270 A Demonstration of Le Gendre’s Theorem 
means serious falls would be greatly avoided. On the presump- 
tion that the machine could thus be guided along any intended 
track, it might perhaps be practicable to change the men at se- 
veral stages like coach horses. Indeed I should not despair of 
yet seeing some such method employed as the most expeditious 
for conveying the mail from one place to another. 
This contrivance is no doubt very inferior to the organs of 
flight with which the feathered race are furnished, and which 
enable them to traverse the air with such admirable facility. But 
it is still a recommendation, that it is free from any reciprocating 
motion, the vanes obviously acting during every part of their re- 
volution; which is a property entirely wanting in those un- 
fortunate artificial wings contrived to act in imitation of the 
birds; since such unwieldy wings are not simply useless whilst 
returning to renew their stroke, but really retard and destroy the 
flight altogether, as the experiment has uniformly proved. 
I have not yet attempted to compute the force to be exerted 
in supporting such a machine. This would be a task of some dif- 
ficulty as well as uncertainty; since our best theories of the re- 
sistance of fluids are still something short of perfection. It might 
however, to a certain extent, be compared with the forces acting 
in the common windmill. 
If the above scheme, which is perhaps as plausible as most of 
the kind that have been proposed, seem to deserve a place in the 
Philosophical Magazine, the insertion of it will oblige 
Yours, &c. 
Edinburgh, Sept. 29, 1821. VoLATOR. 
LXI. A Demonstration of Le GENpRE’s Theorem for solving 
such spherical Triangles as have their Sides very small in 
Proportion to the Radius of the Sphere. By Jamus Ivory, 
M.A. F.R.S. 
Ts E theorem to be demonstrated is one of singular beauty, and 
of great usefulness in geodetical calculations. Although many 
demonstrations have already been given of it, yet the one which 
follows may merit attention on account of its simplicity. 
The theorem is this: 
“ Ina spherical triangle of which the sides are very small re- 
latively to the radius of the sphere, if each of the three angles 
be diminished by one-third part of the excess of their sum above 
two right angles, the remainders will be the angles of a plane 
triangle that has its sides equal in length to those of the spherical 
triangle.” ' 2 
Let r represent the radius of the sphere, and a, l, c, the i 
sides 
