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by Alexander Macfarlane
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Title : Ten British Mathematicians of the 19th Century
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*** START OF THE PROJECT GUTENBERG EBOOK TEN BRITISH MATHEMATICIANS ***
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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 eor the [nter.national Assoceatiox for. Promotixg
THE Study of Quaternions
1916
MATHEMATICAL MONOGRAPHS.
EDITED DY
Mansfield Merritiia.n and Robert S. Woodward.
No. 1. History of Modern Mathematics.
By David Eugene Smith.
No. 2. Synthetic Projective Geometry.
By George Bruce Halsted.
No. 3. Determinants.
By Laenas Gifford Weld.
No. 4. Hyperbolic Functions.
By James McMahon.
No. 5. Harmonic Functions.
By William E. Byerly.
No. 6. Grassmann*s Space Analysis.
By Edward W. Hyde.
No. 7. Probability and Theory of Errors.
By Robert S, Woodward.
No. 8. Vector Analysis and Quaternions.
By Alexander Macfarlane.
No. 9. Differential Equations.
By William Woolsey Johnson.
No. 10. The Solution of Equations.
By Mansfield Merriman.
No. 11. Functions of a Complex Variable.
By Thomas S. Fiske.
No. 12. The Theory of Relativity.
By Robert D. Carmichael.
No. 13. The Theory of Numbers.
By Robert D. Carmichael.
No. 14. Algebraic Invariants.
By Leonard E. Dickson.
No. 15. Mortality Laws and Statistics.
By Robert Hendersgn.
No. 16. Diophantine Analysis.
By Robert D. Carmichael.
No. 17. Ten British Mathematicians.
By Alexander Macfarlane.
PREFACE
During the years 1901-1904 Dr. Alexander Macfarlane delivered, at Dehio;h Uni-
versity, lectures on twenty-five British matliematiciQ.ns 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
sajne 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 ^ven 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 ^ive 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.
Alexander Macfarlane was born April 21, 1851, at Blairgowrie. Scotland.
From 1871 to 1884 he was a student, instructor and examiner in physics at the
University of Edinburgh, from 1SS5 to 1894 professor of physics in the Uni-
versity of Texas, and from 1895 to 1908 lecturer in electrical engineering and
mathematical physics in Lehigh UniA'ersity. He was the author of papers on al-
gebra of logic, vector analysis and quaternions, and of Monograph No. S of this
series. He was twice secretary of the section of physics of the American Asso-
ciation 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 w^ith British mathematicians of the nineteenth century
imparts to many of these lectures a personal touch which greatly adds to their
PREFACE
general interest.
Alexander Macfarlane
From a photograph of 189S
Contents
PREFACE iii
1 George Peacock (1791-1858) 1
2 Augustus De Morgan (1806-1871) 9
3 Sir William Rowan Hamilton (1805-1865) 19
4 George Boole (1815-1864) 30
5 Arthur Cayley (1821-1895) 40
6 William Kingdon Clifford (1845-1879) 49
7 Henry John Stephen Smith (1826-1883) 58
8 James Joseph Sylvester (1814-1897) 68
9 Thomas Penyngton Kirkman (1806-1895) 78
10 Isaac Todhunter (1820-1884) 87
11 PROJECT GUTENBERG "SMALL PRINT"
Chapter 1
GEORGE PEACOCK^
(1791-1858)
George Peacock was born on April 9. 1791, at Denton in the north of Eng-
land. 14 miles from Richmond in Yorkshire. His father, the Rev. Thomas Pea-
cock, was a clerg\^man 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 elemen-
tary 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 Trinit^^ College,
Cambridge.
Here it may be well to give a brief account of that University, as it was the
alma mater of four out of the six mathematicians discussed in this course of
lectures .
At that time the University of Cambridge consisted of seventeen colleges,
each of which had an independent endowment, buildings, master, fellows and
scholars. The endowments, generally in the shape of lands, have come down from
ancient times; for example. Trinity College was founded by Henry VIII in 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 consisted of a pass examination and an
honors examination, the latter called a tripos. Thus, the mathematical tripos
meaiit the examinations of candidates for the degree of Bachelor of Arts who
had made a special study of mathematics. The examination was spread over
^This Lecture wna delivered April 12, 1901. — EDITORS.
"Dr. Mocfarlone's first course included the first six lectures siven in this volume. — EOTTOEtS.
CHAPTER 1. GEORGE PEACOCK (1791-1858) 2
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 developed 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
mathematician 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 except
what would pay in the tripos examination. Modifications have been introduced
to counteract these evils, and the conditions 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 distinction in England.
In 1812 Peacock took the rank of second wrangler, and the second Smith's
prize, the senior wrangler being John Herschel. Tw^o years later he became a
candidate for a fellowship in his college and won it immediately, partly by means
of his extensive 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 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 advancement of science. Sir Isaac Newton
was a Fellow of Trinity College, and its limited terms nearly deprived the world
of the Prificipia.
The year after taking a Fellowship, Peacock was appointed a tutor and lec-
turer 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 out so soon? The true reason was he was worshipped too much as
an authority; the University 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 j they ignored the notation of Leibnitz 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 y] 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
CHAPTER 1. GEORGE PEACOCK (1791-1858) 3
^, y is written ^^■. y' is jydz^ and so on. The result of this Chauvinism on
the part of the British mathematicians of the eighteenth century was that the
developments of the calculus were made by the contemporary mathematicians
of the Continent, namely, the BernouUis, 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 achieA'ements of the Continental mathematicians. Cambridge Univer-
sity 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 Ja-
cobi was then literally true, although it had ceased to be true when he gave it
utterance. He visited Cambridge about 1S42. 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 1815 they formed what they called the Analytical Society^ the object of which
was stated to be to advocate the (^'ism of the Continent versus the doi-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 Govern-
ment, and a scientific commission 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 the diff'erential and integral
calculus: it was published in 181G. 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 afterwards.
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
CHAPTER 1. GEORGE PEACOCK (1791-1858) 4
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 1 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 1 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 sentences 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 deA^elopment 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 of by the following
facts. Baron Maseies, a Fellow of Clare College, Cambridge, and William Frend,
a second wrangler, had both written books protesting against the use of the
negative quantity. Frend published his Principles of Algebra in 179G. 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
comprehension 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 is made 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 metaphor, 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 negatiA'e 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 hate the colour of a serious
thought," So far. Frend. Peacock knew that Argand, Frangais and Warren had
CHAPTER 1. GEORGE PEACOCK (1791-1858) 5
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 moA'ement going on in Great Britain, especially in its more exact de-
partments. Mathematics were at the last gasp^ and Astronomy nearly so — 1
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 experi-
mental 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 1S31 saw the beginning of one of the greatest scientific organiza-
tions 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 intercourse of those who cultivate science in
different parts of the British Empire with one another and with foreign philoso-
phers; 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
persons for the information of the annual meetings, and the first to be placed
on the list was a report on the progress of mathematical science. Dr. Whewell,
the mathematician and philosopher, 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 Low^ndean 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
ScreU'S. 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 statutes of the UniA'ersity; 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.
CHAPTER 1. GEORGE PEACOCK (1791-1858) 6
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 philo-
logical or symbolical school of mathematicians; to which Gregory, De Morgan
and Boole belonged. His answer to Maseies 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 sym-
bols as representing numbers, and the operations to which they are submitted
as included in the same definitions as in common arithmetic; the signs -I- 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 fco them possess values which would render them so in case
they were replaced by digital numbers; thus in expressions such as a -\- b we
must suppose a and h to be quantities of the same kind; in others^ like a — fe, we
must suppose a greater than b and therefore homogeneous with it; in products
and quotients, like ab and t '►ve 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 arithmeti-
cal 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 — 6 is a number only when b is less than a. Again, under the same condi-
tions, ab is always a number, but ^ is really a number only when b is an exact
divisor of a. Hence we are reduced to the following dilemma: Either ^ must be
held to be ai] impossible expression in general, or else the meaning of the funda-
mental 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 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 difficulty, — 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
CHAPTER 1. GEORGE PEACOCK (1791-1858) 7
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 perceive
the thing."
The fact is that even in arithmetic the two processes of multiplication and
division are generalized into a common multiplication; and the difficulty consists
in passing from the original idea of multiplication to the generahzed idea of a
tensor, which idea includes compressing the magnitude as well as stretching
it. Let 771 denote an integer number; the next step is to gain the idea of the
reciprocal of m. not as — but simply as /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 OA^er 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: "Sym-
bolical algebra adopts the rules of arithmetical algebra but removes altogether
their restrictions; thus symbolical subtraction differs from the same operation
in arithmetical algebra in being possible for all relations of value of the sym-
bols 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 symbolical algebra where they are
general in value as well as in form; thus the product of d^ and d^ which is
^i7i-i-]i -^i]^]] j^ Q^y^f^ ji ^^^ 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 -h b)'^ 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^ i^t^y be shown upon the same principle
to the equivalent series for {a -\- 6)" 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 equivalent forms/' and at page 59 of the
Symbolical Algebra it is thus enunciated: ""Whatever algebraical forms are equiv-
alent when the symbols are general in form, but specific in value, will be equiv-
alent likewise when the symbols are general in value as well as in form."
For example, let a, 6, 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 — be.
Peacock's principle says that the form on the left side is equivalent 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^ 6, c^ d may be rational fractions, or surds, or imaginary quantities, or indeed
operators such as -y-. The equivalence is not established by means of the nature
of the quantity denoted: the equiA^alence is assumed to be true, and then it is
attempted to find the different interpretations which may be put on the symbol.
CHAPTER 1. GEORGE PEACOCK (1791-1858) 8
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, fe, c, d denote integer
numbers, of which b is less than a and d less than c, then
(a — b)(c — d) = ac -\- bd — ad — be.
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 prob-
lem 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 jneaning.. 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 showm that
m n ra-\-n
a a = a .
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, ii are to be found by interpretation.
Suppose that a takes the form of the incommensurate quantity e, the base of
the natural system of logarithms. A number is a degraded form^ of a complex
quantity p -h q and a complex quantity is a degraded form of a quaternion;
consequently one meaning which may be assigned to jn and n is that of quater-
nion. Peacock's principle would lead us to suppose that e"^e" = e"^"'"", m and
n denoting quaternions; but that is just what Hamilton^ the inventor of the
quaternion generalization, denies. There are reasons for believing that he was
mistaken, and that the forms remain equivalent eA'en under that extreme gen-
eralization of 771 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?
Chapter 2
AUGUSTUS
DE MORGAN
1
(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 tlie year x ." 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, w^ho 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 Oxford or Cambridge who is not a member of any one
of the Colleges,
When De 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 elemen-
tary 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 demonstration.
De Morgan suffered from a physical defect — one of his eyes was rudimentary
and useless. As a consequence, he did not 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
^ThiH Lecture wqb delivered Apiil 13, 1901. — EDITORS.
CHAPTER 2. AUGUSTUS DE MORGAN (1806-1871) 10
depends on the action of two eyes, but De Morgan testified that so far as lie
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 Engloiid. and
desired that her son should become a clergyman; but by this time De Morgan
had begun to show his non-grooving disposition, due no doubt to some extent
to his physical infirmity. At the age of sixteen he was entered at Trinity College,
Cambridge, 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 fiute. 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 signing 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 University took shape. The two ancient universities were so guarded
by theological tests that no Jew or Dissenter from the 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 discourse 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 Coun-
cil 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 stu-
dents, 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 therea,fter became the continuous center of his labors for thirty years.
The same body of reformers — headed by Lord Brougham, a Scotsman em-
inent 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
CHAPTER 2. AUGUSTUS DE MORGAN (1806-1871) 11
clearly written treatises by the best writers of the time. One of its most volu-
minous and effective writers was De Mor°;an. 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 Soci-
ety, and issued in penny numbers. When De Morgan came to reside in London
he found a congenial friend in William Frend, notwithstanding his mathematical
heresy about negative quantities. 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.
The London University of which De Morgan was a professor was a differ-
ent institution from the University of London. The University of London was
founded about ten years later by the Government for the purpose of grant-
ing degrees after examination, without any qualiflcation 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 aiiy merely analytical dexterity in the application
of half-understood principles to particular cases.
De Morgan had a son George, who acquired great distinction in mathemat-
ics 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 Univer-
sity College: De Morgan was the first president, his son the first secretary. It was
the beginning of the London Mathematical Society. In the year ISGG 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 consid-
ered that the old standard of religious neutrality had been hauled down, and
forthwith resigned. He was now GO years of age. His pupils secured a 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 emi-
nent mathematicians of that name, related as father and son — died. This blow
w^as followed by the death of a daughter. Five years after his resignation from
University College De Morgan died of nervous prostration on March 18, 1871,
in the G5th year of his age.
CHAPTER 2. AUGUSTUS DE MORGAN (1806-1871) 12
De Morgan was a brilliant and witt\^ writer, whether as a con trover siahst
or as a correspondent. In his time there flourished two Sir William Hamiltons
who have often been confounded. 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 con-
tributed 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 interested in the work
of both, and corresponded with both; but the correspondence with the Scots-
man 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 mathematician ex-
tended over twenty-four years: it contains discussions 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 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 meaning. 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 ■ LITER-ARUM.
He disliked the country, and while his family enjoyed the seaside, 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
CHAPTER 2. AUGUSTUS DE MORGAN (1806-1871) 13
ideas or sympathies in common with the physical philosopher. His attitude was
doubtless due to his physical iufirmity, wliich prevented him from being either
an observer or an experimenter. He never voted at an election, and he never
Adsited the House of Commons, or the Tower, or Westminster Abbey.
Were the writings of De Mor°;an published in the form of collected works,
they would form a small library. We have noticed his writings for the Use-
ful Knowledge Society. Mainly through the efforts of Peacock and Whewell, a
Philosophical Society had been inaugurated at Cambridge; and to its Transac-
tions De Morgan contributed four memoirs on the foundations 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 AthentEum 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 -h: when -I- receives its meaning, so also will the word addition. It is most
important that the student should bear in mind that, with one exception, no
word nor sign of arithmetic or algebra has one atom of meaning 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
significant algebras. If any one were to assert that -h and — 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 = 5 . It indicates that the two symbols
have the same resulting meaning, by whatever steps attained. That A and 5,
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 refractory to the sym-
CHAPTER 2. AUGUSTUS DE MORGAN (1806-1871) 14
bolic 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 estab-
lished, 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 iuA^entory of the laws of algebra. The symbols are 0,1,+,
— , X, -i- , ()^ , 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. ++ = +.H — = — , — h = — . = +, xx = x, x-:- = +,
11. Commutative law. a -\- b = b -\- a, ab = ba.
III. Distributive law. a(b -\- c) = ab -\- ac.
IV. Index laws, a^ x a' = a^-^\, {a^Y = a^' , (aby = a'b'.
V. a— a = 0.a-^a = l.
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 preceding
symbols aiid no others, except they be new symbols 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 m^ is not equal to n"" : 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 refractory, 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
impossible 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 w^hich 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 development of algebra is arithmetic,
CHAPTER 2. AUGUSTUS DE MORGAN (1806-1871) 15
where numbers only appear and symbols of operations sucli as +, x , etc. The
next sta^e is universal arithmetic^ where letters appear instead of numbers,
so as to denote numbers universally, ajid the processes are conducted without
knowing the A^alues of the symbols. Let a and b denote any numbers; then
such an expression as a — fe 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 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^/—l 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 — 1 is interpreted as denoting a quadrant. The expression a + b\ —1
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 e^^~ . De Morgan attempted it
by reducing such an expression to the form b + ^y— 1, 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 -I- by —1 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 + 6\/— 1 + c\/— 1
De Morgan and many others worked hard at the problem, but nothing came of it
until 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 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 W' hewell, the Master of Trinity
College, whose principal 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
CHAPTER 2. AUGUSTUS DE MORGAN (1806-1871) 16
of the numerically definite syllo^sm. The followers of Aristotle say and say
truly that from two particular propositions such as Some M's are A^s, and
Some A'/'s are B'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
yl's and B 's may follow of necessity, the middle term must be taken universally
in one of the premises, De Morgan pointed out that from Most M^s are A's
and Most A/'s are Bs it follow^s of necessity that some ^'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 A/ 's is m, of the M's that
are j4's is a, and of the A/ 's that are 5's is 6; then there are at least {a -\-b — m)
A's that are B^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 TOO + 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 fundamental principle in necessary reasoning.
Here then De Morgan had made a great advance by introducing quantifica-
tion 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 A is a part of B for the
Aristotelian form All ^'s are B^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: al-
though 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 AthentEum. aiid 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 practical 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 individuals, 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 something 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
CHAPTER 2. AUGUSTUS DE MORGAN (1806-1871) 17
word we have to old paradox. But there is this difference, that by caihn^ 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 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 discovery 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 complained
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 the authors. He gives the following classification: squarers of the circle,
trisectors of the angle, duplicators of the cube, constructors of perpetual motion,
subverters of gravitation, stagnators 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 aie 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 —
CHAPTER 2. AUGUSTUS DE MORGAN (1806-1871) 18
are illiterate, and a great many waste their means, and are in or approaching
penury. Tliese 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 Liverpool, He found tt = 3g. His mode of reasoning
was a curious caricature of the reductio ad absurdum of Euclid. He said let
TV = 3g, and then showed that on that supposition, every other value of tt must
be absurd; consequently tt = 3g 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 lie hints that I am to answer. In his last of
31 closely written sides of note paper, he informs me, with reference to my
obstinate silence, that though 1 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 snaill 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 would by no means undertake
upon a clock that gains 19 seconds odd in every hour by false quadrative value
of 7J". 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
7J" = 3^ and James Smith, Esq.. of the Mersey Dock Board: and put hors de
cotJibat 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] 1 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 proficient in the field of physics. His
father-in-law was a paradoxer, 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 ordinary material causes. Faraday delivered a lecture
on Spiritualism, 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.
1
Chapter 3
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 exercise 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 GraA'es. 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 ''Hamilton" 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 o^ 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 education 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; at four a good geographer; at five able to read and
translate Latin, Greek, and Hebrew, and liked to recite Dry den, Collins. Milton
and Homer; at eight a reader of Italian and French and ^ving vent to his feelinss
^ThiH Lecture wqb delivered April 10, 1901. — EDITORS.
19
CHAPTER 3. Sm WILLIAM ROWAN HAMILTON (1805-1865) 20
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 soh'ing problems that related to the properties 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, Hamil-
ton 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 textbook, 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 the-
orem 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 investigation 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 the Royal Irish Academy. Dr.
Brinkley is said to have remarked of him at this time: "This young man, I do
not say unll be, but is, the first mathematician of his age/'
At the age of eighteen Hamilton entered Trinity College. Dublin, the Univer-
sity 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 optitne at his examination 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, medi-
ocriter, vix raedi, now, optime means passed with the very highest distinction;
vix means passed but with great difficulty. This scale is still in use in the medical
examinations of the University of Edinburgh. Before entering college 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.
CHAPTER 3. Sm WILLIAM ROWAN HAMILTON (1805-1865) 21
Oh! Ambition hath its hour
Of deep and spirit-stirring power;
Not in the tented field alone.
Nor peer-engiided 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
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.
CHAPTER 3. Sm WILLIAM ROWAN HAMILTON (1805-1865) 22
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 broth-
ers 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 experienced a dis-
appointment. On account of his self-consciousness, inseparable probably from
his genius, he felt the disappointment keenly. He was then known to the pro-
fessor of astronomy, and walking from the College to the Observatory along the
Royal 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 intellectual circle of Dublin.
In his senior year he presented to the Royal Irish Academy a memoir em-
bodying 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 appointed to the bishopric of Cloyne. and in consequence resigned the pro-
fessorship 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 supplement their own application by testimonial letters from com-
petent authorities. In the present case quite a number of candidates 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 ap-
pointment 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^ os-
cillating 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.
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
CHAPTER 3. Sm WILLIAM ROWAN HAMILTON (1805-1865) 23
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. Himself a brilliant computer, with a °;ood 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.
Wordsworth 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 em.ployment may
seduce yon from the path of science which you seem destined to tread with so
much honor to yourself and profit to others. Again and again 1 must repeat that
the composition of verse is infinitely more of an art than men are prepared fco
believe, and absolute success in it depends upon innumerable minutit^ 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 faA'orable 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 less careful and in measures less elaborate."
Hamilton possessed the poetic imagination: what he was deficient in was the
technique of the poet. The imagination 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 mathetical 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 mow 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, that is, 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 aiid philosophy; and was on the eve of proposing when he
gave up because the young lady incidentally 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.
CHAPTER 3. Sm WILLIAM ROWAN HAMILTON (1805-1865) 24
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 hkewise that
under certain conditions a sin°;le emergent ray would appear as a cone of rays.
The prediction was made by Hamilton 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 discoA^ery 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 difficulties
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 previ-
ously 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 satisfied w^ith Peacock's theory of algebra as a "System
of Signs and their Combinations"; nor with De Morgan's improvement of it; he
demanded a more real foundation. In reading Kant's Critique of Pure Reason
he was struck by the following 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 intu-
ition, they render synthetical propositions a priori possible," Thus, according
to Kant, space and time are forms of the intellect; and Hamilton 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
CHAPTER 3. Sm WILLIAM ROWAN HAMILTON (1805-1865) 25
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 Advancement 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,
3D 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: Oxford, 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 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 1 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 Lieu-
tenant 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. '1 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
Hamilton, 1 do not confer a distinction. 1 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 flowing sentences. Then, receiving
the State sword from one of his attendants, he said. ^Kneel down. Professor
Hamilton'; and laying the blade gracefully and gently flrst 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 honor was certainly
merited by him who received it. so the words by which it was conferred were
CHAPTER 3. Sm WILLIAM ROWAN HAMILTON (1805-1865) 26
so graceful and appropriate that they constituted a distinction by themselves,
°;reater 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 hterary 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 oA^er be considered his highest title
to fame. The story of the discovery is told by Hamilton himself in a letter to his
son: "On the ISth day of October, which happened to be a Monday, and Council
day of the Royal Irish Academy, 1 was walking in to attend and preside, and yonr
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 1 felt at once the importance. An electric circuit seemed
to close; and a spark flashed forth, the herald (as 1 foresaw immediately) of
many long years to come of deflnitely directed thought and work, by myself if
spared, and at all events on the part of others, if 1 should even be allowed to
live long enough distinctly to communicate the discovery. Nor could 1 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
i,j,k; namely,
i' = f = k- = 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 flrst general
meeting of the session, which reading took place accordingly on Monday the 13th
of November following."'
Last summer Prof. Joly and 1 took the walk here described. We started
from the ObserA'atory, walked dowm the terraced held, 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. 1 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 scientiflc work.
We have seen how Hamilton gained two optimes^ one in classics, the other
CHAPTER 3. SIR WILLIAM ROWAN HAMILTON (1805-1865) 27
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 so-
ciety 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 Green-
wich astronomer. He broke his good resolution, 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; fre-
quently 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 re-
fresh himself with a quaff of that beverage for which Dublin is famous — porter
labelled ^ ? 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 1S53, under 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
CHAPTER 3. Sm WILLIAM ROWAN HAMILTON (1805-1865) 28
was a paradox — a sound paradox, and of his experience as a par ad oxer Hamilton
wrote: "It required a certain capital of scientific reputation, amassed in former
years, to make it otter 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 1 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 3 x 4 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 imperfec-
tion as a manual of instruction, and he set himself 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 Gist 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. 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 during the years
when he was engaged on the enormous labor of writing the Elements of Quater-
nions^ made him a recluse, and necessarily took away from his power of attend-
ing to the practical affairs of life. Some said that howeA'er great a master of
CHAPTER 3. SIR WILLIAM ROWAN HAMILTON (1805-1865) 29
pure time he might be he was not a master of sublunary time. His neighbors
also took advantage of his °;oodness 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 ci[VT]p
qjLXoTtovo^ Xffl (^i'kctkT\'^T\C,] 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 Quatermons was all finished excepting one chapter.
His eldest son, William Edwin Hamilton, wrote a preface, and the volume was
published at the expense of Trinity College. Dublin. Only 500 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.
Chapter 4
GEORGE BOOLE^
(1815-1864)
George Boole was born at Lincoln, England, on the 2d of November, 1815.
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 mathematical 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 neighborhood 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 m.ade 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 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, afterwards at Waddington near Lincoln;
and the leisure of these years he devoted mainly to the study of the principal
^This Lecture wos delivered April 19. 1901. — EDITORS.
30
CHAPTER 4. GEORGE BOOLE (1815-1864) 31
modern lan^ua^es, 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 decided on opening a school on his own account in his native
city: thenceforth he devoted all the leisure he could command 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 Afecanique 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 substitutions, 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 Cambridge
Mathematical Journal. Gregory and other friends suggested that Boole should
take the regular mathematical course 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 tt. 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 developed by
the French school there were many remarkable phenomena awaiting explanation;
particularly theorems which involved what was called the separation of symbols.
He embodied his results in a paper '"''On the real Nature of symbolical 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, aiid 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
CHAPTER 4. GEORGE BOOLE (1815-1864) 32
with exact analysis, and so embraced formal logic. In 1848, as we have seen, the
controversy arose between 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 Alathematical
Analysis of Logic.
About this time what are denominated the Queen's Colleges of Ireland were
instituted at Belfast, Cork and Galway; and in 1S49 Boole was appointed to
the chair of mathematics in the Queen's College at Cork. In this more suitable
environment he set himself to the preparation of a more elaborate work on the
mathematical analysis of logic. For this purpose 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 Differential 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 dedication, 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 18G4 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. "1 shall be glad to see his work (LaU'S of Thought)
out, for he has. 1 think, got hold of the true connection of algebra and logic."
At another time he wrote to the same as follows: "All metaphysicians except
you and I and Boole consider mathematics as four books of Euclid and algebra
up to quadratic equations." We might infer that these three contemporary
mathematiciaiis who were likewise philosophers would form a triangle of friends.
But it was not so; Hamilton was a friend of De Morgan, and De Morgan a friend
of Boole; but the relation of friend, although convertible, is not necessarily
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, prepared by his friend
Rev. Charles Graves; De Morgan's biography has appeared in one volume, pre-
pared by his widow; of Boole no biography has appeared. A biographical notice
CHAPTER 4. GEORGE BOOLE (1815-1864) 33
of Booie was written for the Proceedings of the Royai Society of London by his
friend the Rev. Robert Horley^ and it is to it that 1 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 husband'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 neces-
sarily 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 aid
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 throughont 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 in the expression
of the data." As regards these conditions it may be observed that they are very
different from the formalist 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 chiefiy calls for study and examination;
respecting it he observes as follows: "The principle in question may be con-
sidered as resting upon a general 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 reason-
ing, in which symbols are employed in obedience to laws founded upon their
interpretation, but without any sustained reference to that interpretation, 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 immediate force of the
evidence is not felt, serve as a verification, a posteriori, of the practical validity
of the principle in question. 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 v — 1 in the intermediate processes of trigonometry
furnishes an illustration of what has been said. I apprehend that there is no
CHAPTER 4. GEORGE BOOLE (1815-1864) 34
mode of explaining that application which does not covertly assume the very
principle in question. But that principle^ 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 equations; 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 dictate 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 principle is the successful employment of the uninterpretable
symbol v — 1 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 num-
bers, 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 introduced to compound expressions which cannot be reduced to sim-
ple 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 symbolical 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 appli-
cation of the mathematical doctrine of Probabilities; and, finally to collect from
the various elements of truth brought to view in the course of these inquiries
some probable intimations concerning the nature and constitution of the human
mind .
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 con-
ceptions: second.. Signs of operation, as -h, — . 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 substantive 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 rea-
CHAPTER 4. GEORGE BOOLE (1815-1864) 35
soning be replaced by the substantive." Let us then agree to represent the class
of individuals to wbicb. a particular name is applicable by a single letter as x . If
the name is tnen for instance, let 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, ail 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 Professor 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 1 ventured to publish my views
on the subject in a small volume called Principles of the Algebra of Logic: one
of the chief points 1 made is the philological and analytical difference between
the substantive and the adjective. What I said was that the 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 other 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 individual 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 ho-
mogeneous 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. 1 will explain the elements of Boole's method on this theory.
CHAPTER 4. GEORGE BOOLE (1815-1864)
36
Fig. 1.
The square is a collection of points: it may serve to represent any collection of
homogeneous units, whether finite or infinite in number, that is, the universe of
the problem. Let z denote inside the left-hand circle, and y inside the right-hand
circle. Uxy will denote the points inside both circles (Fi^, 1). In arithmetical
value X may range from 1 to 0; so also y; while xy cannot be greater than x or
y.. or less than or x -\- y — \. This last is the principle of the syllogism. From
the co-ordinate nature of the operations x and f/, it is evident that Uxy = Uyx\
but this is a different thing from commuting, as Boole does, the relation of U
and r. which is not that of co-ordination, but of subordination of x to f/. and
which is properly denoted by writing U first.
Suppose 7/ to be the same character as x: we will then always have Uxx = f/x;
that is, an elementary selective symbol x is always such that x' = x . These are
but the symbols of ordinary algebra which satisfy this relation, namely 1 and 0;
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.
Fig. 2.
Let Uxy = U z. 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 char-
acter 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 -h and — .
Fig. 3
Fig. 4
CHAPTER 4. GEORGE BOOLE (1815-1864)
37
Let us now consider the expression U(x -hy). 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 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
duplication.
Fig. G.
Similarly, Boole held that the expression U(x — y) does not involve the
condition of the Uy being wholly included in the Ux (Fig. 5). If that condition
is satisfied. U (x — y) denotes a simple class; namely, the Ux's u'ithout 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 f/x's which are not y taken positively and
the Uy^s which are not x taken negatively. In Boole's view U(x — y) was in
general an intermediate uninterpretable 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 subtraction in ordinary algebra. They say
virtually. "How can you throw into a heap the same things twice over; and how
caji 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 aiialytical 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.
\i X -\- y denotes a quality without duplication, it will satisfy the condition
{x-\-yf = T + y^
x^ + 2xy -\-y^ = T -\-y,
but X = x,y = y,
.-. 2xy = 0.
CHAPTER 4. GEORGE BOOLE (1815-1864) 38
Similarly, ii x — y denote a simple quality, then
(x -y)- = 3: -y,
j:- -\-y^ - 2zy = j: -y,
2 2
X = X. y = y.
therefore, y — 2xy = — y,
y = xy.
In other words, the Uy must be included in the Ux (Fi^. 5). Here we have
assumed that the law of si^ns is the same as in ordinary algebra, and the result
comes out correct.
Suppose U z = Uxy; then Ux = U-z. How are the f/x"s related to the Uy's,
and the Uz's? From the diagram (in Fig. 2) we see that the f/j's are identical
with all the Uyz's together with an indefinite portion of the f/'s, which are
neither y nor z. Boole discoA^ered 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,
l = y + (l-y).
Similarly 1 = z + {l - z).
... 1 = j,^ + j,(l _ ^} + (1 _ y)^ + (1 _ j,)(l _ ^),
which means that the [/ 's either have both qualities y and z, or y but not z, or
z but not y. or neither y and z. Let
-z = Ayz + By{l - z) + C{1 - y)z -\- D{1 - y)(l - z),
y
it is required to determine the coefficients A. B. C, D. Suppose y = 1. z = 1;
then \ = A. Suppose y = \. z = Q. then = B. Suppose y = 0, z = 1: then
^ = C and C is infinite: therefore (1 — y)z = 0: which we see to be true from
the diagram. Suppose y = 0, e = 0; then = = D^ or D is indeterminate. Hence
— z = ys + an indefinite portion of(l— y}(l— s).
V
Boole attached great importance to the index law x = x. He held that
it expressed a law of thought, and formed the characteristic distinction of the
operations of the mind in its ordinary discourse and reasoning, as compared
with its operations 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
CHAPTER 4. GEORGE BOOLE (1815-1864) 39
for any being to possess a quality, and at the same time not to possess it. Let x
denote on elementary quality applicable to the uniA'erse U : then 1 — x denotes
the absence of that quahty. But if j:" = j:. then 0= t — z,0 = x(l — x). that
is, from Ux = Ui we deduce Ux{l — x) = 0.
He considers x{l — x) = as an expression of the principle of contradiction.
He proceeds to remark: "The above interpretation has been introduced not on
account of its immediate 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 fundamental axiom of metaphysics is but the
consequence of a law of thought, mathematical in its form. 1 desire to direct
attention also to the circumstance that the equation in which that fundamental
law of thought is expressed is an equation of the second degree. Without spec-
ulating 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. Hamil-
ton of Edinburgh, they are "horrent with mysterious spicul^e." 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 lan-
guage more exact. De Morgan could well appreciate the magnitude of the feat,
and he gave generous testimony to it as follows:
"Boole's system of logic is but one of many proofs of genius and patience
combined. 1 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 calculation, 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."
Chapter 5
ARTHUR CAYLEY^
(1821-1895)
Arthur Cayley was born at Richmond in Surrey. England, on August 16,
1821. His father, Henry Cayley, was descended from an ancient Yorkshire family,
but had settled in St. Petersburg, 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 indicates 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 IT 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
journaK 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 ajid
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 IM.A. degree, and win a
Fellowship by competitive 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 Mathem.atical Journal. On account of
the limited tenure of his fellowship it was necessary to choose a profession; like
^This Lectuie wqs deli\-ered April 20. 1901. — EDITORS.
40
CHAPTER 5. ARTHUR CAY LEY (1821-1895) 41
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 w^as 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 fourteen 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 1860 certain funds bequeathed by Lady Sadleir to the Univer-
sity^ 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 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 him-
self; 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 1880, 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 mathematical science was —
discharged in a handsome manner by the long series of memoirs which he pub-
lished, 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.
CHAPTER 5. ARTHUR CAY LEY (1821-1895) 42
Many mathematicians, of whom Sylvester was an example, find it irksome to
study what others have written, unless, perchance, it is something dealing di-
rectly with their own line of work. Cayley was a man of more cosmopolitan
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 immense amount; and of his kindliness to young in-
vestigators I can speak from personal experience. Several papers which I read
before the Royal Society of Edinburgh on the Analysis of Relationships were re-
ferred to him, and he recommended their publication. Soon after I was invited
by the Anthropological 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 proceedings.
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 GO years
old, considerably bent, and not filling his clothes. What was most remarkable
about him was the active glance of his gray eyes 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 hall 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 Isaa.c Newton 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. 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 nineteenth 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
Goldsmith, 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 portrait fund, wherein the scientific poet with his pen does greater
honor to the mathematician 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!
CHAPTER 5. ARTHUR CAY LEY (1821-1895) 43
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 throuo;h 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 nth root of — l!
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 vulgai' space,
In 11 dimensions flourished unrestricted.
The verses refer to the subjects investigated in several of Cayley's most elab-
orate memoirs: such as. Chapters on the Analytical Geometry of n dimensions:
On the theory of Determinants; Memoir on the theory of Matrices; Memoirs on
skew surfaces, otherwise Scrolls; On the delineation of a Cubic Scroll, etc.
In 1S81 he received from the Johns Hopkins University. Baltimore, where
Sylvester was then professor of mathematics, ai] invitation to deliver a course of
lectures. He accepted the invitation, and lectured at Baltimore during the first
five months of 1SS2 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 possible. Hamilton was the kind
of mathematician to suit such an occasion, but he never got the oflice, on account
of his occasional breaks. Cayley had not the oratorical, the philosophical, 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 Mathematics. 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
CHAPTER 5. ARTHUR CAY LEY (1821-1895) 44
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, 1 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
the meeting is the individual, which on evolution principles, must be sacrificed
for the development of the race," 1 daresay 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,
suggest iveness, and learning.
In 1889 the Cambridge University Press requested him to prepare his math-
ematical papers for publication in a collected form — a request which he appre-
ciated 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. in the T4th year of his age. When the funeral took place, a great
assemblage met in Trinity Chapel, comprising members of the University, of-
ficial representatiA'es 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 Mathematical papers number thirteen
quarto volumes, and contain 9G7 papers. His writings are his best monument,
and certainly no mathematician has ever had his monument in grander style.
Be 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 sometimes in making mathematical diagrams.
To the third edition of Tait's Elementary Treahse on Quaternions, Cayley
contributed a chapter entitled '"''Sketch of the analytical theory of quaternions."
In it the v — 1 reappears in all its glory, and in entire, so it is said, independence
of *, 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 chap-
ter 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 geometry
and are the natural and appropriate basis and method in the science, quater-
nions seemed a particular and very artificial method for treating such parts of
CHAPTER 5. ARTHUR CAY LEY (1821-1895) 45
the science of three-dimensional geometry as are most naturally discussed by
means of the rectangular coordinates x, j/, z. In tlie course of his paper Cay ley
says: "I have the highest admiration for tlie 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 applications.
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 understood, must be translated into coordinates." He goes on to
say, "I remark that the imaginary of ordinary algebra — for distinction call this
— 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 -h Ob, or in other words, say that a scalar quantity is in
general of the form a-\-Ob. 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 t, i/, s."
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 arguments, it may be said, first, y— 1 has a relation to the symbols i,
J, k^ for each of these can be analyzed into a unit axis multiplied by v — 1:
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
unspecified. 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 Iron 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 welding 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 transferred 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 artificial one."
The reply which Tait makes, so far as it is an argument, is: There are two
systems of quaternions, the i. j, k one, and another one which Hamilton de-
veloped from it: Cayley knows the first only, he himself knows the second: the
CHAPTER 5. ARTHUR CAY LEY (1821-1895) 46
former is an intensely artificial system of imaginaries, the latter is the natu-
ral or^an of expression for quantities in space. Should a fourth edition of his
Elementary Treatise be called for i. j, k will 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 Dou-
ble 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 of imaginaries i , j. k] defined by the equations,
i^ = f = k- = ijk = -1,
with the associative, but not the commutative, law for the factors. The novel
and splendid points in it were the treatment of all directions in space as essen-
tially 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 magni-
tude, and of vast importance. And here I thoroughly 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 abso-
lutely natural one" which Tait thus further 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 ap-
plied, without the necessity of appealing to coordinates at all. Coordinates may,
however, easily be read into them: — when anything (such as metrical or numer-
ical detail) is to be gained thereby. Quaternions, in a w^ord, exist in space, and
we have only to recognize them: — but we have to invent or imagine coordinates
of all kinds."
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 conception, 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, fc. were mere excrescences and blots on his improved
method: — but he unfortunately considered that their continued (if only partial)
CHAPTER 5. ARTHUR CAY LEY (1821-1895) 47
recognition was indispensable to the reception of his method by a world steeped
in — Cartesianism ! Through the whole compass of eacli of his tremendous vol-
umes 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, be would be
left without a single reader."
To Cayley's presidential address we are indebted for information about tbe
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 empirical 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 Differential Calculus would be called algebra. Using the word in this
restricted sense. I cannot myself recognize the connection of algebra with the
notion of time: granting that the notion of continuous progression presents itself
and is of importance, 1 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 algebraical couple, or imaginary magnitude,
a + 5v — 1," 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 unidiraensional, 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 geometry 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
CHAPTER 5. ARTHUR CAY LEY (1821-1895) 48
in kind from an experimental truth generalized from experience. Compare, for
instance, the proposition, that the sun, having aheady 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
ififinituiify 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 remarkable instances of propositions 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 Playfair's form of it, has been considered as needing demonstration;
and that Lobatschewsky 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 demonstration, 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 1 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,
Euclideai] 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 propositions 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 geometry is that of imaginary magni-
tude 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 j: or y of a
point of intersection. 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 intersection, 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 giA^en imaginary values? He
seems to say No, and to fall back on the notion of an imaginary space as the
locus in quo of the imaginary point.
1
Chapter 6
WILLIAM
KINGDON CLIFFORD
(1845-1879)
William Kingdon Clifford was born at Exeter, England. May 4, 1S45. His
father was a well-known and active citizen and filled the honorary office of jus-
tice 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 weak-
ness in his physical constitution. He received his elementary education at a
private school in Exeter^ where examinations were annually held by the Board
of Local Examinations of the Universities of Oxford and Cambridge; at these
examinations 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 mathematical abilities, but also gained distinction in
classics and English literature.
When eighteen, he entered Trinity College, Cambridge; the college of Pea-
cock, 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 undergrad-
uate career was distinguished by eminence in mathematics, English literature
and gymnastics. One who was his companion in gymnastics wrote: "His neat-
ness and dexterity were unusually great, but the most remarkable thing was his
great strength as compared with his weight, as show^n 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 year he won the prize awarded by Trinity College
for declamation, his subject being Sir W'alter Raleigh; as a consequence he was
called on to deliver the annual oration at the next Commemoration of Bene-
^Thia Lecture was delivered Apiil 23, 1901. — EDITORS.
49
CHAPTER 6. WILLIAM KINQDON CLIFFORD (1845-1879) 50
factors of the College. He chose for his subject, Dr. WhewelK Master of the
College, eminent for his philosophical and scientific attainments, whose death
had occurred but recently. He treated it in an original and unexpected manner;
Dr. Wheweirs claim to admiration and emulation being put on the ground of
his intellectual life exemplifying 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 him-
self 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 particular ex-
amination. 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 studying fundamental principles, still less for making any philo-
sophical 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 tha,t De Mor-
gan 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
mathematiciai] are WhewelK Sylvester, Kelvin, Maxwell.
In 1808, 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 1S71 he was appointed professor of Applied 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 lsaa.c Newton. About
the middle of the nineteenth century the admission of new members was re-
stricted to fifteen each year; and from applications the Council recommends fif-
teen names which are posted up. and subsequently balloted for by the Fellows.
Hamilton and De Morgan never applied. Clifford did not apply immediately,
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 1S75 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
CHAPTER 6. WILLIAM KINQDON CLIFFORD (1845-1879) 51
a contemporary mathematician, Mr. Dodgson — "Lewis Carroll"' — the author of
Alice in Wonderland. A children's party was one of Chfford'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
accustomed 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. 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 1 spent at Cambridge, I
was not a little exercised by the theorem in question, as 1 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: 1 think that
I can recall the very spot. What he said I do not remember in detail: which is
not surprising, as 1 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 stren-
gth 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 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 in the 34th year
of his age.
On the title page of the volume containing his collected mathematical pa-
pers 1 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
CHAPTER 6. WILLIAM KINQDON CLIFFORD (1845-1879) 52
of the shortness of his life. In his lifetime there appeared Elements of Dy-
namic, Part /. Posthumously there have appeared Elem.ents of Dynam^ic, Part
II; Collected Mathematical Papers; Lectures and Essays; Seeing and Thinking;
Common Sense of the Exact Sciences. The mannsciipt 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 discussed 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 institution was founded by Count Rumfoid, 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 circumstances 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
accustomed 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 consciously 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 definifceness 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 pub-
lished. The following verses were sent to George Eliot, the novelist, with a
presentation copy of The Little People:
CHAPTER 6. WILLIAM KINQDON CLIFFORD (1845-1879) 53
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.
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 mathematical 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 commence-
ment of each subject and carrying it on rapidly to the most advanced stages.
He began with the Elements of Dynatnic, 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 formulae. 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 textbooks would have been.
In Clifford's lifetime it was said in England that he was the only mathemati-
cian who could discourse on mathematics to an audience composed of people of
general culture and make them think that they understood the subject. In 1S72
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 theoretically or
absolutely exact, but only exact enough to correct experiments by. As an in-
stance 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 look-
ing through it. He requires no scientific thought 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
CHAPTER 6. WILLIAM KINQDON CLIFFORD (1845-1879) 54
most distinctly an application of experience gained under certain circumstances
to entirely different circumstances."
In physical science there are two kinds of law — distinguished as '"''enipiricar'
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 engineers. But a rational law states a connection which is accurately
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 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 metaphysics, whereas
it is a very distinct and simple question of fact. I am supposed to know that
the three angles of a rectilinear 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 aiigles: but also I do not know
that the difference would be less than ten degrees or the ninth part of a right
an^le."'
CHAPTER 6. WILLIAM KINQDON CLIFFORD (1845-1879) 55
You will obserA'e that Clifford's philosophy depends on the validity of Lo-
batchewsky's ideas. Now it has been shown by an Italian mathematician, named
Beltrami, that the plane geometry of Lobatchewsky corresponds to trigonom-
etry on a surface called the pseudospkere. Clifford and other followers of Lo-
batchewsky admit Beltrami's interpretation, an interpretation which does not
involve any paradox about geometrical space, and which leaves the trigonom-
etry of the plane alone as a different 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, whateA^er that may mean. If so. then Clif-
ford's argument for the empirical nature of the proposition referred to fails: and
nothing prevents us from falling bock on Kant's position, namely, that there is a
body of knowledge characterized by absolute exactness 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 trigonom-
etry, 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 transition triangle between the spherical triangle and the
equilateral hyperboloidal triangle, the sum of the angles of the plane triangle
must be exactly equal to two right angles.
In 1S73. the British Association met at Bradford; on this occasion the
evening discourse was delivered by Maxwell, the celebrated physicist. He chose
for his subject "Molecules." The application of the method of spectrum-analysis
assures the physicist that he can find out in his laboratory truths of universal
validity in space and time. In fact, the chief maxim of physical science, accord-
ing to Maxwell is, that physical changes are independent of the conditions of
space and time, and depend only on conditions of configuration of bodies, tem-
perature, 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 haA'e passed from one to another; and yet this light, which is to us the sole
evidence of the existence of these distant worlds, tells us also that each of them
is built up of molecules of the same kinds as those which are found on earth. A
molecule of hydrogen, for example, whether in Sirius or in Arcturus, executes its
vibrations in precisely the same time. No theory of evolution can be formed to
account for the similarity of molecules, for evolution necessarily implies contin-
uous 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,
CHAPTER 6. WILLIAM KINQDON CLIFFORD (1845-1879) 56
have produced the slightest difference in the properties of any molecule. We are
therefore unable to ascribe either the existence of the molecules 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
scientific 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 exactness; 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 infinite exactness. We get at
conclusions which are as nearly true as experiment can show, and 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, aiid ex-
tended all the bounds of physical science. Faith in an exact order of Nature was
the characteristic 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 be-
coming 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 depend 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:
a^c^de'^^-f^gh'^j'^l^m^n'^o^pe^s'^t'^'^u^v^-wxy^.
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 physicists 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
CHAPTER 6. WILLIAM KINGDOM CLIFFORD (1845-1879) 57
of physics at Owens College, Manchester. When the fourth edition appeared,
their names were given on the title page.
The kernel of the book is the above so-called discovery, hist published in
the form of an anagram. Preliminary chapters 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 voitex-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 uniA^erse and vice versa.
The soul is a frame which is made of the reflned 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 association with the coarse molecules of the body. In this
way the authors consider 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 flnd in the same book a discussion of Carnot's heat-
engine ai]d extensive quotations from the apostles and prophets. Clifford wrote
an elaborate review which he finished in one sitting occupying 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
consider the 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.
1
Chapter 7
HENRY JOHN
STEPHEN SMITH
(1826-1883)
Henry John Stephen Smith was born in Dublin. Ireland, on NoA'ember 2,
1826. His father, John Smith, was an Irish banister, who had graduated at
Trinity Co]le°;e, Dublin, and had afterwards studied at the Temple, London, as
a pupil of Henry John Stephen, the editor of Blackstone's Cofmneutaries ; hence
the given name of the future mathematician. His mother was Mary Murphy, an
accomplished and clever Irishwoman, tall and beautiful. Henry was the youngest
of four children, and was but tw^o 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 w^as sickly and very near-sighted. When four years of age he
displayed a genius for mastering languages. His first instructor w^as his mother,
who had an accurate knowledge of the classics. 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 impressed 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 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
^ThiH Lecture wqb delivered March 15, 1902. — EDITORS.
58
CHAPTER 7. HENRY JOHN STEPHEN SMITH (1826-1883) 59
Mr. Highton received a mastership at Rugby with a boardinghouse 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 hfe of the Enghsh 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 coA^eted 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 infiuence 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 bachelor of arts he took first-class rank in the classics,
and also first-class rank in the mathematics.
It is not very pleasant to be a double first, for the outwardly envied and dis-
tinguished 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
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 ap-
pointed to inquire into the working of the University. The old system of close
scholarships and fellowships was doomed, and the close preserves 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 unfortu-
nate 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 college, and he reformed
the abuses of privilege and close endowment as far as he legally could. Smith
CHAPTER 7. HENRY JOHN STEPHEN SMITH (1826-1883) 60
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 published anonymously. Whewell
pointed out what he called law of waste traceable in the Divine economy; and
his argument 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 symbohcal 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 exception 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 demonstration, 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 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 difficulty. 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
CHAPTER 7. HENRY JOHN STEPHEN SMITH (1826-1883) 61
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 nevertheless 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 numerorum primorum fornix 4'^ + 1
ex duobus quadratis." In it he proves in an original manner the theorem of
Fermat — "That every prime number of the form 4^" + 1 {n being an integer
number) is the sum of two square numbers." In his second 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: nev-
ertheless the arithmetical books of the elements of Euclid contain the oldest
extant investigations respecting them: and, in particular the celebrated yet sim-
ple 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 essentially 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 arithmeti-
cal 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 repre-
sented by points set off at equal distances along a line, by using a geometrical
compass we can determine without calculation the multiples of any given num-
ber. And in fact, it was by a mechanical contrivance of this nature that M.
Burckhardt 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 arraiiged
to keep house in Oxford, the two spending the terms together, and each being
CHAPTER 7. HENRY JOHN STEPHEN SMITH (1826-1883) 62
allowed complete liberty of moA^ement 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 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 banquets.
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
HVhere 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 1800, when Smith was 34 years old, the Savilian professor of geometry
at Oxford died. At that time the English universities 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 candidate: 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 BallioK 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.
His freedom during vacation gave him the opportunity of attending the meet-
ings 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 1S5S 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
analvzes with remarkable clearness and order the works of mathematicians for
CHAPTER 7. HENRY JOHN STEPHEN SMITH (1826-1883) 63
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 consequence of the re-
searches connected therewith. Smith published several original contributions to
the higher arithmetic. Some were in complete form and appeared in the Philo-
sophical Transactions of the Royal Society of London; others were incomplete,
giving only the results without the extended demonstrations, 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 in determinates,'' enunci-
ates certain general principles by means of which he solves a problem proposed
by Eisenstein, nam^ely, the decomposition of integer numbers into the sum of
fiA'e 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
Lionville w^ere 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 UniA'ersity 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 Educa-
tion; a member of the Commission appointed to reform the University of Oxford;
chairman of the committee of scientists who were given charge of the Meteo-
rological Office, etc. It was not till 1873, when offered a Fellowship by Corpus
Christ! 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
elaborating 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 execution; 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 re-
gion of the theory of numbers, it was perhaps natural that he should take an
ant i- utilitarian view of mathematical science, and that he should express it in
exaggerated terms as a defiance to the grossly utilitarian views then popular.
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." 1 believe that it was at a banquet of the
Red Lions that he proposed the toast "Pure mathematics; may it never be of
CHAPTER 7. HENRY JOHN STEPHEN SMITH (1826-1883) 64
any use to any one."
I may mention some other specimens of his wit. "You take tea in the morn-
ing." 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
Maiischal College, Aberdeen: "^They say; what say they; let them say." "Ah,"
said he, "it expresses the three stages of an undergraduate's career. ^They say' —
in his first year he accepts OA^erything 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, w^ith 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 Molecules." 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 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
exam.ple of the rule he points out that the experiments of Faraday called forth
CHAPTER 7. HENRY JOHN STEPHEN SMITH (1826-1883) 65
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 mathe-
matics. 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 pro-
found scholar, produced a powerful effect: "AH 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 usefulness 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 something of the impenetrable mystery,
of the world in which he 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 important part of the
education which science supplies. But in science sophistry is impossible; science
knows no love of paradox: 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 1805. 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 comparative
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
CHAPTER 7. HENRY JOHN STEPHEN SMITH (1826-1883) 66
towards a country which has, we mi^ht say, been "prospected" for us, and in
which we know beforehand we cannot 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 candidate 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 Conservative Government. Henry
Smith had always been a Liberal in politics, university administration, and reli-
gion. 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 sci-
ences inatkern.atiques was the theory of the decomposition of integer numbers
into a sum of Aa^o squares; and that the attention of competitors was directed
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 IM. Hermite calling his attention to what he had
published: in reply he was assured that the members of the commission did not
know of the existence of his papers, and he was advised to complete his demon-
strations and submit the memoir according to the rules of the competition.
According to the rules each manuscript bears a motto, and the corresponding
envelope containing the name of the successful author is opened. There were
still three months before the closing of the concours (1 June, 1SS2) 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 fran-
chise 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
CHAPTER 7. HENRY JOHN STEPHEN SMITH (1826-1883) 67
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, 1SS3. 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 . Hermite 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 IVI. Minkowski availed himself
of whatever had been published on the subject, including Smith's paper, but to
work up the memoir 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.
Chapter 8
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 sec-
ond wrangler. The senior wrangler of the year did not rise to any eminence; the
fourth wrangler was George Green, celebrated for his contributions to mathe-
matical 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 consequence
he could neither receive the degree of Bachelor of Arts nor compete for the
Smith's prizes, and as a further consequence 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 Cam-
bridge 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 Be 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
^ThiB Lecture wqb delivered March 21, 1902. — EDITORS.
68
CHAPTER 8. JAMES JOSEPH SYLVESTER (1814-1897) 69
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 manipulation. 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 1841 he became professor of mathematics at the University 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
termination. 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 sword-cane; the young
man provided himself with a heavy walking-stick. The brothers lay in wait
for the professor; 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 his 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 actu-
aries, his ultimate aim being to qualify himself to practice conveyancing. He
became a student of the Inner Temple in 1S4G. and was called to the bar in
1850. He chose the same profession as did Cayley; and in fact Cayley and
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 friends; but it is safe to say that the permanence of the friendship
was due to Cayley's kind and patient disposition. Recognized as the leading
mathematiciaiis of their day in England^ they were yet very different both in
nature and talents.
Cayley was patient and equable; Sylvester, fiery and passionate. Cayley
finished off a mathematical memoir with the same care as a legal instrument;
Sylvester never wrote a paper without foot-notes, appendices, supplements; and
the alterations aiid corrections in his proofs were such that the printers found
their task well-nigh impossible. Cayley was well-read in contemporary math-
ematics, and did much useful work as referee for scientific societies: Sylvester
read only what had an immediate 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 mathe-
matician as one lost in reflection, and high above mundane affairs. Cayley was
modest and retiring; Sylvester, courageous and full of his own importaiice. But
while Cayley's papers, almost all, have the stamp of pure logical mathemat-
ics, Sylvester's are full of human interest. Cayley was no orator and no poet;
CHAPTER 8. JAMES JOSEPH SYLVESTER (1814-1897) 70
Sylvester was an orator, and if not a poet, he at least prided himself on his
poetry. It was not long before Cayiey was provided with a chair at Cambridge,
where be im.mediately 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 probationary lecture: with the result that he was not appointed. The profes-
sorship of mathematics in the Royal Military Academy 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 w^as 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 inA^ariants, he was guided by it to take up one of
the most difficult questions in the theory of numbers. Cayiey had reduced the
problem of the enumeration of invariants to that of the partition of numbers;
Syh'ester 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 -\- . . . -\- Id = tn.
Thereafter he attacked the similar problem connected with two such simultane-
ous 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 achievement. 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 establish
either its truth or inadequacy. Sylvester in the paper quoted established the
validity of the rule for algebraic equations as far as the fifth degree inclusive.
Next year in a communication to the Mathematical Society of London^ he fully
established and generalized the rule. '"''I owed my success,"' he said, "chiefly
to merging the theorem to be proved in one of greater scope and generality.
In mathematical research, reversing 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 Mathe-
matical Society. He was the first mathematician to whom that Society awarded
CHAPTER 8. JAMES JOSEPH SYLVESTER (1814-1897) 71
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 math-
ematics and physics. Most of the mathematicians who have occupied that posi-
tion have experienced difficulty in finding 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 AI acmillan'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 utterances of a most distinguished mem-
ber 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, compar-
ison, 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 to grapple with the practical side of certain
mathematical questions; 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 math-
ematician starts with a few simple propositions, 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 proposition
and to construe or parse a sentence are equivalent or identical mental opera-
tions. Such an opinion scarcely seems to need serious refutation. The passage is
taken from an article in MacTniilan^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
CHAPTER 8. JAMES JOSEPH SYLVESTER (1814-1897) 72
number of elementary propositions from "whicli 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 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 defined 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 verification: 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 parenthetically a clause which expresses his theory of mathemat-
ical knowledge. He says that the inner world of thought in each individual man
(which is the world of observation to the mathematician) 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 distinctions, both do-
mestic and foreign, which are usually awarded to a mathematician of the first
rank in his day. But a discontinuity 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 prescribed 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 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 mathe-
matician 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
CHAPTER 8. JAMES JOSEPH SYLVESTER (1814-1897) 73
that the same discovery had been made several years before by M. Peaucelher,
an officer in the French army, but faihn^ to be recognized at its true value had
dropped into obhvion. Sylvester introduced the subject into England in the
form of an evening lecture before the Royal Institution, entitled "On recent dis-
coveries in mechanical 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 lectures are attended
bv fashionable audiences of ladies and gentlemen in full dress.
do-
-OB
Euclid bases his Eletnents 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 first. It is the
mechanical problem of converting motion in a circle into motion in 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.
HoweA'er. he succeeded in finding a contriA'ance which solves the problem very
approximately. 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 point is attached to the middle point
of CD. W^hen 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.
CHAPTER 8. JAMES JOSEPH SYLVESTER (1814-1897) 74
A linkage ghdng a closer approximation to a straight line was also invented
by tlie Russian mathematiciaji before mentioned — 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 con-
necting 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 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 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 Syh'ester worked up the theory of linkages together, and discovered
among other things the skew pantograph. Kempe became so imbued with link-
age 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 Bal-
timore, and Sylvester, at the age of 03, 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 Sylvester's lectures was
enthusiastic love of the thing he was doing. He had in the fullest possible de-
gree, 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 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
CHAPTER 8. JAMES JOSEPH SYLVESTER (1814-1897) 75
cannot be denied. A result announced at one lecture and hailed with loud ac-
claim as a marvel of beauty was by no means sure of not being found before tlie
next lecture to have been erroneous; but tlie 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 Al at hematics, which, at his suggestion, took the
quarto form. He aimed at establishing 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 Am.erican 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 LaU'S of Verse, Sylvester prepared a poem of 400 lines, all rhyming
with the name Rosalind 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 turn-out of ladies and gen-
tlemen. 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 1 visited London to see the Electrical Exhibition in the
Crystal Palace — one of the earliest exhibitions devoted to electricity exclusively.
I had made some investigations 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
iuA'itation to a dinner at his own house to be giA'en 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
CHAPTER 8. JAMES JOSEPH SYLVESTER (1814-1897)
76
scientist. After dinner he conversed veiy eloquently with an elderly lady of
title, while 1 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 Contptes Rendus (in which he published
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 1SS3. 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
elementary reciprocant is j-^, for if ^-^ = then ^-f- = 0. He looked upon this
as the "grub" form, and developed from it the "chrysalis"'
and the "ima^o^'
d^<i>
dxdy
dip
dxdy
d4>
dx
d^i>
dy
dip
'Sy^
d"^
d^^
dxdy
d^-t
dxdr '
d^^
dxdy
d"^
d^^
df
d^^
d^^
dydr '
dxdr
dydr
dr' ■
You will observe that the chrysalis expression is un symmetrical; 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 oe 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?
Thou minds't me much of that presumptuous one^
Who loth, aught less than greatest, to be great.
From Heaven's immensity fell headlong down
To live forlorn, self-centred, desolate:
Or who, new Heraklid. hard exile bore.
Now buoyed by hope, now stretched on rack of fear.
Till throned Astr^ea, wafting to his ear
Words of dim portent through the Atlantic roar^
CHAPTER 8. JAMES JOSEPH SYLVESTER (1814-1897) 77
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 re-
searches 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. Syh^ester removed to London, and
liA'ed 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 relatives were nieces,
but he did not wish to ask their assistance. One day, meeting a mathematical
friend who had a home in London, he complained of the fare at the Club,
and asked his friend to help him flnd 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: Sylvester 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 flre of
his temper had not dimmed with age, and it required all the Christian fortitude
of the ladies to stand his 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 Dalston.
As a theist, Sylvester did not approve of the destructiA^e 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 flrst award has
been made to M. Henri Poincare of Paris, a mathematician for whom Sylvester
had a high professional and personal regard.
Chapter 9
THOMAS
PENYNGTON
KIRKMAN^
(1806-1895)
Thomas Penyn°;ton Kirkman was born on March 31, 1800, 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
Adcar also urged the father, saying that if he would send his son to Cambridge
University, he would guarantee for sixpence that the boy would win a fellowship.
But the father was obdurate; young Kirkman 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, he owed to his own study, having never had a single hour's
^This Lecture wqs delivered Apiil 20. 1903. — EDITORS.
78
CHAPTER 9. THOMAS PENYNGTON KIRKMAN (1806-1895) 79
instruction from any person. To this self-education is due, it appears to me,
both the stren°;th 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 teaching was over
he had the leisure requisite for the great mathematical 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 meditations 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 *. j, k of Hamilton), "not dreading" he says, "the pluperfect
criticism of grammarians, since the convenient barbarism is their own." Hamil-
ton, writing to De Morgan, remarked "Kirkman is a very clever fellow," where
the adjectiA'e 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 Memoria Technica of Grey. In 18-51 he
contributed a paper on the subject to the Literary and Philosophical Society
of Manchester, and in 1852 he published a book. First Alnemonical Lessons in
Geometry, Algebra, and Trigonometry, which is dedicated to his former pupil.
Sir John Blunden. De Morgan pronounced it "the most curious crochet 1 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 distaste
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 unpronounceableness of the formulae: the memory of the tongue and the
ear are 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 "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
CHAPTER 9. THOMAS PENYNGTON KIRKMAN (1806-1895) 80
good English in order to secure the requisites for a useful mneTiiomc. 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 mem-
ory; for instance the following: If in any spherical triangle four parts be taken
in succession, such as AbCa, consisting 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
cos6cosC = sin 6 cot a — sin C cot A.
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 metnoria technica. He distinguishes the large letter from the small by
calling them Aug, Bang. Gang {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 Gang and co h co Gang are cot a si 6
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 judgment: Kirkman's method is
something to be taught by rote. In his book Kirkmai] 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: "De-
termine the number 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 spe-
cial case of ;? = 3 and ^ = 2 and printed his results in the second volume of
the GambrJdge 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
CHAPTER 9. THOMAS PENYNGTON KIRKMAN (1806-1895) 81
when an undergraduate at Cambridge, and that he had given it to fellow un-
dergraduates. Kirkman rephed that up to the time he proposed 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 mem-
oir on the theory of the polyedra: this fact together with his work in combina-
tions 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 combinations of crystallography. His first paper,
"On the representation 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 -I- 2, where P is the number of points or
summits, S the number of plane bounding surfaces and L the number of linear
edges in a geometrical solid. "The question — how many ii-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 ?i-edrons are there whose summits are all trihedral"? He enumerated and
constructed the fourteen 8-edra whose faces are all triangles.
In 1858 the French Institute modified its prize question. As the subject
for the coficours of 1861 was announced: '"''Ferfectionner en quelque point im-
portant la theorie geometrique des polyedres." where the indefiniteness of the
question indicates 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 preface he says, "The following memoir contains a com-
plete solution of the classification and enumeration of the P-edra Q-acra. The
actual construction of the solids is a task impracticable 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 sections: 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 contributions 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
1860: "Quels peuvent etre les nombres de valeurs des fonctions bien definies
qui contiennent un nombre donne de lettres. et comment pent 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 vouchsafed of what the competitors had added to science,
although it was confessed that all had contributed results both new and impor-
tant: and the question, though proposed for the first time for the year 1860,
CHAPTER 9. THOMAS PENYNGTON KIRKMAN (1806-1895) 82
was withdrawn from competition contrary to the usual custom of the Academy.
Kirkman contributed the results of his investigation to the Manchester Soci-
ety under the title ^*The complete theory of groups, being the solution of the
mathematical prize question of the French Academy for ISGO." 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 meet-
ings 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, both were good versifiers if not great po-
ets. 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 18G2 the Educational Times^ a monthly periodical published in Lon-
don, began to devote several columns to the proposing and solving of math-
ematical problems, taking up the work after the demise of the Diary. This
matter was afterwards 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 arraiiged
in five rows on Monday. Shifting the second column one step upward, the third
two steps, the fourth three 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 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.
CHAPTER 9. THOMAS PENYNGTON KIRKMAN (1806-1895) 83
Monday
Tuesday
A
B
C
D
E
A
G
M
s
Y
F
G
H
I
J
F
L
R
X
E
K
L
M
N
K
Q
W
D
J
P
Q
R
S
T
P
V
C
I
O
U
V
W
X
Y
U
B
H
N
T
We
'dnes
day
Thursday
A
L
W
T
T
A
Q
H
X
O
F
Q
c
N
Y
F
V
M
D
T
K
V
H
S
E
K
B
R
I
Y
P
B
M
X
J
P
G
W
N
E
N
G
R
D
U
L
C
S
J
The Rev. Kirkman became at an early period of his life a broad churchman.
About 1803 he came forward in defense of the Bishop of Colenso, a m.athe-
matician, 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: "Tfce 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 NataK 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 according 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 TyndalTs 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 Philosophy
without AssutJiptious, in which he criticises in very vigorous style the materialis-
tic 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 knowledge of
himself as a continuous person, endowed with will; and he infers that there are
will forces around: but he sees no eA'idence of the existence of matter. Matter
is an assumption aiid 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 assumption which
CHAPTER 9. THOMAS PENYNGTON KIRKMAN (1806-1895) 84
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 proposition 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 philosophical toil than all the two volumes on Psy-
chology. 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 nonsense; 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 1.
Shall I spare him who tells me that my moA'ements 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 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 alternative that either the testimony of this
Beings 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 1 have
proved it. And every man must of course talk nonsense who begins his philos-
ophy 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 4aw.' When 1 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
mathematiciaiis. When 1 ask — What does he know about law? — there is neither
beginning nor end to the reply. 1 am advised to read his books about law, and to
master the differentiations and integrations 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 1 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
CHAPTER 9. THOMAS PENYNGTON KIRKMAN (1806-1895) 85
■must he'^
Kirkman points out that the kind of proof offered by these philosophers is
a bold assertion of rtiust-he-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 complex cor-
respondence to be coordinated) must be brouglit into relation with each other.
But this implies some center of communication common to them all, through
which they severally pass; and as they cannot pass tbrougli 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 vari-
ety 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 homogene-
ity to a definite coherent heterogeneity, through continuous differentiations and
integrations." Kirkman ^s paraphrase is "Evolution is a change from a nohow-
ish untalkab out able all-likeness, to a somehowish and in-general-talkaboutable
not-all-likeness^ by continuous somethingelseifications and sticktogetherations."
The tone of Kirkman 's book is distinctly polemical and full of sarcasm. He un-
fortunately wrote as a theologian rather than as a mathematician. 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 construction of the knots which can be formed in an endless
cord — a subject which he was induced to take up on account of its bearing 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
crossings. Kirkman 's investigations on the polyedra were much 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 1 w^as introduced to Rev. Mr. Kirkman: and we
discussed the mathematical analysis of relationships, formal logic, and other
subjects. After I had gone to the University of Texas. Kirkman sent me through
Tait the following 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
CHAPTER 9. THOMAS PENYNGTON KIRKMAN (1806-1895) 86
paper is printed in the Proceedings of the Royal Society of Edinburgh. There are
four solutious, provided Smith and Jones are taken to be mere arbitrary, names:
if the conveution about surnames holds there are only two legal solutions. On
seeiug 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, 1S95. 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 Mathe-
matics. 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 quota-
tion 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 liA^e.
But it is a faint pleasure to think it will one day win a little praise."
Chapter 10
ISAAC TODHUNTERI
(1820-1884)
Isaac Todhunter was born at Rye, Sussex, 23 Nov.^ 1820. He was the second
son of George Todhunter, Congiegationahst minister of the place, and of Mary
his wife, whose maiden name was Hume, a Scottish surname. The minister died
of consumption when Isaac w^as 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 subsequently 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 examining body, winning the exhibition
for mathematics (£30 for two years): in 1S42 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 awarded to the
candidate who gained the greatest distinction in that examination.
Sylvester was then professor of natural philosophy in University College, and
Todhunter studied under him. The writings of Sir John Herschel also had an
influence; for Todhunter wrote as follows [Conflict of Studies^ p. GG): '"''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
^This Lectuie wos delivered April 13, 1904. — EDITORS.
87
CHAPTER 10. ISAAC TODHUNTER (1820-1884) 88
attribute the direction of my own early studies. He says of Astronomy, "Admis-
sion 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 somewhat 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 Elec-
tricity. 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, ^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 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 1S49.
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 1 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 1S53 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 mathematics; his Algebra appeared in 1858, his Trigonometry in
1859, his Theory of Equations in 1801, aiid his Euclid in 1862. Some of his
CHAPTER 10. ISAAC TODHUNTER (1820-1884) 89
textbooks passed througb. 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
elementary 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: 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 18G1; in 1862 he was elected a Fellow
of the Royal Society of London. In 1863 he was a candidate for the Sadlerian
professorship of Mathematics, to which Cayley was appointed. Todhunter was
not a mere mathematical specialist. He was an excellent linguist: besides being a
sound Latin aiid 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 18G3-5 acted as an Examiner for the Moral
Science Tripos, of which the chief founders were himself and Whewell.
By 1804 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 aiid 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 staiiding 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 forearmed, he calmly introduced to the inner circle of their
honeymoon Hamilton on Quaternions.
It was now (18G5) that the London Mathematical Society was organized un-
der 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 in-
volving so much interference with his work as an author that he never accepted
it again. Now (1865) appeared his History of the Afathematical Theory of Prob-
ability, and the same year he was able to edit a new edition of Boole's Treatise
on Differentia! Equations^ the author having succumbed to an untimely death.
Todhunter certainly had a high appreciation of Boole, which he shared in com-
mon 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 inde-
CHAPTER 10. ISAAC TODHUNTER (1820-1884) 90
pendent editions ^Q.ve ample opportunity to correct all errors and misprints. He
now added two more textbooks; Mechanics in 1867 and Mensuration in 18G9.
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 18S9 the subject proposed was "A determination of
the circumstances 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 successfully 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" — an entirely different work from his History of the
Calculus of Variations.
In 1S73 he published his History of the Mathematical Theories of Attraction.
It consists of two volumes of nearly 1000 pages altogether and is probably the
most elaborate of his histories. In the same year (1873) he 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
successive publication in some magazine, but in fact they were published only in
book form. In the first essay, that on the Confiict of Studies — Todhunter gave
his opinion of the educative value in high schools aiid 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 education, 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 barometer in order to
test the infiuence of chaiige 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
CHAPTER 10. ISAAC TODHUNTER (1820-1884) 91
appropriates and destroys its fragrance and its beauty."
At the time when Todhunter was writing the above, tlie Cavendish Labo-
ratory for Experimental Physics was just being built at Cambridge, and Clerk-
Maxwell had just been appointed the professor of the new study: from Tod-
hunter's utterance we can see the state of affairs then prevailing. Consider the
corresponding experiment of Torricelli. which can be performed inside a class-
room; 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 conceptions
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^ y^t. he must admit, they still possess^ to the fresh stu-
dent 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 sovereign 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 transaction. 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 impression 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
blameless character — his suspicion is irrational, and manifests a want of the
power of appreciating evidence, a want fatal to his success in that branch of
science which he is supposed to be cultivating."
Clear physical conceptions cannot be got by tradition, even from a clergy-
man of blameless character; 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 manu-
als are not free from that fault. He deals with the characteristic difficulties
of algebra by authority rather than by scientific explanation. Todhunter goes
on to say: '"''Some considerable 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 baflled by
Eoucault'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."
CHAPTER 10. ISAAC TODHUNTER (1820-1884) 92
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 w^as in the hands of a pure mathematician — Prof. Sylvester.
At the University of Texas there is a fine clear space about 00 feet high inside
the building, very suitable for Foucault's experiment. I fixed up a pendulum,
using a very heavy ball, and the turning of the Earth could be seen in two
successive oscillations. 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 mathematics: 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 Improvement 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 competitive examinations. Great Britain is an examination -rid den 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 mathematicians 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 — Sylvester — as opposed. In his presidential
address before section A of the British Association at Exeter in 18G9, Sylvester
had said: "1 should rejoice to see . . , Euclid honorably shelved or buried 'deeper
than did ever plummet sound' out of the schoolboy's reach: morphology intro-
duced 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 inconceiv-
able." Todhunter replied: "Whatever may have produced the dislike to Euclid
in the illustrious 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 natu-
CHAPTER 10. ISAAC TODHUNTER (1820-1884) 93
rally su^ested by the passage is that nothing preA^ents 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 ex-
ample of this could be wished for than the insane delusion under which they
permit 'Euclid' to be employed in our elementary teaching. They seem vol-
untarily 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 Mathemati-
cal Messenger, and the Philosophical 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. 1 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 dis-
tinguished 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 which Euclid is as-
sailed 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 mathematical prob-
lems 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 mathe-
maticians 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 com-
petitive 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 influ-
ence 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 pro-
cesses resembling examinations before those of Cambridge had become widely
famous, yet there can be little chance of error in regarding our mathematical
CHAPTER 10. ISAAC TODHUNTER (1820-1884) 94
tripos as the model for rigor, justice and importance, of a long succession of
institutions of a similar kind whicli 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 training pupils for ex-
amination. One direct consequence is that the memory is cultivated at the ex-
pense of the understanding; knowledge instead of being assimilated is crammed
for the time beings 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 conundrums. 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 trigonometry
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, professors 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 produced and reproduced only the answer, ^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 professorships — 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 he 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 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
CHAPTER 10. ISAAC TODHUNTER (1820-1884) 95
September, 1883. when he was at Hunstanton, that the worst symptoms came
on. He then partially lost by paralysis the use of the ri^ht 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 04th year of
his a^e.
Todhunter left a History of Elasticity nearly finished. The manuscript was
submitted, to Cayley for report; it was in 1886 pubhshed under the editorship of
Karl Pearson, I believe that he had other histories in contemplation: 1 had the
honor of meetin°; him once, and in the course of conversation on mathematical
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, "Todhunter 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 forsot to see to their wants."
Chapter 11
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