M. Carnot's Theory of Defence. 291 



or fronts, attacked ; each mortar throwing 600 balls at every 

 discharge. 



M, Carnot introduces his theory of the effect of these balls 

 by observing, that of any number which fall in the trenches, the 

 number that take effect will depend upon the proportion which 

 the unoccupied part of the trench bears to the part which is 

 covered by the men posted and working in it. Thus, supposing 

 a man standing upon an horizontal plane, to cover a space of 

 about a foot square, and a man in the attitude of working some- 

 what more, M. Carnot calculates that the projections of the 

 bodies of the men, usually working and posted in the trenches, 

 will occupy about -y^ part of their surface ; from which he 

 infers, that of every 180 balls that fall in the trench, one should, 

 according to the doctrine of chances, hit a man ; and he does 

 not doubt that it will put him " hors de combat." 



M. Carnot's idea of the effect of this " fluie de balles" is 

 founded upon the velocities which he supposes they will acquire 

 in accelerated descent from the vertex of a very elevated curve. 

 This is manifestly the principle upon which he tries to establish 

 his theory ; and this it is which, disregarding for the present the 

 doctrine of chances. Sir Howard Douglas first remarks upon. 



" It is quite clear," says he, " that M. Carnot has formed his 

 theory upon the parabolic hypothesis, which is the theory of a 

 projectile's flight in a non-resisting medium. This theory, con- 

 siderably erroneous in all cases, is particularly and greatly so 

 with small projectiles ; and its deductions, as applied to the 

 velocity of descent of small balls used in very elevated short 

 ranges, are quite fallacious. The velocity of the ball in a hori- 

 zontal direction (which by this theory would be constant, and to 

 the projectile velocity, as radius to the cosine of the angle of 

 elevation), being inconsiderable, it is evident that the effect of 

 vertical fire must depend upon the velocity of descent in the 

 direction of the curve. Estimating this according to the para- 

 bolic theory (as the secant of the angle of elevation), the motion 

 would be slowest at the vertex of the curve, and the velocities 

 of the projectile be equal at equal distances from that point. 

 According to this supposition, we should assign to tlie descent 



