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by Alexander Macfarlane 

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Title : Ten British Mathematicians of the 19th Century 

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No. 17 


OF THE Nineteenth Century 


Late President eor the [nter.national Assoceatiox for. Promotixg 
THE Study of Quaternions 




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. 


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 

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 


general interest. 

Alexander Macfarlane 
From a photograph of 189S 



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 


Chapter 1 



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, 

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 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. 


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 


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. 


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, 


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 


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- 


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, 


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 


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 


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 

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 — 


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. 


Chapter 3 



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. 



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. 


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. 


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 


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. 


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 


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 

■""■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 


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 


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 

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 


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 


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 

Chapter 4 



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 

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. 


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 

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 

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) 


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) 


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 

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), 


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). 

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 

Chapter 5 



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. 


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 

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 

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. 


Chapter 6 




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. 



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 


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 


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: 


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 


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 


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 

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, 


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: 


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 


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. 


Chapter 7 



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. 



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 


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 


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 

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 


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 

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 


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 


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 


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 


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 


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 

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 

Chapter 8 



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. 



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; 


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 


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 


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 


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. 



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. 


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 


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 



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^' 











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^ 


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 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. 



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 


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 


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, 


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. 










































































































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 


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 


■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 

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 


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 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. 


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 ^ 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 

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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by Alexander Maofarlane 


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