Reprinted from Report of the Australasian Association for the Advancement
ol Science, Melbourne Meeting, 1913. Vol. XIV.
[ISSUED DECEMBER, 1913.]
SECTION A.
T-E RvELAT.OV           BETWEEN         PlURE AND
APPFL-LIEA)   3 A.ENEfA_ t.   6 S
BY
PROFESSOR H. S. CARSLAW, Sc.D.
ATBiETRT J, MTi-TLLEi, T OOVERNMETNT PRINTER, IMEaBOURNE.
1913.
15512,








Reprinted from Report of the Australasian Association for the Advancement of Science. Melbourne Meeting, 1913  Vol. XIV.
Section A.
ASTRONOMY, MATHEMATICS,
AND PHYSICS.
ADDRESS BY THE PRESIDENT:
PROFESSOR H. S. CARSLAW, Sc.D.
THE RELATION BETWEEN PURE AND APPLIED
MATHEMATICS.
I propose to-day to speak of the intimate relation between Pure and
Applied Mathematics at the present time, and to refer to some common
but mistaken views on the nature of the Science of Mathematics as a
whole. But before I pass to the subject of my address, I pause to
express in a few words our sense of the loss the world has suffered
within the last few months by the death of Sir George Darwin, the
great mathematical astronomer, and of M. Henri Poincaré, the
greatest mathematician of our time.
Sir George Darwin, by his contributions to science, has worthily
maintained the traditions of his name, and his tenure of the Plumian
Professorship of Astronomy at Cambridge recalls the days when the
other astronomical chair at that University was filled by Adams, who
shares with Leverrier the honour of the discovery of the planet Neptune.
There is probably no part of Applied Mathematics in which the refinements of Analysis can be employed with greater effect than in the
domain of Mathematical Astronomy. Laplace and Lagrange were
consummate mathematicians. To almost every branch of Mathematics
known in their time they made important contributions. And to come
to our own day, Newcomb was a mathematician of the first rank.
Darwin belonged to the same band. Iis researches on the past history
of the earth-moon system and on the practical and theoretical tidal
problems which the oceans present, could only have been carried to a
successful issue by one with a wide knowledge of Pure Mathematics.
Here in this room it is but fitting that we should recall the services
he so fully and gladly rendered to the British Association for the
Advancement of Science, the parent of this our Australasian Association. President of Section A in 1886, he attained the honour of the
presidency of the Association itself in 1905. And we had looked forward
to his taking an active part in the meetings of the Association next
year in Australia.
15512.




7              PRESIDENT S ADDRESS-SECTION A.
One of the last-if not the very last-of the public utterances of
Darwin was a tribute to the unique position occupied in the scientific
world by Poincaré, the other great man whose loss to-day we mourn.
In August of last year Darwin was President of the International
Congress of Mathematicians at Cambridge. At its opening meeting his
earliest words were devoted to the expression of their heartfelt sorrow
at the sudden death of Poincaré a few weeks before in Paris. He
described him as the man who alone among mathematicians could
have occupied the position of President without misgivings as to his
fitness. It brought vividly home to him how great a man Poincaré
was when he reflected that to one incompetent of appreciating fully
one half of his work, he yet appeared a star of the first magnitude.
By universal consent Poincaré was regarded as the greatest
mathematician of his time. Philosophers, mathematicians, and
astronomers looked to him as the leading authority in each of their
domains. Gauss, the famous geometer, the master of analysis, the
great astronomer, had won for himself the title Princeps Mathematicorum. Since his day the title had been vacant. With the coming of
Poincaré his successor appeared. Now, when Poincaré seemed in the
very fulness of his powers, his throne is vacant, and it is impossible to
measure the loss the world has suffered.
There are before me doubtless some who imagine that at the end
of their three years' course in Mathematics our students should have
been brought at any rate within sight of the confines of the subject.
A glance at Poincaré's works would be sufficient to remove such a
thought from their minds.
In Mathematics, as in every other science, one height is reached
only that from it we may press on to regions until then unknown. And
the man is best equipped for this exploratior who has travelled widely
in the regions already discovered. We meet Poincaré first of all as a
Pure Mathematician. To the whole modern Theory of Functions he
made contributions of the highest importance. An instance of the
way in which, even in his younger days, he was able to draw upon one
branch of Mathematics to help him in another is to be found in the
quaint description he has given of the manner in which he was led to
one of his earliest discoveries. At that time he was about 27 or 28 years
of age, had graduated from the School of Mines, and had just received
his Doctorate in Mathematics from the University of Paris. For we
must not forget that Poincaré was one of those mathematicians who had
a complete training in physical science, as well as the course in Pure
Mathematics for which the French School is famous.
He tells us that for about a fortnight he had been trying to
demonstrate the existence of functions analogous to those which he
afterwards discovered and called Fuchsian Functions. Every day he
would sit at his table for an hour or so; attempt a number of




PRESIDENT S ADDRESS-SECTION A.                8
combinations, and reach no result. One evening, contrary to his custom,
he had taken a cup of black coffee, and could not sleep. The ideas
seemed to crowd and jostle one another in his head, and in the
morning he was able in a few hours to establish the existence of a class
of Fuchsian Functions, those which are derived from the Hypergeometric Series.
The next step was to represent these functions as the quotient
of two series. To this the analogy with the elliptic functions guided
him. He investigated what should be the properties of these series,
and established without difficulty the existence of those he called
Theta-Fuchsian Series.
At this stage he had to leave for the country on some official work.
On the steps of an omnibus, the idea flashed into his mind that the
transformations of which he had made use in defining the Fuchsian
Functions were identical with those of the Non-Euclidean Geometry.
He did not verify the conjecture, but resumed the interrupted conversation with his companion. Still, he felt perfectly certain that his
idea was correct; on his return to his home he looked into the
matter, and found that what he had surmised was true.
Then he took up the study of some arithmetical questions without
much success, far from suspecting that they would have anything to
do with his previous investigations. Annoyed at his comparative
failure with his new task, he went to the seaside to spend a few days
and turn his thoughts to other things. One day walking on the cliff,
the thought came to him, suddenly and surely as before, that the
arithmetical transformations of certain quadratic forms were identical
with those of the Non-Euclidean Geometry.
On his return he reflected upon this result and the consequences
to be derived from it. The example of the quadratic forms showed
him that there were Fuchsian groups other than those which correspond
to the Hyper-geometric Series. He saw that he could apply to them
the theory of the Theta-Fuchsian Series, and that consequently there
must exist Fuchsian Functions other than those which were derived
from that series. He set himself to form such functions. He made
a systematic attack upon the position and carried all the outworks
save one. This he could not reduce, however hard his efforts.
Again his work was interrupted; this time that he might put in
his military service. One day, passing down the street, all of a sudden
the solution of the difficulty appeared to him. At the time he made
no attempt to go into the point in detail, but let it stand over till the
end of his service. When the opportunity came, all his material was
there.  He had only to arrange it and the complete memoir was
written, practically, at a sitting.
All this story, Poincaré told in a fascinating lecture entitledL'Invention Mathématique; his view of the matter being that after
A 2




9             PRESIDENT'S ADDRESS-SECTION A.
the preliminary work has been done and the earliest attempts had
their turn and proved unsuccessful, it is best to let the question rest
in the mind and develop itself.  Thus it would appear that one often
works hardest when one is doing nothing.
The field of work to which his attention was first directed always
remained dear to him, but it is probably in the applications of analysis
to the solution of the differential equations of mathematical physics
that he found one of his fàvourite themes. To this point I shall return
later.
I now look at Poincaré, the great astronomer. And first we
notice that neither as a physicist nor as an astronomer did his work
lie in the laboratory or the observatory.  The service which he
rendered these lay in the application of the methods of Analysis and
Geometry to the problems of Physics and Astronomy. His investigation of the form taken by a gravitating fluid mass in rotation led him
to most important theories on the separation of the earth and the
moon.   Of this work Darwin, who had himself made important
discoveries in the same field, records that the memoir wili always mark
an important epoch, not only in the history of Astronomy, but also in
that of the larger domain of General Dynamics.
In his work on the stability of the solar system Poincaré returned
to the problem treated by Laplace, and applied to it the new mathematical instruments now available. His labours in this connexion
are to be found in his book, Les Méthodes Nouvelles de la Mécanique
Céleste. What Newton's Principia did for Astronomy in the 17th
cetttury, this work of Poincaré's has done for the 20th century. We
are assured that all the advances likely to be made in the next 50 years
in Astronomy are certain to rest upon the foundation Poincaié has
laid.
)f the mathematician, the physicist, and the astronomer, we
have spoken. There still remains the philosopher. But to enter
upon a discussion of that part of his work which lies in the borderground between Philosophy and Mathematics is a task for which I
have neither the time nor the qualifications.  The ordinary mathematician meets this section of Poincaré's work when he considers the
Principles of Mathematics in general; and in particular the Principles
of Geometry. No student of Geometry can afford to neglect Poincaré's
contribution to this modern development of Mathematics. It is a
pity that some other mathematical philosophers have not approached
the subject with the clear open vision of Poincaré, and that they have
not placed their views before us in lucid language such as he delighted
to employ.
I now turn to the subject which I have chosen for my address,
The Relation between Pure and Applied Mathematics.




PRESIDENT'S ADDRESS-SECTION A.               10
If the average man were asked what a mathematician is, he might
answer that he is a being possessed of a strange aptitude for, and a
curious delight in, numerical calculation. Some there might be who
would echo the old sayingPurus mathematicus, purus asinus:
but most people would agree that the mathematician is a lucky sort
of fellow with a good head for figures.
It cannot be repeated too often that this idea of the mathematical
mind is quite wrong. The mathematician is not merely a glorified
chess player, who can carry the moves of a prolonged calculation in
his head and be relied upon in the end to return a correct numerical
answer. Certainly there have been mathematicians possessing this
faculty. Gauss had it, but Gauss was the exception, not the rule.
We read that Newton, " though so deep in Algebra and Fluxions
could not readily make up a common account; and when he was
Master of the Mint used to get somebody to make up his accounts for
him."
Poisso? once remarked to Madame Biot that he could nct add as
well as his cook; neither did he understand how Gauss and Bessel
could be at the same time expert calculators and skilled analysts.
Poincaré was not ashamed to say that he was absolutely incapable
of doing an addition sum correctly, and that he was an equally bad
chess player. He could calculate well enough that in making a certain
move he would get into trouble. He would pass in review other
possible moves and give them up for the same reason. Then in the
end he would probably play the move which he had first put aside,
having forgotten the danger which he had then foreseen.
Such instances could be multiplied indefinitely; and we see
that it is not necessarily a good head for figures and a prodigious
memory that make the mathematician. Mathematics and Arithmetic
are not identical. If they were, Mathematics would, in the opinion
of some of us, be a dry and arid science.
A mathematical demonstration is not simply a collection of
syllogisms. It is a series of syllogisms in a certain order, and the order
in which they come is almost as important as their content. The
mathematical mind seems to have an intuitive perception of this
order; it takes in at a glance the whole of the reasoning, and has no
fear of forgetting the elements. These appear to fall into their places
without any special effort of memory. With this mathematical sense
or taste, there is associated the idea of mathematical beauty and
elegance. Only the mathematician appreciates it; probably he alone
would admit its existence; but for it we claim a reality just as actual
as the beauty of the picture, the statue, or the poem.
If this statement of the nature of the mathematical mind be
correct, it is not surprising that the mathematical faculty frequent.




11             PRESIDENT S ADDRESS-SECTION A.
declares itself for the first time, when the youthful mathematician
enters upon the study of Geometry. To Newton the Elements of
Euclid appeared so clear and simple that it was a waste of time to go
through them. A glance at the enunciation of the theorem, and to
him the demonstration was obvious. He passed straight on to such
books as Decartes' Geometry and Kepler's Optics.
Similar stories, if I remember aright, are told of Euler and Lagrange.
Again, Clairaut, at the age of thirteen, had written a paper on the
properties of some new curves, which was presented to the Académie
des Sciences and printed at the end of one of his father's works.
Clerk-Maxwell, it is hardly necessary to remind the members of
this section, published his first mathematical paper at the age of fourteen. For it was at that age that he wrote the paper " On the
Description of some Oval Curves and those having a Plurality of Foci,"
read at the Royal Society of Edinburgh, and published in their
Transactions for 1846.
And, to give one other instance, M. Frederic Masson, in the charming
speech which he delivered on the occasion of Poincaré's admission to
the Académie Française, tells us that his career was settled, when in
the Lycée de Nancy, in the Fourth Class, he opened a book on Geometry.
His astonished master, who had hoped to make of him a student of
letters, hastened to his mother, greeting her with the words" Madame, votre fils sera mathématicien."
And we read that she was not dismayed.
May I be permitted to say, in passing, that the teachers of Mathematics in our schools at the present day must be careful if the study
of Geometry is to retain its value. Without entering into the vexed
question of the extent to which the intuitive method ought to take
the place of the deductive, I would only say that the budding mathematician must sometimes be troubled by the slipshod argument which
he finds in the text-book placed in his hand. Assuming this story of
the youthful Poincaré is true, it is fair to add that it is most unlikely
that the book which roused his ardour was Euclid's Elements. More
probably it was Legendre's Géométrie. But Legendre's book stood the
test of over a century's use on the continent of Europe, and Legendre
was a famous mathematician. Our present trouble is that people are
still to be found teaching mathematics in the schools without a proper
training for their task. That difficulty we rejoice is passing away
with the institution of well-equipped Teachers' Colleges in most of our
States. Another cause of the trouble to which I have referred is to
be traced to the text-book itself. The authors of these texts are men
of a different stamp from Legendre. Now that Heath's great edition
of Euclid's Elements has appeared, and that the story of the rise and
development of the Non-Euclidean Geometries is more widely known,
a more satisfactory state of affairs may arise.




PRESIDENT'S ADDRESS-SECTION A.                  12
The content of the Science of Mathematics has grown so enormously that there are few, even among professed mathematicians,
who can lay claim to a knowledge of more than a part. The physicist,
the engineer, and other practical men are inclined to believe that with
this development the mathematician   is losing sight of what they
believe is the chief reason for his existence: namely, to provide useful
tools which they may employ in the physical sciences.
When one speaks of the growth of Mathematics, it is hardly
necessary to point out that we do not refer to the undergraduate course
at our Universities. Changes in it there have been, and should continue
to be. Doubtless those chiefly concerned are inclined to think that it
has developed past recognition. But the alterations are mostly in
matters of detail or method. In its chief characteristics the course
remains the same. It must range over Geometry in its wider sense,
Analysis, and Applied Mathematics. Its aim is twofold. On the one
hand it seeks to provide a suitable introduction, for the student with a
mathematical mind, into the Science of Mathematics. At its close
he is ready to devote himself to higher study in one or other of the
three main divisions of which I have spoken. The other object before
us is just as definite. Our courses, in greater or less degree, have to
serve as a portion of the training of the physicist, the engineer, the
statistician, or other professional man, of whose equipment the tools
which Mathematics provides form a valuable and necessary part.
However, as scientific men, we must protest against the view that
the path of practical utility is to be that along which mathematical
development is to take place. And the protest is called for in this
country at the present time. To me it seems a matter for great regret
that in several of the younger Universities room has not been found
for a separate Chair of Mathematics, the subject being combined with
Physics, and the professorship being called a Professorship of Mathematics and Physics. Of course, it is well understood    that this
arrangement is simply a temporary one, and rendered necessary by
the funds available not being sufficient for the endowment of separate
chairs. Still, Mathematics is not simply a handmaid to Physics; each
science must stand by itself; and the dignity, both of the University
and of these two branches of knowledge, demands that these temporary
expedients should not be allowed to remain in force any longer than
is absolutely necessary.
But though this protest is necessary in this country and at the
present time, the need for it is a recurrent one, and we find such
remonstrances frequently made in the development and growth of
Mathematics as a Science. And they have been called for at times
even in the house of her friends.
In Jacobi's letters we come upon the following sentences:-" I
have read with pleasure Poisson's report upon my work (the Funda



~13  ~     PRESIDENT'S ADDRESS-SECTION A.
mnenta Nova... ), and I can be well satisfied with what he says.
But M. Poisson should not have repeated in his report a foolish phrase,of the late M. Fourier, where he reproaches us (Abel and me) for not
having turned our attention to the flow of heat. It is true that M.
Fourier believed that the principal aim of Mathematics was practical
utility, and the explanation it could give of natural phenomena. But
a philosopher of his standing should have known that the sole end of
science is the honour of the human intelligence. And, from this point
of view, a problem in the Theory of Numbers is as important as a
question arising in Celestial Mechanics."
Some of the greatest triumphs of Mathematics have no doubt
been won in the conquest of nature and the elucidation of her laws.
In the discoveries which marked the nineteenth century, and changed
the face of the civilized world, the mathematicians were often found
among the pioneers. By many people it is from this stand-point that
Mathematics is regarded. She is the Servant of the Sciences. A
place of honour may be hers; but it is for service rendered and with
the lively expectation of greater benefits in the future.
" I am not making before you a defence of Mathematics," said
Cayley, in his presidential address to the British Association in 1883,
"but, if Iwere, Ishould desire to do it in such manner as in the Republic
Socrates was required to defend justice-quite irrespective of the
worldly advantages which may accompany a life of virtue and justice,
and to show that, independently of all these, justice was a thing
desirable in itself, and for its own sake; not by speaking to you of the
utility of Mathematics in any of the questions of common life or of
physical science... I would, on the contrary, rather consider
the obligations of Mathematics to these different subjects, as the
sources of mathematical theories now as remote from them, and in
as different regions of thought-for, instance, Geometry from  the
measurement of land, or the Theory of Numbers from Arithmetic-as
a river at its mouth is from its mountain source."
And, again, to quote another great Pure Mathematician, Weierstrass: " 1 am not afraid that I shall be blamed for detracting from
the value to which Mathematics as a pure science lays claim, with such
perfect right, when I attach special importance to the fact that it is
only through Mathematics that a true and satisfactory understanding
of natural phenomena can be obtained. Indeed, no one can be readier
than I to admit that we must not seek for the end of a science outside
itself. Such action not only does not add to its dignity; it is an offence
against it. Instead of devoting ourselves to it with our whole heart, we
desire from it only some service, and use it only for some other discipline,
or for the needs of ordinary life..       In this way we would
neglect every path which did not seem immediately to promise results
of practical value. It is my opinion that we must obtain a truer




PRESIDENT'S ADDRESS-SECTION A.                14
appreciation of the relation between Mathematics and Natural Science.
The physicist must no longer look on Mathematics as an auxiliary
discipline, even if he admit that it is an indispensable one. The
mathematician must not continue to regard the questions which the
physicist brings to him simply as a rich collection of problems suited
to his work."
The truth is Mathematics must be treated as any other science.
It does not stand in a class by itself. There ought to be no department of knowledge in which the man of science should feel that he
has the right to ask the author of any discovery-Cui bono? The
only question for him should be whether it is true, and what influence
it will have in the development of the subject of which it forms a part.
The earliest astronomers may have looked upon the stars with their
thoughts upon navigation; but some of them   doubtless pondered
in their hearts the mystery of the Universe. Every botanist does not
live by agriculture; nor is every geologist on the search for precious
stones. It was only in the dark ages that chemistry was confused with
alchemy. The quest for knowledge, in itself and for itself, is the
common heritage of every science. And " the history of natural
philosophy, and even of such a practical science as Medicine, show us
that even from the point of view of utility the subjects must be developed
of themselves, with the single aim of increasing knowledge."
No one could have foretold, when Galvani touched the nerve and
muscle of the frog with two different metals and saw the muscle
contract, that the discovery of the anatomist would lead in 80 years
to the world being traversed by electric cables from end to end. And
it was far from the minds of those who first watched the stream of
sparks bridging the gap of an electric machine or flowing from the
knob of a Leyden Jar, that the phenomena they were watching in a
few years would lead to the marvellous triumphs of Wireless Telegraphy.
To the mathematician the wonderful edifice which the geometer
has created, from the simple practical geometry of the Egyptian and
the theoretical geometry of the Greek, to the great domain of Projective
and Descriptive Geometry, and the realm of Differential Geometry
of Curves and Surfaces, is as much a matter of pride and satisfaction
as any of the theories which have been invented to explain and simplify
the facts of experiment and the wonders of nature.
We are agreed, then, that no branch of Mathematics has a claim
prior to any other. The mathematician turns his attention to the
department which appeals to himself, and in which he feels he can
do the best service. I wish now to give some reasons for my belief
that it is unfortunate that some branches of Applied Mathematics
are not at present attracting English mathematicians as they used to do.
With regard to these, I believe that recent discoveries in Pure Mathematics make it extremely probable that renewed efforts by the Applied
Mathematician would meet with considerable success.




15             PRESIDENT'S ADDRESS —SECTION A.
"The reconstruction in 1909 of the Mathematical Tripos, and the
destruction of many of the distinctive features of the former scheme
must profoundly modify the future history of Mathematics at Cambridge.... The changes in the Tripos regulations have been
accompanied by a curious alteration in the popular subjects, and to-day
but few of the young graduates who desired the change are interesting
themselves in those branches of Applied Mathematics once generally
studied, but rather are turning their attention to subjects like the
theories of functions or groups."
To me the tendency to which Ball calls attention in these words is a
matter for surprise and regret. The English School of Mathematics
certainly had inclined to an excessive degree to the Applied Studies.
It was natural that there should be a reaction; and that the branches
of Pure Mathematics which had been cultivated with such success in
the schools of Paris, Berlin, and Gottingen would find their adherents
in greater numbers on the banks of the Cam. But the traditions of
Cambridge Mathematics are worthy of being maintained, and progress
in Applied Mathematics should not now be left to such an extent to
the Continental Mathematician.
Open the Cambridge Calendar and glance down the Mathematical
Tripos lists from 1837 to 1887. There were very few English mathematicians who flourished in that time whose names we do not find
among the first few wranglers, and in far the greatest number they are
the names of men who are known wherever Mathematics is cultivated
as those who made notable advances in Applied Mathematics.
In 1837 we have Green, the discoverer of Green's Theorem, who
came up to Caius College at the age of 40, with his greatest discoveries
already made. And now in 1913 his name meets us more frequently
than ever in mathematical journals, for Green's Functions have come
to life again in the new branch of Mathematics called Integral
Equations. And in the same year as Green, who was fourth wrangler,
we find Sylvester, as second. Ris name is as remarkable in the history
of Pure Mathematics as Green's is in Applied. We read that " Green
and Sylvester were the first men of the year, but Green's want of
familiarity with  ordinary boy's Mathematics prevented him  from
coming to the top in a time race."
Passing on, we find in rapid succession Stokes, Adams, Thomson
(Lord Kelvin), Tait, Routh, Clerk-Maxwell, Strutt (Lord Rayleigh),
Niven, Darwin, Greenhill, Lamb, Eicks, Poynting, Glazebrook,
Larmor, J. J. Thomson, Turner, Bragg, Love, Bryan, and Michell of
your own University.
And to turn to the Pure Mathematicians of that period, we pass
from Sylvester, in 1837, to Cayley, in 1842; then there is a gap till we
reach Clifford, in 1867; after him  come Glaisher, Burnside, Chrystal,
ilobson, Forsyth, Heath, Mathews, Whitehead, Young, Berry, Richmond,
and Dixon.




PRESIDENT S ADDRESS-SECTION A.                 16
As has been already mentioned, in recent years the ablest men
among the younger Cambridge graduates have turned, oftener than
before, to Pure Mathematics. For this there may be various reasons,
but the decline in Applied Mfathematics at Cambridge is, 1 believe, due
in part to the divorce between   Mathematics and    Experimental
Physics at that place.  '
"Although the mathematician," says Berry, "has given about
naif of his time to Applied Mathematics, he need have, and in fact
frequently has had, no knowledge of Experimental Physics. Normally,
he goes to no experimental lectures, he does no work in a laboratory,
and the experimental facts which he learns in his mathematical textbooks are usually of the simplest character, reduced to an abstract
and almost conventional form, suitable for the direct application of
mathematical analysis. A high wrangler may     be able to   solve
elaborate problems in spherical trigonometry or optics without having
seen a telescope or handled a lens; he may be able to calculate the
potential due to the most curious distributions of electricity, without
the east idea of the mechanism of an electric bell or tram. Physics
learnt in this way is naturally most unreal, and the mathematician
who wishes afterwards to devote himself to Physics is at first at a great
disadvantage, not only by want of familiarity with physical apparatus
and physical data, but by a lack of 'physical instinct,' which enables
the trained physicist to judge what elements are important, or what
unimportant, in any particular investigation."
In the earlier days of Experimental Physics this state of affairs did
not exist, at any rate, to the saine extent. The subject was less
specialized, and it was comparatively easy for any mathematician
who so desired to obtain in a short time a practical acquaintance with
the experimental side of the subjects which interested him.  Nowadays
this is not the case. But with the development of Experimental
Physics and the change in the content of what used to be called
Natural Philosophy, it is just as imperative that Experimental Physics
should enter into the training of the mathematician, as that Mathematics should be part of the course followed by the physicist.
In Australia, 1 think, we have passed from the Cambridge tradition
in this respect, and our Honours graduates in Mathematics will very
seldom be found to have completed their course without one year'sand in many cases two years'-practical and theoretical study of
Physics. In this we follow the example of Germany and France.
The German Ph.D. who takes Mathematics as his chief subject, will
usually combine with it Physics and Chemistry as his minor subjects.
The French mathematician has, I believe, a similar wide curriculum.
Poincaré certainly had no leaning towards experimental work, and
some pure mathematicians have been known to regret that the official
work which fell to him was Mathematical Physics and Astronomy;
for they felt that in him the ideal pure mathematical mind had its




17              PRESIDENT'S ADDRESS-SECTION A.
habitation; but it was his training in the School of Mines that
enabled Poincaré to make the important contributions to Mathematical
Physics which will always be associated with his name. Sommerfeld,
whose name we meet so often in the discussions upon Wireless Telegraphy, has this double qualification. A profound mathematician,
he has also the "physical instinct" of which Berry spoke. As to
Hilbert, I cannot be certain; but my impression is that he also has
had the advantage of a training in Physics. The wide range of his
work must be a perpetual source of wonder to other mathematicians.
Like Poincaré, we find him deep in the discussion of the foundations
of Geometry, and shedding fresh light upon the real meaning of the
Non-Euclidean Geometries. He has also contributed to various
departments of Iigher Algebra and Analysis. And he was the first
to grasp the true significance of Fredlolm's discovery of the solution
of the Integral Equations which   now   bear his name. Hilbert
established the connexion between the Theory of Integral Equations
and that of Quadratic forms. Then he applied his discovery to the
differential equations of Mathematical Physics, and united in one
general theorem all the different questions of the expansion of an
arbitrary function in series, whether the terms be trigonometrical
functions, Bessel's Functions, Spherical  Harmonics,  Ellipsoidal
Harmonics, or Sturm's Functions. And the last two chapters of the
book, in which he has brought togetner his contributions, to this new
branch of analysis, are devoted-one to its applications to the Theory
of Functions, and the other to its applications to the Calculus of
Variations, Geometry, Hydrodynamics, and the Kinetic Theory of
Gases. He now seems to have turned to Mathematical Physics, and
he las recently lectured on the Kinetic Theory of Gases. One of his
courses during last summer Semester was upon the Mathematical
Foundations of Physics.  And during this winter Semester he has
lectured upon the Theory of Partial Differential Equations, and
continued his former course on the Foundations of Physics, his
Seminar being given up to discussions on similar topics.
Another German mathematician, Kneser, in his book " On the
Application of Integral Equations to Mathematical Physics," and in
his published writings upon the same subject, seems to have stepped
right into the region which we might have expected the Cambridge
School to have already occupied, if it had not broken with the traditions
of its past. And the same may be said of part of the work of Stekloff,
the Russian mathematician.
There are, of course, many modern developments of Pure Mathematics with a close bearing upon the problems which meet us in
Applied. In some of my own investigations I have had occasion to
use the Theory of Functions of a Complex Variable, and Riemann's
Surfaces and Space. A year or so ago some work on Non-Euclidean
Geometry compelled me to put aside these questions, and it is only




PRESIDENT'S ADDRESS-SECTION A.                18
recently that I have been able to return to them. And I find that
the Theory of Integral Equations removes several difficulties which
formerly had made my progress difficùlt.
This is but a slight example of the close connexion between the
different branches of Pure and Applied Mathematics. And it is often
in the most unexpected quarters that this relation is revealed. We
saw, in speaking of Poincaré's earlier work, that his acquaintance with
Non-Euclidean Geometry gave him the key to a difficult question in
the Theory of Functions. Curiously enough, in the Theory of Relativity we come upon a similar instance.  Sommerfeld had shown that
the Composition of Velocities in the Theory of Relativity agrees with
the formulae of Spherical Trigonometry when the radius of the sphere
is imaginary.  Now Lobatschewsky and Bolyai long ago established
the connexion between their Non-Euclidean Geometry and this Imaginary Trigonometry. It followed that an interesting field for the
application of the Theory of Relativity was to be found in Non-Euclidean.Geometry. Varicak has proved that the formulae of that theory have a
ready interpretation in that Geometry. His next step was to assume
that the phenomena happen in a Non-Euclidean    Space, and he
obtained the formulae of the Theory of Relativity by very simple
geometrical argument. lie states his results as follows:-" Assuming
the Non-Euclidean terminology, the formulae of the Theory of Relativity
not only become essentially simplified, but they also admit a geometrical
interpretation, which is wholly analogous to the interpretation of the
classical theory in the Euclidean Geometry.  And this analogy often
goes so far as to leave the actual wording of the classical theory
unchanged. We need only replace the Euclidean image by the corresponding image in the Lobatschewsky space, whose parameter c is
equal to 3 x 1010 cm."
By Authority: ALBERT J. MUJLLETT, Government Printer, Melbourne.