82 CARNOT. 



appears. In geometry then, as in algebra, the negative 

 root taken with the sign -J-, 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 



