368 Mr. BapsaceE on the Influence of Signs 
give rise to several observations. The assumption of v to re- 
present the velocity of one body per second, instead of the 
plan pursued in the first solution, was productive of two advan- 
tages: first, it substituted one letter instead of two; and secondly, 
it is so usual in all mechanical problems to make that letter 
denote velocity, that it is in such cases associated with it in 
the mind. The next assumption of v' for the velocity of the 
other body, possessed both these advantages, and tended to make 
the result more apparently symmetrical in case it was susceptible 
of that species of arrangement. The selection of the letters 
s and t, to represent space and. time, was adopted with a 
similar view of making the signs recal the thing signified. 
In pure analysis there is but little room for taking advantage 
of this species of connection, but in all the mixed questions 
to which it is applied, it may be extensively employed. The 
general principle is, that either the initial or some prominent 
letter should be selected from the word which denotes the thing 
we wish to represent. The beneficial effect of this arrangement 
is felt a little in the first stage of the solution; it has no 
influence on the second; but in the last stage it saves consider- 
able trouble by obviating the necessity of constantly referring 
back to enquire what particular letters represent. 
Tn the first solution, there are in fact two distinct cases, in which 
the reasoning is repeated from the beginning to the end; and each 
of these cases has two sub-species: so that there are, in fact, four 
cases treated of in the Arithmetica Universalis, This defect has 
been remedied in the second solution, where it has been shown 
that the four cases are included in one formula, to which it is only 
necessary to give the proper interpretation, and every circumstance 
connected with the problem is brought to light: Two causes seem 
to have contributed to produce this separation into cases: in the 
first place, the extreme generality of the language of Algebra may 
