc.

Peacock's principle says that the form on the left side is equiva-
lent to the form on the right side, not only when the said
restrictions of being less are removed, but when a, b, c, d denote
the most general algebraical symbol. It means that a, b, c, d
may be rational fractions, or surds, or imaginary quantities,

or indeed operators such as — . The equivalence is not estab-

dx

lished by means of the nature of the quantity denoted; the

equivalence is assumed to be true, and then it is attempted

to find the different interpretations which may be put on the

symbol.

• It is not difficult to see that the problem before us

involves the fundamental problem of a rational logic or

theory of knowledge; namely, how are we able to ascend from

particular truths to more general truths. If a, b, c, d denote

18 TEN BRITISH MATHEMATICIANS

integer numbers, of which b is less than a and d less than c,
then

(a — b) {c—d)=acJrbd — ad — bc.

It is first seen that the above restrictions may be removed,
and still the above equation hold. But the antecedent is still
too narrow; the true scientific problem consists in specifying
the meaning of the symbols, which, and only which, will admit
of the forms being equal. It is not to find some meanings, but
the most general meaning, which allows the equivalence to be
true. Let us examine some other cases; we shall find that
Peacock's principle is not a solution of the difficulty; the great
logical process of generalization cannot be reduced to any such
easy and arbitrary procedure. When a, m, n denote integer
numbers, it can be shown that

aman = am+n.

According to Peacock the form on the left is always to be equal to
the form on the right, and the meanings of a, m, n are to be found
by interpretation. Suppose that a takes the form of the incom-
mensurate quantity e, the base of the natural system of logar-
ithms. A number is a degraded form of a complex quantity
(P+q^ — i) and a complex quantity is a degraded form of a
quaternion; consequently one meaning which may be assigned
to m and n is that of quaternion. Peacock's principle would
lead us to suppose that emen = em+n, m and n denoting qua-
ternions; but that is just what Hamilton, the inventor of the
quaternion generalization, denies. There are reasons for believ-
ing that he was mistaken, and that the forms remain equivalent
even under that extreme generalization of m and n; but the
point is this: it is not a question of conventional definition and
formal truth; it is a question of objective definition and real
truth. Let the symbols have the prescribed meaning, does or
does not the equivalence still hold? And if it does not hold,
what is the higher or more complex form which the equivalence
assumes?

AUGUSTUS DE MORGAN*

(1806-1871)

Augustus De Morgan was born in the month of June at
Madura in the presidency of Madras, India; and the year of
his birth may be found by solving a conundrum proposed by
himself, "I was x years of age in the year x2." The problem
is indeterminate, but it is made strictly determinate by the
century of its utterance and the limit to a man's life. His father
was Col. De Morgan, who held various appointments in the service
of the East India Company. His mother was descended from
James Dodson, who computed a table of anti-logarithms, that is,
the numbers corresponding to exact logarithms. It was the time
of the Sepoy rebellion in India, and Col. De Morgan removed
his family to England when Augustus was seven months old.
As his father and grandfather had both been born in India,
De Morgan used to say that he was neither English, nor Scottish,
nor Irish, but a Briton " unattached," using the technical
term applied to an undergraduate of Oxf >rd or Cambridge who
is not a member of any one of the Colleges.

WhenDe Morgan was ten years old, his father died. Mrs.
De Morgan resided at various places in the southwest of England,
and her son received his elementary education at various schools
of no great account. His mathematical talents were unnoticed
till he had reached the age of fourteen. A friend of the family
accidentally discovered him making an elaborate drawing of
a figure in Euclid with ruler and compasses, and explained to
him the aim of Euclid, and gave him an initiation into demon-
stration.

De Morgan suffered from a physical defect — one of his eyes
was rudimentary and useless. As a consequence, he did not

♦This Lecture was delivered April 13, 1901. — Editors.
19

20 TEN BRITISH MATHEMATICIANS

join in the sports of the other boys, and he was even made the
victim of cruel practical jokes by some schoolfellows. Some
psychologists have held that the perception of distance and of
solidity depends on the action of two eyes, but De Morgan
testified that so far as he could make out he perceived with his
one eye distance and solidity just like other people.

He received his secondary education from Mr. Parsons,
a Fellow of Oriel College, Oxford, who could appreciate
classics much better than mathematics. His mother was an
active and ardent member of the Church of England, and
desired that her son should become a clergyman; but by this
time De Morgan had begun to show his non-grooving dispo-
sition, due no doubt to some extent to his physical infirmity.
At the age of sixteen he was entered at Trinity College, Cam-
bridge, where he immediately came under the tutorial influence
of Peacock and Whewell. They became his life-long friends;
from the former he derived an interest in the renovation of
algebra, and from the latter an interest in the renovation of
logic — the two subjects of his future life work.

At college the flute, on which he played exquisitely, was his
recreation. He took no part in athletics but was prominent
in the musical clubs. His love of knowledge for its own sake
interfered with training for the great mathematical race; as
a consequence he came out fourth wrangler. This entitled him
to the degree of Bachelor of Arts; but to take the higher degree
of Master of Arts and thereby become eligible for a fellowship
it was then necessary to pass a theological test. To the sign-
ing of any such test De Morgan felt a strong objection, although
he had been brought up in the Church of England. About
1875 theological tests for academic degrees were abolished in
the Universities of Oxford and Cambridge.

As no career was open to him at his own university, he
decided to go to the Bar, and took up residence in London;
but he much preferred teaching mathematics to reading law.
About this time the movement for founding the London Uni-
versity took shape. The two ancient universities were so
guarded by theological tests that no Jew or Dissenter from the

AUGUSTUS DE MORGAN 21

Church of England could enter as a student; still less be
appointed to any office. A body of liberal-minded men resolved
to meet the difficulty by establishing in London a University
on the principle of religious neutrality. De Morgan, then 22
years of age, was appointed Professor of Mathematics. His
introductory lecture " On the study of mathematics " is a dis-
course upon mental education of permanent value which has
been recently reprinted in the United States.

The London University was a new institution, and the
relations of the Council of management, the Senate of professors
and the body of students were not well defined. A dispute
arose between the professor of anatomy and his students, and in
consequence of the action taken by the Council, several of the
professors resigned, headed by De Morgan. Another professor
of mathematics was appointed, who was accidentally drowned
a few years later. De Morgan had shown himself a prince of
teachers: he was invited to return to his chair, which thereafter
became the continuous center of his labors for thirty years.

The same body of reformers — headed by Lord Brougham,
a Scotsman eminent both in science and politics — who had
instituted the London University, founded about the same time
a Society for the Diffusion of Useful Knowledge. Its object
was to spread scientific and other knowledge by means of cheap
and clearly written treatises by the best writers of the time.
One of its most voluminous and effective writers was De Morgan.
He wrote a great work on The Differential and Integral Calculus
which was published by the Society; and he wrote one-sixth of
the articles in the Penny Cyclopedia, published by the Society,
and issued in penny numbers. When De Morgan came to reside
in London he found a congenial friend in William Frend, not-
withstanding his mathematical heresy- about negative quan-
tities. Both were arithmeticians and actuaries, and their
religious views were somewhat similar. Frend lived in what
was then a suburb of London, in a country-house formerly
occupied by Daniel Defoe and Isaac Watts. De Morgan with
his flute was a welcome visitor; and in 1837 he married Sophia
Elizabeth, one of Frend's daughters.

22 TEN BRITISH MATHEMATICIANS

The London University of which De Morgan was a pro-
fessor was a different institution from the University of London.
The University of London was founded about ten years later
by the Government for the purpose of granting degrees after
examination, without any qualification as to residence. The
London University was affiliated as a teaching college with the
University of London, and its name was changed to University
College. The University of London was not a success as an
examining body; a teaching University was demanded. De
Morgan was a highly successful teacher of mathematics. It was
his plan to lecture for an hour, and at the close of each lecture
to give out a number of problems and examples illustrative
of the subject lectured on; his students were required to sit
down to them and bring him the results, which he looked over
and returned revised before the next lecture. In De Morgan's
opinion, a thorough comprehension and mental assimilation of
great principles far outweighed in importance any merely
analytical dexterity in the application of half-understood prin-
ciples to particular cases.

De Morgan had a son George, who acquired great distinction
in mathematics both at University College and the University
of London. He and another like-minded alumnus conceived
the idea of founding a Mathematical Society in London, where
mathematical papers would be not only received (as by the
Royal Society) but actually read and discussed. The first
meeting was held in University College; De Morgan was the
first president, his son the first secretary. It was the beginning
of the London Mathematical Society. In the year 1866 the
chair of mental philosophy in University College fell vacant.
Dr. Martineau, a Unitarian clergyman and professor of mental
philosophy, was recommended formally by the Senate to the
Council; but in the Council there were some who objected to
a Unitarian clergyman, and others who objected to theistic
philosophy. A layman of the school of Bain and Spencer was
appointed. De Morgan considered that the old standard of
religious neutrality had been hauled down, and forthwith
resigned. He was now 60 years of age. His pupils secured a

AUGUSTUS DE MORGAN 23

pension of $500 for him, but misfortunes followed. Two years
later his son George— the younger Bernoulli, as he loved to
hear him called, in allusion to the two eminent mathematicians
of that name, related as father and son — died. This blow was
followed by the death of a daughter. Five years after his resig-
nation from University College De Morgan died of nervous
prostration on March 18, 187 1, in the 65th year of his age.

De Morgan was a brilliant and witty writer, whether as a
controversialist or as a correspondent. In his time there flour-
ished two Sir William Hamiltons who have often been con-
founded. The one Sir William was a baronet (that is, inherited
the title), a Scotsman, professor of logic and metaphysics in the
University of Edinburgh; the other was a knight (that is, won
the title), an Irishman, professor of astronomy in the University
of Dublin. The baronet contributed to logic the doctrine of the
quantification of the predicate; the knight, whose full name
was William Rowan Hamilton, contributed to mathematics the
geometric algebra called Quaternions. De Morgan was inter-
ested in the work of both, and corresponded with both; but the
correspondence with the Scotsman ended in a public controversy,
whereas that with the Irishman was marked by friendship and
terminated only by death. In one of his letters to Rowan,
De Morgan says, "Be it known unto you that I have discovered
that you and the other Sir W. H. are reciprocal polars with
respect to me (intellectually and morally, for the Scottish
baronet is a polar bear, and you, I was going to say, are a polar
gentleman). When I send a bit of investigation to Edinburgh,
the W. H. of that ilk says I took it from him. When I send
you one, you take it from me, generalize it at a glance, bestow
it thus generalized upon society at large, and make me the second
discoverer of a known theorem."

The correspondence of De Morgan with Hamilton the mathe-
matician extended over twenty-four years; it contains discus-
sions not only of mathematical matters, but also of subjects
of general interest. It is marked by geniality on the part of
Hamilton and by wit on the part of De Morgan. The following
is a specimen: Hamilton wrote, " My copy of Berkeley's work

24 TEN BRITISH MATHEMATICIANS

is not mine; like Berkeley, you know, I am an Irishman." De
Morgan replied, " Your phrase ' my copy is not mine ' is not
a bull. It is perfectly good English to use the same word in
two different senses in one sentence, particularly when there is
usage. Incongruity of language is no bull, for it expresses mean-
ing. But incongruity of ideas (as in the case of the Irishman
who was pulling up the rope, and finding it did not finish, cried
out that somebody had cut off the other end of it) is the genuine
bull."

De Morgan was full of personal peculiarities. We have
noticed his almost morbid attitude towards religion, and the
readiness with which he would resign an office. On the occasion
of the installation of his friend, Lord Brougham, as Rector of
the University of Edinburgh, the Senate offered to confer on
him the honorary degree of LL.D.; he declined the honor as
a misnomer. He once printed his name: Augustus De Morgan,

HOMO • PAUCARUM • L-I-T-E-R-A-R-U-M.

He disliked the country, and while his family enjoyed the sea-
side, and men of science were having a good time at a meeting
of the British Association in the country he remained in the hot
and dusty libraries of the metropolis. He said that he felt
like Socrates, who declared that the farther he got from Athens
the farther was he from happiness. He never sought to become
a Fellow of the Royal Society, and he never attended a meeting
of the Society; he said that he had no ideas or sympathies in
common with the physical philosopher. His attitude was
doubtless due to his physical infirmity, which prevented him
from being either an observer or an experimenter. He never
voted at an election, and he never visited the House of Commons,
or the Tower, or Westminster Abbey.

Were the writings of De Morgan published in the form of
collected works, they would form a small library. We have
noticed his writings for the Useful Knowledge Society. Mainly
through the efforts of Peacock and Whewell, a Philosophical
Society had been inaugurated at Cambridge; and to its Trans-
actions De Morgan contributed four memoirs on the foundations

AUGUSTUS DE MORGAN 25

of algebra, and an equal number on formal logic. The best
presentation of his view of algebra is found in a volume, entitled
Trigonometry and Double Algebra, published in 1849; and his
earlier view of formal logic is found in a volume published in
1847. His most unique work is styled a Budget of Paradoxes]
it originally appeared as letters in the columns of the Athenazum
journal; it was revised and extended by De Morgan in the last
years of his life, and was published posthumously by his widow.
" If you wish to read something entertaining," said Professor
Tait to me, " get De Morgan's Budget of Paradoxes out of the
library." We shall consider more at length his theory of
algebra, his contribution to exact logic, and his Budget of
Paradoxes.

In my last lecture I explained Peacock's theory of algebra.
It was much improved by D. F. Gregory, a younger member
of the Cambridge School, who laid stress not on the permanence
of equivalent forms, but on the permanence of certain formal
laws. This new theory of algebra as the science of symbols
and of their laws of combination was carried to its logical issue
by De Morgan; and his doctrine on the subject is still followed
by English algebraists in general. Thus Chrystal founds his
Textbook of Algebra on De Morgan's theory; although an
attentive reader may remark that he practically abandons it
when he takes up the subject of infinite series. De Morgan's
theory is stated in his volume on Trigonometry and Double
Algebra. In the chapter (of the book) headed " On symbolic
algebra " he writes: " In abandoning the meaning of symbols,
we also abandon those of the words which describe them. Thus
addition is to be, for the present, a sound void of sense. It
is a mode of combination represented by + ; when + receives
its meaning, so also will the word addition. It is most impor-
tant that the student should bear in mind that, with one exception,
no word nor sign of arithmetic or algebra has one atom of mean-
ing throughout this chapter, the object of which is symbols,
and their laws of combination, giving a symbolic algebra which
may hereafter become the grammar of a hundred distinct sig-
nificant algebras. If any one were to assert that -f and —

26 TEN BRITISH MATHEMATICIANS

might mean reward and punishment, and A, B, C, etc., might
stand for virtues and vices, the reader might believe him, or
contradict him, as he pleases, but not out of this chapter. The
one exception above noted, which has some share of meaning,
is the sign = placed between two symbols as in A = B. It indi-
cates that the two symbols have the same resulting meaning,
by whatever steps attained. That A and B, if quantities,
are the same amount of quantity; that if operations, they are
of the same effect, etc."

Here, it may be asked, why does the symbol = prove refrac-
tory to the symbolic theory? De Morgan admits that there
is one exception; but an exception proves the rule, not in the
usual but illogical sense of establishing it, but in the old and
logical sense of testing its validity. If an exception can be
established, the rule must fall, or at least must be modified.
Here I am talking not of grammatical rules, but of the rules
of science or nature.

De Morgan proceeds to give an inventory of the fundamental
symbols of algebra, and also an inventory of the laws of algebra.
The symbols are o, i, -f, — , X, -*-, ()°, and letters; these
only, all others are derived. His inventory of the fundamental
laws is expressed under fourteen heads, but some of them are
merely definitions. The laws proper may be reduced to the
following, which, as he admits, are not all independent of one
another:
I. Law of signs. + + = +, +- = -, - + = -, = +, XX=X,

II. Commutative law. a+b = b+a, ab = ba.

III. Distributive law. a(b+c)=ab+ac.

IV. Index laws. a*XaW+c, (ab)c=abc, (ab)e=aebe. '_
V. a— c = o, a-i-a—i.

The last two may be called the rules of reduction. De Morgan
professes to give a complete inventory of the laws which the
symbols of algebra must obey, for he says, " Any system of
symbols which obeys these laws and no others, except they be
formed by combination of these laws, and which uses the pre-
ceding symbols and no others, except they be new symbols

AUGUSTUS DE MORGAN 27

invented in abbreviation of combinations of these symbols, is
symbolic algebra." From his point of view, none of the above
principles are rules; they are formal laws, that is, arbitrarily
chosen relations to which the algebraic symbols must be subject.
He does not mention the law, which had already been pointed
out by Gregory, namely, (a+b)+c = a+(b+c), (ab)c = a(bc) and
to which was afterwards given the name of the law of association.
If the commutative law fails, the associative may hold good;
but not vice versa. It is an unfortunate thing for the symbolist
or formalist that in universal arithmetic mn is not equal to nm;
for then the commutative law would have full scope. Why
does he not give it full scope? Because the foundations of
algebra are, after all, real not formal, material not symbolic.
To the formalists the index operations are exceedingly refrac-
tory, in consequence of which some take no account of them,
but relegate them to applied mathematics. To give an inventory
of the laws which the symbols of algebra must obey is an impos-
sible task, and reminds one not a little of the task of those
philosophers who attempt to give an inventory of the a priori
knowledge of the mind.

De Morgan's work entitled Trigonometry and Double Algebra
consists of two parts; the former of which is a treatise on
Trigonometry, and the latter a treatise on generalized algebra
which he calls Double Algebra. But what is meant by Double
as applied to algebra? and why should Trigonometry be also
treated in the same textbook? The first stage in the develop-
ment of algebra is arithmetic, where numbers only appear and
symbols of operations such as +, X, etc. The next stage is
universal arithmetic, where letters appear instead of numbers,
so as to denote numbers universally, and the . processes are con-
ducted without knowing the values of the symbols. Let a and
b denote any numbers; then such an expression as a — b may
be impossible; so that in universal arithmetic there is always
a proviso, provided the operation is possible. The third stage is
single algebra, where the symbol may denote a quantity forwards
or a quantity backwards, and is adequately represented by
segments on a straight line passing through an origin. Negative

28

TEN BRITISH MATHEMATICIANS

quantities are then no longer impossible; they are represented
by the backward segment. But an impossibility still remains
in the latter part of such an expression as a+b V — i which arises
in the solution of the quadratic equation. The fourth stage
is double algebra; the algebraic symbol denotes in general a
segment of a line in a given plane; it is a double symbol because
it involves two specifications, namely, length and direction;
and V— i is interpreted as denoting a quadrant. The expres-
sion o+6v— i then represents a line in the plane having an
abscissa a and an ordinate b. Argand and Warren carried
double algebra so far; but they were unable to interpret on
this theory such an expression as eavr=ri. De Morgan attempted
it by reducing such an expression to the form b+qv — i, and
he considered that he had shown that it could be always so
reduced. The remarkable fact is that this double algebra
satisfies all the fundamental laws above enumerated, and as
every apparently impossible combination of symbols has been
interpreted it looks like the complete form of algebra.

If the above theory is true, the next stage of development
ought to be triple algebra and if a+bV — i truly represents a line
in a given plane, it ought to be possible to find a third term which

added to the above would represent a line in space. Argand

and some others guessed that it was a+bV — i-f-cV — iV — i;
although this contradicts the truth established by Euler that
V — iv~l = e~*r. De Morgan and many others worked hard
at the problem, but nothing came of it uutil the problem was
taken up by Hamilton. We now see the reason clearly: the
symbol of double algebra denotes not a length and a direction;
but a multiplier and an angle. In it the angles are confined
to one plane; hence the next stage will be a quadruple algebra,
when the axis of the plane is made variable. And this gives
the answer to the first question; double algebra is nothing but
analytical plane trigonometry, and this is the reason why it
has been found to be the natural analysis for alternating
currents. But De Morgan never got this far; he died
with the belief " that double algebra must remain as the full
development of the conceptions of arithmetic, so far as

AUGUSTUS DE MORGAN 29

those symbols are concerned which arithmetic immediately
suggests."

When the study of mathematics revived at the University
of Cambridge, so also did the study of logic. The moving
spirit was Whewell, the Master of Trinity College, whose prin-
cipal writings were a History of the Inductive Sciences, and
Philosophy of the Inductive Sciences. Doubtless De Morgan
was influenced in his logical investigations by Whewell; but
other contemporaries of influence were Sir W. Hamilton of
Edinburgh, and Professor Boole of Cork. De Morgan's work
on Formal Logic, published in 1847, is principally remarkable
for his development of the numerically definite syllogism. The
followers of Aristotle say and say truly that from two par-
ticular propositions such as Some M's are A's, and Some M's
are 5's nothing follows of necessity about the relation of the
A's and B's. But they go further and say in order that any
relation about the A's and B's may follow of necessity, the mid-
dle term must be taken universally in one of the premises. De
Morgan pointed out that from Most M's are A's and Most M's
are B's it follows of necessity that some A's are B's and he
formulated the numerically definite syllogism which puts this
principle in exact quantitative form. Suppose that the number
of the M's is m, of the M's that are ^4's is a, and of the M's that
are B's is b; then there are at least (a+b—m) A's that area's.
Suppose that the number of souls on board a steamer was 1000,
that 500 were in the saloon, and 700 were lost; it follows of
necessity, that at least 700+500 — 1000, that is, 200, saloon
passengers were lost. This single principle suffices to prove
the validity of all the Aristotelian moods; it is therefore a funda-
mental principle in necessary reasoning.

Here then De Morgan had made a great advance by intro-
ducing quantification of the terms. At that time Sir W. Hamilton
was teaching at Edinburgh a doctrine of the quantification of
the predicate, and a correspondence sprang up. However, De
Morgan soon perceived that Hamilton's quantification was of
a different character; that it meant for example, substituting
the two forms The whole of A is the whole of B, and The whole of

30 TEN BRITISH MATHEMATICIANS

A is a part of B for the Aristotelian form All A's> are 5's.
Philosophers generally have a large share of intolerance; they
are too apt to think that they have got hold of the whole truth,
and that everything outside of their system is error. Hamilton
thought that he had placed the keystone in the Aristotelian
arch, as he phrased it; although it must have been a curious
arch which could stand 2000 years without a keystone. As
a consequence he had no room for De Morgan's innovations.
He accused De Morgan of plagiarism, and the controversy
raged for years in the columns of the Athenceum, and in the
publications of the two writers.

The memoirs on logic which De Morgan contributed to the
Transactions of the Cambridge Philosophical Society subsequent
to the publication of his book on Formal Logic are by far the
most important contributions which he made to the science,
especially his fourth memoir, in which he begins work in the
broad field of the logic of relatives. This is the true field for the
logician of the twentieth century, in which work of the greatest
importance is to be done towards improving language and
facilitating thinking processes which occur all the time in prac-
tical life. Identity and difference are the two relations which
have been considered by the logician; but there are many
others equally deserving of study, such as equality, equivalence,
consanguinity, affinity, etc.

In the introduction to the Budget of Paradoxes De Morgan
explains what he means by the word. " A great many indi-
viduals, ever since the rise of the mathematical method, have,
each for himself, attacked its direct and indirect consequences.
I shall call each of these persons a paradoxer, and his system a
paradox. I use the word in the old sense: a paradox is some-
thing which is apart from general opinion, either in subject
matter, method, or conclusion. Many of the things brought
forward would now be called crotchets, which is the nearest word
we have to old paradox. But there is this difference, that by
calling a thing a crotchet we mean to speak lightly of it; which
was not the necessary sense of paradox. Thus in the 16th
century many spoke of the earth's motion as the paradox of

AUGUSTUS DE MORGAN 31

Copernicus and held the ingenuity of that theory in very high
esteem, and some I think who even inclined towards it. In the
seventeenth century the depravation of meaning took place,
in England at least."

How can the sound paradoxer be distinguished from the false
paradoxer? De Morgan supplies the following test: "The
manner in which a paradoxer will show himself, as to sense
or nonsense, will not depend upon what he maintains, but upon
whether he has or has not made a sufficient knowledge of what
has been done by others, especially as to the mode of doing it,
a preliminary to inventing knowledge for himself. . . . New
knowledge, when to any purpose, must come by contemplation
of old knowledge, in every matter which concerns thought;
mechanical contrivance sometimes, not very often, escapes this
rule. All the men who are now called discoverers, in every
matter ruled by thought, have been men versed in the minds
of their predecessors and learned in what had been before them.
There is not one exception."

I remember that just before the American Association met at
Indianapolis in 1890, the local newspapers heralded a great discov-
ery which was to be laid before the assembled savants — a young
man living somewhere in the country had squared the circle.
While the meeting was in progress I observed a young man going
about with a roll of paper in his hand. He spoke to me and com-
plained that the paper containing his discovery had not been
received. I asked him whether his object in presenting the
paper was not to get it read, printed and published so that
everyone might inform himself of the result; to all of which
he assented readily. But, said I, many men have worked at
this question, and their results have been tested fully, and
they are printed for the benefit of anyone who can read; have
you informed yourself of their results? To this there was no
assent, but the sickly smile of the false paradoxer.

The Budget consists of a review of a large collection of
paradoxical books which De Morgan had accumulated in his
own library, partly by purchase at bookstands, partly from
books sent to him for review, partly from books sent to him by

32 TEN BRITISH MATHEMATICIANS

the authors. He gives the following classification: squarers of
the circle, trisectors of the angle, duplicators of the cube, con-
structors of perpetual motion, subverters of gravitation, stag-
nators of the earth, builders of the universe. You will still
find specimens of all these classes in the New World and in the
new century.

De Morgan gives his personal knowledge of paradoxers. " I
suspect that I know more of the English class than any man in
Britain. I never kept any reckoning: but I know that one
year with another — and less of late years than in earlier time —
I have talked to more than five in each year, giving more than
a hundred and fifty specimens. Of this I am sure, that it is
my own fault if they have not been a thousand. Nobody
knows how they swarm, except those to whom they naturally
resort. They are in all ranks and occupations, of all ages and
characters. They are very earnest people, and their purpose
is bona fide, the dissemination of their paradoxes. A great many
— the mass, indeed — are illiterate, and a great many waste their
means, and are in or approaching penury. These discoverers
despise one another."

A paradoxer to whom De Morgan paid the compliment
which Achilles paid Hector — to drag him round the walls again
and again — was James Smith, a successful merchant of Liver-
pool. He found x = 3§. His mode of reasoning was a curious
caricature of the reductio ad absurdum of Euclid. He said let
ir=2>s, and then showed that on that supposition, every other
value of 7r must be absurd ; consequently t = 3! is the true value.
The following is a specimen of De Morgan's dragging round
the walls of Troy: " Mr. Smith continues to write me long
letters, to which he hints that I am to answer. In his last of
31 closely written sides of note paper, he informs me, with refer-
ence to my obstinate silence, that though I think myself and
am thought by others to be a mathematical Goliath, I have
resolved to play the mathematical snail, and keep within my
shell. A mathematical snail! This cannot be the thing so called
which regulates the striking of a clock; for it would mean that
I am to make Mr. Smith sound the true time of day, which I

AUGUSTUS DE MORGAN 33

would by no means undertake upon a clock that gains 19 seconds
odd in every hour by false quadrative value of tt. But he
ventures to tell me that pebbles from the sling of simple truth
and common sense will ultimately crack my shell, and put
me hors de combat. The confusion of images is amusing:
Goliath turning himself into a snail to avoid tt = 3| and James
Smith, Esq., of the Mersey Dock Board: and put hors de combat
by pebbles from a sling. If Goliath had crept into a snail shell,
David would have cracked the Philistine with his foot. There
is something like modesty in the implication that the crack-shell
pebble has not yet taken effect; it might have been thought
that the slinger would by this time have been singing — And
thrice [and one-eighth] I routed all my foes, And thrice [and one-
eighth] I slew the slain."

In the region of pure mathematics De Morgan could detect
easily the false from the true paradox; but he was not so pro-
ficient in the field of physics. His father-in-law was a para-
doxer, and his wife a paradoxer; and in the opinion of the
physical philosophers De Morgan himself scarcely escaped.
His wife wrote a book describing the phenomena of spiritualism,
table-rapping, table-turning, etc.; and De Morgan wrote a
preface in which he said that he knew some of the asserted
facts, believed others on testimony, but did not pretend to know
whether they were caused by spirits, or had some unknown
and unimagined origin. From this alternative he left out ordi-
nary material causes. Faraday delivered a lecture on Spirit-
ualism, in which he laid it down that in the investigation we
ought to set out with the idea of what is physically possible,
or impossible; De Morgan could not understand this.

SIR WILLIAM ROWAN HAMILTON*

(1805-1865)

William Rowan Hamilton was born in Dublin, Ireland,
on the 3d of August, 1805. His father, Archibald Hamilton,
was a solicitor in the city of Dublin; his mother, Sarah Hutton,
belonged to an intellectual family, but she did not live to exer-
cise much influence on the education of her son. There has
been some dispute as to how far Ireland can claim Hamilton;
Professor Tait of Edinburgh in the Encyclopaedia Brittanica
claims him as a Scotsman, while his biographer, the Rev. Charles
Graves, claims him as essentially Irish. The facts appear to
be as follows: His father's mother was a Scotch woman; his
father's father was a citizen of Dublin. But the name "Hamil-
ton" points to Scottish origin, and Hamilton himself said that his
family claimed to have come over from Scotland in the time
of James I. Hamilton always considered himself an Irishman;
and as Burns very early had an ambition to achieve something
for the renown of Scotland, so Hamilton in his early years had
a powerful ambition to do something for the renown of Ireland.
In later life he used to say that at the beginning of the century
people read French mathematics, but that at the end of it they
would be reading Irish mathematics.

Hamilton, when three years of age, was placed in the charge
of his uncle, the Rev. James Hamilton, who was the curate of
Trim, a country town, about twenty miles from Dublin, and
who was also the master of the Church of England school.
From his uncle he received all his primary and secondary edu-
cation and also instruction in Oriental languages. As a child
Hamilton was a prodigy; at three years of age he was a superior
reader of English and considerably advanced in arithmetic;

* This Lecture was delivered April 16, 1901. — Editors.
34

SIR WILLIAM ROWAN HAMILTON 35

at four a good geographer; at five able to read and translate
Latin, Greek, and Hebrew, and liked to recite Dryden, Collins,
Milton and Homer; at eight a reader of Italian and French
and giving vent to his feelings in extemporized Latin; at ten
a student of Arabic and Sanscrit. When twelve years old he
met Zerah Colburn, the American calculating boy, and engaged
with him in trials of arithmetical skill, in which trials Hamilton
came off with honor, although Colburn was generally the victor.
These encounters gave Hamilton a decided taste for arithmetical
computation, and for many years afterwards he loved to perform
long operations in arithmetic in his mind, extracting the square
and cube root, and solving problems that related to the proper-
ties of numbers. When thirteen he received his initiation into
algebra from Clairault's Algebra in the French, and he made
an epitome, which he ambitiously entitled " A Compendious
Treatise on Algebra by William Hamilton."

When Hamilton was fourteen years old, his father died and
left his children slenderly provided for. Henceforth, as the elder
brother of three sisters, Hamilton had to act as a man. This
year he addressed a letter of welcome, written in the Persian
language, to the Persian Ambassador then on a visit to Dublin;
and he met again Zerah Colburn. In the interval Zerah had
attended one of the great public schools of England. Hamilton
had been at a country school in Ireland, and was now able to
make a successful investigation of the methods by which Zerah
made his lightning calculations. When sixteen, Hamilton
studied the Differential Calculus by the help of a French text-
book, and began the study of the Mecanique celeste of Laplace,
and he was able at the beginning of this study to detect a flaw
in the reasoning by which Laplace demonstrates the theorem of
the parallelogram of forces. This criticism brought him to the
notice of Dr. Brinkley, who was then the professor of astronomy
in the University of Dublin, and resided at Dunkirk, about
five miles from the centre of the city. He also began an inves-
tigation for himself of equations which represent systems of
straight lines in a plane, and in so doing hit upon ideas which
he afterwards developed into his first mathematical memoir to

36 TEN BRITISH MATHEMATICIANS

the Royal Irish Academy. Dr. Brinkley is said to have
remarked of him at this time: "This young man, I do not
say will be, but is, the first mathematician of his age."

At the age of eighteen Hamilton entered Trinity College,
Dublin, the University of Dublin founded by Queen Elizabeth,
and differing from the Universities of Oxford and Cambridge
in having only one college. Unlike Oxford, which has always
given prominence to classics, and Cambridge, which has always
given prominence to mathematics, Dublin at that time gave
equal prominence to classics and to mathematics. In his first
year Hamilton won the very rare honor of optime at his exami-
nation in Homer. In the old Universities marks used to be and
in some cases still are published, descending not in percentages
but by means of the scale of Latin adjectives: optime, valdebene,
bene, satis, mediocriter , vix medi, non; optime means passed
with the very highest distinction; vix means passed but with
great difficulty. This scale is still in use in the medical exami-
nations of the University of Edinburgh. Before entering col-
lege Hamilton had been accustomed to translate Homer into
blank verse, comparing his result with the translations of Pope
and Cowper; and he had already produced some original
poems. In this, his first year he wrote a poem " On college
ambition " which is a fair specimen of his poetical attainments.

Oh! Ambition hath its hour

Of deep and spirit-stirring power;

Not in the tented field alone,

Nor peer-engirded court and throne;

Nor the intrigues of busy life ;

But ardent Boyhood's generous strife,

While yet the Enthusiast spirit turns

Where'er the light of Glory burns,

Thinks not how transient is the blaze,

But longs to barter Life for Praise.

Look round the arena, and ye spy
Pallid cheek and faded eye ;
Among the bands of rivals, few
Keep their native healthy hue:
Night and thought have stolen away

SIR WILLIAM ROWAN HAMILTON 37

Their once elastic spirit's play.
A few short hours and all is o'er,
Some shall win one triumph more;
Some from the place of contest go
Again defeated, sad and slow.

What shall reward the conqueror then

For all his toil, for all his pain,

For every midnight throb that stole

So often o'er his fevered soul?

Is it the applaudings loud

Or wond'ring gazes of the crowd;

Disappointed envy's shame,

Or hollow voice of fickle Fame?

These may extort the sudden smile,

May swell the heart a little while;

But they leave no joy behind,

Breathe no pure transport o'er the mind,

Nor will the thought of selfish gladness

Expand the brow of secret sadness.

Yet if Ambition hath its hour
Of deep and spirit-stirring power,
Some bright rewards are all its own,
And bless its votaries alone:
The anxious friend's approving eye;
The generous rivals' sympathy;
And that best and sweetest prize
Given by silent Beauty's eyes!
These are transports true and strong,
Deeply felt, remembered long:
Time and sorrow passing o'er
Endear their memory but the more.

The "silent Beauty" was not an abstraction, but a young
lady whose brothers were fellow-students of Trinity College.
This led to much effusion of poetry; but unfortunately while
Hamilton was writing poetry about her another young man
was talking prose to her; with the result that Hamilton experi-
enced a disappointment. On account of his self-consciousness,
inseparable probably from his genius, he felt the disappointment
keenly. He was then known to the professor of astronomy,
and walking from the College to the Observatory along the Royal

38 TEN BRITISH MATHEMATICIANS

Canal, he was actually tempted to terminate his life in the
water.

In his second year he formed the plan of reading so as
to compete for the highest honors both in classics and in
mathematics. At graduation two gold medals were awarded,
the one for distinction in classics, the other for distinction in
mathematics. Hamilton aimed at carrying off both. In his
junior year he received an optime in mathematical physics;
and, as the winner"of two optimes, the one in classics, the other
in mathematics, he immediately became a celebrity in the intel-
lectual circle of Dublin.

In his senior year he presented to the Royal Irish Academy
a memoir embodying his research on systems of lines. He now
called it a "Theory of Systems of Rays " and it was printed
in the Transactions. About this time Dr. Brinkley was ap-
pointed to the bishopric of Cloyne, and' in consequence resigned
the professorship of astronomy. In the United Kingdom it is
customary when a post becomes vacant for aspirants to lodge
a formal application with the appointing board and to sup-
plement their own application by testimonial letters from com-
petent authorities. In the present case quite a number of can-
didates appeared, among them Airy, who afterwards became
Astronomer Royal of England, and several Fellows of Trinity
College, Dublin. Hamilton did not become a formal candidate,
but he was invited to apply, with the result that he received
the appointment while still an undergraduate, and not twenty-
two years of age. Thus was his undergraduate career signalized
much more than by the carrying off of the two gold medals.
Before assuming the duties of his chair he made a tour through
England and Scotland, and met for the first time the poet
Wordsworth at his home at Rydal Mount, in Cumberland.
They had a midnight walk, oscillating backwards and forwards
between Rydal and Ambleside, absorbed in converse on high
themes, and finding it almost impossible to part. Wordsworth
afterwards said that Coleridge and Hamilton were the two
most wonderful men, taking all their endowments together,
that he had ever met.

SIR WILLIAM ROWAN HAMILTON 39

In October, 1827, he came to reside at the place which was
destined to be the scene of his scientific labors. I had the
pleasure of visiting it last summer as the guest of his successor.
The Observatory is situated on the top of a hill, Dunsink, about
five miles from Dublin. The house adjoins the observatory;
to the east is an extensive lawn; to the west a garden with stone
wall and shaded walks; to the south a terraced field ; at the foot
of the hill is the Royal Canal; to the southeast the city of
Dublin; while the view is bounded by the sea and the Dublin
and Wicklow Mountains ; a fine home for a poet or a philosopher
or a mathematician, and in Hamilton all three were combined.
Settled at the Observatory he started out diligently as an
observer, but he found it difficult to stand the low temperatures
incident to the work. He never attained skill as an observer,
and unfortunately he depended on a very poor assistant. Him-
self a brilliant computer, with a good observer for assistant,
the work of the observatory ought to have flourished. One of
the first distinguished visitors at the Observatory was the poet
Wordsworth, in commemoration of which one of the shaded
walks in the garden was named Wordsworth's walk. Words-
worth advised him to concentrate his powers on science; and,
not long after, wrote him as follows: "You send me showers
of verses which I receive with much pleasure, as do we all: yet
have we fears that this employment may seduce you from the
path of science which you seem destined to tread with so much
honor to yourself and profit to others. Again and again I must
repeat that the composition of verse is infinitely more of an
art than men are prepared to believe, and absolute success in
it depends upon innumerable minutiae which it grieves me you
should stoop to acquire a knowledge of. . . Again I do venture
to submit to your consideration, whether the poetical parts of
your nature would not find a field more [favorable to their
exercise in the regions of prose; not because those regions are
humbler, but because they may be gracefully and profitably
trod, with footsteps lesscareful and in measures less elaborate."
Hamilton possessed the poetic imagination; what he was
deficient in was the technique of the poet. The imagination

40 TEN BRITISH MATHEMATICIANS

of the poet is kin to the imagination of the mathematician;
both extract the ideal from a mass of circumstances. In this
connection De Morgan wrote: " The moving power of mathe-
tical invention is not reasoning but imagination. We no longer
apply the homely term maker in literal translation of poet; but
discoverers of all kinds, whatever may be their lines, are makers,
or, as we now say, have the creative genius." Hamilton spoke
of the Mecanique analytique of Lagrange as a "scientific poem";
Hamilton himself was styled the Irish Lagrange. Engineers
venerate Rankine, electricians venerate Maxwell; both were
scientific discoverers and likewise poets, thatjs, amateur poets.
The proximate cause of the shower of verses was that Hamilton
had fallen in love for the second time. The young lady was
Miss de Vere, daughter of an accomplished Irish baronet, and
who like Tennyson's Lady Clara Vere de Vere could look back
on a long and illustrious descent. Hamilton had a pupil in
Lord Adare, the eldest son of the Earl of Dunraven, and it was
while visiting Adare Manor that he was introduced to the De
Vere family, who lived near by at Curragh Chase. His suit
was encouraged by the Countess of Dunraven, it was favorably
received by both father and mother, he had written many sonnets
of which Ellen de Vere was the inspiration, he had discussed
with her astronomy, poetry and philosophy; and was on the
eve of proposing when he gave up because the young lady inci-
dentally said to him that "she could not live happily anywhere
but at Curragh." His action shows the working of a too self-
conscious mind, proud of his own intellectual achievements,
and too much awed by her long descent. So he failed for the
second time; but both of these ladies were friends of his to the
last.

At the age of 27 he contributed to the Irish Academy a
supplementary paper on his Theory of Systems of Rays, in which
he predicted the phenomenon of conical refraction; namely,
that under certain conditions a single ray incident on a biaxial
crystal would be broken up into a cone of rays, and likewise
that under certain conditions a single emergent ray would
appear as a cone of rays. The prediction was made by Hamilton

SIR WILLIAM ROWAN HAMILTON 41

on Oct. 22nd; it was experimentally verified by his colleague
Prof. Lloyd on Dec. 14th. It is not experiment alone or
mathematical reasoning alone which has built up the splendid
temple of physical science, but the two working together; and
of this we have a notable exemplification in the discovery of
conical refraction.

Twice Hamilton chose well but failed; now he made another
choice and succeeded. The lady was a Miss Bayly, who
visited at the home of her sister near Dunsink hill. The lady
had serious misgivings about the state of her health; but the
marriage took place. The kind of wife which Hamilton needed
was one who could govern him and efficiently supervise all
domestic matters; but the wife he chose was, from weakness
of body and mind, incapable of doing it. As a consequence,
Hamilton worked for the rest of his life under domestic dif-
ficulties of no ordinary kind.

At the age of 28 he made a notable addition to the theory
of Dynamics by extending to it the idea of a Characteristic
Function, which he had previously applied with success to
the science of Optics in his Theory of Systems of Rays. It
was contributed to the Royal Society of London, and printed
in their Philosophical Transactions. The Royal Society of
London is the great scientific society of England, founded
in the reign of Charles II, and of which Newton was one of
the early presidents ; Hamilton was invited to become a fellow
but did not accept, as he could not afford the expense.

At the age of 29 he read a paper before the Royal Irish
Academy, which set forth the result of long meditation and
investigation on the nature of Algebra as a science; the paper
is entitled " Algebra as the Science of Pure Time." The main
idea is that as Geometry considered as a science is founded
upon the pure intuition of space, so algebra as a science is
founded upon the pure intuition of time. He was never satis-
fied with Peacock's theory of algebra as a " System of Signs
and their Combinations"; nor with De Morgan's improve-
ment of it; he demanded a more real foundation. In reading
Kant's Critique of Pure Reason he was struck by the follow-

42 TEN BRITISH MATHEMATICIANS

ing passage: "Time and space are two sources of knowledge
from which various a priori synthetical cognitions can be
derived. Of this, pure mathematics gives a splendid example
in the case of our cognitions of space and its various relations.
As they are both pure forms of sensuous intuition, they render
synthetical propositions a priori possible." Thus, according
to Kant, space and time are forms of the intellect; and Ham-
ilton reasoned that, as geometry is the science of the former,
so algebra must be the science of the latter. When algebra
is based on any unidimensional subject, such as time, or a
straight line, a difficulty arises in explaining the roots of a
quadratic equation when they are imaginary. To get over
this difficulty Hamilton invented a theory of algebraic couplets,
which has proved a conundrum in the mathematical world.
Some 20 years ago there nourished in Edinburgh a mathematician
named Sang who had computed the most elaborate tables
of logarithms in existence — which still exist in manuscript.
On reading the theory in question he first judged that either
Hamilton was crazy, or else that he (Sang) was crazy, but
eventually reached the more comforting alternative. On the
other hand, Prof. Tait believes in its soundness, and endeavors
to bring it down to the ordinary comprehension.

We have seen that the British Association for the Advance-
ment of Science was founded in 1831, and that its first meeting
was in the ancient city of York. It was a policy of the founders
not to meet in London, but in the provincial cities, so that
thereby greater interest in the advance of science might be
produced over the whole land. The cities chosen for the place
of meeting in following years were the University towns: Ox-
ford, Cambridge, Edinburgh, Dublin. Hamilton was the only
representative of Ireland present at the Oxford meeting; and
at the Oxford, Cambridge, and Edinburgh meetings he not
only contributed scientific papers, but he acquired renown
as a scientific orator. In the case of the Dublin meeting
he was chief organizer beforehand, and chief orator when
it met. The week of science was closed by a grand
dinner given in the library of Trinity College; and an

SIR WILLIAM ROWAN HAMILTON 43

incident took place which is thus described by an American
scientist:

" We assembled in the imposing hall of Trinity Library,
two hundred and eighty feet long, at six o'clock. When the
company was principally assembled, I observed a little stir
near the place where I stood, which nobody could explain,
and which, in fact, was not comprehended by more than two
or three persons present. In a moment, however, I perceived
myself standing near the Lord Lieutenant and his suite, in
front of whom a space had been cleared, and by whom was
Professor Hamilton, looking very much embarrassed. The
Lord Lieutenant then called him by name, and he stepped
into the vacant space. ' I am,' said his Excellency, ' about
to exercise a prerogative of royalty, and it gives me great
pleasure to do it, on this splendid public occasion, which has
brought together so many distinguished men from all parts
of the empire, and from all parts even of the world where
science is held in honor. But, in exercising it, Professor Ham-
ilton, I do not confer a distinction. I but set the royal, and
therefore the national mark on a distinction already acquired
by your genius and labors.' He went on in this way for
three of four minutes, his voice very fine, rich and full; his
manner as graceful and dignified as possible; and his language
and allusions appropriate and combined into very ample flow-
ing sentences. Then, receiving the State sword from one of
his attendants, he said, ' Kneel down, Professor Hamilton ';
and laying the blade gracefully and gently first on one shoulder,
and then on the other, he said, ' Rise up, Sir William Rowan
Hamilton.' The Knight rose, and the Lord Lieutenant then
went up, and with an appearance of great tact in his manner,
shook hands with him. No reply was made. The whole
scene was imposing, rendered so, partly by the ceremony itself,
but more by the place in which it passed, by the body of very
distinguished men who were assembled there, and especially
by the extraordinarily dignified and beautiful manner in which
it was performed by the Lord Lieutenant. The effect at the
time was great, and the general impression was that, as the

44 TEN BRITISH MATHEMATICIANS

honor was certainly merited by him who received it, so the
words by which it was conferred were so graceful and appro-
priate that they constituted a distinction by themselves, greater
than the distinction of knighthood. I was afterwards told
that this was the first instance in which a person had been
knighted by a Lord Lieutenant either for scientific or literary
merit."

Two years after another great honor came to Hamilton
— the presidency of the Royal Irish Academy. While holding
this office, in the year 1843, when 38 years old, he made the
discovery which will ever be considered his highest title to
fame. The story of the discovery is told by Hamilton him-
self in a letter to his son: " On the 16th day of October, which
happened to be a Monday, and Council day of the Royal
Irish Academy, I was walking in to attend and preside, and
your mother was walking with me along the Royal Canal,
to which she had perhaps driven; and although she talked with
me now and then, yet an undercurrent of thought was going
on in my mind, which gave at last a result, whereof it is not
too much to say that I felt at once the importance. An electric
circuit seemed to close; and a spark flashed forth, the herald
(as I foresaw immediately) of many long years to come of
definitely directed thought and work, by myself if spared, and
at all events on the part of others, if I should even be allowed
to live long enough distinctly to communicate the discovery.
Nor could I resist the impulse — unphilosophical as it may
have been — to cut with a knife on a stone of Brougham Bridge,
as we passed it, the fundamental formula with the symbols
ij, k; namely,

i2 —j2 = k2 = ijk = — 1,

which contains the solution of the problem, but of course as
an inscription has long since mouldered away. A more durable
notice remains, however, in the Council Book of the Academy
for that day, which records the fact that I then asked for and
obtained leave to read a paper on Quaternions, at the first
general meeting of the session, which reading took place accord-
ingly on Monday the 13th of November following."

SIR WILLIAM ROWAN HAMILTON 45

Last summer Prof. Joly and I took the walk here described.
We started from the Observatory, walked down the terraced
field, then along the path by the side of the Royal Canal
towards Dublin until we came to the second bridge spanning
the canal. The path of course goes under the Bridge, and
the inner side of the Bridge presents a very convenient surface
for an inscription. I have seen this incident quoted as an
example of how a genius strikes on a discovery all of a sudden.
No doubt a problem was solved then and there, but the problem
had engaged Hamilton's thoughts and researches for fifteen
years. It is rather an illustration of how genius is patience,
or a faculty for infinite labor. What was Hamilton struggling
to do all these years? To emerge from Flatland into Space;
in other words, Algebra had been extended so as to apply to
lines in a plane; but no one had been able to extend it so as to
apply to lines in space. The greatness of the feat is made
evident by the fact that most analysts are still crawling in
Flatland. The same year in which he discovered Quaternions
the Government granted him a pension of £200 per annum
for life, on account of his scientific work.

We have seen how Hamilton gained two optimes, one in clas-
sics, the other in physics, the highest possible distinction in his
college course; how he was appointed professor of astronomy
while yet an undergraduate ; how he was a scientific chief in the
British Association at 27; how he was knighted for his scientific
achievements at 30; how he was appointed president of the
Royal Irish Academy at 32; how he discovered Quaternions
and received a Government pension at 38; can you imagine
that this brilliant and successful genius would fall a victim to
intemperance? About this time at a dinner of a scientific society
in Dublin he lost control of himself, and was so mortified that,
on the advice of friends he resolved to abstain totally. This
resolution he kept for two years; when happening to be a
member of a scientific party at the castle of Lord Rosse, an
amateur astronomer then the possessor of the largest telescope
in existence, he was taunted for sticking to water, particularly
by Airy the Greenwich astronomer. He broke his good reso-

46 TEN BRITISH MATHEMATICIANS

lution, and from that time forward the craving for alcoholic
stimulants clung to him. How could Hamilton with all his
noble aspirations fall into such a vice? The explanation lay-
in the want of order which reigned in his home. He had no
regular times for his meals; frequently had no regular meals
at all, but resorted to the sideboard when hunger compelled
him. What more natural in such condition than that he should
refresh himself with a quaff of that beverage for which Dublin
is famous — porter labelled X3? After Hamilton's death the
dining-room was found covered with huge piles of manuscript,
with convenient walks between the piles; when these literary
remains were wheeled out and examined, china plates with the
relics of food upon them were found between the sheets of
manuscript, plates sufficient in number to furnish a kitchen.
He used to carry on, says his eldest son, long trains of algebraical
and arithmetical calculations in his mind, during which he was
unconscious of the earthly necessity of eating; "we used to bring
in a ' snack ' and leave it in his study, but a brief nod of
recognition of the intrusion of the chop or cutlet was often the
only result, and his thoughts went on soaring upwards."

In 1845 Hamilton attended the second Cambridge meeting
of the British Association; and after the meeting he was lodged
for a week in the rooms in Trinity College which tradition
points out as those in which Sir Isaac Newton composed the
Principia. This incident was intended as a compliment and it
seems to have impressed Hamilton powerfully. He came back
to the Observatory with the fixed purpose of preparing a work
on Quaternions which might not unworthily compare with the
Principia of Newton, and in order to obtain more leisure for this
undertaking he resigned the office of president of the Royal
Irish Academy. He first of all set himself to the preparation
of a course of lectures on Quaternions, which were delivered in
Trinity College, Dublin, in 1848, and were six in number. Among
his hearers were George Salmon, now well known for his highly
successful series of manuals on Analytical Geometry; and Arthur
Cayley, then a Fellow of Trinity College, Cambridge. These
lectures were afterward expanded and published in 1853, under

SIR WILLIAM 'ROWAN HAMILTON 47

the title of Lectures on Quaternions, at the expense of Trinity
College, Dublin. Hamilton had never had much experience
as a teacher; the volume was criticised for diffuseness of style,
and certainly Hamilton sometimes forgot the expositor in the
orator. The book was a paradox — a sound paradox, and of
his experience as a paradoxer Hamilton wrote: "It required a
certain capital of scientific reputation, amassed in former years,
to make it other than dangerously imprudent to hazard the
publication of a work which has, although at bottom quite
conservative, a highly revolutionary air. It was part of the
ordeal through which I had to pass, an episode in the battle
of life, to know that even candid and friendly people secretly
or, as it might happen, openly, censured or
ridiculed me, for what appeared to them
my monstrous innovations." One of these
monstrous innovations was the principle that
ij is not =ji but = — ji; the truth of which
is evident from the diagram. Critics said
that he held that 3X4 is not =4X3; which proceeds on the
assumption that only numbers can be represented by letter
symbols.

Soon after the publication of the Lectures, he became aware
of its imperfection as a manual of instruction, and he set him-
self to prepare a second book on the model of Euclid's Elements.
He estimated that it would fill 400 pages and take two years
to prepare; it amounted to nearly 800 closely printed pages
and took seven years. At times he would work for twelve
hours on a stretch; and he also suffered from anxiety as to the
means of publication. Trinity College advanced £200, he paid
£50 out of his own pocket, but when illness came upon him the
expense of paper and printing had mounted up to £400. He
was seized by an acute attack of gout, from which, after
several months of suffering, he died on Sept. 2, 1865, in the
61st year of his age.

It is pleasant to know that this great mathematician received
during his last illness an honor from the United States, which
made him feel that he had realized the aim of his great labors.

48 TEN BRITISH MATHEMATICIANS

While the war between the North and South was in progress,
the National Academy of Sciences was founded, and the news
which came to Hamilton was that he had been elected one of
ten foreign members, and that his name had been voted to
occupy the specially honorable position of first on the list. Sir
William Rowan Hamilton was thus the first foreign associate
of the National Academy of Sciences of the United States.

As regards religion Hamilton was deeply reverential in
nature. He was born and brought up in the Church of England,
which was then the established Church in Ireland. He lived
in the time of the Oxford movement, and for some time he
sympathized with it; but when several of his friends, among
them the brother of Miss De Vere, passed over into the Roman
Catholic Church, he modified his opinion of the movement and
remained Protestant to the end.

The immense intellectual activity of Hamilton, especially dur-
ing the years when he was engaged on the enormous labor of
writing the Elements of Quaternions, made him a recluse, and
necessarily took away from his power of attending to the prac-
tical affairs of life. Some said that however great a master of
pure time he might be he was not a master of sublunary time.
His neighbors also took advantage of his goodness of heart.
Surrounding the house there is an extensive lawn affording good
pasture, and on it Hamilton pastured a cow. A neighbor advised
Hamilton that his cow would be much better contented by
having another cow for company and bargained with Hamilton
to furnish the companion provided Hamilton paid something
like a dollar per month.

Here is Hamilton's own estimate of himself. " I have
very long admired Ptolemy's description of his great astronomical
master, Hipparchus, as avrjp <£i\o7rovos ko.1 (fn\akridr)$; a labor-
loving and truth-loving man. Be such my epitaph."

Hamilton's family consisted of two sons and one daughter.
At the time of his death, the Elements of Quaternions was
all finished excepting one chapter. His eldest son, William
Edwin Hamilton, wrote a preface, and the volume was pub-
lished at the expense of Trinity College, Dublin. Only 500

SIR WILLIAM ROWAN HAMILTON 49

copies were printed, and many of those were presented. In
consequence it soon became a scarce book, and as much as
$35.00 has been paid for a copy. A new edition, in two volumes,
is now being published by Prof. Joly, his successor in Dunsink
Observatory.

GEORGE BOOLE*

(1815-1864)

George Boole was born at Lincoln, England, on the 2d
of November, 18 15. His father, a tradesman of very limited
means, was attached to the pursuit of science, particularly
of mathematics, and was skilled in the construction of optical
instruments. Boole received his elementary education at the
National School of the city, and afterwards at a commercial
school; but it was his father who instructed him in the elements
of mathematics, and also gave him a taste for the construction
and adaptation of optical instruments. However, his early
ambition did not urge him to the further prosecution of mathe-
mathical studies, but rather to becoming proficient in the
ancient classical languages. In this direction he could receive no
help from his father, but to a friendly bookseller of the neigh-
borhood he was indebted for instruction in the rudiments of
the Latin Grammar. To the study of Latin he soon added
that of Greek without any external assistance; and for some
years he perused every Greek or Latin author that came within
his reach. At the early age of twelve his proficiency in Latin
made him the occasion of a literary controversy in his native
city. He produced a metrical translation of an ode of Horace,
which his father in the pride of his heart inserted in a local
journal, stating the age of the translator. A neighboring
school-master wrote a letter to the journal in which he denied,
from internal evidence, that the version could have been the
work of one so young. In his early thirst for knowledge of
languages and ambition to excel in verse he was like Hamilton,
but poor Boole was much more heavily oppressed by the res
angusta domi — the hard conditions of his home. Accident

♦This Lecture was delivered April 19, 1901. — Editors.
50

GEORGE BOOLE 51

discovered to him certain defects in his methods of classical
study, inseparable from the want of proper early training, and
it cost him two years of incessant labor to correct them.

Between the ages of sixteen and twenty he taught school
as an assistant teacher, first at Doncaster in Yorkshire, after-
wards at Waddington near Lincoln; and the leisure of these
years he devoted mainly to the study of the principal modern
languages, and of patristic literature with the view of studying
to take orders in the Church. This design, however, was not
carried out, owing to the financial circumstances of his parents
and some other difficulties. In his twentieth year he de-
cided on opening a school on his own account in his native
city; thenceforth he devoted all the leisure he could com-
mand to the study of the higher mathematics, and solely with
the aid of such books as he could procure. Without other
assistance or guide he worked his way onward, and it was his
own opinion that he had lost five years of educational progress
by his imperfect methods of study, and the want of a helping
hand to get him over difficulties. No doubt it cost him much
time; but when he had finished studying he was already not
only learned but an experienced investigator.

We have seen that at this time (1835) the great masters
of mathematical analysis wrote in the French language; and
Boole was naturally led to the study of the Mecanique celeste
of Laplace, and the Mecanique analytique of Lagrange. While
studying the latter work he made notes from which there
eventually emerged his first mathematical memoir, entitled,
" On certain theorems in the calculus of variations." By the
same works his attention was attracted to the transformation
of homogeneous functions by linear substitu ions, and in the
course of his subsequent investigations he was led to results
which are now regarded as the foundation of the modern
Higher Algebra. In the publication of his results he received
friendly assistance from D. F. Gregory, a younger member of
the Cambridge school, and editor of the newly founded Cam-
bridge Mathematical Journal. Gregory and other friends sug-
gested that Boole should take the regular mathematical course

52 TEN BRITISH MATHEMATICIANS

at Cambridge, but this he was unable to do; he continued to
teach school for his own support and that of his aged parents,
and to cultivate mathematical analysis in the leisure left by
a laborious occupation.

Duncan F. Gregory was one of a Scottish family already
distinguished in the annals of science. His grandfather was
James Gregory, the inventor of the refracting telescope and
discoverer of a convergent series for r. A cousin of his father
was David Gregory, a special friend and fellow worker of Sir
Isaac Newton. D. F. Gregory graduated at Cambridge, and
after graduation he immediately turned his attention to the
logical foundations of analysis. He had before him Peacock's
theory of algebra, and he knew that in the analysis as devel-
oped by the French school there were many remarkable phe-
nomena awaiting explana on; particularly theorems which
involved what was cal ed the separation of symbols. He
embodied his results in a paper " On the real Nature of sym-
bolical Algebra " which was printed in the Transactions of
the Royal Society of Edinburgh.

Boole became a master of the method of separation of
symbols, and by attempting to apply it to the solution of
differential equations with variable coefficients was led to devise
a general method in analysis. The account of it was printed
in the Transactions of the Royal Society of London, and brought
its author a Royal medal. Boole's study of the separation
of symbols naturally led him to a study of the foundations
of analysis, and he had before him the writings of Peacock,
Gregory and De Morgan. He was led to entertain very wide
views of the domain of mathematical analysis; in fact that it
was coextensive with exact analysis, and so embraced formal
logic. In 1848, as we have seen, the controversy arose be-
tween Hamilton and De Morgan about the quantification of
terms; the general interest which that controversy awoke in
the relation of mathematics to logic induced Boole to prepare
for publication his views on the subject, which he did that
same year in a small volume entitled Mathematical Analysis
of Logic.

GEORGE BOOLE 53

About this time what are denominated the Queen's Colleges
of Ireland were instituted at Belfast, Cork and Galway; and
in 1849 Boole was appointed to the chair of mathematics in
the Queen's College at Cork. In this more suitable environ-
ment he set himself to the preparation of a more elaborate
work on the mathematical analysis of logic. For this pur-
pose he read extensively books on psychology and logic, and
as a result published in 1854 the work on which his fame
chiefly rests — " An Investigation of the Laws of Thought, on
which are founded the mathematical theories of logic and
probabilities." Subsequently he prepared textbooks on Dif-
ferential Equations and Finite Differences; the former of which
remained the best English textbook on its subject until the
publication of Forsyth's Differential Equations.

Prefixed to the Laws of Thought is a dedication to Dr.
Ryall, Vice-President and Professor of Greek in the same
College. In the following year, perhaps as a result of the ded-
ication, he married Miss Everest, the niece of that colleague.
Honors came: Dublin University made him an LL.D., Oxford
a D.C.L.; and the Royal Society of London elected him a
Fellow. But Boole's career was cut short in the midst of his
usefulness and scientific labors. One day in 1864 he walked
from his residence to the College, a distance of two miles, in
a drenching rain, and lectured in wet clothes. The result
was a feverish cold which soon fell upon his lungs and terminated
his career on December 8, 1864, in the 50th year of his age.

De Morgan was the man best qualified to judge of the value
of Boole's work in the field of logic; and he gave it generous
praise and help. In writing to the Dublin Hamilton he said,
" I shall be glad to see his work (Laws of Thought) out, for
he has, I think, got hold of the true connection of algebra and
logic." At another time he wrote to the same as follows:
" Ail metaphysicians except you and I and Boole consider
mathematics as four books of Euclid and algebra up to quad-
ratic equations." We might infer that these three contem-
porary mathematicians who were likewise philosophers would
form a triangle of friends. But it was not so; Hamilton was

54 ' TEN BRITISH MATHEMATICIANS

a friend of De Morgan, and De Morgan a friend of Boole;
but the relation of friend, although convertible, is not neces-
sarily transitive. Hamilton met De Morgan only once in his
life, Boole on the other hand with comparative frequency;
yet he had a voluminous correspondence with the former
extending over 20 years, but almost no correspondence with
the latter. De Morgan's investigations of double algebra and
triple algebra prepared him to appreciate the quaternions,
whereas Boole was too much given over to the symbolic theory
to appreciate geometric algebra.

Hamilton's biography has appeared in three volumes, pre-
pared by his friend Rev. Charles Graves; De Morgan's biography
has appeared in one volume, prepared by his widow; of Boole
no biography has appeared. A biographical notice of Boole
was written for the Proceedings of the Royal Society of London
by his friend the Rev. Robert Harley, and it is to it that I
am indebted for most of my biographical data. Last summer
when in England I learned that the reason why no adequate
biography of Boole had appeared was the unfortunate temper
and lack of sound judgment of his widow. Since her hus-
band's death Mrs. Boole has published a paradoxical book
of the false kind worthy of a notice in De Morgan's Budget.

The work done by Boole in applying mathematical analysis
to logic necessarily led him to consider the general question
of how reasoning is accomplished by means of symbols. The
view which he adopted on this point is stated at page 68 of the
Laws of Thought. " The conditions of valid reasoning by the
cid of symbols, are : First, that a fixed interpretation be assigned
to the symbols employed in the expression of the data; and
that the laws of the combination of those symbols be correctly
determined from that interpretation; Second, that the formal
processes of solution or demonstration be conducted throughout
in obedience to all the laws determined as above, without
regard to the question of the interpretability of the particular
results obtained; Third, that the final result be interpretable
in form, and that it be actually interpreted in accordance
with that system of interpretation which has been employed

GEORGE BOOLE 55

in the expression of the data." As regards these conditions
it may be observed that they are very different from the for-
malist view of Peacock and De Morgan, and that they incline
towards a realistic view of analysis, as held by Hamilton. True
he speaks of interpretation instead of meaning, but it is a
fixed interpretation; and the rules for the processes of solution
are not to be chosen arbitrarily, but are to be found out from
the particular system of interpretation of the symbols.

It is Boole's second condition which chiefly calls for study
and examination; respecting it he observes as follows: "The
principle in question may be considered as resting upon a gen-
eral law of the mind, the knowledge of which is not given to
us a priori, that is, antecedently to experience, but is derived,
like the knowledge of the other laws of the mind, from the
clear manifestation of the general principle in the particular
instance. A single example of reasoning, in which symbols
are employed in obedience to laws founded upon their inter-
pretation, but without any sustained reference to that inter-
pretation, the chain of demonstration conducting us through
intermediate steps which are not interpretable to a final result
which is interpretable, seems not only to establish the validity
of the particular application, but to make known to us the
general law manifested therein. No accumulation of instances
can properly add weight to such evidence. It may furnish
us with clearer conceptions of that common element of truth
upon which the application of the principle depends, and so
prepare the way for its reception. It may, where the imme-
diate force of the evidence is not felt, serve as a verification,
a posteriori, of the practical validity of the principle in ques-
tion. But this does not affect the position affirmed, viz., that
the general principle must be seen in the particular instance —
seen to be general in application as well as true in the special
example. The employment of the uninterpretable symbol VT^"
in the intermediate processes of trigonometry furnishes an
illustration of what has been said. I apprehend that there is
no mode of explaining that application which does not covertly
assume the very principle in question. But that principle,

56 TEN BRITISH MATHEMATICIANS

though not, as I conceive, warranted by formal reasoning
based upon other grounds, seems to deserve a place among those
axiomatic truths which constitute in some sense the foundation
of general knowledge, and which may properly be regarded
as expressions of the mind's own laws and constitution."

We are all familiar with the fact that algebraic reasoning
may be conducted through intermediate equations without
requiring a sustained reference to the meaning of these equa-
tions; but it is paradoxical to say that these equations can, in
any case, have no meaning or interpretation. It may not be
necessary to consider their meaning, it may even be difficult
to find their meaning, but that they have a meaning is a dic-
tate of common sense. It is entirely paradoxical to say that,
as a general process, we can start from equations having a
meaning, and arrive at equations having a meaning by passing
through equations which have no meaning. The particular
instance in which Boole sees the truth of the paradoxical prin-
ciple is the successful employment of the uninterpretable sym-
bol V — i in the intermediate processes of trigonometry. So
soon then as this symbol is interpreted, or rather, so soon as
its meaning is demonstrated, the evidence for the principle
fails, and Boole's transcendental logic falls.

In the algebra of quantity we start from elementary symbols
denoting numbers, but are soon led to compound forms which
do not reduce to numbers; so in the algebra of logic we start
from elementary symbols denoting classes, but are soon intro-
duced to compound expressions which cannot be reduced to
simple classes. Most mathematical logicians say, Stop, we do
not know what this combination means. Boole says, It may
be meaningless, go ahead all the same. The design of the Laws
of Thought is stated by the author to be to investigate the
fundamental laws of those operations of the mind by which
reasoning is performed; to give expression to them in the sym-
bolical language 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,

GEORGE BOOLE 57

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.

Boole's inventory of the symbols required in the algebra of
logic is as follows: first, Literal symbols, as x, y, etc., representing
things as subjects of our conceptions; second, Signs of operation,
as +, — , X, standing for those operations of the mind by which
the conceptions of things are combined or resolved so as to form
new conceptions involving the same elements; third, The sign
of identity = ; not equality merely, but identity which involves
equality. The symbols x, y, etc., are used to denote classes;
and it is one of Boole's maxims that substantives and adjectives
alike denote classes. " They may be regarded," he says, " as
differing only in this respect, that the former expresses the sub-
stantive existence of the individual thing or things to which
it refers, the latter implies that existence. If we attach to the
adjective the universally understood subject, " being " or
" thing," it becomes virtually a substantive, and may for all the
essential purposes of reasoning be replaced by the substantive."
Let us then agree to represent the class of individuals to which a
particular name is applicable by a single letter as x. If the
name is men for instance, Jet x represent all men, or the class
men. Again, if an adjective, as good, is employed as a term of
description, let us represent by a letter, as y, all things to which
the description good is applicable, that is, all good things or the
class good things. Then the combination yx will represent
good men.

Boole's symbolic logic was brought to my notice by Pro-
fessor Tait, when I was a student in the physical laboratory of
Edinburgh University. I studied the Laws of Thought and I
found that those who had written on it regarded the method
as highly mysterious; the results wonderful, but the processes
obscure. I reduced everything to diagram and model, and I
ventured to publish my views on the subject in a small volume
called Principles of the Algebra of Logic; one of the chief points
I made is the philological and analytical difference between
the substantive and the adjective. What I said was that the

58 TEN BRITISH MATHEMATICIANS

word man denotes a class, but the word white does not; in the
former a definite unit-object is specified, in the latter no unit-
object is specified. We can exhibit a type of a man, we cannot
exhibit a type of a white.

The identification of the substantive and adjective on
the one hand and their discrimination on the ether hand,
lead to different conceptions of what De Morgan called the
universe. Boole's conception of the Universe is as follows
(Laws of Thought, p. 42) : "In every discourse, whether of
the mind conversing with its own thoughts, or of the indi-
vidual in his intercourse with others, there is an assumed or
expressed limit within which the subjects of its operation are
confined. The most unfettered discourse is that in which the
words we use are understood in the widest possible application,
and for them the limits of discourse are coextensive with those
of the universe itself. But more usually we confine ourselves
to a less spacious field. Sometimes in discoursing of men we
imply (without expressing the limitation) that it is of men only
under certain circumstances and conditions that we speak, as
of civilized men, or of men in the vigor of life, or of men under
some other condition or relation. Now, whatever may be the
extent of the field within which all the objects of our discourse
are found, that field may properly be termed the universe of
discourse."

Another view leads to the conception of the Universe as a
collection of homogeneous units, which may be finite or infinite
in number; and in a particular problem the mind considers the
relation of identity between different groups of this collection.
This universe corresponds to the series of events, in the theory
of Probability; and the characters correspond to the different
ways in which the event may happen. The difference is that
the Algebra of Logic considers necessary data and relations;
while the theory of Probability considers probable data and
relations. I will explain the elements of Boole's method on this
theory.

The square is a collection of points: it may serve to represent
any collection of homogeneous units, whether finite or infinite

GEORGE BOOLE

59

Fig. i.

in number, that is, the universe of the problem. Let x denote

inside the left-hand circle, and y inside the right-hand circle.

Uxy will denote the points inside both circles

(Fig. i). In arithmetical value x may range from

i to o; so also y; while xy cannot be greater than

x or y, or less than o or x+y—i. This last is the

principle of the syllogism. From the co-ordinate

nature of the operations x and y, it is evident

that Uxy = Uyx; but this is a different thing from commuting,

as Boole does, the relation of U and x, which is not that of

co-ordination, but of subordination of x to U, and which is

properly denoted by writing U first.

Suppose y to be the same character as x; we will then always
have Uxx=Ux; that is, an elementary selective symbol x is
always such that x2 = x. These are but the symbols of ordinary
algebra which satisfy this relation, namely i and o; these are
also the extreme selective symbols all and none. The law in
question was considered Boole's paradox; it plays a very great
part in the development of his method.

Let Uxy = TJz, where = means identical with,
not equal to; we may write xy = z, leaving the U
to be understood. It does not mean that the
combination of characters xy is identical with the
character z; but that those points which have the
characters x and y are identical with the points
which have the character z (Fig. 2). From xy = z, we derive

x = -z\ what is the meaning of this expression? We shall return

to the question, after we have considered -f and — .

Let us now consider the expression U{x-\-y).
If the x points and the y points are outside of one
another, it means the sum of the x points and the
y points (Fig. 3). So far all are agreed. But
suppose that the x points and the y points are
partially identical (Fig. 4); then there arises
difference of opinion. Boole held that the common points must
be taken twice over, or in other words that the symbols x and

Fig. 2.

Fig. 3.

60 TEN BRITISH MATHEMATICIANS

y must be treated all the same as if they were independent of
one another; otherwise, he held, no general analysis is possible.
U(x+y) will not in general denote a single class
of points; it will involve in general a duplica-
tion.

Similarly, Boole held that the expression
U(x—y) does not involve the condition of the
FlG-4- Uy being wholly included in the Ux (Fig. 5).

If that condition is satisfied, U(x-y) denotes a simple class;
namely, the Z7x's without the Uy's. But when there is partial
coincidence (as in Fig. 4), the common points will be cancelled,
and the result will be the Ux's which are not y taken posi-
tively and the Uy's which are not x taken negatively. In
Boole's view U(x—y) was in general an intermediate uninter-
pretable form, which might be used in reasoning the same way
as analysts used V — 1.

Most of the mathematical logicians who have come after
Boole are men who would have stuck at the impossible sub-
traction in ordinary algebra. They say virtually, " How can
you throw into a heap the same things twice over; and how
can you take from a heap things that are not there." Their
great principle is the impossibility of taking the pants from a
Highlander. Their only conception of the analytical processes
of addition and subtraction is throwing into a heap and taking
out of a heap. It does not occur to them that the processes of
algebra are ideal, and not subject to gross material restrictions.
If x-\-y denotes a quality without duplication, it will sat-
isfy the condition

(x-\-y)2 = x+y,

x2-\-2xy -\-y2 = x -\-y,
but x2=x, y2 = y,

2xy = o.
Similarly, if x— y denote a simple quality, then

(x — y)2 = x— y,
x2 -\-y2 ~-2xy =x—y,

x2=x, y2z=y,

GEORGE BOOLE 61

therefore, y — 2xy=—y,

.'. y = xy.

In other words, the Uy must be included in the Ux (Fig. 5).
Here we have assumed that the law of signs is
the same as in ordinary algebra, and the result
comes out correct.

Suppose Uz - Uxy ; then Ux = U-z. How are

y

the Ux's related to the Uy's and the Uz's? From FlG- s<

the diagram (in Fig. 2) we see that the Ux's are identical with
all the Uyz's together with an indefinite portion of the U's, which
are neither y nor z. Boole discovered a general method for
finding the meaning of any function of elementary logical
symbols, which applied to the above case, is as follows:
When y is an elementary symbol,

j^y + fr-y).

Similarly 1 = z + ( 1 — z) .

/. 1 =yz+y(i -z) + (1 -y)z+(i -y)(i -z),

which means that the Z7's either have both qualities y and 2,
or y but not z, or z but not y, or neither y and 2. Let

-z = Ayz+By(i -z) +C(i -y)z+D{i - y)(i -z),

y

it is required to determine the coefficients A, B, C, D. Sup-
pose y = i, 3 = 1; theni=^4. Suppose y = i, 2=0, then 0 = B.

Suppose y = o, 2 = 1; then - = C, and C is infinite; therefore

o

(1— v)2 = o; which we see to be true from the diagram. Sup-
pose y = o, z = o; then - = D, or D is indeterminate. Hence
o

-2 = ^2 +an indefinite portion of (1—3;) (1—2).
******

Boole attached great importance to the index law x2 = x.
He held that it expressed a law of thought, and formed the

62 TEN BRITISH MATHEMATICIANS

characteristic distinction of the operations of the mind in its
ordinary discourse and reasoning, as compared with its oper-
ations when occupied with the general algebra of quantity.
It makes possible, he said, the solution of a quintic or equation
of higher degree, when the symbols are logical. He deduces from
it the axiom of metaphysicians which is termed the principle
of contradiction, and which affirms that it is impossible for
any being to possess a quality, and at the same time not to
possess it. Let x denote an elementary quality applicable
to the universe U; then i— x denotes the absence of that
quality. But if x2 = x, then o=x — x2, o = x(i—x), that is,
from Ux2 = Ux we deduce Ux(i —x) =o.

He considers x(i— #)=o as an expression of the prin-
ciple of contradiction. He proceeds to remark: "The above
interpretation has been introduced not on account of its imme-
diate value in the present system, but as an illustration of a
significant fact in the philosophy of the intellectual powers,
viz., that what has been commonly regarded as the funda-
mental axiom of metaphysics is but the consequence of a law
of thought, mathematical in its form. I desire to direct atten-
tion also to the circumstance that the equation in which that
fundamental law of thought is expressed is an equation of
the second degree. Without speculating at all in this chapter
upon the question whether that circumstance is necessary in
its own nature, we may venture to assert that if it had not
existed, the whole procedure of the understanding would have
been different from what it is."

We have seen that De Morgan investigated long and
published much on mathematical logic. His logical writings
are characterized by a display of many symbols, new alike
to logic and to mathematics; in the words of Sir W. Hamilton
of Edinburgh, they are " horrent with mysterious spiculae."
It was the great merit of Boole's work that he used the immense
power of the ordinary algebraic notation as an exact language,
and proved its power for making ordinary language more
exact. De Morgan could well appreciate the magnitude of
the feat, and he gave generous testimony to it as follows;

GEORGE BOOLE 63

" Boole's system of logic is but one of many proofs of genius
and patience combined. I might legitimately have entered
it among my paradoxes, or things counter to general opinion:
but it is a paradox which, like that of Copernicus, excited
admiration from its first appearance. That the symbolic
processes of algebra, invented as tools of numerical calcula-
tion, should be competent to express every act of thought,
and to furnish the grammar and dictionary of an all-containing
system of logic, would not have been believed until it was
proved. When Hobbes, in the time of the Commonwealth,
published his " Computation or Logique " he had a remote
glimpse of some of the points which are placed in the light of
day by Mr. Boole. The unity of the forms of thought in all
the applications of reason, however remotely separated, will
one day be matter of notoriety and common wonder: and
Boole's name will be remembered in connection with one of
the most important steps towards the attainment of this
knowledge."

ARTHUR CAYLEY*
(1821-1895)

Arthur Cayley was born at Richmond in Surrey, England,
on August 16, 182 1. His father, Henry Cayley, was descended
from an ancient Yorkshire family, but had settled in St. Peters-
burg, Russia, as a merchant. His mother was Maria Antonia
Doughty, a daughter of William Doughty; who, according
to some writers, was a Russian; but her father's name indi-
cates an English origin. Arthur spent the first eight years of
his life in St. Petersburg. In 1829 his parents took up their
permanent abode at Blackheath, near London; and Arthur
was sent to a private school. He early showed great liking
for, and aptitude in, numerical calculations. At the age of
14 he was sent to King's College School, London; the master
of which, having observed indications of mathematical genius,
advised the father to educate his son, not for his own business,
as he had at first intended, but to enter the University of
Cambridge.

At the unusually early age of 17 Cayley began residence
at Trinity College, Cambridge. As an undergraduate he had
generally the reputation of being a mere mathematician;
his chief diversion was novel-reading. He was also fond of
travelling and mountain climbing, and was a member of
the Alpine Club. The cause of the Analytical Society had
now triumphed, and the Cambridge Mathematical Journal
had been instituted by Gregory and Leslie Ellis. To this
journal, at the age of twenty, Cayley contributed three
papers, on subjects which had been suggested by reading the
Mecanique analytique of Lagrange and some of the works of
Laplace. We have already noticed how the works of Lagrange

* This Lecture was delivered April 20, 1901. — Editors.
64

ARTHUR CAYLEY

65

and Laplace served to start investigation in Hamilton and Boole.
Cayley finished his undergraduate course by winning the place
of Senior Wrangler, and the first Smith's prize. His next step
was to take the M.A. degree, and win a Fellowship by com-
petitive examination. He continued to reside at Cambridge
for four years; during which time he took some pupils, but his
main work was the preparation of 28 memoirs to the Mathe-
matical Journal. On account of the limited tenure of his
fellowship it was necessary to choose a profession; like De
Morgan, Cayley chose the law, and at 25 entered at Lincoln's
Inn, London. He made a specialty of conveyancing and became
very skilled at the work; but he regarded his legal occupation
mainly as the means of providing a livelihood, and he reserved
with jealous care a due portion of his time for mathematical
research. It was while he was a pupil at the bar that he went
over to Dublin for the express purpose of hearing Hamilton's
lectures on Quaternions. He sat alongside of Salmon (now
provost of Trinity College, Dublin) and the readers of Salmon's
books on Analytical Geometry know how much their author
was indebted to his correspondence with Cayley in the matter
of bringing his textbooks up to date. His friend Sylvester,
his senior by five years at Cambridge, was then an actuary,
resident in London; they used to walk together round the
courts of Lincoln's Inn, discussing the theory of invariants and
covariants. During this period of his life, extending over four-
teen years, Cayley produced between two and three hundred
papers.

At Cambridge University the ancient professorship of pure
mathematics is denominated the Lucasian, and is the chair
which was occupied by Sir Isaac Newton. About i860 certain
funds bequeathed by Lady Sadleir to the University, having
become useless for their original purpose, were employed to
establish another professorship of pure mathematicas, called
the Sadlerian. The duties of the new professor were defined
to be " to explain and teach the principles of pure mathematics
and to apply himself to the advancement of that science." To
this chair Cayley was elected when 42 years old. He gave

63 TEN BRITISH MATHEMATICIANS

up a lucrative practice for a modest salary; but he never
regretted the exchange, for the chair at Cambridge enabled
him to end the divided allegiance between law and mathematics,
and to devote his energies to the pursuit which he liked best.
He at once married and settled down in Cambridge. More
fortunate than Hamilton in his choice, his home life was one
of great happiness. His friend and fellow investigator, Sylvester,
once remarked that Cayley had been much more fortunate
than himself; that they both lived as bachelors in London,
but that Cayley had married and settled down to a quiet and
peaceful life at Cambridge; whereas he had never married,
and had been fighting the world all his days. The remark was
only too true (as may be seen in the lecture on Sylvester).

At first the teaching duty of the Sadlerian professorship was
limited to a course of lectures extending over one of the terms
of the academic year; but when the University was reformed
about 1886, and part of the college funds applied to the better
endowment of the University professors, the lectures were
extended over two terms. For many years the attendance was
small, and came almost entirely from those who had finished
their career of preparation for competitive examinations; after
the reform the attendance numbered about fifteen. The subject
lectured on was generally that of the memoir on which the
professor was for the time engaged.

The other duty of the chair — the advancement of mathe-
matical science — was discharged in a handsome manner by the
long series of memoirs which he published, ranging over every
department of pure mathematics. But it was also discharged
in a much less obtrusive way; he became the standing referee
on the merits of mathematical papers to many societies
both at home and abroad. Many mathematicians, of whom
Sylvester was an example, find it irksome to study what others
have written, unless, perchance, it is something dealing directly
with their own line of work. Cayley was a man of more cos-
mopolitan spirit; he had a friendly sympathy with other workers,
and especially with young men making their first adventure in
the field of mathematical research. Of referee work he did an

ARTHUR CAYLEY" 67

immense amount; and of his kindliness to young investigators
I can speak from personal experience. Several papers which
I read before the Royal Society of Edinburgh on the Analysis
of Relationships were referred to him, and he recommended
their publication. Soon after I was invited by the Anthropo-
logical Society of London to address them on the subject,
and while there, I attended a meeting of the Mathematical
Society of London. The room was small, and some twelve
mathematicians were assembled round a table, among whom
was Prof. Cayley, as became evident to me from the proceed-
ings. At the close of the meeting Cayley gave me a cordial
handshake and referred in the kindest terms to my papers
which he had read. He was then about 60 years old, con-
siderably bent, and not filling his clothes. What was most
remarkable about him was the active glance of his gray eye?
and his peculiar boyish smile.

In 1876 he published a Treatise on Elliptic Functions, which
was his only book. He took great interest in the movement
for the University education of women. At Cambridge the
women's colleges are Girton and Newnham. In the early days
of Girton College he gave direct help in teaching, and for some
years he was chairman of the council of Newnham College,
in the progress of which he took the keenest interest to the
last. His mathematical investigations did not make him a
recluse; on the contrary he was of great practical usefulness,
especially from his knowledge of law, in the administration of
the University.

' In 1872 he was made an honorary fellow of Trinity College,
and three years later an ordinary fellow, which meant stipend
as well as honor. About this time his friends subscribed for
a presentation portrait, which now hangs on the side wall
of the dining hal'. of Trinity College, next to the portrait of
James Clerk Maxwell, while on the end wall, behind the
high table, hang the more ancient portraits of Sir Isaac New-
ton and Lord Bacon of Verulam. In the portrait Cayley is
represented as seated at a desk, quill in hand, after the mode
in which he used to write out his mathematical investigations.

68 TEN BRITISH MATHEMATICIANS

The investigation, however, was all thought out in his mind
before he took up the quill.

Maxwell was one of the greatest electricians of the nine-
teenth century. He was a man of philosophical insight and
poetical power, not unlike Hamilton, but differing in this, that
he was no orator. In that respect he was more like Gold-
smith, who " could write like an angel, but only talked like
poor poll." Maxwell wrote an address to the committee of
subscribers who had charge of the Cayley protrait fund, wherein
the scientific poet with his pen does greater honor to the mathe-
matician than the artist, named Dickenson, could do with his
brush. Cayley had written on space of n dimensions, and the
main point in the address is derived from the artist's business
of depicting on a plane what exists in space :

O wretched race of men, to space confined!

What honor can ye pay to him whose mind

To that which lies beyond hath penetrated?

The symbols he hath formed shall sound his praise,

And lead him on through unimagined ways

To conquests new, in worlds not yet created.

First, ye Determinants, in ordered row
And massive column ranged, before him go,
To form a phalanx for his safe protection.
Ye powers of the wth root of — i !
Around his head in endless cycles run,
As unembodied spirits of direction.

And you, ye undevelopable scrolls!

Above the host where your emblazoned rolls,

Ruled for the record of his bright inventions.

Ye cubic surfaces ! by threes and nines

Draw round his camp your seven and twenty lines

The seal of Solomon in three dimensions.

March on, symbolic host! with step sublime,
Up to the flaming bounds of Space and Time!
There pause, until by Dickenson depicted
In two dimensions, we the form may trace
Of him whose soul, too large for vulgar space,
In n dimensions flourished unrestricted.

ARTHUR CAYLEY 69

The verses refer to the subjects investigated in several of
Cayley's most elaborate memoirs; such as, Chapters on the
Analytical Geometry of n dimensions; On the theory of De-
terminants; Memoir on the theory of Matrices; Memoirs
on skew surfaces, otherwise Scrolls; On the delineation of a
Cubic Scroll, etc.

In 1 88 1 he received from the Johns Hopkins University,
Baltimore, where Sylvester was then professor of mathematics,
an invitation to deliver a course of lectures. He accepted
the invitation, and lectured at Baltimore during the first five
months of 1882 on the subject of the Abelian and Theta Functions.

The next year Cayley came prominently before the world,
as President of the British Association for the Advancement
of Science. The meeting was held at Southport, in the north
of England. As the President's address is one of the great
popular events of the meeting, and brings out an audience of
general culture, it is usually made as little technical as pos-
sible. Hamilton was the kind of mathematician to suit such
an occasion, but he never got the office, on account of his
occasional breaks. Cayley had not the oratorical, the philo-
sophical, or the poetical gifts of Hamilton, but then he was an
eminently safe man. He took for his subject the Progress of
Pure Mathematics; and he opened his address in the following
naive manner: " I wish to speak to you to-night upon Mathe-
matics. I am quite aware of the difficulty arising from the
abstract nature of my subject; and if, as I fear, many or some
of you, recalling the providential addresses at former meetings,
should wish that you were now about to have from a different
President a discourse on a different subject, I can very well
sympathize with you in the feeling. But be that as it may,
I think it is more respectful to you that I should speak to you
upon and do my best to interest you in the subject which has
occupied me, and in which I am myself most interested. And
in another point of view, I think it is right that the address
of a president should be on his own subject, and that different
subjects should be thus brought in turn before the meetings.
So much the worse, it may be, for a particular meeting: but

70 TEN BRITISH MATHEMATICIANS

the ii.ee ting is the individual, which on evolution principles,
must be sacrificed for the development of the race." I dare-
say that after this introduction, all the evolution philosophers
listened to him attentively, whether they understood him or
not. But Cayley doubtless felt that he was addressing not
only the popular audience then and there before him, but the
mathematicians of distant places and future times; for the
address is a valuable historical review of various mathematical
theories, and is characterized by freshness, independence of
view, suggestiveness, and learning.

In 1889 the Cambridge University Press requested him to
prepare his mathematical papers for publication in a collected
form — a request which he appreciated very much. They are
printed in magnificent quarto volumes, of which seven appeared
under his own editorship. While editing these volumes, he
was suffering from a painful internal malady, to which he
succumbed on January 26, 1895, ^n the 74th year of his age.
When the funeral took place, a great assemblage met in Trinity
Chapel, comprising members of the University, official rep-
resentatives of Russia and America, and many of the most
illustrious philosophers of Great Britain.

The remainder of his papers were edited by Prof. Forsyth,
his successor in the Sadlerian chair. The Collected Mathe-
matical papers number thirteen quarto volumes, and con-
tain 967 papers. His writings are his best monument, and
certainly no mathematician has ever had his monument in
grander style. De Morgan's works would be more extensive,
and much more useful, but he did not have behind him a
University Press. As regards fads, Cayley retained to the
last his fondness for novel-reading and for travelling. He
also took special pleasure in paintings and architecture, and
he practised water-color painting, which he found useful some-
times in making mathematical diagrams.

To the third edition of Tait's Elementary Treatise on Qua-
ternions, Cayley contributed a chapter entitled " Sketch of
the analytical theory of quaternions." In it the V~^i re-
appears in all its glory, and in entire, so it is said, independence

ARTHUR CAYLEY 71

of i,j, k. The remarkable thing is that Hamilton started with
a quaternion theory of analysis, and that Cayley should present
instead an analytical theory of quaternions. I daresay that
Prof. Tait was sorry that he allowed the chapter to enter
his book, for in 1894 there arose a brisk discussion between
himself and Cayley on " Coordinates versus Quaternions,"
the record of which is printed in the Proceedings of the Royal
Society of Edinburgh. Cayley maintained the position that
while coordinates are applicable to the whole science of geom-
etry and are the natural and appropriate basis and method
in the science, quaternions seemed a particular and very arti-
ficial method for treating such parts of the science of three-
dimensional geometry as are most naturally discussed by means
of the rectangular coordinates x, y, z. In the course of his
paper Cayley says: " I have the highest admiration for the
notion of a quaternion; but, as I consider the full moon far
more beautiful than any moonlit view, so I regard the notion
of a quaternion as far more beautiful than any of its applica-
tions. As another illustration, I compare a quaternion formula
to a pocket-map — a capital thing to put in one's pocket, but
which for use must be unfolded: the formula, to be under-
stood, must be translated into coordinates." He goes on
to say, " I remark that the imaginary of ordinary algebra —
for distinction call this 0 — has no relation whatever to the
quaternion symbols i, j, k; in fact, in the general point of
view, all the quantities which present themselves, are, or may
be, complex values a + db, or in other words, say that a scalar
quantity is in general of the form a-\-6b. Thus quaternions do
not properly present themselves in plane or two-dimensional
geometry at all; but they belong essentially to solid or three-
dimensional geometry, and they are most naturally applicable
to the class of problems which in coordinates are dealt with
by means of the three rectangular coordinates x, y, z."

To the pocketbook illustration it may be replied that a set
of coordinates is an immense wall map, which you cannot carry
about, even though you should roll it up, and therefore is
useless for many important purposes. In reply to the argu-

72 TEN BRITISH MATHEMATICIANS

ments, it may be said, first, V^T has a relation to the symbols
*, /, k, for each of these can be analyzed into a unit axis mul-
tiplied by V — i; second, as regards plane geometry, the
ordinary form of complex quantity is a degraded form of the
quaternion in which the constant axis of the plane is left un-
specified. Cayley took his illustrations from his experience as
a traveller. Tait brought forward an illustration from which
you might imagine he had visited the Bethlehem lion
Works, and hunted tigers in India. He says, " A much more
natural and adequate comparison would, it seems to me,
liken Coordinate Geometry to a steam-hammer, which an
expert may employ on any destructive or constructive work of
one general kind, say the cracking of an eggshell, or the weld-
ing of an anchor. But you must have your expert to manage
it, for without him it is useless. He has to toil amid the heat,
smoke, grime, grease, and perpetual din of the suffocating
engine-room. The work has to be brought to the hammer,
for it cannot usually be taken to its work. And it is not in
general, transferable; for each expert, as a rule, knows, fully
and confidently, the working details of his own weapon only.
Quaternions, on the other hand, are like the elephant's trunk,
ready at any moment for anything, be it to pick up a crumb
or a field-gun, to strangle a tiger, or uproot a tree; portable
in the extreme, applicable anywhere — alike in the trackless
jungle and in the barrack square — directed by a little native
who requires no special skill or training, and who can be trans-
ferred from one elephant to another without much hesitation.
Surely this, which adapts itself to its work, is the grander
instrument. But then, it is the natural, the other, the arti-
ficial one."

The reply which Tait makes, so far as it is an argument,
is: There are two systems of quaternions, the i, jt k one, and
another one which Hamilton developed from it; Cayley knows
the first only, he himself knows the second; the former is an
intensely artificial system of imaginaries, the latter is the
natural organ of expression for quantities in space. Should a
fourth edition of his Elementary Treatise be called for i,j, k will

ARTHUR CAYLEY ^ 73

disappear from it, excepting in Cayley's chapter, should it
be retained. Tait thus describes the first system : "Hamilton's
extraordinary Preface to his first great book shows how from
Double Algebras, through Triplets, Triads, and Sets, he finally
reached Quaternions. This was the genesis of the Quaternions
of the forties, and the creature thus produced is still essentially
the Quaternion of Prof. Cayley. It is a magnificent analytical
conception; but it is nothing more than the full development
of the system o_f imaginaries i,j, k; defined by the equations,

i2 =j2 = k2 = ijk — — i,

with the associative, but not the commutative, law for the
factors. The novel and splendid points in it were the treat-
ment of all directions in space as essentially alike in character,
and the recognition of the unit vector's claim to rank also as
a quadrantal versor. These were indeed inventions of the
first magnitude, and of vast importance. And here I thor-
oughly agree with Prof. Cayley in his admiration. Considered
as an analytical system, based throughout on pure imaginaries,
the Quaternion method is elegant in the extreme. But, unless
it had been also something more, something very different
and much higher in the scale of development, I should have
been content to admire it: — and to pass it by."

From ' the most intensely artificial of systems, arose, as
if by magic, an absolutely natural one " which Tait thus fur-
ther describes. " To me Quaternions are primarily a Mode
of Representation: — immensely superior to, but of essentially
the same kind of usefulness as, a diagram or a model. They
are, virtually, the thing represented; and are thus antecedent
to, and independent of, coordinates; giving, in general, all
the main relations, in the problem to which they are applied,
without the necessity of appealing to coordinates at all. Co-
ordinates may, however, easily be read into them: — when any-
thing (such as metr cal or numerical detail) is to be gained
thereby. Quaternions, in a word, exist in space, and we have
only to recognize them: — but we have to invent or imagine
coordinates of all kinds."

74 TEN BRITISH MATHEMATICIANS

To meet the objection why Hamilton did not throw i,j, k
overboard, and expound the developed system, Tait says:
" Most unfortunately, alike for himself and for his grand con-
ception, Hamilton's nerve failed him in the composition of
his first great volume. Had he then renounced, for ever, all
dealings with i, j, k, his triumph would have been complete.
He spared Agog, and the best of the sheep, and did not utterly
destroy them. He had a paternal fondness for i, j, k ; perhaps
also a not unnatural liking for a meretricious title such as
the mysterious word Quaternion; and, above all, he had an
earnest desire to make the utmost return in his power for the
liberality shown him by the authorities of Trinity College,
Dublin. He had fully recognized, and proved to others, that
his i, j, k, were mere excrescences and blots on his improved
method: — but he unfortunately considered that their continued
(if only partial) recognition was indispensable to the reception
of his method by a world steeped in — Cartesianism! Through
the whole compass of each of his tremendous volumes one can
find traces of his desire to avoid even an allusion to i, j, k,
and along with them, his sorrowful conviction that, should he
do so, he would be left without a single reader."

To Cayley's presidential address we are indebted for in-
formation about the view which he took of the foundations of
exact science, and the philosophy which commended itself to
his mind. He quoted Plato and Kant with approval, J. S.
Mill with faint praise. Although he threw a sop to the empiri-
cal philosophers at the beginning of his address, he gave them
something to think of before he finished.

He first of all remarks that the connection of arithmetic
and algebra with the notion of time is far less obvious than
that of geometry with the notion of space; in which he, of
course, made a hit at Hamilton's theory of Algebra as the
science of pure time. Further on he discusses the theory
directly, and concludes as follows: " Hamilton uses the term
algebra in a very wide sense, but whatever else he includes
under it, he includes all that in contradistinction to the Dif-
ferential Calculus would be called algebra. Using the word

ARTHUR CAYLEY 75

in this restricted sense, I cannot myself recognize the con-
nection of algebra with the notion of time; granting that the
notion of continuous progression presents itself and is of im-
portance, I do not see that it is in anywise the fundamental
notion of the science. And still less can I appreciate the manner
in which the author connects with the notion of time his alge-
braical couple, or imaginary magnitude, a-\-bV — i." So you
will observe that doctors differ — Tait and Cayley — about the
soundness of Hamilton's theory of couples. But it can be shown
that a couple may not only be represented on a straight line,
but actually means a portion of a straight line; and as a line
is unidimensional, this favors the truth of Hamilton's theory.

As to the nature of mathematical science Cayley quoted
with approval from an address of Hamilton's:

" These purely mathematical sciences of algebra and geom-
etry are sciences of the pure reason, deriving no weight and
no assistance from experiment, and isolated or at least isolable
from all outward and accidental phenomena. The idea of order
with its subordinate ideas of number and figure, we must not
call innate ideas, if that phrase be defined to imply that all
men must possess them with equal clearness and fulness; they
are, however, ideas which seem to be so far born with us that
the possession of them in any conceivable degree is only the
development of our original powers, the unfolding of our proper
humanity."

It is the aim of the evolution philosopher to reduce all
knowledge to the empirical status; the only intuition he grants
is a kind of instinct formed by the experience of ancestors and
transmitted cumulatively by heredity. Cayley first takes him up
on the subject of arithmetic: " Whatever difficulty be raisable
as to geometry, it seems to me that no similar difficulty applies
to arithmetic; mathematician, or not, we have each of us, in
its most abstract form, the idea of number; we can each of us
appreciate the truth of a proposition in numbers ; and we cannot
but see that a truth in regard to numbers is something different
in kind from an experimental truth generalized from experience.
Compare, for instance, the proposition, that the sun, having

76 TEN BRITISH MATHEMATICIANS

already risen so many times, will rise to-morrow, and the next
day, and the day after that, and so on; and the proposition
that even and odd numbers succeed each other alternately ad
infinitum; the latter at least seems to have the characters of
universality and necessity. Or again, suppose a proposition
observed to hold good for a long series of numbers, one thousand
numbers, two thousand numbers, as the case may be: this is
not only no proof, but it is absolutely no evidence, that the
proposition is a true proposition, holding good for all numbers
whatever; there are in the Theory of Numbers very remark-
able instances of proposit'ons observed to hold good for very
long series of numbers which are nevertheless untrue."

Then he takes him up on the subject of geometry, where
the empiricist rather boasts of his success. "It is well known
that Euclid's twelfth axiom, even in Playf air's form of it, has
been considered as needing demonstration; and that Lobat-
schewsky constructed a perfectly consistent theory, wherein
this axiom was assumed not to hold good, or say a system of
non-Euclidean plane geometry. My own view is that Euclid's
twelfth axiom in Playfair's form of it does not need demonstra-
tion, but is part of our notion of space, of the physical space of
our experience — the space, that is, which we become acquainted
with by experience, but which is the representation lying at the
foundation of all external experience. Riemann's view before
referred to may I think be said to be that, having in intellectu
a more general notion of space (in fact a notion of non-Euclidean
space), we learn by experience that space (the physical space
of our experience) is, if not exactly, at least to the highest degree
of approximation, Euclidean space. But suppose the physical
space of our experience to be thus only approximately Euclidean
space, what is the consequence which follows? Not that the prop-
ositions of geometry are only approximately true, but that they
remain absolutely true in regard to that Euclidean space which
has been so long regarded as being the physical space of our
experience."

In his address he remarks that the fundamental notion
which underlies and pervades the whole of modern analysis and

ARTHUR CAYLEY 77

geometry is that of imaginary magnitude in analysis and of
imaginary space (or space as a locus in quo of imaginary points
and figures) in geometry. In the case of two given curves
there are two equations satisfied by the coordinates (x, y) of the
several points of intersection, and these give rise to an equation
of a certain order for the coordinate x or y of a point of inter-
section. In the case of a straight line and a circle this is a
quadratic equation; it has two roots real or imaginary. There
are thus two values, say of x, and to each of these corresponds
a single value of y. There are therefore two points of inter-
section, viz., a straight line and a circle intersect always in
two points, real or imaginary. It is in this way we are led
analytically to the notion of imaginary points in geometry.
He asks, What is an imaginary point? Is there in a plane a
point the coordinates of which have given imaginary values?
He seems to say No, and to fall back on the notion of an imagi-
nary space as the locus in quo of the imaginary point.

WILLIAM KINGDON CLIFFORD*
(1845-1879)

William Kingdon Clifford was born at Exeter, England,
May 4, 1845. His father was a well-known and active citizen
and filled the honorary office of justice of the peace; his mother
died while he was still young. It is believed that Clifford
inherited from his mother not only some of his genius, but a
weakness in his physical constitution. He received his ele-
mentary education at a private school in Exeter, where examina-
tions were annually held by the Board of Local Examinations
of the Universities of Oxford and Cambridge; at these examina-
tions Clifford gained numerous distinctions in widely different
subjects. When fifteen years old he was sent to King's College,
London, where he not only demonstrated his peculiar mathe-
matical abilities, but also gained distinction in classics and
English literature.

When eighteen, he entered Trinity College, Cambridge; the
college of Peacock, De Morgan, and Cayley. He already had
the reputation of possessing extraordinary mathematical powers;
and he was eccentric in appearance, habits and opinions. He
was reported to be an ardent High Churchman, which was then
a more remarkable thing at Cambridge than it is now. His
undergraduate career was distinguished by eminence in mathe-
matics, English literature and gymnastics. One who was his com-
panion in gymnastics wrote : " His neatness and dexterity were
unusually great, but the most remarkable thing was his great
strength as compared with his weight, as shown in some exercises.
At one time he would pull up on the bar with either hand, which
is well known to be one of the greatest feats of strength. His
nerve at dangerous heights was extraordinary." In his third

* This Lecture was delivered April 23, 1901. — Editors.
78

WILLIAM KINGDON CLIFFORD 79

year he won the prize awarded by Trinity College for decla-
mation, his subject being Sir Walter Raleigh; as a consequence
he was called on to deliver the annual oration at the next Com-
memoration of Benefactors of the College. He chose for his
subject, Dr. Whewell, Master of the College, eminent for his
philosophical and scientific attainments, whose death had
occurred but recently. He treated it in an original and un-
expected manner; Dr. Whewell's claim to admiration and
emulation being put on the ground of his intellectual life exem-
plifying in an eminent degree the active and creating faculty.
" Thought is powerless, except it make something outside of
itself; the thought which conquers the world is not contemplative
but active. And it is this that I am asking you to worship
to-day."

To obtain high honors in the Mathematical Tripos, a student
must put himself in special training under a mathematican,
technically called a coach, who is not one of the regular college
instructors, nor one of the University professors, but simply
makes a private business of training men to pass that par-
ticular examination. Skill consists in the rate at which one can
solve and more especially write out the solution of problems.
It is excellent training of a kind, but there is no time for study-
ing fundamental principles, still less for making any philosoph-
ical investigations. Mathematical insight is something higher
than skill in solving problems; consequently the senior wrangler
has not always turned out the most distinguished mathematician
in after life. We have seen that De Morgan was fourth wrangler.
Clifford also could not be kept to the dust of the race-course;
but such was his innate mathematical insight that he came
out second wrangler. Other instances of the second wrangler
turning out the better mathematician are Whewell, Sylvester,
Kelvin, Maxwell.

In 1868, when he was 23 years old, he was elected a Fellow
of his College; and while a resident fellow, he took part in the
eclipse expedition of 1870 to Italy, and passed through the
experience of a shipwreck near Catania on the coast of the
island of Sicily. In 1871 he was appointed professor of Ap-

80 TEN BRITISH MATHEMATICIANS

plied Mathematics and Mechanics in University College,
London; De Morgan's college, but not De Morgan's chair.
Henceforth University College was the centre of his labors.

He was now urged by friends to seek admission into
the Royal Society of London. This is the ancient scientific
society of England, founded in the time of Charles II, and
numbering among its first presidents Sir Isaac Newton. About
the middle of the nineteenth century the admission of new
members was restricted to fifteen each year; and from appli-
cations the Council recommends fifteen names which are posted
up, and subsequently balloted for by the Fellows. Hamilton
and De Morgan never applied. Clifford did not apply imme-
diately, but he became a Fellow a few years later. He joined
the London Mathematical Society — for it met in University
College- — and he became one of its leading spirits. Another
metropolitan Society in which he took much interest was the
Metaphysical Society; like Hamilton, De Morgan, and Boole,
Clifford was a scientific philosopher.

In 1875 Clifford married; the lady was Lucy, daughter of
Mr. John Lane, formerly of Barbadoes. His home in London
became the meeting-point of a numerous body of friends, in
which almost every possible variety of taste and opinion was
represented, and many of whom had nothing else in common.
He took a special delight in amusing children, and for their
entertainment wrote a collection of fairy tales called The Little
People. In this respect he was like a contemporary mathe-
matician, Mr. Dodgson — " Lewis Carroll " — the author of
Alice in Wonderland. A children's party was one of Clifford's
greatest pleasures. At one such party he kept a waxwork
show, children doing duty for the figures; but I daresay he
drew the line at walking on all fours, as Mr. Dodgson was ac-
customed to do. A children's party was to be held in a house
in London and it happened that there was a party of adults
held simultaneously in the neighboring house; to give the
children a surprise Dodgson resolved to walk in on all fours;
unfortunately he crawled into the parlor of the wrong house !

Clifford possessed unsurpassed power as a teacher. Mr.

WILLIAM KINGDON CLIFFORD 81

Pollock, a fellow student, gives an instance of Clifford's theory
of what teaching ought to be, and his constant way of carrying
it out in his discourses and conversations on mathematical
and scientific subjects. " In the analytical treatment of statics
there occurs a proposition called Ivory's Theorem concerning
the attractions of an ellipsoid. The textbooks demonstrate
it by a formidable apparatus of coordinates and integrals,
such as we were wont to call a grind. On a certain day in the
Long Vacation of 1866, which Clifford and I spent at Cambridge,
I was not a little exercised by the theorem in question, as I
suppose many students have been before and since. The chain
of symbolic proof seemed artificial and dead; it compelled the
understanding, but failed to satisfy the reason. After reading
and learning the proposition one still failed to see what it was
all about. Being out for a walk with Clifford, I opened my
perplexities to him; I think that I can recall the very spot.
What he said I do not remember in detail, which is not sur-
prising, as I have had no occasion to remember anything about
Ivory's Theorem these twelve years. But I know that as he
spoke he appeared not to be working out a question, but
simply telling what he saw. Without any diagram or symbolic
aid he described the geometrical conditions on which the
solution depended, and they seemed to stand out visibly in
space. There were no longer consequences to be deduced,
but real and evident facts which only required to be seen."

Clifford inherited a constitution in which nervous energy
and physical strength were unequally balanced. It was in
his case specially necessary to take good care of his health,
but he did the opposite; he would frequently sit up most of
the night working or talking. Like Hamilton he would work
twelve hours on a stretch; but, unlike Hamilton, he had
laborious professional duties demanding his personal attention
at the same time. The consequence was that five years after
his appointment to the chair in University College, his health
broke down; indications of pulmonary disease appeared. To
recruit his health he spent six months in Algeria and Spain,
and came back to his professional duties again. A year and

82 TEN BRITISH MATHEMATICIANS

a half later his health broke down a second time, and he was
obliged to leave again for the shores of the Mediterranean.
In the fall of 1878 he returned to England for the last time,
when the winter came he left for the Island of Madeira; all
hope of recovery was gone; he died March 3, 1879 m the 34th
year of his age.

On the title page of the volume containing his collected
mathematical papers I find a quotation, "If he had lived we
might have known something." Such is the feeling one has
when one looks at his published works and thinks of the short-
ness of his life. In his lifetime there appeared Elements of Dy-
namic, Part I. Posthumously there have appeared Elements of
Dynamic, Part II; Collected Mathematical Papers; Lectures and
Essays; Seeing and Thinking; Common Sense of the Exact
Sciences. The manuscript of the last book was left in a very
incomplete state, but the design was filled up and completed
by two other mathematicians.

In a former lecture I had occasion to remark on the relation
of Mathematics to Poetry — on the fact that in mathematical
investigation there is needed a higher power of imagination
akin to the creative instinct of the poet. The matter is dis-
cussed by Clifford in a discourse on " Some of the conditions
of mental development," which he delivered at the Royal
Institution in 1868 when he was 23 years of age. This insti-
tution was founded by Count Rumford, an American, and is
located in London. There are Professorships of Chemistry,
Physics, and Physiology; its professors have included Davey,
Faraday, Young, Tyndall, Rayleigh, Dewar. Their duties are
not to teach the elements of their science to regular students,
but to make investigations, and to lecture to the members
of the institution, who are in general wealthy and titled people.

In this discourse Clifford said " Men of science have to
deal with extremely abstract and general conceptions. By
constant use and familiarity, these, and the relations between
them, become just as real and external as the ordinary objects
of experience, and the perception of new relations among them
is so rapid, the correspondence of the mind to external circum-

WILLIAM KINGDON CLIFFORD 83

stances so great, that a real scientific sense is developed, by
which things are perceived as immediately and truly as I see
you now. Poets and painters and musicians also are so accus-
tomed to put outside of them the idea of beauty, that it becomes
a real external existence, a thing which they see with spiritual
eyes and then describe to you, but by no means create, any
more than we seem to create the ideas of table and forms and
light, which we put together long ago. There is no scientific
discoverer, no poet, no painter, no musician, who will not tell
you that he found ready made his discovery or poem or picture —
that it came to him from outside, and that he did not con-
sciously create it from within. And there is reason to think
that these senses or insights are things which actually increase
among mankind. It is certain, at least, that the scientific sense
is immensely more developed now than it was three hundred
years ago; and though it may be impossible to find any absolute
standard of art, yet it is acknowledged that a number of minds
which are subject to artistic training will tend to arrange
themselves under certain great groups and that the members
of each group will give an independent and yet consentient
testimony about artistic questions. And this arrangement
into schools, and the definiteness of the conclusions reached in
each, are on the increase, so that here, it would seem, are actually
two new senses, the scientific and the artistic, which the mind
is now in the process of forming for itself."

Clifford himself wrote a good many poems, but only a few
have been published. The following verses were sent to George
Eliot, the novelist, with a presentation copy of The Little People:

Baby drew a little house,

Drew it all askew;
Mother saw the crooked door

And the window too.

Mother heart, whose wide embrace

Holds the hearts of men,
Grows with all our growing hopes,

Gives them birth again,

84 TEN BRITISH MATHEMATICIANS

Listen to this baby-talk:
'Tisn't wise or clear;
But what baby-sense it has
Is for you to hear.

An amusement in which Clifford took pleasure even in his
maturer years was the flying of kites. He made some mathe-
matical investigations in the subject, anticipating, as it were, the
interest which has been taken in more recent years in the subject
of motion through the atmosphere. Clifford formed a project
of writing a series of textbooks on Mathematics beginning at
the very commencement of each subject and carrying it on
rapidly to the most advanced stages. He began with the
Elements of Dynamic, of which three books were printed in his
lifetime, and a fourth book, in a supplementary volume, after
his death. The work is unique for the clear ideas given of the
science; ideas and principles are more prominent than symbols
and formulas. He takes such familiar words as spin, twist,
squirt, whirl, and gives them an exact meaning. The book is
an example of what he meant by scientific insight, and from its
excellence we can imagine what the complete series of text-
books would have been.

In Clifford's lifetime it was said in England that he was the
only mathematician who could discourse on mathematics to
an audience composed of people of general culture and make
them think that they understood the subject. In 1872 he was
invited to deliver an evening lecture before the members of the
British Association, at Brighton; he chose for his subject " The
aims and instruments of scientific thought." The main theses
of the lecture are First, that scientific thought is the application
of past experience to new circumstances by means of an observed
order of events. Second, this order of events is not th oreti ally
or absolutely exact, but only exa.ct enough to correct experi-
ments by. As an instance of what is, and what is not scientific
thought, he takes the phenomenon of double refraction. " A
mineralogist, by measuring the angles of a crystal, can tell you
whether or no it possesses the property of double refraction
without looking through it. He requires no scientific thought

WILLIAM KINGDON CLIFFORD 85

to do that. But Sir William Rowan Hamilton, knowing these
facts and also the explanation of them which Fresnel had given,
thought about the subject, and he predicted that by looking
through certain crystals in a particular direction we should see
not two dots but a continuous circle. Mr. Lloyd made the
experiment, and saw the circle, a result which had never been
even suspected. This has always been considered one of the
most signal instances of scientific thought in the domain of
physics. It is most distinctly an application of experience
gained under certain circumstances to entirely different cir-
circumstances."

In physical science there are two kinds of law — distinguished
as "empirical " and " rational." The former expresses a relation
which is sufficiently true for practical purposes and within
certain limits; for example, many of the formulas used by engi-
neers. But a rational law states a connection which is accu-
rately true, without any modification of limit. In the theorems
of geometry we have examples of scientific exactness; for
example, in the theorem that the sum of the three interior
angles of a plane triangle is equal to two right angles. The
equality is one not of approximation, but of exactness. Now
the philosopher Kant pointed to such a truth and said : We know
that it is true not merely here and now, but everywhere and for
all time; such knowledge cannot be gained by experience; there
must be some other source of such knowledge. His solution
was that space and time are forms of the sensibility ; that truths
about them are not obtained by empirical induction, but by
means of intuition; and that the characters of necessity and
universality distinguished these truths from other truths. This
philosophy was accepted by Sir William Rowan Hamilton, and
to him it was not a barren philosophy, for it served as the
starting point of his discoveries in algebra which culminated in
the discovery of quaternions.

This philosophy was admired but not accepted by Clifford ;
he was, so long as he lived, too strongly influenced by the
philosophy which has been built upon the theory of evolution.
He admits that the only way of escape from Kant's conclusions

86 TEN BRITISH MATHEMATICIANS

is by denying the theoretical exactness of the proposition referred
to. He says, "About the beginning of the present century the
foundations of geometry were criticised independently by two
mathematicians, Lobatchewsky and Gauss, whose results have
been extended and generalized more recently by Riemann and
Helmholtz. And the conclusion to which these investigations
lead is that, although the assumptions which were very properly
made by the ancient geometers are practically exact — that is
to say, more exact than experiment can be — for such finite things
as we have to deal with, and such portions of space as we can
reach; yet the truth of them for very much larger things, or
very much smaller things, or parts of space which are at present
beyond our reach, is a matter to be decided by experiment,
when its powers are considerably increased. I want to make
as clear as possible the real state of this question at present,
because it is often supposed to be a question of words or meta-
physics, whereas it is a very distinct and simple question of
fact. I am supposed to know that the three angles of a recti-
linear triangle are exactly equal to two right angles. Now
suppose that three points are taken in space, distant from one
another as far as the Sun is from a Centauri, and that the
shortest distances between these points are drawn so as to form
a triangle. And suppose the angles of this triangle to be very
accurately measured and added together; this can at present
be done so accurately that the error shall certainly be less than
one minute, less therefore than the five-thousandth part of a
right angle. Then I do not know that this sum would differ
at all from two right angles; but also I do not know that the
difference would be less than ten degrees or the ninth part of
a right angle."

You will observe that Clifford's philosophy depends on the
validity of Lobatchewsky's ideas. Now it has been shown by
an Italian mathematician, named Beltrami, that the plane
geometry of Lobatchewsky corresponds to trigonometry on a
surface called the pseudosphere. Clifford and other followers of
Lobatchewsky admit Beltrami's interpretation, an interpretation
which does not involve any paradox about geometrical space,

WILLIAM KINGDON CLIFFORD 87

and which leaves the trigonometry of the plane alone as a dif-
ferent thing. If that interpretation is true, the Lobatchewskian
plane triangle is after all a triangle on a special surface, and the
straight lines joining the points are not the shortest absolutely,
but only the shortest with respect to the surface, whatever that
may mean. If so, then Clifford's argument for the empirical
nature of the proposition referred to fails; and nothing pre-
vents us from falling back on Kant's position, namely, that
there is a body of knowledge characterized by absolute exact-
ness and possessing universal application in time and space;
and as a particular case thereof we believe that the sum of the
three angles of Clifford's gigantic triangle is precisely two right
angles.

Trigonometry on a spherical surface is a generalized form of
plane trigonometry, from the theorems of the former we can
deduce the theorems of the latter by supposing the radius of
the sphere to be infinite. The sum of the three angles of a
spherical triangle is greater than two right angles; the sum of
the angles of a plain triangle is equal to two right angles; we
infer that there is another surface, complementary to the
sphere, such that the angles of any triangle on it are less than
two right angles. The complementary surface to which I refer
is not the pseudosphere, but the equilateral hyperboloid. As
the plane is the transition surface between the sphere and
the equilateral hyperboloid, and a triangle on it is the transi-
tion triangle between the spherical triangle and the equilateral
hyperboloidal triangle, the sum of the angles of the plane tri-
angle must be exactly equal to two right angles.

In 1873, the British Association met at Bradford; on this
occasion the evening discourse was delivered by Maxwell,
the celebrated physicist. He chose for his subject " Mole-
cules." The application of the method of spectrum-ana ysis
assures the physicist that he can find out in his laboratory
truths of universal validity in space and t me. In fact, the
chief maxim of physical science, according to Maxwell is,
that physical changes are independent of the conditions of
space and time, and depend only on conditions of configuration

88 TEN BRITISH MATHEMATICIANS

of bodies, temperature, pressure, etc. The address closed
with a celebrated passage in striking contrast to Clifford's
address: " In the heavens we discover by their light, and
by their light alone, stars so distant from each other that no
material thing can ever have passed from one to another; and
yet this light, which is to us the sole evidence of the exis ence
of these distant worlds, tells us also that each of th m is built
up of molecules of the same kinds as those which are found on
earth. A mol cule of hydrogen, for example, whether in S rius
or in Arcturus, executes its vibrations in precisely the fame
time. No theory of evolution can be formed to account for
the similarity of molecules, for evolution necessarily implies
continuous change, and the molecule is incapable of growth
or decay, of generation or destruction. None of the processes
of Nature since the time when Nature began, have produced
the slightest difference in the properties of any molecule. We
are therefore unable to ascribe either the existence of the mole-
cules or the identity of their properties to any of the causes
which we call natural. On the other hand, the exact equality
of each molecule to all others of the same kind gives it, as
Sir John Herschel has well said, the essential character of a
manufactured article, and precludes the idea of its being eternal
and self -existent."

What reply could Clifford make to this? In a discourse
on the " First and last catastrophe " delivered soon afterwards,
he said " If anyone not possessing the great authority of
Maxwell, had put forward an argument, founded upon a sci-
entific basis, in which there occurred assumptions about what
things can and what things cannot have existed from eternity,
and about the exact similarity of two or more things established
by experiment, we would say: ' Past eternity; absolute exact-
ness; won't do '; and we should pass on to another book.
The experience of all scientific culture for all ages during which
it has been a light to men has shown us that we never do get
at any conclusions of that sort. We do not get at conclusions
about infinite time, or "nfinite exactness. We get at conclu-
sions which are as nearly true as exper'ment can show, and

WILLIAM KINGDON CLIFFORD 89

sometimes which are a great deal more correct than direct
experiment can be, so that we are able actually to correct
one experiment by deductions from another, but we never
get at conclusions which we have a right to say are absolutely
exact."

Clifford had not faith in the exactness of mathematical science
nor faith in that maxim of physical science which has built
up the new astronomy, and extended all the bounds of physical
science. Faith in an exact order of Nature was the charac-
teristic of Faraday, and he was by unanimous consent the
greatest electrician of the nineteenth century. What is the
general direction of progress in science? Physics is becoming
more and more mathematical; chemistry is becoming more
and more physical, and I daresay the biological sciences are
moving in the same direction. They are all moving towards
exactness; consequently a true philosophy of science will de-
pend on the principles of mathematics much more than
upon the phenomena of biology. Clifford, I believe, had he
lived longer, would have changed his philosophy for a more
mathematical one. In 1874 there appeared in Nature among
the letters from correspondents one to the following effect:

An anagram: The practice of enclosing discoveries in sealed
packets and sending them to Academies seems so inferior to
the old one of Huyghens, that the following is sent you for
publication in the old conservated form :

A8C*DE12F*GH6J&L3M3N50&PR4S5TUU6V2WXY2.

This anagram was explained in a book entitled The Unseen
Universe, which was published anonymously in 1875; and is
there translated, " Thought conceived to affect the matter of
another universe simultaneously with this may explain a future
state." The book was evidently a work of a physicist or
physicists, and as phys'cists were not so numerous then as
they are now, it was not difficult to determine the authorship
from internal evidence. It was attributed to Tait, the professor
of physics at Edinburgh University, and Balfour Stewart, the
professor of physics at Owens College, Manchester. When

90 TEN BRITISH MATHEMATICIANS

the fourth edition appeared, their names were given on the
title page.

The kernel of the book is the above so-called discovery,
first published in the form of an anagram. Preliminary chap-
ters are devoted to a survey of the beliefs of ancient peoples
on the subject of the immortality of the soul; to physical
axioms; to the physical doctrine of energy, matter, and ether;
and to the biological doctrine of development; in the last
chapter we come to the unseen universe. What is meant by
the unseen universe? Matter is made up of molecules, which
are supposed to be vortex-rings of an imperfect fluid, namely,
the luminiferous ether; the luminous ether is made up of much
smaller molecules, which are vortex-rings in a second ether.
These smaller molecules with the ether in which they float
are the unseen universe. The authors see reason to believe
that the unseen universe absorbs energy from the visible uni-
verse and vice versa. The soul is a frame which is made of
the refined molecules and exists in the unseen universe. In
life it is attached to the body. Every thought we think is
accompanied by certain motions of the coarse molecules of
the brain, these motions are propagated through the visible
universe, but a part of each motion is absorbed by the fine
molecules of the soul. Consequently the soul has an organ
of memory as well as the body; at death the soul with its
organ of memory is simply set free from associat'on with the
coarse molecules of the body. In this way the authors con-
sider that they have shown the physical possibility of the
immortality of the soul.

The curious part of the book follows: the authors change
their possibility into a theory and apply it to explain the main
doctrines of Christianity; and it is certainly remarkable to find
in the same book a discussion of Carnot's heat-engine and ex-
tensive quotations from the apostles and prophets. Clifford
wrote an elaborate review which he finished in one sitting occu-
pying twelve hours. He pointed out the difficulties to which
the main speculation, which he admitted to be ingenious, is
liable; but his wrath knew no bounds when he proceeded to con-

WILLIAM KINGDON CLIFFORD 91

siderthe application to the doctrines of Christianity; for from
being a High Churchman in youth he became an agnostic in
later years ; and he could not write on any religious question
without using language which was offensive even to his friends.
The Phaedo of Plato is more satisfying to the mind than the
Unseen Universe of Tait and Stewart. In it, Socrates discusses
with his friends the immortality of the soul, just before taking
the draught of poison. One argument he advances is, How can
the works of an artist be more enduring than the artist himself?
This is a question which comes home in force when we peruse
the works of Peacock, De Morgan, Hamilton, Boole, Cayley
and Clifford.

HENRY JOHN STEPHEN SMITH*

(1826-1883)

Henry John Stephen Smith was born in Dublin, Ireland,
on November 2, 1826. His father, John Smith, was an Irish
barrister, who had graduated at Trinity College, Dublin, and
had afterwards studied at the Temple, London, as a pupil of
Henry John Stephen, the editor of Blackstone's Commentaries;
hence the given name of the future mathematician. His mother
was Ma-y Murphy, an accomplished and clever Irishwoman,
tall and beautiful. Henry was the youngest of four children,
and was but two years old when his father died. His mother
would have been left in straitened circumstances had she not
been successful in claiming a bequest of £10,000 which had been
left to her husband but had been disputed. On receiving this
money, she migrated to England, and finally settled in the Isle
of Wight.

Henry as a child was sickly and very near-sighted. When
four years of age he displayed a genius for mastering languages.
His first instructor was his mother, who had an accurate knowl-
edge of the class 'cs. When eleven years of age, he, along with
his brother and sisters, was placed in the charge of a private
tutor, who was strong in the classics; in one year he read a
large portion of the Greek and Latin authors commonly studied.
His tutor was mpressed with his power of memory, quickness
of perception, indefatigable diligence, and intuitive grasp of
whatever he studied. In their leisure hours the children would
improvise plays from Homer, or Robinson Crusoe; and they
also became diligent students of animal and insect life. Next
year a new tutor was strong in the mathematics, and with his
aid Henry became acquainted with advanced arithmetic, and

* This Lecture was delivered March 15, 1902. — Editors.
92

HENRY JOHN STEPHEN SMITH 93

the elements of algebra and geometry. The year following,
Mrs. Smith moved to Oxford, and placed Henry under the
care of Rev. Mr. Highton, who was not only a sound scholar,
but an exceptionally good mathematician. The year following
Mr. Highton received a mastership at Rugby with a boarding-
house attached to it (which is important from a financial point
of view) and he took Henry Smith with him as his first boarder.
Thus at the age of fifteen Henry Smith was launched into the
life of the English public school, and Rugby was then under the
most famous headmaster of the day, Dr. Arnold. Schoolboy life
as it was then at Rugby has been depicted by Hughes in " Tom
Brown's Schooldays."

Here he showed great and all-around ability. It became

his ambition to crown his school career by carrying off an

entrance scholarship at Balliol College, Oxford. But as a sister

and brother had already died of consumption, his mother did

not allow him to complete his third and final year at Rugby, but

took him to Italy, where he continued his reading privately.

Notwithstanding this manifest disadvantage, he was able to

carry off the coveted scholarship; and at the age of nineteen

he began residence as a student of Balliol College. The next

long vacation was spent in Italy, and there his health broke

down. By the following winter he had not recovered enough

to warrant his return to Oxford; instead, he went to Paris,

and took several of the courses at the Sorbonne and the College

de France. These studies abroad had much influence on his

future career as a mathematician. Thereafter he resumed his

undergraduate studies at Oxford, carried off what is considered

the highest classical honor, and in 1849, when 23 years old,

finished his undergraduate career with a double-first; that is,

in the honors examination for bache or of arts he took first-class

rank in the classics, and also first-class rank in he math matics.

It is not very pleasant to be a double fi st, for the outwardly

envied and distinguished recipient is apt to find himself in the

position of the ass between two equally inviting bundles of hay,

unless indeed there is some external attraction superior to both.

In the case of Smith, the external attraction was the bar, for

94 TEN BRITISH MATHEMATICIANS

which he was in many respects well suited; but the feebleness
of his constitution led him to abandon that course. So he had
a difficulty in deciding between classics and mathematics, and
there is a story to the effect that he finally solved the difficulty
by tossing up a penny. He certainly used the expression: but
the reasons which determined his choice in favor of mathematics
were first, his weak sight, which made thinking preferable to
reading, and secondly, the opportunity which presented itself.

At that time Oxford was recovering from the excitement
which had been produced by the Tractarian movement, and
which had ended in Newman going over to the Church of Rome.
But a Parliamentary Commission had been appointed to inquire
into the working of the University. The old system of close
scholarships and fellowships was doomed, and the close pre-
serves of the Colleges were being either extinguished or thrown
open to public competition. Resident professors, married tutors
or fellows were almost or quite unknown; the heads of the
several colleges, then the governing body of the University,
formed a little society by themselves. Balliol College (founded
by John Balliol, the unfortunate King of Scotland who was
willing to sell its independence) was then the most distinguished
for intellectual eminence; the master was singular among his
compeers for keeping steadily in view the true aim of a col-
lege, and he reformed the abuses of privilege and close endow-
ment as far as he legally could. Smith was elected a fellow
with the hope that he would consent to reside, and take the
further office of tutor in mathematics, which he did. Soon
after he became one of the mathematical tutors of Balliol he
was asked by his college to deliver a course of lectures on
chemistry. For this purpose he took up the study of chemical
analysis, and exhibited skill in manipulation and accuracy in
work. He had an idea of seeking numerical relations connecting
the atomic weights of the elements, and some mathematical
basis for their properties which might enable experiments to
be predicted by the operation of the mind.

About this time Whewell, the master of Trinity College,
Cambridge, wrote The Plurality of Worlds, which was at first

HENRY JOHN STEPHEN SMITH 95

published anonymously. Whewell pointed out what he called
law of waste traceable in the Divine economy; and his argu-
ment was that the other planets were waste effects, the Earth
the only oasis in the desert of our system, the only world
inhabited by intelligent beings; Sir David Brewster, a Scottish
physicist, inventor of the kaleidoscope, wrote a fiery answer
entitled " More worlds than one, the creed of the philosopher
and the hope of the Christian." In 1855 Smith wrote an essay
on this subject for a volume of Oxford and Cambridge Essays
in which the fallibility both of men of science and of theologians
was impartially exposed. It was his first and only effort at
popular writing.

His two earliest mathematical papers were on geometrical
subjects, but the third concerned that branch of mathematics
in which he won fame — the theory of numbers. How he was
led to take up this branch of mathematics is not stated on
authority, but it was probably as follows: There was then no
school of mathematics at Oxford; the symbolical school was
flourishing at Cambridge; and Hamilton was lecturing on
Quaternions at Dublin. Smith did not estimate either of these
very highly; he had studied at Paris under some of the great
French analysts; he had lived much on the Continent, and was
familiar with the French, German and Italian languages. As
a scholar he was drawn to the masterly disquisitions of Gauss,
who had made the theory of numbers a principal subject of
research. I may quote here his estimate of Gauss and of his
work: " If we except the great name of Newton (and the excep-
tion is one which Gauss himself would have been delighted to
make) it is probable that no mathematician of any age or country
has ever surpassed Gauss in the combination of an abundant
fertility of invention with an absolute vigorousness in demonstra-
tion, which the ancient Greeks themselves might have envied.
It may be admitted, without any disparagement to the eminence
of such great mathematicians as Euler and Cauchy that they
were so overwhelmed with the exuberant wealth of their own
creations, and so fascinated .by the interest attaching to the
results at which they arrived, that they did not greatly care

96 TEN BRITISH MATHEMATICIANS

to expend their time in arranging their ideas in a strictly logical
order, or even in establishing by irrefragable proof propositions
which they instinctively felt, and could almost see to be true.
With Gauss the case was otherwise. It may seem paradoxical,
but it is probably nevertheless true that it is precisely the effort
after a logical perfection of form which has rendered the writings
of Gauss open to the charge of obscurity and unnecessary diffi-
culty. The fact is that there is neither obscurity nor difficulty
in his writings, as long as we read them in the submissive spirit
in which an intelligent schoolboy is made to read his Euclid.
Every assertion that is made is fully proved, and the assertions
succeed one another in a perfectly just analogical order; there
nothing so far of which we can complain. But when we have
finished the perusal, we soon begin to feel that our work is but
begun, that we are still standing on the threshold of the temple,
and that there is a secret which lies behind the veil and is as
yet concealed from us. No vestige appears of the process by
which the result itself was obtained, perhaps not even a trace
of the considerations which suggested the successive steps of
the demonstration. Gauss says more than once that for brevity,
he gives only the synthesis, and suppresses the analysis of his
propositions. Pauca sed matura — few but well-matured — were
the words with which he delighted to describe the character which
he endeavored to impress upon his mathematical writings.
If, on the other hand, we turn to a memoir of Euler's, there
is a sort of free and luxuriant gracefulness about the whole
performance, which tells of the quiet pleasure which Euler must
have taken in each step of his work ; but we are conscious never-
theless that we are at an immense distance from the severe
grandeur of design which is characteristic of all Gauss's greater
efforts."

Following the example of Gauss, he wrote his first paper
on the theory of numbers in Latin: " De compositione nume-
rorum primorum formae 4W+1 ex duobus quadratis." In it
he proves in an original manner the theorem of Fermat — " That
every prime number of the form 4.W+1 in being an integer
number) is the sum of two square numbers." In his second

HENRY JOHN STEPHEN SMITH 97

paper he gives an introduction to the theory of numbers. " It
is probable that the Pythagorean school was acquainted with
the definition and nature of prime numbers; nevertheless the
arithmetical books of the elements of Euclid contain the oldest
extant investigations respecting them; and, in particular the
celebrated yet simple demonstration that the number of the
primes is infinite. To Eratosthenes of Alexandria, who is for
so many other reasons entitled to a place in the history of the
sciences, is attributed the invention of the method by which the
primes may successively be determined in order of magnitude.
It is termed, after him, ' the sieve of Eratosthenes '; and is es-
sentially a method of exclusion, by which all composite numbers
are successively erased from the series of natural numbers, and
the primes alone are left remaining. It requires only one kind
of arithmetical operation; that is to say, the formation of the
successive multiples of given numbers, or in other words,
addition only. Indeed it may be said to require no arithmetical
operation whatever, for if the natural series of numbers be
represented by points set off at equal distances along a line,
by using a geometrical compass we can determine without cal-
culation the multiples of any given number. And in fact, it
was by a mechanical contrivance of this nature that M. Burck-
hardt calculated his table of the least divisors of the first three
millions of numbers.

In 1857 Mrs. Smith died; as the result of her cares and
exertions she had seen her son enter Balliol College as a
scholar, graduate a double-first, elected a fellow of his college,
appointed tutor in mathematics, and enter on his career as an
independent mathematician. The brother and sister that were
left arranged to keep house in Oxford, the two spending the
terms together, and each being allowed complete liberty of move-
ment during the vacations. Thereafter this was the domestic
arrangement in which Smith lived and worked ; he never married.
As the owner of a house, instead of living in rooms in college
he was able to satisfy his fondness for pet animals, and also
to extend Irish hospitality to visiting friends under his own
roof. He had no household cares to destroy the needed serenity

98 TEN BRITISH MATHEMATICIANS

for scientific work, excepting that he was careless in money
matters, and trusted more to speculation in mining shares than
to economic management of his income. Though addicted to
the theory of numbers, he was not in any sense a recluse; on
the contrary he entered with zest into every form of social
enjoyment in Oxford, from croquet parties and picnics to ban-
quets. He had the rare power of utilizing stray hours of leisure,
and it was in such odd times that he accomplished most of his
scientific work. After attending a picnic in the afternoon, he
could mount to those serene heights in the theory of numbers

" Where never creeps a cloud or moves a wind,
Nor ever falls the least white star of snow,
Nor ever lowest roll of thunder moans,
Nor sound of human sorrow mounts, to mar
Their sacred everlasting calm."

Then he could of a sudden come down from these heights
to attend a dinner, and could conduct himself there, not as a
mathematical genius lost in reverie and pointed out as a poor
and eccentric mortal, but on the contrary as a thorough man
of the world greatly liked by everybody.

In i860, when Smith was 34 years old, the Savilian professor
of geometry at Oxford died. At that time the English uni-
versities were so constituted that the teaching was done by the
college tutors. The professors were officers of the University;
and before reform set in, they not only did not teach, they did
not even reside in Oxford. At the present day the lectures of
the University professors are in general attended by only a few
advanced students. Henry Smith was the only Oxford candi-
date; there were other candidates from the outside, among
them George Boole, then professor of mathematics at Queens
College, Cork. Smith's claims and talents were considered so
conspicuous by the electors, that they did not consider any
other candidates. He did not resign as tutor at Balliol, but
continued to discharge the arduous duties, in order that the
income of his Fellowship might be continued. With proper
financial sense he might have been spared from labors which
militated against the discharge of the higher duties of professor.

HENRY JOHN STEPHEN SMITH

99

His freedom during vacation gave him the opportunity of
attending the meetings of the British Association, where he
was not only a distinguished savant, but an accomplished
member of the social organization known as the Red Lions. In
1858 he was selected by that body to prepare a report upon the
Theory of Numbers. It was prepared in five parts, extending
over the years 1859-1865. It is neither a history nor a treatise,
but something intermediate. The author analyzes with remark-
able clearness and order the works of mathematicians for the
preceding century upon the theory of congruences, and upon
that of binary quadratic forms. He returns to the original
sources, indicates the principle and sketches the course of the
demonstrations, and states the result, often adding something
of his own. The work has been pronounced to be the most
complete and elegant monument ever erected to the theory of
numbers, and the model of what a scientific report ought to be.
During the preparation of the Report, and as a logical con-
sequence of the researches connected therewith, Smith pub-
lished several original contributions to the higher rithmetic.
Some were in complete form and appeared in the Philosophical
Transactions of the Royal Society of London; others were
incomplete, giving only the results without the extended demon-
strations, and appeared in the Proceedings of that Society. One
of the latter, entitled " On the orders and genera of quadratic
forms containing more than three indeterminates," enunciates
certain general principles by means of which he solves a problem
proposed by Eisenstein, namely, the decomposition of integer
numbers into the sum of five squares; and further, the analogous
problem for seven squares. It was also indicated that the four,
six, and eight-square theorems of Jacobi, Eisenstein and Lion-
ville were deducible from the principles set forth.

In 1868 he returned to the geometrical researches which had
first occupied his attention. For a memoir on " Certain cubic
and biquadratic problems " the Royal Academy of Sciences of
Berlin awarded him the Steiner prize. On account of his ability
as a man of affairs, Smith was in great demand for University
and scientific work of the day. He was made Keeper of the

100 TEN BRITISH MATHEMATICIANS

University Museum; he accepted the office of Mathematical
Examiner to the University of London; he was a member of a
Royal Commission appointed to report on Scientific Education ;
a member of the Commission appointed to reform the University
of Oxford; chairman of the committee of scientists who were
given charge of the Meteorological Office, etc. It was not till
1873, when offered a Fellowship by Corpus Christi College,
that he gave up his tutorial duties at Balliol. The demands
of these offices and of social functions upon his time and energy
necessarily reduced the total output of mathematical work of
the highest order; the results of long research lay buried in
note-books, and the necessary time was not found for elabora-
ting them into a form suitable for publication. Like his master,
Gauss, he had a high ideal of what a scientific memoir ought
to be in logical order, vigor of demonstration and literary execu-
tion; and it was to his mathematical friends matter of regret
that he did not reserve more of his energy for the work for
which he was exceptionally fitted.

He was a brilliant talker and wit. Working in the purely
speculative region of the theory of numbers, it was perhaps
natural that he should take an anti-utilitarian view of mathe-
matical science, and that he should express it in exaggerated
terms as a defiance to the grossly utilitarian views then pop-
ular. It is reported that once in a lecture after explaining
a new solution of an old problem he said, "It is the peculiar
beauty of this method, gentlemen, and one which endears it
to the really scientific mind, that under no circumstances can it
be of the smallest possible utility." I believe that it was at
a banquet of the Red Lions that he proposed the toast " Pure
mathematics; may it never be of any use to any one."

I may mention some other specimens of his wit. " You
take tea in the morning," was the remark with which he once
greeted a friend; "if I did that I should be awake all day."
Some one mentioned to him the enigmatical motto of Marischal
College, Aberdeen: " They say; what say they; let them say."
"Ah," said he, " it expresses the three stages of an undergradu-
ate's career. ' They say ' — in his first year he accepts every-

HENRY JOHN STEPHEN SMITH 101

thing he is told as if it were inspired. ' What say they '—in
his second year he is skeptical and asks that question. ' Let
them say ' expresses the attitude of contempt characteristic of
his third year." Of a brilliant writer but illogical thinker he
said " He is never right and never wrong; he is never to the
point." Of Lockyer, the astronomer, who has been for many
years the editor of the scientific journal Nature, he said,
" Lockyer sometimes forgets that he is only the editor, not
the author, of Nature." Speaking to a newly elected fellow
of his college he advised him in a low whisper to write a little
and to save a little, adding " I have done neither."

At the jubilee meeting of the British Association held at
York in 1881, Prof. Huxley and Sir John Lubbock (now Lord
Avebury) strolled down one afternoon to the Minster, which
is considered the finest cathedral in England. At the main
door they met Prof. Smith coming out, who made a mock
movement of surprise. Huxley said, " You seem surprised to
see me here." " Yes," said Smith, "going in, you know; I would
not have been surprised to see you on one of the pinnacles."
Once I was introduced to him at a garden party, given in the
grounds of York Minster. He was a tall man, with sandy hair
and beard, decidedly good-looking, with a certain intellectual
distinction in his features and expression. He was everywhere
and known to everyone, the life and soul of the gathering. He
retained to the day of his death the simplicity and high spirits
of a boy. Socially he was an embodiment of Irish blarney
modified by Oxford dignity.

In 1873 the British Association met at Bradford; at which
meeting Maxwell delivered his famous " Discourse on Mole-
cules." At the same meeting Smith was the president of the
section of mathematics and physics. He did not take up any
technical subject in his address; but confined himself to matters
of interest in the exact sciences. He spoke of the connection
between mathematics and physics, as evidenced by the dual
province of the section. " So intimate is the union between
mathematics and physics that probably by far the larger part
of the accessions to our mathematical knowledge have been

102 TEN BRITISH MATHEMATICIANS

obtained by the efforts of mathematicians to solve the problems
set to them by experiment, and to create for each successive
class of phenomena a new calculus or a new geometry, as the
case might be, which might prove not wholly inadequate to the
subtlety of nature. Sometimes indeed the mathematician has
been before the physicist, and it has happened that when some
great and new question has occurred to the experimenter or the
observer, he has found in the armory of the mathematician
the weapons which he has needed ready made to his hand. But
much oftener the questions proposed by the physicist have
transcended the utmost powers of the mathematics of the time,
and a fresh mathematical creation has been needed to supply
the logical instrument required to interpret the new enigma."
As an example of the rule he points out that the experiments
of Faraday called forth the mathematical theory of Maxwell;
as an example of the exception that the work of Apollonius
on the conic sections was ready for Kepler in investigating the
orbits of the planets.

At the time of the Bradford meeting, education in the public
schools and universities of England was practically confined
to the classics and pure mathematics. In his address Smith
took up the importance of science as an educational discipline
in schools; and the following sentences, falling as they did from
a profound scholar, produced a powerful effect : " All knowledge
of natural science that is imparted to a boy, is, or may be, useful
to him in the business of his after-life; but the claim of natural
science to a place in education cannot be rested upon its useful-
ness only. The great object of education is to expand and to
train the mental faculties, and it is because we believe that the
study of natural science is eminently fitted to further these two
objects that we urge its introduction into school studies. Science
expands the minds of the young, because it puts before them
great and ennobling objects of contemplation; many of its
truths'are such as a child can understand, and yet such that while
in a measure he understands them, he is made to feel something
of the greatness, something of the sublime regularity and some-
thing of the impenetrable mystery, of the world in which he

HENRY JOHN STEPHEN SMITH 103

is placed. But science also trains the growing faculties, for
science proposes to itself truth as its only object, and it presents
the most varied, and at the same time the most splendid
examples of the different mental processes which lead to the
attainment of truth, and which make up what we call reasoning.
In science error is always possible, often close at hand; and the
constant necessity for being on our guard against it is one im-
portant part of the education which science supplies. But in
science sophistry is impossible; science knows no love of para-
dox; science has no skill to make the worse appear the better
reason; science visits with a not long deferred exposure all our
fondness for preconceived opinions, all our partiality for views
which we have ourselves maintained; and thus teaches the two
best lessons that can well be taught — on the one hand, the love
of truth; and on the other, sobriety and watchfulness in the use
of the understanding."

The London Mathematical Society was founded in 1865.
By going to the meetings Prof. Smith was induced to prepare
for publication a number of papers from the materials of his
notebooks. He was for two years president, and at the end
of his term delivered an address " On the present state and
prospects of some branches of pure mathematics." He began
by referring to a charge which had been brought against the
Society, that its Proceedings showed a partiality in favor of
one or two great branches of mathematical science to the com-
parative neglect and possible disparagement of others. He
replies in the language of a miner. " It may be rejoined with
great plausibility that ours is not a blamable partiality, but a
well-grounded preference. So great (we might contend) have
been the triumphs achieved in recent times by that combination
of the newer algebra with the direct contemplation of space
which constitutes the modern geometry — so large has been the
portion of these triumphs, which is due to the genius of a few
great English mathematicians ; so vast and so inviting has been
the field thus thrown open to research, that we do well to press
along towards a country which has, we might say, been ' pros-
pected ' for us, and in which we know beforehand we cannot

104 TEN BRITISH MATHEMATICIANS

fail to find something that will repay our trouble, rather than
adventure ourselves into regions where, soon after the first step,
we should have no beaten tracks to guide us to the lucky spots,
and in which (at the best) the daily earnings of the treasure-
seeker are small, and do not always make a great show, even
after long years of work. Such regions, however, there are in
the realm of pure mathematics, and it cannot be for the interest
of science that they should be altogether neglected by the rising
generation of English mathematicians. I propose, therefore,
in the first instance to direct your attention to some few of these
comparatively neglected spots." Since then quite a number of
the neglected spots pointed out have been worked.

In 1878 Oxford friends urged him to come forward as a candi-
date for the representation in Parliament of the University of
Oxford, on the principle that a University constituency ought
to have for its representative not a mere party politician, but
an academic man well acquainted with the special needs of the
University. The main question before the electors was the
approval or disapproval of the Jingo war policy of the Con-
servative Government. Henry Smith had always been a Liberal
in politics, university administration, and religion. The voting
was influenced mainly by party considerations — Beaconsfield
or Gladstone — with the result that Smith was defeated by more
than 2 to 1; but he had the satisfaction of knowing that his
support came mainly from the resident and working members of
the University. He did not expect success and he hardly desired
it, but he did not shrink when asked to stand forward as the
representative of a principle in which he believed. The election
over, he devoted himself with renewed energy to the publication
of his mathematical researches. His report on the theory of
numbers had ended in elliptic functions; and it was this subject
which now engaged his attention.

In February, 1882, he was surprised to see in the Comptes
rendus that the subject proposed by the Paris Academy of
Science for the Grand prix des sciences mathematiques was the
theory of the decomposition of integer numbers into a sum of
five squares; and that the attention of competitors was directed

HENRY JOHN STEPHEN SMITH 105

to the results announced without demonstration by Eisenstein,
whereas nothing was said about his papers dealing with
the same subject in the Proceedings of the Royal Society. He
wrote to M. Hermite calling his attention to what he had pub-
lished; in reply he was assured that the members of the com-
mission did not know of the existence of his papers, and he
was advised to complete his demonstrations and submit the
memoir according to the rules of the competition. According
to the rules each manuscript bears a motto, and the correspond-
ing envelope containing the name of the successful author is
opened. There were still three months before the closing of the
concours (i June, 1882) and Smith set to work, prepared the
memoir and despatched it in time.

Meanwhile a political agitation had grown up in favor of
extending the franchise in the county constituencies. In the
towns the mechanic had received a vote; but in the counties
that power remained with the squire and the farmer; poor
Hodge, as he is called, was left out. Henry Smith was not merely
a Liberal; he felt a genuine sympathy for the poor of his own
land. At a meeting in the Oxford Town Hall he made a speech
in favor of the movement, urging justice to all classes. From
that platform he went home to die. When he spoke he was
suffering from a cold. The exposure and excitement were
followed by congestion of the liver, to which he succumbed on
February 9, 1883, in the 57th year of his age.

Two months after his death the Paris Academy made their
award. Two of the three memoirs sent in were judged worthy
of the prize. When the envelopes were opened, the authors
were found to be Prof. Smith and M. Minkowski, a young
mathematician of Koenigsberg, Prussia. No notice was taken
of Smith's previous publication on the subject, and M. Heimite
on being written to, said that he forgot to bring the matter to
the notice of the commission. It was admitted that there was
considerable similarity in the course of the investigation in the
two memoirs. The truth seems to be that M. Minkowski
availed himself of whatever had been published on the sub-
ject, including Smith's paper, but to work up the memoir

106 TEN BRITISH MATHEMATICIANS

from that basis cost Smith himself much intellectual labor,
and must have cost Minkowski much more. Minkowski is
now the chief living authority in that high region of the theory
of numbers. Smith's work remains the monument of one of
the greatest British mathematicians of the nineteenth century.

JAMES JOSEPH SYLVESTER*
(1814-1897)

James Joseph Sylvester was born in London, on the 3d
of September, 1814. He was by descent a Jew. His father
was Abraham Joseph Sylvester, and the future mathematician
was the youngest but one of seven children. He received his
elementary education at two private schools in London, and his
secondary education at the Royal Institution in Liverpool. At
the age of twenty he entered St. John's College, Cambridge;
and in the tripos examination he came out second wrangler.
The senior wrangler of the year did not rise to any eminence;
the fourth wrangler was George Green, celebrated for his con-
tributions to mathematical physics; the fifth wrangler was
Duncan F. Gregory, who subsequently wrote on the foundations
of algebra. On account of his religion Sylvester could not sign
the thirty-nine articles of the Church of England; and as a con-
sequence he could neither receive the degree of Bachelor of
Arts nor compete for the Smith's prizes, and as a further conse-
quence he was not eligible for a fellowship. To obtain a degree
he turned to the University of Dublin. After the theological
tests for degrees had been abolished at the Universities of
Oxford and Cambridge in 1872, the University of Cambridge
granted him his well-earned degree of Bachelor of Arts and also
that of Master of Arts.

On leaving Cambridge he at once commenced to write papers,
and these were at first on applied mathematics. His first paper
was entitled " An analytical development of Fresnel's optical
theory of crystals," which was published in the Philosophical
Magazine. Ere long he was appointed Professor of Physics in
University College, London, thus becoming a colleague of De

* A Lecture delivered March 21, 1902. — Editors.
107

108 TEN BRITISH MATHEMATICIANS

Morgan. At that time University College was almost the only-
institution of higher education in England in which theological
distinctions were ignored. There was then no physical laboratory
at University College, or indeed at the University of Cambridge;
which was fortunate in the case of Sylvester, for he would have
made a sorry experimenter. His was a sanguine and fiery
temperament, lacking the patience necessary in physical manipu-
lation. As it was, even in these pre-laboratory days he felt
out of place, and was not long in accepting a chair of pure
mathematics.

In 184 1 he became professor of mathematics at the Uni-
versity of Virginia. In almost all notices of his life nothing
is said about his career there; the truth is that after the short
space of four years it came to a sudden and rather tragic ter-
mination. Among his students were two brothers, fully imbued
with the Southern ideas about honor. One day Sylvester
criticised the recitation of the younger brother in a wealth of
diction which offended the young man's sense of honor; he
sent word to the professor that he must apologize or be chastised.
Sylvester did not apologize, but provided himself with a ^word-
cane; the young man provided himself with a heavy walking-
stick. The brothers lay in wait for the prof essor ; and when he
came along the younger brother demanded an apology, almost
immediately knocked off Sylvester's hat, and struck him a blow
on the bare head with hi , heavy stick. Sylvester drew his sword-
cane, and pierced the young man just over the heart; who fell
back into his brother's arms, calling out " I am killed." A
spectator, coming up, urged Sylvester away from the spot.
Without waiting to pack his books the professor left for New
York, and took the earliest possible passage for England. The
student was not seriously hurt; fortunately the point of the
sword had struck fair against a rib.

Sylvester, on his return to London, connected himself with
a firm of actuaries, his ultimate aim being to qualify himself
to practice conveyancing. He became a student of the Inner
Temple in 1846, and was called to the bar in 1850. He chose
the same profession as did Cayley; and in fact Cayley and

JAMES JOSEPH SYLVESTER 109

Sylvester, while walking the law-courts, discoursed more on
mathematics than on conveyancing. Cayley was full of the
theory of invariants; and it was by his discourse that Sylvester
was induced to take up the subject. These two men were life-
long fri nds; but it is safe to say that the permanence of the
friendship was due to Cay ley's kind and patient disposition.
Recognized as the leading mathematicians of their day in Eng-
land, they were yet very different both in nature and talents.

Cayley was patient and equable; Sylvester, fiery and passion-
ate. Cayley finished off a mathematical memoir with the same
care as a legal instrument; Sylvester never wrote a paper with-
out foot-notes, appendices, supplements; and the alterations
and corrections in his proofs were such that the printers found
their task weli-nigh impossible. Cayley was well-read in con-
temporary mathematics, and did much useful work as referee
for scientific societies; Sylvester read only what had an immedi-
ate bearing on his own researches, and did little, if any, work
as a referee. Cayley was a man of sound sense, and of great
service in University administration; Sylvester satisfied the
popular idea of a mathematician as one lost in reflection, and
high above mundane affairs. Cayley was modest and retiring;
Sylvester, courageous and full of his own importance. But
while Cayley's papers, almost all, have the stamp of pure logical
mathematics, Sylvester's are full of human interest. Cayley
was no orator and no poet; Sylvester was an orator, and if
not a poet, he at least prided himself on his poetry. It was
not long before Cayley was provided with a chair at Cam-
bridge, where he immediately married, and settled down to
work as a mathematician in the midst of the most favorable
environment. Sylvester was obliged to continue what he called
" fighting the world " alone and unmarried.

There is an ancient foundation in London, named after its
founder, Gresham College. In 1854 the lectureship of geometry
fell vacant and Sylvester applied. The trustees requested him
and I suppose also the other candidates, to deliver a probation-
ary lecture; with the result that he was not appointed. The
professorship of mathematics in the Royal Military Academy

110 TEN BRITISH MATHEMATICIANS

at Woolwich fell vacant; Sylvester was again unsuccessful; but

the appointee died in the course of a year, and then Sylvester

succeeded on a second application. This was in 1855, when he

was 41 years old.

He was a professor at the Military Academy for fifteen years;

and these years constitute the period of his greatest scientific

activity. In addition to continuing his work on the theory of

invariants, he was guided by it to take up one of the most

difficult questions in the theory of numbers. Cayley had

reduced the problem of the enumeration of invariants to that

of the partition of numbers; Sylvester may be said to have

revolutionized this part of mathematics by giving a complete

analytical solution of the problem, which was in effect to

enumerate the solutions in positive integers of the indeterminate

equation :

ax-\-by-\-cz-\- .... -\-ld = m.

Thereafter he attacked the similar problem connected with two
such simultaneous equations (known to Euler as the problem
of the Virgins) and was partially and considerably successful. In
June, 1859, he delivered a series of seven lectures on compound
partition in general at King's College, London. The outlines
of these lectures have been published by the Mathematical
Society of London.

Five years later (1864) he contributed to the Royal Society
of London what is considered his greatest mathematical achieve-
mant. Newton, in his lectures on algebra, which he called
" Universal Arithmetic " gave a rule for calculating an inferior
limit to the number of imaginary roots in an equation of any
degree, but he did not give any demonstration or indication
of the process by which he reached it. Many succeeding
mathematicians such as Euler, Waring, Maclaurin, took up the
problem of investigating the rule, but they were unable to estab-
lish either its truth or inadequacy. Sylvester in the paper
quoted established the validity of the rule for algebraic equa-
tions as far as the fifth degree inclusive. Next year in a com-
munication to the Mathematical Society of London, he fully
established and generalized the rule. " I owed my success," he

JAMES JOSEPH SYLVESTER 111

said, " chiefly to merging the theorem to be proved in one of
greater scope and generality. In mathematical research, revers-
ing the axiom of Euclid and controverting the proposition of
Hesiod, it is a continual matter of experience, as I have found
myself over and over again, that the whole is less than its part."
Two years later he succeeded De Morgan as president of
the London Mathematical Society. He was the first mathe-
matician to whom that Society awarded the Gold medal founded
in honor of De Morgan. In 1869, when the British Association
met in Exeter, Prof. Sylvester was president of the section of
mathematics and physics. Most of the mathematicians who
have occupied that position have experienced difficulty in find-
ing a subject which should satisfy the two conditions of being
first, cognate to their branch of science; secondly, interesting
to an audience of general culture. Not so Sylvester. He took
up certain views of the nature of mathematical science which
Huxley the great biologist had just published in Macmillan's
Magazine and the Fortnightly Review. He introduced his subject
by saying that he was himself like a great party leader and
orator in the House of Lords, who, when requested to make a
speech at some religious or charitable, at-all-events non-political
meeting declined the honor on the ground that he could not
speak unless he saw an adversary before him. I shall now
quote from the address, so that you may hear Sylvester's own

words.

" In obedience," he said, "to a somewhat similar combative
instinct, I set to myself the task of considering certain utter-
ances of a most distinguished member of the Association, one
whom I no less respect for his honesty and public spirit, than
I admire for his genius and eloquence, but from whose opinions
on a subject he has not studied I feel constrained to differ. I
have no doubt that had my distinguished friend, the probable
president-elect of the next meeting of the Association, applied
his uncommon powers of reasoning, induction, comparison,
observation and invention to the study of mathematical science,
he would have become as great a mathematician as he is now
a biologist; indeed he has given public evidence of his ability

112 TEN BRITISH MATHEMATICIANS

to grapple with the practical side of certain mathematical ques-
tions; but he has not made a study of mathematical science
as such, and the eminence of his position, and the weight justly
attaching to his name, render it only the more imperative that
any assertion proceeding from such a quarter, which may appear
to be erroneous, or so expressed as to be conducive to error?
should not remain unchallenged or be passed over in silence.

" Huxley says ' mathematical training is almost purely
deductive. The mathematician starts with a few simple prop-
ositions, the proof of which is so obvious that they are called
self-evident, and the rest of his work consists of subtle deduc-
tions from them. The teaching of languages at any rate as
ordinarily practised, is of the same general nature — authority
and tradition furnish the data, and the mental operations are
deductive.' It would seem from the above somewhat singularly
juxtaposed paragraphs, that according to Prof. Huxley, the
business of the mathematical student is, from a limited number
of propositions (bottled up and labelled ready for use) to deduce
any required result by a process of the same general nature
as a student of languages employs in declining and conjugating
his nouns and verbs — that to make out a mathematical propo-
sition and to construe or parse a sentence are equivalent or
identical mental operations. Such an opinion scarcely seems
to need serious refutation. The passage is taken from an article
in Macmillan's Magazine for June last, entitled, ' Scientific
Education — Notes of an after-dinner speech'; and I cannot
but think would have been couched in more guarded terms by
my distinguished friend, had his speech been made before dinner
instead of after.

" The notion that mathematical truth rests on the narrow
basis of a limited number of elementary propositions from
which all others are to be derived by a process of logical inference
and verbal deduction has been stated still more strongly and
explicitly by the same eminent writer in an article of even date
with the preceeding in the Fortnightly Review; where we are
told that ' Mathematics is that study which knows nothing of
observation, nothing of experiment, nothing of induction, nothing

JAMES JOSEPH SYLVESTER 113

of causation.' I think no statement could have been made
more opposite to the undoubted facts of the case, which are
that mathematical analysis is constantly invoking the aid of
new principles, new ideas and new methods not capable of
being denned by any form of words, but springing direct from
the inherent powers and activity of the human mind, and from
continually renewed introspection of that inner world of thought
of which the phenomena are as varied and require as close
attention to discern as those of the outer physical world; that
it is unceasingly calling forth the faculties of observation and
comparison; that one of its principal weapons is induction;
that is has frequent recourse to experimental trial and veri-
fication; and that it affords a boundless scope for the exercise
of the highest efforts of imagination and invention."

Huxley never replied; convinced or not, he had sufficient
sagacity to see that he had ventured far beyond his depth.
In the portion of the address quoted, Sylvester adds paren-
thetically a clause which expresses his theory of mathematical
knowledge. He says that the inner world of thought in each
individual man (which is the world of observation to the mathe-
matician) may be conceived to stand in somewhat the same
general relation of correspondence to the outer physical world
as an object to the shadow projected from it. To him the
mental order was more real than the world of sense, and the
foundation of mathematical science was ideal, not experimental.
By this time Sylvester had received most of the high dis-
tinctions, both domestic and foreign, which are usually awarded
to a mathematician of the first rank in his day. But a dis-
continuity was at hand. The War Office issued a regulation
whereby officers of the army were obliged to retire on half pay
on reaching the age of 55 years. Sylvester was a professor in
a Military College; in a few, months, on his reaching the pre-
scribed age, he was retired on half pay. He felt that though
no longer fit for the field he was still fit for the classroom. And
he felt keenly the diminution in his income. It was about this
time that he issued a small volume— the only book he ever
published; not on mathematics, as you may suppose, but

114 TEN BRITISH MATHEMATICIANS

entitled The Laws of Verse. He must have prided himself a
good deal on this composition, for one of his last letters in Nature
is signed " J. J. Sylvester, author of The Laws of Verse." He
made some excellent translations from Horace and from German
poets; and like Sir W. R. Hamilton he was accustomed to express
his feelings in sonnets.

The break in his life appears to have discouraged Sylvester
for the time being from engaging in any original research. But
after three years a Russian mathematician named Tschebicheff,
a professor in the University of Saint Petersburg, visiting
Sylvester in London, drew his attention to the discovery by a
Russian student named Lipkin, of a mechanism for drawing
a perfect straight line. Mr. Lipkin received from the Russian
Government a substantial award. It was found that the same
discovery had been made several years before by M. Peaucellier,
an officer in the French army, but failing to be recognized at
its true value had dropped into oblivion. Sylvester introduced
the subject into England in the form of an evening lecture before
the Royal Institution, entitled " On recent discoveries in mechan-
ical conversion of motion." The Royal Institution of London
was founded to promote scientific research; its professors have
been such men as Davy, Faraday, Tyndall, Dewar. It is not
a teaching institution, but it provides for special courses of
lectures in the afternoons and for Friday evening lectures by
investigators of something new in science. The evening lec-
tures are attended by fashionable audiences of ladies and gentle-
men in full dress.

Euclid bases his Elements on two postulates; first, that a
straight line can be drawn, second, that a circle can be described.
It is sometimes expressed in this way; he postulates a ruler and
compass. The latter contrivance is not difficult to construct,
because it does not involve the use of a ruler or a compass in
its own construction. But how is a ruler to be made straight,
unless you already have a ruler by which to test it? The
problem is to devise a mechanism which shall assume the second
postulate only, and be able to satisfy the firsts It is the mechan-
ical problem of converting motion in a circle into motion in

JAMES JOSEPH SYLVESTER 115

a straight line, without the use of any guide. James Watt,
the inventor of the steam-engine, tackled the problem with all
his might, but gave it up as impossible. However, he succeeded
in finding a contrivance which solves the problem very approxi-
mately. Watt's parallelogram, employed in nearly every beam-
engine, consists of three links; of which AC and BD are
equal, and have fixed pivots at A and B respectively. The
link CD is of such a length that AC and BD are parallel
when horizontal. The tracing

ao <?c

point is attached to the middle

point of CD. When C and D \

move round their pivots, the

tracing point describes a straight line very approximately, so
long as the arc of displacement is small. The complete figure
which would be described is the figure of 8, and the part utilized
is near the point of contrary flexure.

A linkage giving a closer approximation to a straight line
was also invented by the Russian mathematician before men-
tioned— Tschebicheff; it likewise made use of three links. But
the linkage invented by Peaucellier and later by Lipkin had
seven pieces. The arms AB and AC are of equal length, and

have a fixed pivot at A. The links
DB, BE, EC, CD are of equal length.
EF is an arm connecting E with the
fixed pivot F and is equal in length to
the distance between A and F. It is
readily shown by geometry that, as
--jr.-- the point E describes a circle around
the center F, the point D describes
an exact straight line perpendicular to the line joining it and F.
The exhibition of this contrivance at work was the climax of
Sylvester's lecture.

In Sylvester's audience were two mathematicians, Hart and
Kempe, who took up the subject for further investigation. Hart
perceived that the contrivances of Watt and of Tschebicheff
consisted of three links, whereas Peaucellier's consisted of seven.
Accordingly he searched for a contrivance of five links which

116 TEN BRITISH MATHEMATICIANS

would enable a tracing point to describe a perfect straight line;
and he succeeded in inventing it. Kempe was a London barrister
whose specialty was ecclesiastical law. He and Sylvester worked
up the theory of linkages together, and discovered among other
things the skew pantograph. Kempe became so imbued with
linkage that he contributed to the Royal Society of London
a paper on the " Theory of Mathematical Form," in which he
explains all reasoning by means of linkages.

About this time (1877) the Johns Hopkins University was
organized at Baltimore, and Sylvester, at the age of 6t,} was
appointed the first professor of mathematics. Of his work
there as a teacher, one of his pupils, Dr. Fabian Franklin, thus
spoke in an address delivered at a memorial meeting in that
University: "The one thing which constantly marked Syl-
vester's lectures was enthusiastic love of the thing he was doing.
He had in the fullest possible degree, to use the French phrase,
the defect of this quality; for as he almost always spoke with
enthusiastic ardor, so it was almost never possible for him to
speak on matters incapable of evoking this ardor. In other
words, the substance of his lectures had to consist largely of his
own work, and, as a rule, of work hot from the forge. The
consequence was that a continuous and systematic presentation
of any extensive body of doctrine already completed was not to
be expected from him. Any unsolved difficulty, any suggested
extension, such as would have been passed by with a mention
by other lecturers, became inevitably with him the occasion of
a digression which was sure to consume many weeks, if indeed
it did not take him away from the original object permanently.
Nearly all of the important memoirs which he published, while
in Baltimore, arose in this way. We who attended his lectures
may be said to have seen these memoirs in the making. He
would give us on the Friday the outcome of his grapplings with
the enemy since the Tuesday lecture. Rarely can it have
fallen to the lot of any class to follow so completely the workings
of the mind of the master. Not only were all thus privileged to
see ' the very pulse of the machine,' to learn the spring and
motive of the successive steps that led to his results, but we

JAMES JOSEPH SYLVESTER 117

were set aglow by the delight and admiration which, with perfect
naivete and with that luxuriance of language peculiar to him,
Sylvester lavished upon these results. That in this enthusiastic
admiration he sometimes lacked the sense of proportion cannot
be denied. A result announced at one lecture and hailed with
loud acclaim as a marvel of beauty was by no means sure of
not being found before the next lecture to have been erroneous;
but the Esther that supplanted this Vashti was quite certain
to be found still more supremely beautiful. The fundamental
thing, however, was not this occasional extravagance, but the
deep and abiding feeling for truth and beauty which underlay
it. No young man of generous mind could stand before that
superb grey head and hear those expositions of high and dear-
bought truths, testifying to a passionate devotion undimmed
by years or by arduous labors, without carrying away that
which ever after must give to the pursuit of truth a new and
deeper significance in his mind."

One of Sylvester's principal achievements at Baltimore was
the founding of the American Journal of Mathematics, which,
at his suggestion, took the quarto form. He aimed at estab-
lishing a mathematical journal in the English language, which
should equal Liouville's Journal in France, or Crelle's Journal
in Germany. Probably his best contribution to the American
Journal consisted in his "Lectures on Universal Algebra";
which, however, were left unfinished, like a great many other
projects of his.

Sylvester had that quality of absent-mindedness which is
popularly supposed to be, if not the essence, at least an invariable
accompaniment, of a distinguished mathematician. Many
stories are related on this point, which, if not all true, are at
least characteristic. Dr. Franklin describes an instance which
actually happened in Baltimore. To illustrate a theory of
versification contained in his book The Laws of Verse, Sylvester
prepared a poem of 400 lines, all rhyming with the name Rosa-
lind or Rosalind; and it was announced that the professor would
read the poem on a specified evening at a specified hour at the
Peabody Institute. At the time appointed there was a large

118 TEN BRITISH MATHEMATICIANS

turn-out of ladies and gentlemen. Prof. Sylvester, as usual,
had a number of footnotes appended to his production; and he
announced that in order to save interruption in reading the
poem itself, he would first read the footnotes. The reading of
the footnotes suggested various digressions to his imagination;
an hour had passed, still no poem; an hour and a half passed
and the striking of the clock or the unrest of his audience
reminded him of the promised poem. He was astonished to
find how time had passed, excused all who had engagements,
and proceeded to read the Rosalind poem.

In the summer of 1881 I visited London to see the Electrical
Exhibition in the Crystal Palace — one of the earliest exhibitions
devoted to electricity exclusively. I had made some investi-
gations on the electric discharge, using a Holtz machine where
De LaRue used a large battery of cells. Mr. De LaRue was
Secretary of the Royal Institution; he gave me a ticket to a
Friday evening discourse to be delivered by Mr. Spottiswoode,
then president of the Royal Society, on the phenomena of the
intensive discharge of electricity through gases; also an invi-
tation to a dinner at his own house to be given prior to the
lecture. Mr. Spottiswoode, the lecturer for the evening, was
there; also Prof. Sylvester. He was a man rather under the
average height, with long gray beard and a profusion of gray
locks round his head surmounted by a great dome of forehead.
He struck me as having the appearance of an artist or a poet
rather than of an exact scientist. After dinner he conversed
very eloquently with an elderly lady of title, while I conversed
with her daughter. Then cabs were announced to take us to
the Institution. Prof. Sylvester and I, being both bachelors,
were put in a cab together. The professor, who had been so
eloquent with the lady, said nothing; so I asked him how he
liked his work at the Johns Hopkins University. " It is very
pleasant work indeed," said he, " and the young men who
study there are all so enthusiastic." We had not exhausted
that subject before we reached our destination. We went up
the stairway together, then Sylvester dived into the library to
see the last number of Comptes Rendus (in which he published

JAMES JOSEPH SYLVESTER

119

many of his results at that time) and I saw him no more. I have

always thought it very doubtful whether he came out to hear

Spottiswoode's lecture.

We have seen that H. J. S. Smith, the Savilian professor of

Geometry at Oxford, died in 1883. Sylvester's friends urged

his appointment, with the result that he was elected. After two

years he delivered his inaugural lecture; of which the subject

was differential invariants, termed by him reciprocants. An

dP"\i d2v d x

elementary reciprocant is ~~, for if — = 0 then — = 0. He

looked upon this as the " grub " form, and developed from it
the " chrysalis "

d2<!> d2cj> dcf>

dx2

d24>

and the " imago

dxdy

d4
dx

d2$
dx2

d2$

dxdy
d2$

dxdy

d?±
dy2

d£
dy

d2$

dxdy

d2$

dy2

d2$

dx

d$
dy'

d2$

dxdr

d2$
dydr

d2$>
dr2'

dxdr dydr

You will observe that the chrysalis expression is unsymmetrical;
the place of a ninth term is vacant. It moved Sylvester's poetic
imagination, and into his inaugural lecture he interjected the
following sonnet:

To a Missing Member of a Family Group of Terms in
an Algebraical Formula:

Lone and discarded one ! divorced by fate,
Far from thy wished-for fellows — whither art flown?
Where lingerest thou in thy bereaved estate,
Like some lost star, or buried meteor stone?

120 TEN BRITISH MATHEMATICIANS

Thou minds't me much of that presumptuous one,

Who loth, aught less than greatest, to be great.

From H.aven's immensity fell headlong down

To live forlorn, self-centred, desolate:

Or who, new Heraklid, hard exile bore,

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

Till throned Astraea, 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."

This inaugural lecture was the beginning of his last great
contribution to mathematics, and the subsequent lectures of
that year were devoted to his researches in that line. Smith
and Sylvester were akin in devoting attention to the theory of
numbers, and also in being eloquent speakers. But in other
respects the Oxonians found a great difference. Smith had
been a painstaking tutor; Sylvester could lecture only on his
own researches, which were not popular in a place so wholly
given over to examinations. Smith was an incessantly active
man of affairs; Sylvester became the subject of melancholy
and complained that he had no friends.

In 1872 a deputy professor was appointed. Sylvester re-
moved to London, and lived mostly at the Athenaeum Club.
He was now 78 years of age, and suffered from partial loss of
sight and memory. He was subject to melancholy, and his
condition was indeed " forlorn and desolate." His nearest rela-
tives were nieces, but he did not wish to ask their assistance.
One day, meeting a mathematical friend who had a home in Lon-
don, he complained of the fare at the Club, and asked his friend
to help him find suitable private apartments where he could have
better cooking. They drove about from place to place for a
whole afternoon, but none suited Sylvester. It grew late: Syl-
vester said, " You have a pleasant home: take me there," and
this was done. Arrived, he appointed one daughter his reader
and another daughter his amanuensis. " Now," said he, "I feel
comfortably installed; don't let my relatives know where I
am." The fire of his temper had not dimmed with age, and it
required all the Christian fortitude of the ladies to stand his

JAMES JOSEPH SYLVESTER 121

exactions. Eventually, notice had to be sent to his nieces to
come and take charge of him. He died on the 15th of March,
1897, in the 83d year of his age, and was buried in the Jewish
cemetery at Dais ton.

As a theist, Sylvester did not approve of the destructive
attitude of such men as Clifford, in matters of religion. In the
early days of his career he suffered much from the disabilities
attached to his faith, and they were the prime cause of so much
" fighting the world." He was, in all probability, a greater
mathematical genius than Cayley; but the environment in
which he lived for some years was so much less favorable that
he was not able to accomplish an equal amount of solid work.
Sylvester's portrait adorns St. John's College, Cambridge. A
memorial fund of £1500 has been placed in the charge of the
Royal Society of London, from the proceeds of which a medal
and about £100 in money is awarded triennially for work
done in pure mathematics. The first award has been made to
M. Henri Poincare of Paris, a mathematician for whom Sylvester
had a high professional and personal regard.

THOMAS PENYNGTON KIRKMAN*
(1806-1895)

Thomas Penyngton Kirkman was born on March 31, 1806,
at Bolton in Lancashire. He was the son of John Kirkman,
a dealer in cotton and cotton waste; he had several sisters but
no brother. He was educated at the Grammar School of Bolton,
where the tuition was free. There he received good instruction
in Latin and Greek, but no instruction in geometry or algebra;
even Arithmetic was not then taught in the headmaster's upper
room. He showed a decided taste for study and was by far
the best scholar in the school. His father, who had no taste
for learning and was succeeding in trade, was determined that
his only son should follow his own business, and that without
any loss of time. The schoolmaster tried to persuade the father
to let his son remain at school; and the vicar also urged the
father, saying that if he would send his son to Cambridge Uni-
versity, he would guarantee for sixpence that the boy would
win a fellowship. But the father was obdurate; young Kirk-
man was removed from school, when he was fourteen years of
age, and placed at a desk in his father's office. While so engaged,
he continued of his own accord his study of Latin and Greek,
and added French and German.

After ten years spent in the counting room, he tore away
from his father, secured the tuition of a young Irish baronet,
Sir John Blunden, and entered the University of Dublin with
the view of passing the examinations for the degree of B.A.
There he never had instruction from any tutor. It was not
until he entered Trinity College, Dublin, that he opened any
mathematical book. He was not of course abreast with men
who had good preparation. What he knew of mathematics,

* This Lecture was delivered April 20, 1903. — Editors.
122

THOMAS PENYNGTON KIRKMAN 123

he owed to his own study, having never had a single hour's
instruction from any person. To this self-education is due, it
appears to me, both the strength and the weakness to be found
in his career as a scientist. However, in his college course he
obtained honors, or premiums as they are called, and graduated
as a moderator, something like a wrangler.

Returning to England in 1835, when he was 29 years old,
he was ordained as a minister in the Church of England. He
was a curate for five years, first at Bury, afterwards at Lymm;
then he became the vicar of a newly-formed parish— Croft with
Southworth in Lancashire. This parish was the scene of his
life's labors. The income of the benefice was not large, about
£200 per annum; for several years he supplemented this by
taking pupils. He married, and property which came to his
wife enabled them to dispense with the taking of pupils. His
father became poorer, but was able to leave some property to
his son and daughters. His parochial work, though small, was
discharged with enthusiasm; out of the roughest material he
formed a parish choir of boys and girls who could sing at sight
any four-part song put before them. After the private teach-
ing was over he had the leisure requisite for the great mathe-
matical researches in which he now engaged.

Soon after Kirkman was settled at Croft, Sir William Rowan
Hamilton began to publish his quaternion papers and, being a
graduate of Dublin University, Kirkman was naturally one of
the first to study the new analysis. As the fruit of his medi-
tations he contributed a paper to the Philosophical Magazine^
" On pluquaternions and homoid products of sums of n squares."
He proposed the appellation " pluquaternions " for a linear
expression involving more than three imaginaries (the i, j, k of
Hamilton), " not dreading " he says, " the pluperfect criticism
of grammarians, since the convenient barbarism is their own."
Hamilton, writing to De Morgan, remarked "Kirkman is a
very clever fellow," where the adjective has not the American
colloquial meaning but the English meaning.

For his own education and that of his pupils he devoted
much attention to mathematical mnemonics, studying the

124 TEN BRITISH MATHEMATICIANS

Memoria Technica of Grey. In 1851 he contributed a paper on
the subject to the Literary and Philosophical Society of Man-
chester, and in 1852 he published a book, First Mnemonical Les-
sons in Geometry, Algebra, and Trigonometry, which is dedicated
to his former pupil, Sir John Blunden. De Morgan pronounced
it " the most curious crochet I ever saw," which was saying a
great deal, for De Morgan was familiar with many quaint books
in mathematics. In the preface he says that much of the dis-
taste for mathematical study springs largely from the difficulty
of retaining in the memory the previous results and reasoning.
" This difficulty is closely connected with the unpronounceable-
ness of the formulae; the memory of the tongue and the ear arc
not easily turned to account; nearly everything depends on the
thinking faculty or on the practice of the eye alone. Hence
many, who see hardly anything formidable in the study of a
language, look upon mathematical acquirements as beyond
their power, when in truth they are very far from being so.
My object is to enable the learner to l talk to himself,' in rapid,
vigorous and suggestive syllables, about the matters which he
must digest and remember. I have sought to bring the memory
of the vocal organs and the ear to the assistance of the reasoning
faculty and have never scrupled to sacrifice either good grammar
or good English in order to secure the requisites for a useful
mnemonic, which are smoothness, condensation, and jingle.

As a specimen of his mnemonics we may take the cotangent
formula in spherical trigonometry:

cot A sin C+cos b cos C = cot a sin b

To remember this formula most masters then required some aid
to the memory; for instance the following: If in any spherical
triangle four parts be taken in succession, such as AbCa, consist-
ing of two means bC and two extremes Aa, then the product of
the cosines of the two means is equal to the sine of the mean
side X cotangent of the extreme side minus sine of the mean
angle X cotangent of the extreme angle, that is

cos b cos C = sin b cot a — sin C cot A .

THOMAS PENYNGTON KIRKMAN 125

This is an appeal to the reason. Kirkman, however, proceeds
on the principle of appealing to the memory of the ear, of the
tongue, and of the lips altogether; a true memoria technica.
He distinguishes the large letter from the small by calling them
Ang, Bang, Cang {ang from angle in contrast to side). To
make the formula more euphoneous he drops the s from cos
and the n from sin. Hence the formula is

cot Ang si Cang and co b co Cang are cot a si b

which is to be chanted till it becomes perfectly familiar to the
ear and the lips. The former rule is a hint offered to the judg-
ment; Kirkman's method is something to be taught by rote.
In his book Kirkman makes much use of verse, in the turning
of which he was very skillful.

In the early part of the nineteenth century a publication
named the Lady's and Gentlemen's Diary devoted several columns
to mathematical problems. In 1844 the editor offered a prize
for the solution of the following question: " Determine the num-
ber of combinations that can be made out of n symbols, each
combination having p symbols, with this limitation, that no
combination of q symbols which may appear in any one of
them, may be repeated in any other." This is a problem of
great difficulty; Kirkman solved it completely for the special
case of ^ = 3 and q = 2 and printed his results in the second
volume of the Cambridge and Dublin Mathematical Journal.
As a chip off this work he published in the Diary for 1850 the
famous problem of the fifteen schoolgirls as follows: " Fifteen
young ladies of a school walk out three abreast for seven days
in succession; it is required to arrange them daily so that no two
shall walk abreast more than once." To form the schedules
for seven days is not difficult; but to find all the possible
schedules is a different matter. Kirkman found all the possible
combinations of the fifteen young ladies in groups of three to
be 35, and the problem was also considered and solved by Cayley,
and has been discussed by many later writers; Sylvester gave
91 as the greatest number of days; and he also intimated that
the principle of the puzzle was known to him when an under-

126 TEN BRITISH MATHEMATICIANS

graduate at Cambridge, and that he had given it to fellow
undergraduates. Kirkman replied that up to the time he pro-
posed the problem he had neither seen Cambridge nor met
Sylvester, and narrated how he had hit on the question.

The Institute of France offered several times in succession
a prize for a memoir on the theory of the polyedra; this fact
together with his work in combinations led Kirkman to take
up the subject. He always writes polyedron not polyhedron;
for he says we write periodic not perihodic. When Kirkman
began work nothing had been done beyond the very ancient
enumeration of the five regular solids and the simple combi-
nations of crystallography. His first paper, " On the represen-
tation and enumeration of the polyedra," was communicated
in 1850 to the Literary and Philosophical Society of Manchester.
He starts with the well-known theorem P+S = L+2, where P
is the number of points or summits, 5 the number of plane
bounding surfaces and L the number of linear edges in a geo-
metrical solid. " The question — how many w-edrons are
there? — has been asked, but it is not likely soon to receive a
definite answer. It is far from being a simple question, even
when reduced to the narrower compass — how many w-edrons
are there whose summits are all trihedral "? He enumerated
and constructed the fourteen 8-edra whose faces are all tri-
angles.

In 1858 the French Institute modified its prize question.
As the subject for the concours of 1861 was announced: "Per-
fectionner en quelque point important la theorie geometrique
des polyedres," where the indefiniteness of the question indi-
cates the very imperfect state of knowledge on the subject.
The prize offered was 3000 francs. Kirkman appears to have
worked at it with a view of competing, but he did not send in
his memoir. Cayley appears to have intended to compete.
The time was prolonged for a year, but there was no award and
the prize was taken down. Kirkman communicated his results
to the Royal Society through his friend Cayley, and was soon
elected a Fellow. Then he contributed directly an elaborate
paper entitled " Complete theory of the Polyedra." In the

THOMAS PENYNGTON KIRKMAN 127

preface he says, " The following memoir contains a complete
solution of the classification and enumeration of the P-edra
Q-acra. The actual construction of the solids is a task imprac-
ticable from its magnitude, but it is here shown that we can
enumerate them with an accurate account of their symmetry
to any values of P and Q." The memoir consisted of 21 sec-
tions; only the two introductory sections, occupying 45 quarto
pages, were printed by the Society, while the others still remain
in manuscript. During following years he added many con-
tributions to this subject.

In 1858 the French Academy also proposed a problem in the
Theory of Groups as the subject for competition for the grand
mathematical prize in i860: " Quels peuvent 6tre les nombres
de valeurs des fonctions bien definies qui contiennent un nombre
donne de lettres, et comment peut on former les fonctions pour
lesquelles il existe un nombre donne de valeurs? " Three
memoirs were presented, of which Kirkman's was one, but no
prize was awarded. Not the slightest summary was vouch-
safed of what the competitors had added to science, although
it was confessed that all had contributed results both new and
important; and the question, though proposed for the first
time for the year i860, was withdrawn from competition con-
trary to the usual custom of the Academy. Kirkman con-
tributed the results of his investigation to the Manchester
Society under the title " The complete theory of groups, being
the solution of the mathematical prize question of the French
Academy for i860." In more recent years the theory of groups
has engaged the attention of many mathematicians in Germany
and America; so far as British contributors are concerned
Kirkman was the first and still remains the greatest.

In 1861 the British Association met at Manchester; it was
the last of its meetings which Sir William Rowan Hamilton
attended. After the meeting Hamilton visited Kirkman at his
home in the Croft rectory, and that meeting was no doubt a
stimulus to both. As regards pure mathematics they were
probably the two greatest in Britain; both felt the loneliness of
scientific work, both were metaphysicians of penetrating power,

128 TEN BRITISH MATHEMATICIANS

both were good versifiers if not great poets. Of nearly the same
age, they were both endowed with splendid physique; but the
care which was taken of their health was very different; in
four years Hamilton died but Kirkman lived more than 30 years
longer.

About 1862 the Educational Times, a monthly periodical
published in London, began to devote several columns to the
proposing and solving of mathematical problems, taking up
the work after the demise of the Diary. This matter was after-
wards reprinted in separate volumes, two for each year. In
these reprints are to be found many questions proposed by
Kirkman; they are generally propounded in quaint verse, and
many of them were suggested by his study of combinations. A
good specimen is " The Revenge of Old King Cole "

" Full oft ye have had your fiddler's fling,

For your own fun over the wine;

And now " quoth Cole, the merry old king,

" Ye shall have it again for mine.

My realm prepares for a week of joy

At the coming of age of a princely boy —

Of the grand six days procession in square,

In all your splendour dressed,

Filling the city with music rare

From fiddlers five abreast," etc.

The problem set forth by this and other verses is that of
25 men arranged in five rows on Monday. Shifting the second
column one step upward, the third two steps, the fourth three

Monday Tuesday Wednesday Thursday

ABCDE AGMSY ALWIT AQHXO

FGHIJ FLRXE FQCNY FVMDT

KLMNO KQWDJ KVHSE K B R I Y

PQRST PVCIO PBMXJ PGWNE

UVWXY UBHNT NGRDO ULCSJ

steps, and the fifth four steps gives the arrangement for Tuesday.
Applying the same rule to Tuesday gives Wednesday's array,
and similarly are found those for Thursday and Friday. In

THOMAS PENYNGTON KIRKMAN 129

none of these can the same two men be found in one row. But
the rule fails to work for Saturday, so that a special arrangement
must be brought in which I leave to my hearers to work out.
This problem resembles that of the fifteen schoolgirls.

The Rev. Kirkman became at an early period of his life a
broad churchman. About 1863 he came forward in defense of
the Bishop of Colenso, a mathematician, and later he contributed
to a series of pamphlets published in aid of the cause of "Free
Enquiry and Free Expression." In one of his letters to me
Kirkman writes as follows: " The Life of Colenso by my friend
Rev. Sir George Cox, Bart., is a most charming book; and
the battle of the Bishops against the lawyers in the matter of
the vacant see of Natal, to which Cox is the bishop-elect, is
exciting. Canterbury refuses to ask, as required, the Queen's
mandate to consecrate him. The Natal churchmen have just
petitioned the Queen to make the Primate do his duty accord-
ing to law. Natal was made a See with perpetual succession,
and is endowed. The endowment has been lying idle since
Colenso's death in 1883; and the bishops who have the law
courts dead against them here are determined that no successor
to Colenso shall be consecrated. There is a Bishop of South
African Church there, whom they thrust in while Colenso lived,
on pretense that Colenso was excommunicate. We shall soon
see whether the lawyers or the bishops are to win." It was
Kirkman's own belief that his course in this matter injured
his chance of preferment in the church; he never rose above
being rector of Croft.

While a broad churchman the Rev. Mr. Kirkman was very
vehement against the leaders of the materialistic philosophy.
Two years after Tyndall's Belfast address, in which he announced
that he could discern in matter the promise and potency of
every form of life, Kirkman published a volume entitled Philoso-
phy without Assumptions, in which he criticises in very vigorous
style the materialistic and evolutional philosophy advocated by
Mill, Spencer, Tyndall, and Huxley. In ascribing everything
to matter and its powers or potencies he considers that they
turn philosophy upside down. He has, he writes, first-hand

130 TEN BRITISH MATHEMATICIANS

knowledge of himself as a continuous person, endowed with
will; and he infers that there are will forces around; but he
sees no evidence of the existence of matter. Matter is an
assumption and forms no part of his philosophy. He relies on
Boscovich's theory of an atom as simply the center of forces.
Force he understands from his knowledge of will, but any other
substance he does not understand. The obvious difficulty in
this philosophy is to explain the belief in the existence of other
conscious beings — other will forces. Is it not the great assump-
tion which everyone is obliged to make; verified by experience,
but still in its nature an assumption? Kirkman tries to get
over this difficulty by means of a syllogism, the major premise
of which he has to manufacture, and which he presents to his
reason for adoption or rejection. How can a universal propo-
sition be easier to grasp than the particular case included in it?
If the mind doubts about an individual case, how can it be sure
about an infinite number of such cases? It is a petitio principii.
As a critic of the materialistic philosophy Kirkman is more
successful. He criticises Herbert Spencer on free will as follows :
" The short chapter of eight pages on Will cost more philosophi-
cal toil than all the two volumes on Psychology. The author
gets himself in a heat, he runs himself into a corner, and brings
himself dangerously to bay. Hear him: ' To reduce the general
question to its simplest form; psychical changes either conform
to law, or they do not. If they do not conform to law, this work,
in common with all other works on the subject, is sheer non-
sense; no science of Psychology is possible. If they do conform
to law, there cannot be any such thing as free will' Here we
see the horrible alternative. If the assertors of free will refuse
to commit suicide, they must endure the infinitely greater pang
of seeing Mr. Spencer hurl himself and his books into that
yawning gulf, a sacrifice long devoted, and now by pitiless Fate
consigned, to the abysmal gods of nonsense. Then pitch him
down say I. Shall I spare him who tells me that my movements
in this orbit of conscious thought and responsibility are made
under ' parallel conditions ' with those of yon driven moon?
Shall I spare him who has juggled me out of my Will, my noblest

THOMAS PENYNGTON KIRKMAN 131

attribute; who has hocuspocused me out of my subsisting
personality; and then, as a refinement of cruelty, has frightened
me out of the rest of my wits by forcing me to this terrific alterna-
tive that either the testimony of this Being, this Reason and this
Conscience is one ever-thundering lie, or else he, even he, has
talked nonsense? He has talked nonsense, I say it because I
have proved it. And every man must of course talk nonsense
who begins his philosophy with abstracts in the clouds instead
of building on the witness of his own self-consciousness. ' If
they do conform to law,' says Spencer, ' there cannot be any such
thing as free will.' The force of this seems to depend on his
knowledge of ' law.' When I ask, What does this writer know
of law — definite working law in the Cosmos? — the only answer
I can get is — Nothing, except a very little which he has picked
up, often malappropriately, as we have seen, among the mathe-
maticians. When I ask — What does he know about law? — there
is neither beginning nor end to the reply. I am advised to read
his books about law, and to master the differentiations and inte-
grations of the coherences, the correlations, the uniformities,
and universalities which he has established in the abstract over
all space and all time by his vast experience and miraculous
penetration. I have tried to do this, and have found all pretty
satisfactory, except the lack of one thing — something like proof
of his competence to decide all that scientifically. When I
persist in my demand for such proof, it turns out at last — that he
knows by heart the whole Hymn Book, the Litanies, the Missal,
and the Decretals of the Must-be-ite religion! ' Conform to
law.' Shall I tell you what he means by that? Exactly ninety-
nine hundredths of his meaning under the word law is must be."
Kirkman points out that the kind of proof offered by these
philosophers is a bold assertion of must-be-so. For instance
he mentions Spencer's evolution of consciousness out of the
unconscious: " That an effectual adjustment may be made
they (the separate impressions or constituent changes of a com-
plex correspondence to be coordinated) must be brought into
relation with each other. But this implies some center of com-
munication common to them all, through which they severally

132 TEN BRITISH MATHEMATICIANS

pass; and as they cannot pass through it simultaneously, they
must pass through it in succession. So that as the external
phenomena responded to become greater in number and more
complicated in kind, the variety and rapidity of the changes
to which this common center of communication is subject must
increase, there must result an unbroken series of those changes,
there must arise a consciousness."

The paraphrase which Kirkman gave of Spencer's definition
of Evolution commended itself to such great minds as Tait and
Clerk-Maxwell. Spencer's definition is: " Evolution is a change
from an indefinite incoherent homogeneity to a definite coherent
heterogeneity, through continuous differentiations and integra-
tions." Kirkman's' paraphrase is " Evolution is a change from
a nohowish untalkaboutable all-likeness, to a somehowish and
in-general-talkaboutable not-all-likeness, by continuous some-
thingelseifications and sticktogetherations." The tone of Kirk-
man's book is distinctly polemical and full of sarcasm. He
unfortunately wrote as a theologian rather than as a mathe-
matician. The writers criticised did not reply, although they
felt the edge of his sarcasm; and they acted wisely, for they
could not successfully debate any subject involving exact science
against one of the most penetrating mathematicians of the
nineteenth century.

We have seen that Hamilton appreciated Kirkman's genius;
so did Cayley, De Morgan, Clerk-Maxwell, Tait. One of Tait's
most elaborate researches was the enumeration and construc-
tion of the knots which can be formed in an endless cord — a
subject which he was induced to take up on account of its bear-
ing on the vortex theory of atoms. If the atoms are vortex
filaments their differences in kind, giving rise to differences in
the spectra of the elements, must depend on a greater or less
complexity in the form of the closed filament, and this difference
would depend on the knottiness of the filament. Hence the
main question was " How many different forms of knots are
there with any given small number of crossings?" Tait made
the investigation for three, four, five, six, seven, eight cross-
ings. Kirkman's investigations on the polyedra were much

THOMAS PENYNGTON KIRKMAN 133

allied. He took up the problem and, with some assistance
from Tait, solved it not only for nine but for ten crossings. An
investigation by C. N. Little, a graduate of Yale University,
has confirmed Kirkman's results.

Through Professor Tait I was introduced to Rev. Mr. Kirk-
man; and we discussed the mathematical analysis of relation-
ships, formal logic, and other subjects. After I had gone to the
University of Texas, Kirkman sent me through Tait the follow-
ing question which he said was current in society: " Two boys,
Smith and Jones, of the same age, are each the nephew of the
other; how many legal solutions? " I set the analysis to work,
wrote out the solutions, and the paper is printed in the Proceed-
ings of the Royal Society of Edinburgh. There are four solu-
tions, rovided Smith and Jones are taken to be mere arbitrary,
names; if the convention about surnames holds there are only
two legal solutions. On seeing my paper Kirkman sent the
question to the Educational Times in the following improved

form:

Baby Tom of baby Hugh

The nephew is and uncle too ;

In how many ways can this be true?

Thomas Penyngton Kirkman died on February 3, 1895,
having very nearly reached the age of 89 years. I have found
only one printed notice of his career, but all his writings are
mentioned in the new German Encyclopaedia of Mathematics.
He was an honorary member of the Literary and Philosophical
Societies of Manchester and of Liverpool, a Fellow of the Royal
Society, and a foreign member of the Dutch Society of Sciences
at Haarlem. I may close by a quotation from one of his letters:
" What I have done in helping busy Tait in knots is, like the
much more difficult and extensive things I have done in polyedra
or groups, not at likely to be talked about intelligently by people
so long as I live. But it is a faint pleasure to think it will one
day win a little praise."

ISAAC TODHUNTER*

(1820-1884)

Isaac Todhunter was born at Rye, Sussex, 23 Nov., 1820.
He was the second son of George Todhunter, Congregationalist
minister of the place, and of Mary his wife, whose maiden name
was Hume, a Scottish surname. The minister died of con-
sumption when Isaac was six years old, and left his family,
consisting of wife and four boys, in narrow circumstances. The
widow, who was a woman of strength, physically and mentally,
moved to the larger town of Hastings in the same county, and
opened a school for girls. After some years Isaac was sent to
a boys' school in the same town kept by Robert Carr, and sub-
sequently to one newly opened by a Mr. Austin from London;
for some years he had been unusually backward in his studies,
but under this new teacher he made rapid progress, and his
career was then largely determined.

After his school days were over, he became an usher or
assistant master with Mr. Austin in a school at Peckham; and
contrived to attend at the same time the evening classes at
University College, London. There he came under the great
educating influence of De Morgan, for whom in after years he
always expressed an unbounded admiration; to De Morgan
" he owed that interest in the history and bibliography of
science, in moral philosophy and logic which determined the
course of his riper studies." In 1839 he passed the matriculation
examination of the University of London, then a merely examin-
ing body, winning the exhibition for mathematics (£30 for two
years); in 1842 he passed the B.A. examination carrying off
a mathematical scholarship (of £50 for three years) ; and in 1844
obtained the degree of Master of Arts with the gold medal

* This Lecture was delivered April 13, 1904. — Editors.
134

ISAAC TODHUNTER 135

awarded to the candidate who gained the greatest distinction
in that examination.

Sylvester was then professor of natural philosophy in Uni-
versity College, and Todhunter studied under him. The
writings of Sir John Herschel also had an influence; for Tod-
hunter wrote as follows {Conflict of Studies, p. 66): "Let me
at the outset record my opinion of mathematics; I cannot do
this better than by adopting the words of Sir J. Herschel,
to the influence of which I gratefully attribute the direction of
my own early studies. He says of Astronomy, ' Admission to
its sanctuary can only be gained by one means, — sound and
sufficient knowledge of mathematics, the great instrument of
all exact inquiry, without which no man can ever make such
advances in this or any other of the higher departments of
science as can entitle him to form an independent opinion on
any subject of discussion within their range.' "

When Todhunter graduated as M.A. he was 24 years of age.
Sylvester had gone to Virginia, but De Morgan remained. The
latter advised him to go through the regular course at Cambridge;
his name was now entered at St. John's College. Being some-
what older, and much more brilliant than the honor men of
his year, he was able to devote a great part of his attention to
studies beyond those prescribed. Among other subjects he
took up Mathematical Electricity. In 1848 he took his B.A.
degree as senior wrangler, and also won the first Smith's prize.

While an undergraduate Todhunter lived a very secluded
life. He contributed along with his brothers to the support
of their mother, and he had neither money nor time to spend
on entertainments. The following legend was applied to him,
if not recorded of him: " Once on a time, a senior wrangler
gave a wine party to celebrate his triumph. Six guests took
their seats round the table. Turning the key in the door, he
placed one bottle of wine on the table asseverating with unction,
1 None of you will leave this room while a single drop remains.' "

At the University of Cambridge there is a foundation which
provides for what is called the Burney prize. According to the
regulations the prize is to be awarded to a graduate of the

136 TEN BRITISH MATHEMATICIANS

University who is not of more than three years' standing from
admission to his degree and who shall produce the best English
essay " On some moral or metaphysical subject, or on the
existence, nature and attributes of God, or on the truth and
evidence of the Christian religion." Todhunter in the course
of his first postgraduate year submitted an essay on the thesis
that " The doctrine of a divine providence is inseparable from
the belief in the existence of an absolutely perfect Creator."
This essay received the prize, and was printed in 1849.

Todhunter now proceeded to the degree of M.A., and unlike
his mathematical instructors in University College, De Morgan
and Sylvester, he did not parade his non-conformist principles,
but submitted to the regulations with as good grace as possible.
He was elected a fellow of his college, but not immediately,
probably on account of his being a non-conformist, and appointed
lecturer on mathematics therein; he also engaged for some time
in work as a private tutor, having for one of his pupils P. G.
Tait, and I believe E. J. Routh also.

For a space of 15 years he remained a fellow of St. John's
College, residing in it, and taking part in the instruction. He
was very successful as a lecturer, and it was not long before he
began to publish textbooks on the subjects of his lectures. In
1853 he published a textbook on Analytical Statics; in 1855
one on Plane Coordinate Geometry; and in 1858 Examples of
Analytical Geometry of Three Dimensions. His success in these
subjects induced him to prepare manuals on elementary mathe-
matics; his Algebra appeared in 1858, his Trigonometry in 1859,
his Theory of Equations in 1861, and his Euclid in 1862. Some
of his textbooks passed through many editions and have been
widely used in Great Britain and North America. Latterly
he was appointed principal mathematical lecturer in his college,
and he chose to drill the freshmen in Euclid and other elemen-
tary mathematics.

Within these years he also labored at some works of a more
strictly scientific character. Professor Woodhouse (who was
the forerunner of the Analytical Society) had written a history
of the calculus of variations, ending with the eighteenth century ;

ISAAC TODHTJNTER 137

this work was much admired for its usefulness by Todhunter,
and as he felt a decided taste for the history of mathematics,
he formed and carried out the project of continuing the history
of that calculus during the nineteenth century. It was the
first of the great historical works which has given Todhunter
his high place among the mathematicians of the nineteenth
century. This history was published in 1861; in 1862 he was
elected a Fellow of the Royal Society of London. In 1863 he
was a candidate for the Sadlerian professorship of Mathe-
matics, to which Cayley was appointed. Todhunter was not
a mere mathematical specialist. He was an excellent linguist;
besides being a sound Latin and Greek scholar, he was familiar
with French, German, Spanish, Italian and also Russian, Hebrew
and Sanskrit. He was likewise well versed in philosophy, and
for the two years 1863-5 acted as an Examiner for the Moral
Science Tripos, of which the chief founders were himself and
Whewell.

By 1864 the financial success of his books was such that he
was able to marry, a step which involved the resigning of his
fellowship. His wife was a daughter of Captain George Davies
of the Royal Navy, afterwards Admiral Davies.

As a fellow and tutor of St. John's College he had lived a
very secluded life. His relatives and friends thought he was a
confirmed bachelor. He had sometimes hinted that the grapes
were sour. For art he had little eye; for music no ear. " He
used to say he knew two tunes; one was ' God save the Queen,'
the other wasn't. The former he recognized by the people
standing up." As owls shun the broad daylight he had shunned
the glare of parlors. It was therefore a surprise to his friends
and relatives when they were invited to his marriage in 1864.
Prof. Mayor records that Todhunter wrote to his fiancee, " You
will not forget, I am sure, that I have always been a student,
and always shall be; but books shall not come into even distant
rivalry with you," and Prof. Mayor insinuated that thus fore-
armed, he calmly introduced to the inner circle of their honey-
moon Hamilton on Quaternions.

It was now (1865) that the London Mathematical Society

138 TEN BRITISH MATHEMATICIANS

was organized under the guidance of De Morgan, and Todhunter
became a member in the first year of its existence. The same
year he discharged the very onerous duties of examiner for the
mathematical tripos — a task requiring so much labor and
involving so much interference with his work as an author
that he never accepted it again. Now (1865) appeared his
History of the Mathematical Theory of Probability, and the same
year he was able to edit a new edition of Boole's Treatise on
Differential Equations, the author having succumbed to an
untimely death. Todhunter certainly had a high appreciation
of Boole, which he shared in common with De Morgan. The
work involved in editing the successive editions of his elementary
books was great; he did not proceed to stereotype until many
independent editions gave ample opportunity to correct all
errors and misprints. He now added two more textbooks;
Mechanics in 1867 and Mensuration in 1869.

About 1847 the members of St. John's College founded a
prize in honor of their distinguished fellow, J. C. Adams. It
is awarded every two years, and is in value about £225. In
1869 the subject proposed was " A determination of the circum-
stances under which Discontinuity of any kind presents itself
in the solution of a problem of maximum or minimum in the
Calculus of Variations." There had been a controversy a few
years previous on this subject in the pages of Philosophical
Magazine and Todhunter had there advocated his view of the
matter. " This view is found in the opening sentences of his
essay : ' We shall find that, generally speaking, discontinuity is
introduced, by virtue of some restriction which we impose, either
explicitly or implicitly in the statement of the problems which
we propose to solve.' This thesis he supported by considering
in turn the usual applications of the calculus, and pointing out
where he considers the discontinuities which occur have been
introduced into the conditions of the problem. This he success-
fully proves in many instances. In some cases, the want of a
distinct test of what discontinuity is somewhat obscures the
argument." To his essay the prize was awarded ; it is published
under the title " Researches in the Calculus of Variations " —

ISAAC TODHUNTER 139

an entirely different work from his History of the Calculus of
Variations.

In 1873 ne published his History of the Mathematical
Theories of Attraction. It consists of two volumes of nearly iooo
pages altogether and is probably the most elaborate of his
histories. In the same year (1873) ne published in book form
his views on some of the educational questions of the day,
under the title of The Conflict of Studies, and other essays on
subjects connected with education. The collection contains
six essays; they were originally written with the view of suc-
cessive publication in some magazine, but in fact they were
published only in book form. In the first essay, that on the
Conflict of Studies — Todhunter gave his opinion of the edu-
cative value in high schools and colleges of the different kinds
of study then commonly advocated in opposition to or in
addition to the old subjects of classics and mathematics. He
considered that the Experimental Sciences were little suitable,
and that for a very English reason, because they could not be
examined on adequately. He says:

" Experimental Science viewed in connection with educa-
tion, rejoices in a name which is unfairly expressive. A real
experiment is a very valuable product of the mind, requiring
great knowledge to invent it and great ingenuity to carry it
out. When Perrier ascended the Puy de Dome with a barom-
eter in order to test the influence of change of level on the
height of the column of mercury, he performed an experiment,
the suggestion of which was worthy of the genius of Pascal
and Descartes. But when a modern traveller ascends Mont
Blanc, and directs one of his guides to carry a barometer, he
cannot be said to perform an experiment in any very exact or
very meritorious sense of the word. It is a repetition of an
observation made thousands of times before, and we can never
recover any of the interest which belonged to the first trial,
unless indeed, without having ever heard of it, we succeeded
in reconstructing the process of ourselves. In fact, almost
always he who first plucks an experimental flower thus ap-
propriates and destroys its fragrance and its beauty."

140 TEN BRITISH MATHEMATICIANS

At the time when Todhunter was writing the above, the
Cavendish Laboratory for Experimental Physics was just being
built at Cambridge, and Clerk-Maxwell had just been appointed
the professor of the new study; from Todhunter's utterance
we can see the state of affairs then prevailing. Consider the
corresponding experiment of Torricelli, which can be performed
inside a classroom; to every fresh student the experiment
retains its fragrance; the sight of it, and more especially the
performance of it imparts a kind of knowledge which cannot be
got from description or testimony; it imparts accurate concep-
tions and is a necessary preparative for making a new and
original experiment. To Todhunter it may be replied that the
flowers of Euclid's Elements were plucked at least 2000 years
ago, yet, he must admit, they still possess, to the fresh student
of mathematics, even although he becomes acquainted with
them through a textbook, both fragrance and beauty."

Todhunter went on to write another passage which roused
the ire of Professor Tait. " To take another example. We
assert that if the resistance of the air be withdrawn a sovereing
and a feather will fall through equal spaces in equal times.
Very great credit is due to the person who first imagined the
well-known experiment to illustrate this; but it is not obvious
what is the special benefit now gained by seeing a lecturer repeat
the process. It may be said that a boy takes more interest in
the matter by seeing for himself, or by performing for himself,
that is, by working the handle of the air-pump; this we admit,
while we continue to doubt the educational value of the trans-
action. The boy would also probably take much more interest
in football than in Latin grammar; but the measure of his
interest is not identical with that of the importance of the
subjects. It may be said that the fact makes a stronger impres-
sion on the boy through the medium of his sight, that he believes
it the more confidently. I say that this ought not to be the case.
If he does not believe the statements of his tutor — probably
a clergyman of mature knowledge, recognized ability and blame-
less character — his suspicion is irrational, and manifests a want
of the power of appreciating evidence, a want fatal to his sue-

ISAAC TODHUNTER 141

cess in that branch of science which he is supposed to be cul-
tivating."

Clear physical conceptions cannot be got by tradition, even
from a clergyman of blameless charater; they are best got
directly from Nature, and this is recognized by the modern
laboratory instruction in physics. Todhunter would reduce
science to a matter of authority; and indeed his mathematical
manuals are not free from that fault. He deals with the charac-
teristic difficulties of algebra by authority rather than by sci-
entific explanation. Todhunter goes on to say: " Some con-
siderable drawback must be made from the educational value
of experiments, so called, on account of their failure. Many
persons must have been present at the exhibitions of skilled
performers, and have witnessed an uninterrupted series of
ignominious reverses, — they have probably longed to imitate the
cautious student who watched an eminent astronomer baffled
by Foucault's experiment for proving the rotation of the Earth;
as the pendulum would move the wrong way the student retired,
saying that he wished to retain his faith in the elements of
astronomy."

It is not unlikely that the series of ignominious reverses
Todhunter had in his view were what he had seen in the physics
classroom of University College when the manipulation was
in the hands of a pure mathematician — Prof. Sylvester. At
the University of Texas there is a fine clear space about 60
feet high inside the building, very suitable for Foucault's experi-
ment. I fixed up a pendulum, using a very heavy ball, and
the turning of the Earth could be seen in two successive oscilla-
tions. The experiment, although only a repetition according to
Todhunter, was a live and inspiring lesson to all who saw it,
whether they came with previous knowledge about it or no.
The repetition of any such great experiment has an educative
value of which Todhunter had no conception.

Another subject which Todhunter discussed in these essays
is the suitability of Euclid's Elements for use as the elementary
textbook of Geometry. His experience as a college tutor for
25 years; his numerous engagements as an examiner in mathe-

142 TEN BRITISH MATHEMATICIANS

matics; his correspondence with teachers in the large schools
gave weight to the opinion which he expressed. The question
was raised by the first report of the Association for the Improve-
ment of Geometrical Teaching; and the points which Todhunter
made were afterwards taken up and presented in his own unique
style by Lewis Carroll in " Euclid and his modern rivals." Up
to that time Euclid's manual was, and in a very large measure
still is, the authorized introduction to geometry; it is not as in
this country where there is perfect liberty as to the books and
methods to be employed. The great difficulty in the way of
liberty in geometrical teaching is the universal tyranny of com-
petitive examinations. Great Britain is an examination-ridden
country. Todhunter referred to one of the most distinguished
professors of Mathematics in England; one whose pupils had
likewise gained a high reputation as investigators and teachers;
his " venerated master and friend," Prof. De Morgan; and
pointed out that he recommended the study of Euclid with all
the authority of his great attainments and experience.

Another argument used by Todhunter was as follows: In
America there are the conditions which the Association desires;
there is, for example, a textbook which defines parallel lines as
those which have the same direction. Could the American mathe-
maticians of that day compare with those of England? He
answered no.

While Todhunter could point to one master— De Morgan —
as in his favor, he was obliged to quote another master — Syl-
vester— as opposed. In his presidential address before section A
of the British Association at Exeter in 1869, Sylvester had said :
" I should rejoice to see . . . Euclid honorably shelved or buried
' deeper than did ever plummet sound ' out of the schoolboy's
reach; morphology introduced into the elements of algebra;
projection, correlation, and motion accepted as aids to geometry;
the mind of the student quickened and elevated and his faith
awakened by early initiation into the ruling ideas of polarity,
continuity, infinity, and familiarization with the doctrine of
the imaginary and inconceivable." Todhunter replied: " What-
ever may have produced the dislike to Euclid in the illustrious

ISAAC TODHUNTER

143

mathematician whose words I have quoted, there is no ground
for supposing that he would have been better pleased with the
substitutes which are now offered and recommended in its place.
But the remark which is naturally suggested by the passage,
is that nothing prevents an enthusiastic teacher from carrying
his pupils to any height he pleases in geometry, even if he starts
with the use of Euclid."

Todhunter also replied to the adverse opinion, delivered by
some professor (doubtless Tait) in an address at Edinburgh
which was as follows: " From the majority of the papers in our
few mathematical journals, one would almost be led to fancy
that British mathematicians have too much pride to use a simple
method, while an unnecessarily complex one can be had. No
more telling example of this could be wished for than the insane
delusion under which they permit ' Euclid ' to be employed in
our elementary teaching. They seem voluntarily to weight
alike themselves and their pupils for the race." To which
Todhunter replied: "The British mathematical journals with
the titles of which I am acquainted are the Quarterly Journal of
Mathematics, the Mathematical Messenger, and the Philosoph-
ical Magazine; to which may be added the Proceedings of the
Royal Society and the Monthly Notices of the Astronomical
Society. I should have thought it would have been an adequate
employment, for a person engaged in teaching, to read and master
these periodicals regularly; but that a single mathematician
should be able to improve more than half the matter which is
thus presented to him fills me with amazement. I take down
some of these volumes, and turning over the pages I find article
after article by Profs. Cayley, Salmon and Sylvester, not to
mention many other highly distinguished names. The idea of
amending the elaborate essays of these eminent mathematicians
seems to me something like the audacity recorded in poetry
with which a superhuman hero climbs to the summit of
the Indian Olympus and overturns the thrones of Vishnu,
Brahma and Siva. While we may regret that such ability
should be exerted on the revolutionary side of the question,
here is at least one mournful satisfaction: the weapon with

144 TEN BRITISH MATHEMATICIANS

which Euclid is assailed was forged by Euclid himself. The
justly celebrated professor, from whose address the quotation
is taken, was himself trained by those exercises which he now
considers worthless; twenty years ago his solutions of mathe-
matical problems were rich with the fragrance of the Greek
geometry. I venture to predict that we shall have to wait
some time before a pupil will issue from the reformed school,
who singlehanded will be able to challenge more than half the
mathematicians of England." Professor Tait, in what he said,
had, doubtless, reference to the avoidance of the use of the
Quaternion method by his contemporaries in mathematics.

More than half of the Essays is taken up with questions
connected with competitive examinations. Todhunter explains
the influence of Cambridge in this matter: " Ours is an age
of examination; and the University of Cambridge may claim
the merit of originating this characteristic of the period. When
we hear, as we often do, that the Universities are effete bodies
which have lost their influence on the national character, we may
point with real or affected triumph to the spread of examinations
as a decisive proof that the humiliating assertion is not absolutely
true. Although there must have been in schools and elsewhere
processes resembling examinations before those of Cambridge
had become widely famous, yet there can be little chance of
error in regarding our mathematical tripos as the model for
rigor, justice and importance, of a long succession of insti-
tutions of a similar kind which have since been constructed."
Todhunter makes the damaging admission that " We cannot
by our examinations, create learning or genius; it is uncertain
whether we can infallibly discover them ; what we detect is simply
the examination-passing power."

In England education is for the most part directed to train-
ing pupils for examination. One direct consequence is that
the memory is cultivated at the expense of the understanding;
knowledge instead of being assimilated is crammed for the time
being, and lost as soon as the examination is over. Instead of
a rational study of the principles of mathematics, attention is
directed to problem-making, — to solving ten-minute conun-

ISAAC TODHUNTER 145

drums. Textbooks are written with the view not of teaching
the subject in the most scientific manner, but of passing certain
specified examinations. I have seen such a textbook on trigo-
nometry where all the important theorems which required the
genius of Gregory and others to discover, are put down as so
many definitions. Nominal knowledge, not real, is the kind
that suits examinations.

Todhunter possessed a considerable sense of humour. We
see this in his Essays; among other stories he tells the following:
A youth who was quite unable to satisfy his examiners as to a
problem, endeavored to mollify them, as he said, " by writing
out book work bordering on the problem." Another youth
who was rejected said " if there had been fairer examiners and
better papers I should have passed ; I knew many things which
were not set." Again: "A visitor to Cambridge put himself
under the care of one of the self -constituted guides who obtrude
their services. Members of the various ranks of the academical
state were pointed out to the stranger — heads of colleges, pro-
fessors and ordinary fellows; and some attempt was made to
describe the nature of the functions discharged by the heads
and professors. But an inquiry as to the duties of fellows pro-
duced and reproduced only the answer, l Them's fellows I say.'
The guide had not been able to attach the notion of even the
pretense of duty to a fellowship."

In 1874 Todhunter was elected an honorary fellow of his
college, an honor which he prized very highly. Later on he
was chosen as an elector to three of the University professor-
ships—Moral Philosophy, Astronomy, Mental Philosophy and
Logic. " When the University of Cambridge established its
new degree of Doctor of Science, restricted to those who have
made original contributions to the advancement of science or
learning, Todhunter was one of those whose application was
granted within the first few months." In 1875 ne published
his manual Functions of Laplace, Bessel and Legendre. Next
year he finished an arduous literary task — the preparation
of two volumes, the one containing an account of the writings
of Whewell, the other containing selections from his literary

146 TEN BRITISH MATHEMATICIANS

and scientific correspondence. Todhunter's task was marred
to a considerable extent by an unfortunate division of the
matter: the scientific and literary details were given to him,
while the writing of the life itself was given to another.

In the summer of 1880 Dr. Todhunter first began to suffer
from his eyesight, and from that date he gradually and
slowly became weaker. But it was not till September, 1883,
when he was at Hunstanton; that the worst symptoms came on.
He then partially lost by paralysis the use of the right arm;
and, though he afterwards recovered from this, he was left
much weaker. In January of the next year he had another
attack, and he died on March 1, 1884, in the 64th year of his
age.

Todhunter left a History of Elasticity nearly finished. The
manuscript was submitted to Cay ley for report; it was in 1886
published under the editorship of Karl Pearson. I believe that
he had other histories in contemplation; I had the honor of
meeting him once, and in the course of conversation on mathe-
matical logic, he said that he had a project of taking up the
history of that subject; his interest in it dated from his study
under De Morgan. Todhunter had the same ruling passion as
Airy — love of order — and was thus able to achieve an immense
amount of mathematical work. Prof. Mayor wrote, " Tod-
hunter had no enemies, for he neither coined nor circulated
scandal; men of all sects and parties were at home with him,
for he was many-sided enough to see good in every thing. His
friendship extended even to the lower creatures. The canaries
always hung in his room, for he never forgot to see to their
wants."

INDEX

Adams, J. C, 138
Airy, G. B., 38, 45, 146
Apollonius, 102
Argand, J. R., 13, 56, 138
Arnold, T., 93

Babbage, C, 10, 13

Ball, R., 14

Beltrami, E., 86

Boole, G., 50-63, 14, 29, 80, 98, 138

Boscovich, R. J., 130

Brewster, D., 95

Brinkley, N., 35, 36, 38

Burkhardt, J. C, 97

Cauchy, A. L., 95

Cayley, A., 64-77, 46, 78, 108, 109, no,

121, 126, 132, 137, 143, 146

Chrystal, G., 25, 35
Clairault, A. C., 10
Clifford, W. K., 78-91, 121
Colburn, Z., 35
Colenso, J. W., 129

Davy, H., 82, 114

Delambre, J. B. J., 10

DeLaRue, W., 118

De Morgan, A., 19-33, 8, 14, 40, 41,
52,53.54, 58, 62,63,65, 70, 78, 79,
80, 108, in, 123, 124, 132, 134, 135,
138, 142, 146

De Morgan, G., 22

Dewar, J., 82, 114

Dodgson, C. L., 80, 142

Dodson, J., 19

Eisenstein, F. G., 99
Ellis, L., 64

Eratosthenes, 97

Euclid, 19, no, in, 114, 140, 142

Euler, L., 10, 96, no

Faraday, M., 82, 102, 114
Fermat, P. de, 96
Forsyth, A. R., 53, 70
Foucault, J. B. L., 141
Francois, 13
Franklin, F., 116, 117
Frend, W., 12, 13, 21

Gauss, K. F., 86, 95, 96, 100

Graves, C., 54

Green, G., 107

Gregory, D., 52

Gregory, D. F., 14, 25, 27, 51, 52, 64,

65, 107
Gregory, J., 52

Hamilton, J., 34

Hamilton, W., 23, 29

Hamilton, W. E., 48

Hamilton, W. R., 34"49, J4, 18, 23, 28,
52,53,54, 62,69,71, 73, 74, 75, 80,
81, 85, 95, 114, 123, 127, 128, 132

Harley, R., 54

Hart, H., 115

Helmholtz, H. L. F., 86

Hermite, C., 105

Herschel, J. F. W., 8, 10, 13, 88, 135

Highton, H., 93

Hipparchus, 48

Hobbes, T., 63

Huxley, T. H., 101, in, 112, 113, 129

Ivory, J., 10, 81

147

148

INDEX

Jacoby, K. G. J., io, 99
Joly, J-, 45, 49

Kant, E., 41, 42, 74, 85, 87
Kelvin, Lord, 79
Kempe, A. B., 115, 116
Kepler, J., 102
Kirkman, T. P., 122-133

Lacroix, S. F., 10
Lagrange, C. F. L., 10, 4c
Laplace, P. S., 10, 35, 65
Legendre, A. M., 10
Leibnitz, G. W., 9
Little, C. N., 133
Lipkin, 114
Lloyd, H., 41, 85
Lobatchewsky, N. I., 86
Lockyer, J. N., 101
Lubbock, J., 101

Macfarlane,. A., 3, 4, 31, 45, 57, 67,

101, 118, 133, 141, 146
Maclaurin, C, no
Martineau, J., 22
Maseres, F., 12, 14

Maxwell, J. C, 40, 67, 68, 79, 87, 101,

102, 132, 140
Mill, J. S., 74, 129
Minkowski, 105, 106

Newton, I., 9, 41, 46, 65, 67, 80, no

Peacock, G., 7-18, 20, 24, 25, 41, 52,

55,78
Peaucellier, C. N., 114, 115

Plato, 74, 91
Poincar6, J. H., 121
Pollock, F., 81

Rankine, W. J. M., 40
Rayleigh, Lord, 82
Record, R., 16
Riemann, G. F. B., 86
Rosse, Lord, 45
Routh, E. J., 136
Rumford, Count, 82

Salmon, G., 46, 65, 143
Smith, H. J. S., 92-106, 119, 120
Smith, J., 32, 33
Socrates, 24, 91

Spencer, H., 129, 130, 131, 132
Spottiswoode, W., 119
Stewart, E., 89, 91

Sylvester, J. J., 107-121, 66, 69, 79, 135,
141, i43

Tait, P. G., 34, 42, 57, 71, 72, 73, 89, 91,

132, 133, 136, 140, 143, 144, 146
Todhunter, I., 134-146
Tschebicheff, 114, 115
Tyndall, J., 82, 114, 129

Waring, E., no

Warren, J., 13, 28

Watt, J., 115

Whewell, W., 14, 20, 24, 29, 79, 94

Woodhouse, R., 136

Wordsworth, W., 38, 39

Young, T., 82

DATE

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WELLESLEY COLLEGE LIBRARY

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Science QA 28 . M2 1916a

Hacfarlane, Alexander, 1851
1913.

Lectures on ten British
mathematicians of the