parallel Lines in Geometry. 163 
to the second, by the substitution of a negative in place of a po- 
sitive quantity; that is, merely by changing the signs of some 
of the terms; which changes are made not in an arbitrary man- 
ner, but by fixt rules derived from the mechanism of analytical 
language. This conclusion was not perhaps perceived in all its 
extent by the first algebraists, but it has been firmly established 
in the progress of the science, and is indeed a necessary conse- 
quence of the general rules about which all are agreed. Jt is in 
this manner that an algebraic expression, the structure of which 
remains essentially the same, adapts itself to all the possible 
cases of a problem, while in geometry the same cases are only 
connected by a vague similarity not reducible to precise rules. 
When an equation is solved, the result may either be posi- 
tive, that is, a quantity to be added; or it may be negative, that 
is, a quantity to be subtracted ; but in both cases the meaning 
is equally clear when we go back to the primitive hypothesis, 
and consider the algebraic signs as notes of reference to the 
different views that may be taken of the same problem. 
Algebra therefore, by means of the doctrine of negative quan- 
tities, possesses a great advantage over geometry. In the for- 
mer science, a problem is comprehended in one expression ca~ 
pable of adapting itself, by the regular changes it admits of, to 
every particular case; in the latter, all the subordinate cases 
remain detached, and must be separately considered. By the 
comprehensive spirit of the first science, the investigation of 
truth is shortened and facilitated. Nor does it necessarily fol- 
low that the generalizations of algebra must be attended with 
obscurity. Jt may be affirmed that the ideas of the algebraist 
are clear in many instances where, by using the language of his’ 
predecessors, he has expressed himself in terms the most ex- 
ceptionable. 
The mathematicians, who would reject negative quantities, 
would introduce into algebra the procedure, necessarily followed 
in geometry, of minutely subdividing every proposition into all 
its particular cases. By this means no doubt the same clearness 
would be obtained in the one science which is so commendable 
in the other; but at the same time algebra would be stript of 
its greatest and most peculiar excellence as an instrument for 
the investigation of truth. What the purists recommend is, in 
reality, to cut the knot of the difficulty, in order to avoid the 
trouble of unravelling it. The proper remedy seems to be, to 
mount up to the cause of the imperfections complained of, and 
by an enlarged view of the nature and scope of the science, to 
preserve all the generality. of which it is capable, at the same 
time that its rules are deduced with the evidence required in 
mathematical reasoning, 
X2 But, 
