in Mathematical Reasoning’. 369 
not have been sufliciently noticed: the ancients were accustomed 
to divide their problems into cases, and the habit of treating these 
separately, may have produced the same cautious treatment of a 
question when resolved by methods of a far more general and com- 
prehensive nature: such indeed would naturally be the case, until 
the degree of generalization introduced by the new method, was fully 
ascertained. In some instances, the action of another cause may 
be traced, and one that is more easily removed than that which 
arises from a want of confidence in the method employed: it may 
be referred to the imperfect manner, in which the very first elements 
of the application of Algebra te Geometry and to Mechanics, have 
been communicated to us: im order to explain clearly the result, it 
is quite indispensible that we should be perfectly familiar with the 
principles which regulate these: without them, we may frequently 
put the question into an algebraic form, because we view it only in 
one light; and from the nature of the language made use of, that 
one, whichever it may be, virtually embodies the whole; but when 
we require to retranslate our conclusions, as they contain every 
possible case, we must of course be able to translate each indi- 
vidual. To display in a more prominent point of view, the reason 
why there are so few failures in expressing the conditions of a 
problem in Algebra, and so many in giving the full account of 
all which the solution informs us, let us imagine a problem pro- 
posed that admits of ten cases; if we are capable of translating 
any one of these into an equation, such is the comprehensive nature 
of Algebraic language, that though they are all contained in that 
single expression, we may be ignorant of the manner of treating 
nine cases, and competent to manage one of them. If this happen- 
ed, we might succeed in resolving the equation, and discovering all 
its roots, which would answer all the ten cases; yet it is scarcely 
probable, with such moderate knowledge, we should succeed in 
explaining in common Janguage, the meaning of many of them. 
