GEOMETRY OF POSITION. 81 
opher himself established in principle, that in geometry 
these quantities only differ from the positive in the direc- 
tion of the lines on which they ought to be reckoned. 
This profound and simple view is unfortunately subject 
to some exceptions. Let us suppose, for example, that 
it is proposed to draw from a point without a circle, a 
straight line so situated that the portion comprised within 
the circle shall have a given length. If the distance be- 
tween the point from which the line is to be drawn, and 
the point in the circumference which it will first meet, be 
taken as unknown, the calculation gives two values: the 
one, positive, corresponds with the first point of the inter- 
section of the straight line sought with the circle; the 
other, negative, determines the place of the second inter- 
section. Now who does not see* that these two lengths, 
the one positive, the other negative, must be measured 
from the same side of the point from which the straight 
line was drawn ? 
Carnot proposed to himself to cause these exceptions 
to disappear. He does not admit isolated negative solu- 
tions in geometry any more than in algebra. To him 
these solutions, taking away their signs, are the differ- 
ences of two other absolute quantities ; the one of those 
quantities which was the greatest in the case reasoned 
on, only becomes the smallest when the negative root 
* “ Who does not see ?’’ We cannot say that we do, nor can any- 
body else, perhaps, who has not the calculation before him. There 
are many ways of measuring distances about a circle; and two differ- 
ent lines in it amounting to the same effect can be so often drawn, 
that those wishing to be convinced would prefer hearing more about 
it; at any rate it is easier to suppose there is some thing misunderstood 
in the working of the problem, or in the meaning of its solution, than 
that the whole system of notation, on which all former results depend, 
should be wrong.— Translator. 
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