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\Large{ISAAC TODHUNTER}\footnote{This Lecture was delivered April 13,
1904.---\textsc{Editors}} \\
\Large{(1820--1884)}
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\textsc{Isaac Todhunter} was born at Rye, Sussex, 23 Nov., 1820.  He was
the second son of George Todhunter, Congregationalist minister of the
place, and of Mary his wife, whose maiden name was Hunic, a Scottish
surname. The minister died of consumption when Isaac was six years
old, and left his family, consisting of wife and four boys, in narrow
circumstances. The widow, who was a woman of strength, physically and
mentally, moved to the larger town of Hastings in the same county, and
opened a school for girls. After some years Isaac was sent to a boys'
school in the same town kept by Robert Carr, and 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 I839 he passed the matriculation examination of the University of
London, then a merely examining body, winning the exhibition for
mathematics ($\pounds$30 for two  years); in 1842 he passed the B.A.
examination carrying off a mathematical scholarship (of $\pounds$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 Tod\-hunter studied under him. The writings of Sir John
Herschel also had an influence; for Todhunter wrote as follows
(\textit{Conflict of Studies}, p.\ 66): Let me at the outset record my
opinion of mathematics; I cannot do this better than by adopting, the
words of Sir J.~Herschel, to the influence of which I gratefully
attribute the direction of my own early studies. He says of Astronomy,
Admission to its sanctuary can only be gained by one means,---sound and
sufficient knowledge of mathematics, the great instrument of all exact
inquiry, without which no man can ever make such advances in this or
any, other of the higher departments of science as can entitle him to
form an independent opinion on any subject of discussion within their
range.' ''

When Todhunter graduated as M.A.\ he was 24 years of age. Sylvester had
gone to Virginia, but De Morgan remained. The latter advised him to go
through the regular course at Cambridge; his name was now entered at St.
John's College. Being 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 Electricity. In I848 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 I849.

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

For a space of 15 years he remained a fellow of St.\ John's College,
residing in it, and taking part in the instruction. He was very
successful as a lecturer, and it was not long before he began to publish
textbooks on the subjects of his lectures. In 1853 he published a
textbook on \textit{Analytical Statics}; in 1855 one on \textit{Plain
Coordinate Geometry}; and in 1858 \textit{Examples of Analytical
Geometry of Three Dimensions}. His success in these subjects induced him
to prepare manuals on elementary mathematics; his \textit{Algebra}
appeared in 1858, his \textit{Trigonometry} in 1859, his \textit{Theory
of Equations} in 1861, and his \textit{Euclid} in 1862. Some of his
textbooks passed through many editions and have been widely used in
Great Britain and North America. Latterly he was appointed principal
mathematical lecturer in his college, and he chose to drill the freshmen
in Euclid and other 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 1861; in 1862 he was elected a Fellow of the Royal Society
of London. In 1863 he was a candidate for the Sadlerian professorship of
Mathematics, to which Cayley was appointed. Todhunter was not a mere
mathematical specialist. He was an excellent linguist; besides being a
sound Latin and Greek scholar, he was familiar with French, German,
Spanish, Italian and also Russian, Hebrew and Sanskrit. He was likewise
well versed in philosophy, and for the two years 1863--5 acted as an
Examiner for the Moral Science Tripos, of which the chief founders were
himself and Whewell.

By 1864 the financial success of his books was such that he was able to
marry, a step which involved the resigning of his fellowship. His wife
was a daughter of Captain George Davies of the Royal Navy, afterwards
Admiral Davies. As a fellow and tutor of St.\ John's College he had
lived a secluded life. His relatives and friends thought he was a very
confirmed bachelor. He had sometimes hinted that the grapes were sour.
For art he had little eye; for music no ear.  He used to say he knew two
tunes; one was God save the Queen,' the other wasn't. The former he
recognized by the people standing up.'' As owls shun the broad daylight
he had shunned the glare of parlors. It was therefore a surprise to his
friends and relatives when they were invited to his marriage in 1864.
Prof.\ Mayor records that Todhunter wrote to his fianc\'ee, 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 \textit{Quaternions}.

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

About 1847 the members of St. John's College founded a prize in honor of
their distinguished fellow, S.~C.~Adams. It is awarded every two years,
and is in value about $\pounds$225. In 1869 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 \textit{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
\textit{History of the Calculus of Variations}.

In 1873 he published his \textit{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 \textit{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 Conflict of Studies---Todhunter gave his opinion of the educative
value in high schools and colleges of the different kinds of study then
commonly advocated in opposition to or in addition to the old subjects
of classics and mathematics. He considered that the Experimental
Sciences were little suitable, and that for a very English reason,
because they could not be examined on adequately. He says:

Experimental Science viewed in connection with 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 D\^ome
with a barometer in order to test the influence of change of level on
the height of the column of mercury, he performed an experiment, the
suggestion of which was worthy of the genius of Pascal and Descartes.
But when a modern traveller ascends Mont Blanc, and directs one of his
guides to carry a barometer, he cannot be said to perform an experiment
in any very exact or very meritorious sense of the word. It is a
repetition of an observation made thousands of times before, and we can
never recover any of the interest which belonged to the first trial,
unless indeed, without having ever heard of it, we succeeded in
reconstructing the process of ourselves.  In fact, almost always he
who first plucks an experimental flower thus appropriates and
destroys its fragrance and its beauty.''

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

Todhunter went on to write another passage which roused the ire of
Professor Tait.  To take another example. We assert that if the
resistance of the air be withdrawn a 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 clear
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 manuals 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 ignomimous reverses,---they have probably longed
to imitate the cautious student who watched an eminent astronomer
baffled by Foucault's experiment for proving the rotation of the Earth;
as the pendulum would move the wrong way the student retired, saying
that he wished to retain his faith in the elements of astronomy.''

It is not unlikely that the series of ignominious reverses Todhunter
had in his view were what he had seen in the physics classroom of
University College when the manipulation was in the hands of a pure
mathematician---Prof.\ Sylvester. At the University of Texas there is a
fine clear space about 60 feet high inside the building very suitable
for Foucault's 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

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-ridden country. Todhunter referred to one of the most
distinguished professors of Mathematics in England; one whose pupils had
likewise gained a high reputation as investigators and teachers; his
venerated master and friend,'' Prof.\ De Morgan; and pointed out that
he recommended the study of Euclid with all the authority of his great
attainments and experience.

Another argument used by Todhunter was as follows: In America there are
the conditions which the Association desires; there is, for example, a
textbook which defines parallel lines as those which have the same
direction. Could the American 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 i869, Sylvester had said: I should rejoice to see \dots
Euclid honourably shelved or buried deeper than did ever plummet sound'
out of the schoolboy's reach; morphology introduced into the elements of
algebra; projection, correlation, and motion accepted as aids to
geometry; the mind of the student quickened and elevated and his faith
awakened by early initiation into the ruling ideas of polarity,
continuity, infinity, and familiarization with the doctrine of the
imaginary and inconceivable.'' Todhunter replied: 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 naturally suggested by
the passage. is that nothing prevents an enthusiastic teacher from
carrying his pupils to any height he pleases in geometry, even if he
starts with the use of Euclid.''

Todhunter also replied to the adverse opinion, delivered by some
professor (doubtless Tait) in an address at Edinburgh which was as
follows: From the majority of the papers in our few mathematical
journals, one would almost be led to fancy that British mathematicians
have too much pride to use a simple method, while an unnecessarily
complex one can be had. No more telling example of this could be wished
for than the insane delusion under which they permit Euclid' to be
employed in our elementary teaching. They seem voluntarily to weight
alike themselves and their pupils for the race.'' To which Todhunter
replied: The British mathematical journals with the titles of which I
am acquainted are the Quarterly Journal of Mathematics, the Mathematical
Messenger, and the 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
master these periodicals regularly; but that a single mathematician
should be able to improve more than half the matter which is thus
presented to him fills me with amazement.  I take down some of these
volumes, and turning over the pages I find article after article by
Profs.\ Cayley, Salmon and Sylvester, not to mention many other highly
distinguished names.  The idea of amending the elaborate essays of these
eminent mathematicians seems to me something like the audacity recorded
in poetry with which a superhuman hero climbs to the summit of the
Indian Olympus and overturns the thrones of Vishnu, Brahma and Siva.
While we may regret that such ability should be exerted on the
revolutionary side of the question, here is at least one mournful
satisfaction: the weapon with which Euclid is assailed was forged by
Euclid himself. The justly celebrated professor, from whose address the
quotation is taken, was himself trained by those exercises which he now
considers worthless; twenty years ago his solutions of mathematical
problems were rich with the fragrance of the Greek geometry.  I venture
to predict that we shall have to wait some time before a pupil will
issue from the reformed school, who singlehanded will be able to
challenge more than half the mathematicians of England.'' Professor Tait,
in what he said, had, doubtless, reference to the avoidance of the use
of the Quaternion method by his contemporaries in mathematics.

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

In England education is for the most part directed to training pupils
for examination. One direct consequence is that the memory is cultivated
at the expense of the understanding; knowledge instead of being
assimilated is crammed for the time being, and lost as soon as the
examination is over. Instead of a rational study of the principles of
mathematics, attention is directed to problem-making,---to solving
ten-minute 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 \textit{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 September, 1883, when he was at Hunstanton; that the
worst symptoms came on. He then partially lost by paralysis the use of
the right arm; and, though he afterwards recovered from this, he was
left much weaker. In January of the next year he had another attack, and
he died on March 1, 1884, in the 64th year of his  age.

Todhunter left a \textit{History of Elasticity} nearly finished. The
manuscript was submitted to Cayley for report; it was in 1886 published
under the editorship of Karl Pearson.  I believe that he had other
histories in contemplation; I had the honor of  meeting him once, and in
the course of conversation on 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 forgot to
see to their wants.''
\begin{flushleft}
From A~Macfarlane, \textit{Lectures on Ten British Mathematicians of the
Nineteenth Century}, New York: Wiley and London: Chapman and Hall 1916,
pp.\,134--146.
\end{flushleft}

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