AN INQUIRY INTO THE PROPERTIES OF SOLIDS. 323 



ment of s. For since one solid angle K has been cut off, one Demonstr. 

 surface G H M N has been added to the number c (by 

 Prop. 2, and its Cor.); therefore the increment of c=:l ; 

 but G H M N touches p solid angles with its angles G, H, 

 M, N, by the same, and one solid angle (K) has been cut 

 off; therefore the increment of t=p~l; and the incre- 

 ment of c -|- the increment of t=p: but no edge has been 

 cut off; therefore the number of the edges G H, H M, MN, 

 NG, has been added tos: but this numbers/) also (by 

 Prop. 2nd.) Q. E. D. 



Prop. Aih. If a solid of any kind, A B C D (Fig. 7.), Prop. 4. ' 

 have any number of its solid angles, as B, C, eut off" by a 

 surface of any kind, M K G N, so as to change the number 

 of its faces ; the increment of c together with the increment 

 of i = the increment of 4. For, put/= number of lines Demonstr. 

 and angles bounding the additional face M K G N; (/= the 

 number of solid angles, B, C, &c., cutoff: then the re- 

 maining solid, AMK.DGN, has one surface more than 

 the solid A B C D ; hence the increment of c = 1 ; but the 

 surface M K G N touches/ new solid angles in M, K, G, N; 

 and d solid angles have been cut away at B, C, &c. ; there- 

 fore the increment of c -|- the increment of t z::zf -\-\ — d. 

 Again, the additional edges of the solid A M K D G N are/ 

 in number, being the lines bounding the face M K DN; but 

 when two solid angles are cut away, as B, C, one edge, 

 B C, is lo.st; and in general when d solid angles are cut 

 away, so as to increase the number of faces belonging to 

 the new solid, AMKDGN, rf— ledges are lost; hence 

 the increment of s=.f-\-\—d. (-J. E. D. 



Prop, 5th. Let c, t, s, denote the number of faces, solid Prop- 5. 

 angles, and edges of any solid whatever, and we have the fol- 

 lowing general expression; c-\-t~'2=s. For let the solid Demonstr, 

 AMKDGN be cut by one operation from the tetrae- 

 dron A B C D (Fig. 7), or the solid A B C D G II M N 

 (Fig, 6 ) he formed by repeated operations from a similar 

 figure ; both of which suppositions are possible (hy A.v. Ind): 

 then the sum of the additions made to c and t by one or 

 several operations is equal to the addition or additions made 

 to s in the same manner ; therefore the difference of c-{-t 

 and s in the tctraedron is equal to the same difference in 

 Y 2 the 



