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TRANSACTIONS OF SECTION A. 72) 
never know more of the nature of the pure sciences than they can derive 
from the processes and methods which they learned at the very outset of their 
mathematical studies. This is necessarily the case with many of the wranglers, as 
the first part of the mathematical tripos includes no true mathematical theory. 
Most of the mathematical text-books in use at Cambridge are so admirably adapted 
to the purposes for which they are intended that it seems ungracious to make an 
adverse criticism of a general kind. But I cannot help feeling regret that their 
writers have had so much in view the immediate application of the principles of 
the pure subjects to the treatment of physical problems, In the case of the differen- 
tial and integral calculus, for example, there seems an increasing tendency to intro- 
duce into the bookwork and examples propositions which really belong to the 
physical subjects. This is an important tribute to the growth and influence of 
physical mathematics in this country, and a zealous physicist might even consider 
it satisfactory that the student should not be required to encumber himself with 
Imowledge which was not directly applicable to the theory of matter. But from 
the mathematician’s point of view it is unfortunate, for, while shortening by very 
little the path of the student, it cannot fail to give an incomplete, if not erroneous, 
idea of the relations of the pure to the applied sciences. How can he help feeling 
that the former are merely ancillary to the latter when he finds that the mathe- 
matical problems which arise naturally in physical investigations have been already 
dealt with out of their place in the treatises which should have been devoted 
solely to the sciences of quantity and position? 
Perhaps few persons who have not had the matter forced upon their attention 
fully realise how fragmentary and unsatisfactory is the treatment of even those 
fundamental subjects in pure mathematics which form the groundwork of any 
course of mathematical study. Algebra is necessarily the first subject set before 
the student ; it has therefore to be adapted to the beginner, who at that time is only 
learning the first elements of the language of analysis. It is customary to regard 
trigonometry as primarily concerned with the solution of triangles; the geometrical 
definitions of the sine and cosine are therefore adopted, and after the application of 
the formulze to practical measurement and calculation a new departure is made with 
De Moivre’s theorem. The elementary portions of the theory of equations and the 
differential and integral calculus and differential equations are valuable collections 
of miscellaneous principles, processes, and theorems, useful either as results or as 
instruments of research, but possessing no great interest of their own. Analytical 
geometry fares the best, for it includes one small subject—curves of the second 
order—which is treated scientifically and with thoroughness. It is true, however, 
that the course of reading just mentioned includes one theory which, though itself 
an imperfect one, receives a tolerably complete development—I mean the theory of 
single periodic functions: but it is dispersed in such small fragments among the 
various subjects that it does not naturally present itself to the mind as a whole. If 
we could commence this theory by considering analytically the forms and necessary 
properties of functions of one period (thus obtaining their definitions as series and 
products), and could then proceed to a detailed discussion of the functions so 
defined—including their derivatives, the integrals involving them, the representa- 
tion of functions by their means in series (Fourier’s theorem), &c.—we should 
obtain a connected system of results relating to a definite branch of knowledge 
which would give a good idea of the orderly development of a mathematical 
theory ; but the fact that the student at the time of his introduction to sines and 
cosines is supposed to be ignorant of all but the most elementary algebra places 
great difficulties in the way of any such systematic treatment of the subject. 
Passing now to the consideration of pure mathematics itself, that is to say, of 
the abstract sciences which can only be conquered and explored by mathematical 
methods, it is difficult not to feel somewhat appalled by the enormous development 
they have received in the last fifty years. The mass of investigation, as measured by 
the pages in Transactions and Journals, which are annually added to the literature 
of the subject, is so great that it is fast becoming bewildering from its mere magni- 
tude, and the extraordinary extent to which many special lines of study have been 
carried. To those who believe, if any such there are, that mathematics exists for 
