TRANSACTIONS OF SECTION A. 317 
matical physics and pure mathematics have given much to each other in the 
past and will give much to each other in the future; in doing so, they will take 
harmonised action in furthering the progress of knowledge. But neither science 
must pretend to absorb the activity of the other. It is almost an irony of 
circumstance that a theorem, initiated by Fourier in the treatise just mentioned, 
has given rise to a vast amount of discussion and attention, which, while of 
supreme value in the development of one branch of pure mathematics, have 
hitherto offered little, if anything, by way of added explanation of natural 
phenomena. 
The century that has gone has witnessed a wonderful development of pure 
mathematics. The bead-roll of names in that science—Gauss; Abel, Jacobi; 
Cauchy, Riemann, Weierstrass, Hermite; Cayley, Sylvester; Lobatchewsky, 
Lie—will on only the merest recollection of the work with which their names 
are associated show that an age has been reached where the development of 
human thought is deemed as worthy a scientific occupation of the human mind 
as the most profound study of the phenomena of the material universe. 
The last feature of the century that will be mentioned has been the increase 
in the number of subjects, apparently dissimilar from one another, which are now 
being made to use mathematics to some extent. Perhaps the most surprising is 
the application of mathematics to the domain of pure thought ; this was effected by 
George Boole in his treatise ‘Laws of Thought,’ published in 1854; and though 
the developments have passed considerably beyond Boole’s researches, his work 
is one of those classics that mark a new departure. Political economy, on the 
initiative of Cournot and Jevons, has begun to employ symbols and to develop 
the graphical methods ; but there the present use seems to be one of suggestive 
record and expression, rather than of positive construction. Chemistry, in a 
modern spirit, is stretching out into mathematical theories; Willard Gibbs, in 
his memoir on the equilibrium of chemical systems, has led the way ; and, though 
his way is a path which chemists find strewn with the thorns of analysis, his 
work has rendered, incidentally, a real service in co-ordinating experimental results 
belonging to physics and to chemistry. A new and generalised theory of 
statistics is being constructed; and a school has grown up which is applying 
them to biological phenomena. Its activity, however, has not yet met with 
the sympathetic goodwill of all the pure biologists; and those who remember the 
quality of the discussion that took place last year at Cambridge between the 
biometricians ard some of the biologists will agree that, if the new school should 
languish, it will not be for want of the tonic of criticism. 
If I have dealt with the past history of some of the sciences with which our 
Section is concerned, and have chosen particular epochs in that history with the 
aim of concentrating your attention upon them, you will hardly expect me to 
plunge into the future, Being neither a prophet nor the son of a prophet, not 
being possessed of the knowledge which enabled Halley to don the prophet’s mantle 
with confidence, I shall venture upon no prophecy even so cautious as Bacon’s— 
‘As for the mixed mathematics I may only make this prediction, that there 
cannot fail to be more kinds of them as Nature grows further disclosed’—a 
declaration that is sage enough, though a trifle lacking in precision. Prophecy, 
unless based upon confident knowledge, has passed out of vogue, except perhaps 
in controversial politics; even in that domain, it is helpless to secure its own 
fulfilment. Let me rather exercise the privilege of one who is not entirely 
unfamiliar with the practice of geometry, and let me draw the proverbial line 
before indulgence in prophetic estimates. The names that have flitted through 
my remarks, the discoveries and the places associated with those names, definitely 
indicate that, notwithstanding all appearance of divergence and in spite of 
scattered isolation, the sum of human knowledge, which is an inheritance common 
to us all, grows silently, sometimes slowly, yet (as we hope) safely and surely, 
through the ages. You who are in South Africa have made an honourable and 
an honoured contribution to that growing knowledge, conspicuously in your 
astronomy and through a brilliant succession of astronomers. Here, not as 
an individual but as a. representative officer of our brotherhood in the British 
