Roberts — Modern MatliematicB. 153 
fiysics and geometry, and the modern mathematical conceptions could 
course be hardly perceived, or have any attention paid to them by 
LOse who considered these subjects exclusively as their object. 
In these earlier times the study of mathematics, perhaps, affordedmore 
easure than it does now, because new discoveries were more easily 
lade, and with a less extent of reading. Now, no doubt, the methods 
attacking questions are both more numerous and more powerful ; 
It it is necessary to read, and hold together in the mind, the greater 
;.rt of what has been previously done in a subject, before we can 
! tempt to discover any new results. Still, however, it is remarkable 
iw many new and interesting theorems can be obtained when we 
ough a narrow field such as the geometry of circles and triangles. 
The modern mathematical idea consists chiefly in the theory of 
;ojection, combined with the principle of continuity, and the recog- 
. tion of the fact that angles and lengths in the Euclidian geometry of 
iperience depend upon a certain absolute curve of the second degree. 
I the algebraic side it has developed the theory of linear transforma- 
itns and invariants, to which we may add the recognition of the value 
^homogeneity, and the symmetry derived therefrom. These prin- 
(;)les have been only applied later on in the transcendental analysis; 
I t it is evident what advantages are obtained by keeping in their 
{iQeral form the quantities involved in elliptic and hyper-elliptic 
ioegrals, as we are enabled thence by symmetry to write down 
5 feral formulae by interchanging the roots. Also, in the theory of 
( iptic integrals, it may be observed that we have a decided gain 
i 1 symmetry by considering the functions sn, cn, dn as the ratios of 
i ir functions, the absolute magnitudes of which are, and remain, 
i determinate. 
In the other branches of mathematics, such as the theory of Diffe- 
] iitial Equations, the modern methods do not appear to have effected 
J much progress. This is, no doubt, partly due to the nature of the 
5 )ject. In fact, it is evident that before we can proceed much 
\ ther in the solution of differential equations, we must have a more 
( nplete knowledge of the various functions which are introduced to 
< : notice by the Integral Calculus. However, it may be admitted 
1 it we cannot expect any systematic and regular progress in the 
\ eory of Differential Equations until the modern methods are applied 
{ least to some extent. This has at length been done in a very com- 
] te and suggestive manner by Professor Sylvester in his idea of 
' leciprocants ; " so that now, the attention of mathematicians being 
1 'ned in this direction, we may hope for the opening up of a large 
E. I. A. PROC, SEE. III., VOL. I. M 
