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XIV. 
JODERIS" MATHEMATICS. By R. A. EGBERTS, M.A., Trinity 
College^ Dnblin. 
[Read November 8, 1886.] 
'] EEEE have been such vast strides made in mathematical knowledge 
i this century, and especially in the latter half of it, that it is much 
e ier now than heretofore to form an opinion as to the position of 
I thematics. From this great extension of mathematics, we can now 
D re exactly assign the points where it touches on other subjects, 
a I its relative position among the different branches of human know- 
Ifge. And from the internal point of view, we can now more 
r idily classify all the various divisions and subdivisions, and are 
t fLs enabled to enumerate all the principal results which have been 
0 ^ained, and form some idea as to where something remains still to 
b Idone. 
1 : In opening any mathematical treatise of the last century or begin- 
n g of the present, and another belonging to the modern school, we 
a at once struck by the great difference in the appearance of the 
n &hematical work. We see new notations, old ones greatly altered, 
a i a number of new words arising from the growth of new mathe- 
n tical ideas. In fact, we see a complete transformation effected in 
t whole nature of the subject. This is due to the modern mathe- 
u cical idea. The first recognition of the modern idea was on the 
g metrical side, and is to be found in Poncelet's Treatise on Projec- 
t\ Properties, published in 1822. Afterwards, on the algebraic side, 
I iile first came upon the germs of the theory of invariants, but did 
L seem to perceive its importance. It was then Cayley took up this 
b ach of the subject, grasped the idea in all its entirety, and worked 
ii lut to its fullest extent, thus laying the foundation of the present 
g it structure of mathematical knowledge, and opening up a wide 
fi i of labour for the mathematicians of the future. 
In the days of Newton, Leibnitz, Euler, and others, mathematics 
\ 1 a vast unexplored world. In whatever direction investigations 
1 'e made, the searcher was constantly rewarded by discoveries. 
