in Mathematical Reasoning. 371 
consequently, the smallest end next to the smallest quantity. Ata 
period when the more frequently occurring signs were not per- 
manently settled, one method of denoting equality was by the 
following sign, 2|2, and the same author P. Herigone employed 
3|2, and 2|3 to represent respectively, greater than, and less than. 
In each of these, the principle we have been contending for, has 
undoubtedly had its influence, and the signs themselves are objec- 
tionable, on entirely different grounds. The same writer made 
use of the common sign of equality to denote parallelism; a pur- 
pose for which it was then well adapted; since that time, 
however, its long continued use in another sense, has compelled 
geometers to change its position; and when it is proposed to state 
that two lines 4B and CD are parallel, instead of putting AB = 
CD as Herigone would have done, they merely change the posi- 
tion, and write 4B || CB; thus preserving the advantage, without 
infringing another rule, which ought never to be violated, that 
of avoiding the use of any sign in two senses. 
In the doctrine of triangles, Lagrange has introduced a species 
of symmetry, which has been found productive of very advan- 
tageous results; it consists in denoting the three sides by the 
letters a, b,c, and the angles respectively opposite to them by the 
capitals 4, B, C: by this arrangement, not merely the quantities 
themselves are indicated, but in some measure also their position, 
and the transition from any relation between one side and given 
data, to other sides and the corresponding data, are made with 
the greatest ease. 
The more complicated the enquiries on which we enter, and 
the more numerous the quantities which it becomes necessary to 
represent symbolically, the more essentially necessary it will be 
found to assist the memory by contriving such signs as may 1imme- 
diately recal the thing which they are intended to represent. The 
notation which M. Carnot has contrived, for the purpose of illus- 
Vol. II, Part Il. 3B 
