82 CARNOT. 
appears. In geometry then, as in algebra, the negative 
root taken with the sign +-, is the solution of a different 
question from that which was put, or, at any rate, from 
that which it was exclusively desired to put, in the equa- 
tion. How is it now that problems foreign to the par- 
ticular one which the geometer wished to resolve, mix 
themselves up with it: that analysis answers with deplo- 
rable fertility to questions which have not been put to it ;_ 
that if its aid is sought, for example, to determine the 
ellipse whose area is a maximum amongst all those which 
can be drawn through four stated points, it gives three 
solutions, whilst evidently there is only one good, admis- 
sible, and capable of application; that without the knowl- 
edge and against the will of the calculator, it thus groups, 
in this particular case, a problem relating to the limited 
area of the ellipse with one concerning the hyperbola, a 
curve with indefinite branches, and therefore with indefi- 
nite area? Here is what required clearing up, here is 
that of which the theory of the co-relation of figures and 
the Geometry of Position, which Carnot has connected 
with his very ingenious views on negative quantities, give 
generally easy solutions. 
Since these labours of our member, every one thus 
applies without scruple, the formula established on one 
particular state of any curve, to all the different forms 
which that curve may take. Those who will read the 
works of the ancient mathematicians, the collection of . 
Pappus, for example ; those who will observe, even in 
the last century, two celebrated geometers, Simson and 
Stewart, giving as many demonstrations of a proposition 
as the figure to which it related could take different posi- 
tions or forms by the disarrangement of its parts; they 
will, I say, estimate Carnot’s service to geometry as very 
