in Mathematical Reasoning. 331 
in which it terminated, but every stage of its progress. At first 
it appeared probable that this triumph of signs over words would 
have limits to its extent: a time it might be feared would arrive, 
when oppressed by the multitude of its productions, the language 
of signs would sink under the obscurity produced by its own 
multiplication: had these expectations been realized, still its 
utility would have been extensive, and mankind, whilst they felt 
grateful for the many stages it had advanced them, must have 
sought some more powerful auxiliary for their ulterior progress. 
Fortunately however such anticipations have proved unfounded ; 
in whatever department of analysis the number of symbols has 
encreased to a troublesome extent, contrivances have soon occurred 
for diminishing it without any sacrifice of perspicuity : the incon- 
venience has always been temporary, the advantage permanent. 
In later times the generalization and contraction introduced 
by the use of signs, seems even to have outstepped the discoveries 
which have resulted from them; and reasoning from the past 
course of science to its future advances, we may fairly presume 
that our power of condensing symbols will at least keep pace 
with the demands of the science. 
Examples of the power of a well contrived notation to con- 
dense into small space, a meaning which would in ordinary lan- 
guage require several lines or even pages, can hardly have 
escaped the notice of most of my readers: in the calculus of 
functions this condensation is carried to a far greater extent than 
in any other branch of analysis, and yet instead of creating any 
obscurity, the expressions are far more readily understood than if 
they were written at length: the imstance I shall choose as an 
example is the equation 
V(x, y) = (a, y)*. 
* Transactions of Cambridge Philosophical Society, Vol. I. p.68. 
Vol. If. Part II. Uv 
