450 
not only belongs wholly to mathematics, but which taxes to the 
utmost the resources of the mathematics which we now possess. 
So intimate is the union between mathematics and physics that 
probably by far the larger part of the accessions to our mathe- 
matical knowledge have been obtained by the efforts of mathe- 
maticians 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 
experimentalist or the observer, he has found in the armoury 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 afresh mathematical creation has been needed to 
supply the logicalinstrument requisite to interpret thenew enigma. 
* Perhaps I may be allowed to mention an example of each of 
these two ways in which mathematical and physical discovery 
have acted and re-acted on each other. I purposely choose 
examples which are well known and belong, the one to the 
oldest, the other to the latest times of scientific history. 
The early Greek geometers, considerably before the time of 
Euclid, applied themselves to the study of the various curve 
lines, in which a conical figure may be cut by a plane—curve 
lines to which they gave the name, never since forgotten, 
of conic sections. It is difficult to imagine that any pro- 
blem ever had more completely the character of a ‘* problem 
of mere curiosity,” than this problem of the conic sections must 
have had in those earlier times. Not a single natural pheno- 
menon which in the state of science at that time could have been 
intelligently observed was likely to require for its explanation a 
knowledge of the nature of these curves. Still less can any 
application to the arts have seemed possible ; a nation which did 
not even use the arch were not likely to use the ellipse in any 
work of construction, The difficulties of the inquiry, the 
pleasure of grappling with the unknown, the love of abstract 
truth, can alone have furnished the charm which attracted some 
of the most powerful minds in antiquity to this research. If 
Euclid and Apollonius had been told by any of their contem- 
poraries that they were giving a wholly wrong direction to their 
energies, and that instead of dealing with the problems pre- 
sented to them by nature were applying their minds to in- 
quiries which not only were of no use, but which never could 
come to be of any use, I do not know what answer they could 
have given which might not now be given with equal, or even 
with greater justice, to the similar reproaches which it is 
not uncommon to address to those mathematicians of our 
own day who study quantics of #-indeterminates, curves of 
the th order, and (it may be) spaces of #-dimensions. And 
not only so, but for pretty nearly two thousand years, the expe- 
rience of mankind would have justified the objection : for there 
is no record that during that long period which intervened 
between the first invention of the conic sections and the time of 
Galileo and Kepler, the knowledge of these curves possessed by 
geometers was of the slightest use to natural science. And yet, 
when the fulness of time was come, these seeds of knowledge, 
that had waited so long, bore splendid fruit in the discoveries 
of Kepler. If we may use the great names of Kepler and 
Newton to signify stages in the progress of human discovery, it 
is not too much to say that without the treatises of the Greek 
geometers on the conic sections there could have been no Kepler, 
without Kepler no Newton, and without Newton no science in 
our modern sense of the term, or at least no such conception of 
nature as now lies at the basis of all our science, of nature as 
subject in its smallest as well as in its greatest phenomena, 
to exact quantitative relations, and to definite numerical laws. 
This is an old story; but it has always seemed to me to 
convey a lesson, occasionally needed even in our own time, 
against a species of scientific utilitarianism which urges the 
scientific man to devote himself to the less abstract parts of 
science, as being more likely to bear immediate fruit in the 
augmentation of our knowledge of the world without. I admit, 
however, that the ultimate good fortune of the Greek geo- 
meters can hardly be expected by all the abstract speculations 
which, in the form of mathematical memoirs, crowd the Tran- 
sactions of the learned societies ; and I would venture to add 
that, on the part of the mathematician there is room for the exer- 
cise of good sense, and, I would almost say, of a kind of tact, 
in the selection of those branches of mathematical inquiry which 
(‘NATURE — 
[Sepe. 25, 1873 
are likely to be conducive to the advancement of his own or any 
other science: ; ‘ 
I pass to my second example, of which I may treat very briefly. 
In the course of the present year a treatise on electricity has 
been published by Prof. Maxwell, giving a complete account of 
the mathematical theory of that science, as we owe it to the 
labours of a long series of distinguished men, beginning with 
Coulomb and ‘ending with contemporaries of our own, in- 
cluding Prof. Maxwell himself. No mathematician can turn 
over the pages of these volumes without very speedily con- 
vincing himself that they contain the; first outlines (and 
something more than the first outlines) of a theory which 
has already added largely to the methods and resources of pure 
mathematics, and which may one day render to that abstract 
science services no less than those which it owes to astronomy. 
For electricity now, like astronomy of old, has placed before 
the mathematician an entirely new set of questions, requiring the 
creation of entirely new methods for their solution, while the great 
practical importance of telegraphy has enabled the methods of 
electrical measurement to be rapidly perfected to an extent which 
renders their accuracy comparable to that of astronomical observa- 
tions, and thus makes it possible to bring the most abstract deduc- 
tions of theory at every moment to the test of fact. It must be con- 
sidered fortunate for the mathematicians that such a vast field of 
research in the application of mathematics to physical inquiries 
should be thrown open to them, at the very time when the scien- 
tific interest in the old mathematical astronomy has for the 
moment flagged, and when the very name of pes astronomy, 
so long appropriated to the mathematical development of the 
theory of gravitation, appears likely to be handed over to that 
wonderful series of discoveries which have already taught us so 
much concerning the physical constitution of the heavenly bodies 
themselves. 
Having now stated, from the point of view of a mathematician, 
the reasons which appear to me sto justify the existence of so 
composite an institution as Section A, and the advantages which 
that very compositeness sometimes brings to those who attend 
its meetings, I wish to refer very briefly to certain definite ser- 
vices which this section has rendered and may yet render to 
Science. The improvement and extension of scientific educa- 
tion is to many of us one of the most urgent questions of the 
day ; and the British Association has already exerted itself more 
than once to press the question on the public attention. Perhaps 
the time has arrived when some further efforts of the same kind 
may be desirable, Without a rightly organised scientific edu- 
cation we cannot hope to maintain our supply of scientific men ; 
since the increasing complexity and difficulty of science renders 
it more and more difficult for untaught men, by mere power of 
genius, to force their way to the front. Every improvement, 
therefore, which tends to render scientific knowledge more acces- 
sible to the learner, is a real step towards the advancement of 
science, because it tends to increase the number of well quali- 
fied workers in science. 
For some years past this section has appointed a committee to 
aid in the improvement of geometrical teaching in this country. 
The report of this committee will be laid before the section in 
due course; and without anticipating any discussion that may 
arise on that report, I think I may say that it will show that we 
have advanced at least one step in the direction of an important 
and long-needed reform, The action of this section led to the 
formation of an Association for the improvement of geometrical 
teaching, and the members of that Association have now com- 
pleted the first part of their work. They seem to me, and to 
other judges much more competent than myself, to have been 
guided by a sound judgment in the execution of their difficult 
task, and to have held, not unsuccessfully, a middle course be- 
tween the views of the conservatives who would uphold the 
absolute monarchy of Euclid, or, more properly, of Euclid as 
edited by Simeon, and the radicals who would dethrone him alto- 
gether. One thing at least they have not forgotten, that geome- 
try is nothing if it be not rigorous, and that the whole educa- 
tional value of the study is lost, if strictness of demonstration be 
trifled with, The methods of Euclid are, by almost universal 
consent, unexceptionable in point of rigour. Of this perfect 
rigorousness his doctrine of parallels, and his doctrine of propor- 
tion, are perhaps the most striking examples. That Euclid’s 
treatment of the doctrine of parallels is an example of perfect 
rigorousness, is an assertion which sounds almost paradoxical, 
but which I, nevertheless, believe to be true, Euclid has based 
his theory on an axiom (in the Greek text it is one of the postu- 
