324^ 



AN INQUIRY INTO tHE PliOPERTIKS Or SOl-tm 



-Demonstr. 



the solid ABCDGHMN, or any other solid; that is 

 c-^t~2=s, (hyCor.2, Prop. 1.) Q. E. D. 

 Prop. 6. Prop. 6th. Let A B C D (Fig. 6) be a solid bounded by 



polygons of the same denomination; j)ut ?/ = the number 

 of sides of each polygon: p = the number of polygons 

 which touch each solid angle: then c=:4/-'-f-2 j>-j-2;i— w;j; 

 t = 4?i~2p-{-2 7i — np; s—'2np~2p-}-'2n'-np. For 

 s = nc~2=pt~'i (Prop. 1st.) =zc-\-t--2 (Prop. 5th.) ; 

 a.t\d t ==7i c-^p c=p t-r-Ji (Cor. 1, Prop.!.); therefore 

 n c-J-2=c+Mc-f-;j— 2 ; hence c=:4 p—l p-\-2 n~np : again 

 ' pt~1r^pt-^7iyt — '2,i and i'lt^i «4-2 px2 ?i-^n p; again 

 p t~^z:zp t-i-n-{-t^'2. ; and t—l n~^ ;;+2 n-n p ; but 

 6=r:r-{-?-2— 2 wp-^2;;-f 2 n~ up. Q. E. D. 

 Corol. 1. Cor. 1st. p, which is constant relative to any given solid, 



denotes the number of polygons, which surround each 

 solid angle of that body; the«refore the plain angles of these 

 polygons are equal among themselves ; that is, the polygons 

 arc ordinate figures (by Def. 2nd.), and the preceding pro- 

 position relates to the regular bodies only. 

 Corol. 2. Cor. Ind. If there be two regular bodies, one of which 



'; is bounded by ordinate polygons of the dcnominatiou n, 



each solid angle of it being contained under p planes, and the 

 other is bounded by polygons of the denomination p; each 

 solid angle of li being contained under ?? planes ; c in the 

 former =:^ in the latter; and c in the latter rz:^ in the 

 former ; but s is the same in both. This is evident from the 

 equations in the proposition. 

 Gorol. 3. Cor. 3rd. '2,p-\-2 n—np, the common divisor of the three 



preceding equations, is equal to one, or greater than one; 

 but w is at Ieastr=:3 (Euclid, Ax. 10, Bk. 1); also/) is at 

 least = 3 (Def. 9, Bk. 11). Now to find the limit of n, 

 put2n-f-2p— wp=0; and p=2+4-rn — 2— :a, w, ?i, which 

 gives OT— fi ; therefore n can only be, 3, 4, or 5 ; that is, 

 the regular bodies are bounded by triangles, squares, and 

 pentagons only. 

 Corol 4, Cor. 4th. PutM=3(and hy prep.) ^ = 1 ' p; hence 



^ = 3, 4, or 5 ; put j:=z3 ; and c:r4 ; ^=4 ; s=6, the pro- 

 perties of a tetraedron; put fi=:i; and ^=16—8 — 2jt» ; 

 p~3; c=G;t=iS; now put «n:3, p:r:4 ; and cms ; /— 6; 

 the former of these is the cube ; and the latter the octae- 

 dron; and. *zi:12 ia both cases (by Cor. 2.) Pat nzr5. 



