322 ^^^ INQUIRY INTO THE PROPJEIITIES OJf SOMCS. 



polygons of the denomination tj ; and let p denote the number 

 of these polygons, which bound each solid angle of the 



Demonstration, body; then we have ^=— =—-. For there are c faces 

 and each face has n edges (Dcf. 3rd.); but each edge is com- 

 mon to two faces (Def. I.) ; therefore s= — . In like man- 

 ner there arc t solid angles, and each solid angle is bounded 

 by p faces, and therefore is touched by /J edges ; hence 



s=''J.. Q. E D. 



At ^ p t 71 C 



(;;or. 1. Cor. 1a/. w c = ;j ^, and c = ; / = . 



Cor 2. ^'^''- 2??(/. Now let the given solid be a pyramid under 



four triangles ; and ?r-=3 ; r=4; j?=3; because it cannot 

 be greater or less than 3; hence (by Proiy.) s^=ZX'i-r-'^ = Q', 

 and by (Cor. \.) ;=3x 4-^3=4; therefore iu this case 



Prop. 2. Prop, ^nd If a solid angle K (Plate VIII, Fig. 6,) be 



contained under p planes, r/=. A KB, B K C, C K D, 

 D K A, &c. ; and the solid angle K be cut off by any plane 

 GIIMN; the section will be a plane bounded by j9 right 



Demonstr. lines and /> plain angles. For GR MN cuts each of the 

 planes A K B, B K C, &:c. ; because it cuts off tlic solid 

 angle Iv ; but its intersections with these planes, viz.GIly 

 II M, M N, NG, &c., are right lines (Euclid 3.11), and 

 any two of thena are in the same ])Iaiie (Kuclid 2.11); there- 

 fore they are all in the »a)nc j)lane, and their number 

 =p= the number of planes con'tainiiig the solid angle K ; 

 but the number of plane angles G, II, M, N=the number 

 of sides G ir, H M, kc.=p. Q. E. D. 



Corol. Cor. If the solid angle K be cut off by a curved surface, 



the section will be a curved surface, bounded bypcurves 

 and. p angles; and this will happen, if all or any of the 

 faces A K B, &c., be curved. 



Prop. 3. Prop. 3rd. Let A B C D K, Fig. 6, be a solid bounded 



by surfaces of any kind, and let c, /, *, be the numbers of 

 its faces, solid angles, and edges ; and let one of its solid 

 angles, K,be cutoff by a surface of any kind, GHMN; the 

 Micrcraent of c together with Iha increment of t =. the incre- 

 ment 



