84 CARNOT. 



perpendicular to one another which intersect a sphere, 

 the sum of the areas of the three circles forming the inter- 

 sections will always be the same, whatever direction be 

 given to these planes : provided that they all three cut 

 the sphere." 



"In every trapezium, the sum of the squares of the 

 diagonals is equal to the sum of the squares of the sides 

 which are not parallel, plus twice the product of the par- 

 allel sides." 



" In every plane or uneven quadrilateral figure, the 

 sum of the square of the two diagonals is double the sum 

 of the squares of the two straight lines which join the 

 centres of the opposite sides." 



I shall have attained my end if these quotations, which 

 I could multiply to any amount, inspire professors of 

 mathematics with the desire of seeing for themselves, in 

 Carnot's Geometry of Position, how easily all these curi- 

 ous theorems flow from the methods of our illustrious 

 member. 



CARNOT INVENTOR OF A NEW SYSTEM OF FORTIFICA- 

 TION. 



There would be a gap in this biography which would 

 justly attract your criticism, if, notwithstanding the many 

 different points of view from which I have already con- 

 sidered the imposing figure of Carnot, I should neglect to 

 speak to you of the military engineer, of the inventor of 

 a new system of fortification. 



You doubtless recollect the violent arguments which 

 Carnot had to sustain, from the time of his entering on 

 the military career, with the chiefs of the army to which 

 he belonged. An upright and inflexible character already 

 made him repel the heavy yoke of esprit de corps. 



