320 
classificatory, are even now struggling to assimilate a 
mathematical method. But if it is just to claim that 
other. sciences, nowadays even the biological, aspire 
with increasing success to become mathematical—that 
is, exact—in structure, there is, on. the other hand, a 
duty enjoined on mathematicians to see to it that the 
main stream of their discipline is kept accessible—free 
from specialities and complexities, which, valuable 
and promising as they may be, and usually are,, on 
their own account, to those capable of cultivating 
them, are yet for.the present outside the current of the 
main .advances of human knowledge: The play of 
human thought knows of no boundaries; it can pursue 
and clarify itself without limitation into endless mazes. 
All the more, we must be careful, in reclaiming and 
cultivating our boundless domains of mental evolution, 
not to lose touch of one another; if a theorist cannot 
command ‘the attention of his own generation, he is 
scarcely likely to attract the interest or serve the pur- 
poses of posterity. The one criterion that is available 
of the value of an addition to pure knowledge is the 
human mental interest it can excite. We have our 
very being inside a well-ordered cosmos, intellectual 
and material, which it is our highest mental pleasure 
to explore in all directions and learn to comprehend ; 
and we have a not unsafe guide in trained instinct and 
sense of fitness and symmetry, industriously applied, 
to appraise aright the value of each new departure. 
Knowledge thus cultivated on a broad basis for its 
own sake, so far from obstructing industrial applica- 
tions, is their profound source. The study of curves, 
especially the conic sections, by the Greeks, at home 
and afterwards at Alexandria, is not, as is sometimes 
asserted, an example of mere useless mental ramifica- 
tions happening to receive an application in later 
ages; it was on the direct path of progress, and formed 
the material, adequate and effective because not un- 
duly complex or abstract, on which the ideas of the 
infinitesimal calculus—and may we add the mechanics 
of Archimedes and Galileo?—were gradually matured. 
And if it became in Newton’s hands the weapon for 
the elucidation of the doctrine of universal gravitation, 
whereby human science first reached out securely into 
the illimitable universe, what analyst will deny the 
preordained fitness of the association ? 
There was a time, when the annual output of the 
Mathematical Society was smaller in bulk than it is 
now, that many of us made a point of taking an 
interest in all the papers that it published. It would 
be a great thing if we could get back again towards 
that state of affairs. At least two of our most dis- 
tinguished analysts have in my hearing traced the 
aloofness, and even aridity, of much recent work to 
the neglect of geometrical ideas, the potent source in 
the past of mathematical progress and consolidation, 
and the vehicle for the diffusion of our science. It 
seems a strange phase of development, when we con- 
sider the preponderant graphical, tentative, and prac- 
tical bent of the national intellect, and remember how 
much of our most characteristic progress and origin- 
ality in theoretical physics has been, for the sake of 
being comprehensively grasped and mastered by the 
mind, so concisely wrapped up in geometrical imagery, 
and so freed from analytical technicalities, as to have 
been even obscure to communities trained in more 
formal and syllogistic methods. 
There is always risk in getting too far from the 
main currents of our times; there is the danger, not 
always avoided, that in the fog of ignorance and the 
lack of interest we may encourage expansion in arti- 
ficial and unfruitful. and even tedious, ramifications, 
while criticising and suppressing with rigour worthy, 
but immature, attempts in the well-explored regions 
of our science, where improvements are so important 
and originality is so difficult. The contrast with the 
No. 2460, vot. 98] 
NATURE 
[DECEMBER 21, 1916 
difficulty of obtaining publication at all a century ago,, 
except in brief summary, gives ground for reflection. 
Of recent years the question must have presented 
itself to not a few of our authors whether the Pro-_ 
ceedings, developing in so abstract a direction, are, 
now quite as suitable a place for the publication of. 
mathematical physics as they were in the days when, 
Maxwell and Kelvin, and Rayleigh and Routh, were 
frequent contributors. Yet the potent source of even 
the most abstract branches of modern analysis has. 
lain in the seizure and orderly cultivation of the in-. 
tuitional idea's, largely cast in geometrical mould, that. 
are forged by physical science in the effort to sys-+ 
tematise its observations of the uniformities of the 
rational world around us. To renew our strength for 
wider flights we must return frequently to mother 
earth. The main feature of the technique of physical 
mathematics is that we are seldom dealing with a com- 
pleted; and therefore strictly limited, logical complex; 
it is. of its essence that the specification of the problem: 
is fluent and provisional, always ready to take on new 
features as the discussion opens out. The student of- 
mathematical physics cannot with safety afford to be- 
a specialist; every department of physics is dovetailed, 
into the other departments and progresses by their 
aid; knowledge must be so far as possible on an in- 
tuitive basis, to prevent it from becoming top-heavy, 
and all the threads must be in hand. For intuition 
sees, however imperfectly, all round a problem at a 
single glance; while analysis afterwards consolidates 
a permanent structure by fitting brick to brick. Even 
the most abstract of analysts must work at a dis-. 
advantage if he has no informed interest in the 
problems of external nature for which his analysis - 
might be of assistance; and conversely, even the most 
recondite constructions of pure analysis would be of 
interest to a wider audience if they could be expounded” 
in a non-technical manner, without the great detail that 
is sometimes thought to be essential to the necessary 
degree of precision. Nature is never. irrational, but 
our main intellectual aim is the redemption of our 
views of her operations from that reproach; it is the” 
freshly detected and systematically traced concatena- 
tions of her working that enlarge our stock of ideas, 
and become for us a source of new generalisations in~ 
abstract procedure, giving fresh points of view to be- 
developed and to react in their turn. It is sufficient 
to cite the names of Cauchy and Riemann, not to— 
mention the supreme examples of Lagrange and 
Gauss, to show that the most brilliant originality in_ 
abstract analysis, and habitude in the.intuitions of 
physical science, can go together, to great mutual’ 
advantage, 
Fortunately there are signs, abundant on both sides, 
that the repulsion which somehow arose with us in 
the last decades between the tentative, yet essentially 
progressive, though concise, prospecting of mathe- 
matical physics, and the stern but limited rigours 
associated with undiluted pure analysis, is now be- 
ginning to be recognised as cramping and unnatural; 
it may thus melt away in a better mutual understand- 
ing, and may one even say mutual interest, to the 
great advantage of both disciplines. Our analysts 
have been turning with success, and with a zest of a. 
kind that seems familiar to their more physical col- 
leagues, to semi-empirical methods in the theory of 
numbers; speculative interest has again arisen even in 
divergent series, such as would have rejoiced the soul 
of de Morgan, logician though he was; and the time- 
worn problems of partitions and combinations. have 
been yielding their secrets to the powerful leverage of 
an apparatus of arrays and lattices, that may remind 
us of crystallography and even of thermodynamics. 
Our society has lost by death not a few of her 
veteran members during my two years of office. 
