782] 
417 
782. 
ON MONGE’S “MÉMOIRE SUR LA THÉORIE DES DÉBLAIS ET 
DES REMBLAIS.” 
[From the Proceedings of the London Mathematical Society, vol. xiv. (1883), 
pp. 139—142. Read March 8, 1883.] 
The Memoir referred to, published in the Mémoires de VAcadémie, 1781, pp. 666— 
704, is a very remarkable one, as well for the problem of earthwork there considered 
as because the author was led by it to his capital discovery of the curves of curva 
ture of a surface. The problem is, from a given area, called technically the Déblai, 
to transport the earth to a given equal area, called the Remblai, with the least 
amount of carriage. Taking the earth to be of uniform infinitesimal thickness over 
the whole of each area (and therefore of the same thickness for both areas), the 
problem is a plane one; viz. stating it in a purely geometrical form, the problem is: 
Given two equal areas, to transfer the elements of the first area to the second area 
in such wise that the sum of the products of each element into the traversed 
distance may be a minimum ; the route of each element is, of course, a straight line. 
And we have the corresponding solid problem : Given two equal volumes, to transfer 
the elements of the first volume to the second volume in such wise that the sum 
of the products of each element into the traversed distance may be a minimum ; the 
route of each element is, of course, a straight line. The Memoir is divided into two 
parts : the first relating to the plane problem (and to some variations of it) : the 
second part contains a theorem as to congruences, the general theory of the curvature 
of surfaces, and finally a solution of the solid problem; in regard to this, I find a 
difficulty which will be referred to further on. 
I have said that Monge gives a theorem as to congruences. This is not stated 
quite in the best form,—viz. instead of speaking of a singly infinite system of lines, 
or even of the lines drawn according to a given law from the several points of a 
surface, he speaks of the lines drawn according to a given law from the several points 
c. xi. 53