Aug. 21, 1885.] 



♦ KNOM^LEDGE ♦ 



CHATS OX GEOMETRICAL MEASUREMENT, 

 By Richard A. Peoctok. 



THE SPUERE. 

 (Continued from page 120.) 

 A. The demonstration of the volume of the sphere seems as 

 complete as it is simple. Of course by making the triangles ab c, 

 bed, in Fig. 3, &c. sufficiently small, we make the pyramid 

 C a b ca.B nearly equal as we please to the solid sector C abc. In 

 fact, it is obyious that all the portions left over in this way will 

 be included within a spherical shell (between the surfaces of the 

 sphere A B D, and a concentric sphere within it touching the 

 plane of the largest of the triangles a be, b c d, &c.), and this sphere 

 can be made as thin as we please by making all the triangles 

 sufficientlv small. 



M. That seems to you obvious ? I confess I think you are 

 right ; for it seems obvious to me also. Yet Euclid made this 

 the theme of the most difficult — at least the longest— of all his 

 problems, viz. the last problem but one of the Twelfth Book. 



A. How was this? 



M. Because Euclid would not allow of any constraction the pre- 

 cise method of which had not been determined. 



A. Can you indicate an actual demonstration, which shall not 

 be quite so complicated as the " spider's web " proposition to which 

 you refer ? 



Fig. 5. 



shall lie wholly without a concentric sphere F G H K, of radius 

 C :; (Fig. 5). Around C as centre (Fig. 5) describe the circular arc 

 a B a', and through g draw a g a' square to C B. Now let abc, 

 Fig. 4, be a small plane triangle having sides ab, ac equal, and 

 be not greater than either. Suppose a 6 c Fig. 6 to represent this 

 triangle, ab c the circumscribing circle, g its centre, and a g 

 Fig. G equal toag Fig. 5. Then if gC be drawn in Fig. 6, square 

 to the plane abc, and equal to Cg Fig. 5, it is obvious (since 

 a!7 = ''3 = f I/, and C;; is square to each) that C a = Cb = C c=C a of 

 Fig. 5. Hence a sphere having centre C and radius Ca will pass 

 through a, }>, and c ; and a sphere having centre C and radius C g 

 will touch the plane abc a.t g (since Cg is square to the plane abc, 

 and therefore is the shortest distance to that plane). A sphere 

 then F G U K having centre C Fig. 4 and the same radius C g will 

 not cut the plane abc, and will only touch it if fc c = a b, ora c ; for 

 if 6 c is less than a b the circumscribing circle abc Fig. 4 

 will be lees than the circle abc Fig. 6, and its plane 

 will therefore be at a distance from C exceeding C g 

 (Figs. 5 and 6). A fortiori, any plane triangle with its 

 angnlar points as a, b, c, on the spherical surface A B D, but having 

 two equal sides each less than a b, a c, and the third side not ex- 

 ceeding either, will lie outside the inner sphere F G H K. Now it 

 is easy to divide the whole surface of the sphere into such spherical 

 triangles that the plane triangles having the same angular points 

 will— as thus shown, — not touch the interior sphere. For let the 

 great circle B D be divided into any number of equal parts, c d, df, 

 fi, i k, each less than a b (which can be done, of c .^rse, by con- 

 tinually halving the arcs all round, starting first with quadrants) ; 

 and let the half quadrant AB be divided into any number of equal 

 parts BL, LN, N P, &c., each less than an, the arc through a 

 (Fig. 4) bisecting b c iu n. Let LL', N N', P P', Ac, be parallel 

 small circles through L, N, P, &c., the successive points of division 

 on the quadrant B A. On c d, df, fi, i k, Ac, as bases, let a series 

 of isosceles spherical triangles ced, dhf, fji, il k, &c., be described, 

 having their vertices e, h, j, I, &c., on LL'; on eh, hj, j I, &c., 

 another series of isosceles triangles emh, hnj, jo I, &c., having 

 their vertices m, n, o, &c., on N N' ; on m n, m o, &c., another seri s 

 of isosceles triangles mpn, n qo, &c., having their vertices p, q, &c., 

 on P Q ; and so on continually, until A is reached. It is manifest 

 that the vertices of each circuit of triangles will draw nearer and 

 nearer together, the farther we pass from B D ; for the number £ 

 them is the same on each successive circle, and the circles con- 

 tinually diminish with increasing distance from B D. Thus the 

 bases of the successive series of triangles grow less and less, as do 

 their equal sides. Moreover the same is true, not only of the 

 triangles as ced, dhf, fji, &c., but of the triangles, edh, hfj, 

 jil, Ac, which are also isosceles, but have their vertices turned 

 downwards instead of upwards. Hence, doing the like with the 

 other half, BED, of the sphere, we have finally, the surface of 

 the whole sphere divided into a number of isosceles spherical 

 triangles all less than abc, and none having the centre of its cir- 

 cumscribing circle so near to the centre as g. Hence the series of 

 corresponding plane triangles form a polyhedron enclosed within 

 the sphere ABED, but wholly without the sphere F G H K. 



A. Is that an abridgment of Eucliil's proof ? 



M. No : Euclid's proof is different. He divides the sphere into 

 strips, by a series of great circles all passing through the poles 

 A, E, and each strip into spherical quadrangles by arcs of circles 

 parallel to B D (except at the poles, where, of course, the strips 

 end in spherical triangles). But the demonstration is very com- 

 plicated, and the result does not seem to me so simple and satis- 

 factory as when the sphere is divided into triangles. 



A. You do not think, however, that any demonstration was 

 necessary ? 



M. No. Take an orange— or, better, a croquet-ball — and mark 

 dots over it so as to form a number of little triangles, beginning 

 with a small equilateral triangle, as abc in Fig. 7 ; and working 

 round it with the points d, e,f, g,h, Ac, and you will feel it to be 

 simply obvious that you can cover over the ^ 



whole surface of the sphere with acute-angled , \ <"' 



triangles (isosceles if yon like, but not all ''' V >/ 



equilateral) as small as you please, and there- , , \^. J\^_\ 

 fore having the planes of the cn-iv.fponding \ /\/\/ " 



plane triangles at distanr.-i d !:.■ ■ i tie of .V -Ail — \' 



the sphere as nearly ri|i,.: ; lure's " 



radius as you may pi. i i ': : • r as Fig. 7. 



little 







\<,u /i.,-. I :,r,, 'led triangles? 



'ill,. ,-. nil. . i . , ; .: ::;..iibing an acute angled 

 NMlhin ll;.. Iii:i: '-■, :.ii'l ■:.'<- makes the study of the 

 >1lt. But as tlie ilianioior uf a circle circumscribed 

 angle cannot bo greater than the longest side of the 



