1820.] Royal Ac ademi/ of Sciences. 217 



also procured, nothing is required but the use of our eyes to b^ 

 convinced of the perfect accuracy of these methods ; v,e may 

 then easily comprehend the figures traced by Hypsicles, and the 

 demonstrations become clear. M. Peyrard accuses these 

 demonstrations of being deficient, both in exactness and in 

 elegance. We allow they are much too long, but that fault 

 ought not to be thrown on Euclid, who, we know not why, has 

 determined the inclination to be the acute angle formed by two 

 contiguous surfaces. In reality the inclination is never an acute 

 angle, except in the tetrahedron : it is a right angle in the hexa- 

 hedron, and an obtuse angle in the other three sohds, so that an 

 acute angle is not to be found in the hexahedron, and in the 

 three other solids, it is the angles between one surface and the 

 prolongation of the neighbouring surfaces that are acute. Now 

 half the demonstrations of Hypsicles are devoted to settling the 

 species of angles, wliilst the constructions of Isidorus always give 

 the true angle, whether acute, or obtuse, so as to preclude the 



• possibility of any mistake. 



We may add that these demonstrations, although they are 

 different for each of the five regular solids, yet depend upon one 

 single principle, which would render them clear, even indepen- 

 dently of the figure in relief. This principle consists in suppos- 

 ing in each solid a line to be drawn, which would serve as a 

 common basis to two isosecles triangles, whose sides are known^ 

 In one of these triangles the angle at the summit is always 

 known ; in the other, it is the inclination which we seek for; a 

 Tery simple relation between the cosines of the two angles is the 

 result of this ; and if we apply to these triangles, one of the rules 

 of our modern trigonometry, we immediately obtain an equation 

 exactly similar to that which is furnished by spherical trigono- 

 metry. 



But this modern rule was entirely unknown to Euclid, to Isi- 

 dorus, and to Hypsicles, who, in the very defective solution 

 which he has elsewhere given us of a problem resolved nearly at 

 the same time by Hippaichus, has left us a striking proof of his 

 complete ignorance of both plane and spherical trigonometry. 



It is rather remarkable that this theory of regular solids, per- 

 ■ plexed and imperfect as it was with the Greeks and their 

 continuators, should depend entirely upon a rectangular spherical 

 triangle, traced on the surface of the sphere upon which we wish 

 to inscribe at once all these solids. The angles of this triangle 

 are always given, and the formulae resulting from them for the 

 three sides furnish the most simple expressions of the edges of 

 the polar distances of all these planes of their nmtual inclinations, 

 of their distances from the centre of the sphere ; and, lastly, of 

 the methods of measuring with equal facility the surfaces either 

 partial or total, and the bulks of the five regular solids in parts of 



• the radius of the sphere taken as unity. 



"This triaugksnot only gives the precise and numerical quantity 



