in Mathematical Reasoning. 339 
tangent is also a separate investigation. Now in the algebraic 
mode of treating these questions, the three cases are comprised 
in one formula*. 
The indication of the extraction of roots by means of an 
appropriate sign, instead of actually performing the operation, is 
one of the circumstances which add generality to the conclusions 
of Algebra, and the same principle of indicating operations, in- 
stead of executing them, when employed with judgement, con- 
tributes frequently in no small degree to the perspicuity of the 
result, and sometimes enables us to read in the conclusion every 
stage which has been passed through in the progress towards it. 
Any general rules to direct us in the application of this principle 
will be difficult to form, because they ought in a great measure 
to depend on the objects we have in view: it may, however, be 
stated generally that it is improper to adhere to it, when by an 
opposite course any reduction or contraction can be made in the 
formula; thus generally speaking it would be better to write 
(a+ a)? — Aax+ B’, 
than 
and on the other hand, wherever in the course of any reasoning 
the actual execution of operations would add to the length of 
the formula, it is preferable merely to indicate them. 
Some of the advantages which arise from the use of letters 
to denote known quantities, have already been adverted to; but 
there are others of considerable value which may now be noticed, 
and which relate in a great measure to the higher departments 
of analysis. If a player bet a certain sum of money u, he may 
either win it and become possessed of +z, or he may lose it and 
possess —u. If we now suppose that he regulates the amount of his 
* Book III. Prop. 36. and 37. 
Vol. Il. Part Il. xX x 
