338 Mr. BappaceE on the Influence of Signs 
rests on any of these individualities, itself requires a considerable 
effort. The substances of these observations may be expressed 
in this conclusion. The reasonings employed in Geometry and 
in Algebra are both of them general, but the signs which we use 
in the former, are of an* individual nature, whilst those which are 
employed in the lutter, are as abstract as any of the terms in 
which the reasoning is expressed. 
The signs used in Geometry, are frequently merely individuals 
of the species they represent; whilst those employed in Algebra 
having a connection purely arbitrary with the species for which 
they stand, do not force on the attention one individual in pre- 
ference to any other. 
An example of the limitation which geometrical considera- 
tions introduce, we shall select from a very well known author. 
In determining the relation between the rectangle under the 
parts of two lines intersecting each other and cutting a circle, 
Euclid considers separately the two cases of the point of inter- 
section being situated within and without the circle, and he 
shows that in the two figures 
the rectangle under CP and CQ, is in both cases equal to that 
under CP, and CQ: the case of one of the lines becoming a 
* Halley’s paper on the determination of the foci of lenses, would furnish a very 
apposite example of this principle, and probably few of my readers will fail to recollect 
instances where the same identical words of a proposition, and the same letters apply to 
two, three, or more different geometrical figures. 
