162 On the Theory of 
Some mathematicians of this country, founding their objec- 
tions chiefly on verbal inaccuracies, contend that the doctrine 
of negative quantities should be banished from algebra. With- 
out stopping to inquire into the cause of the obscurities of which 
they complain, these purists will give no quarter to any thing 
that has even the appearance of infringing the clearness and 
evidence which is the boast of mathematical science. They 
rather choose to obviate the difficulties they meet with, by 
breaking down every proposition into its particular cases, than, 
by following the spirit of the algebraic analysis, to comprehend 
them all in one investigation. In order to obtain the same 
clearness in algebra for which the ancient geometry is admired, 
they would neglect the distinction between the two sciences, and 
would cramp the former by the restrictions to which the latter is 
necessarily subject. 
In geometry every proposition, even the most general, is de- 
monstrated with reference to a particular diagram. In the 47th 
of the first book of Euclid, all the reasoning is directed to the 
particular triangle represented in the scheme. But as no part 
of the demonstration depends upon any peculiar relations of the 
sides or angles of that triangle, it is clearly seen that the pro- 
perty proved will belong to any one of the same species. In 
the instance now mentioned there are no subordinate cases that 
require an alteration of the diagram. But as one geometrical 
figure can properly represent all those only that are exactly 
similar to it, it often happens that the different cases of the same 
proposition require several diagrams, to each of which a sepa- 
rate demonstration must be applied. ‘The geometer may per- 
ceive a great similitude between the subordinate cases; inso- 
much that, when one is understood, all the rest are readily de- 
duced from it; but the science he cultivates furnishes no me- 
thod of bringing the observed analogy under precise and general 
rules. A geometrical demonstration is never deemed complete, 
unless all the cases be fully enumerated, and separately investi- 
gated. 
The algebraist can no more translate a problem from the 
common into the analytical language, without conceiving a par- 
ticular state of the quantities concerned, than the geometer can 
demonstrate without reference to a particular diagram. But — 
when an equation has been obtained from one particular case, it 
necessarily comprehends under it every possible case of the same 
problem. In algebra there can be no variation in the state of a 
problem excepting as the quantities concerned are greater or 
less. If the quantity sought be greater than some known quan- 
tity, an addition is implied; if less, we must conceive a subtrac- 
tion, But an equation obtained on the first hypothesis, applies 
to 
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