254 
Of the three the triangle CFE will manifestly generate a series of 
round So- 
—\—— 
cylinders circumscribing the, cone; the igs ‘in- 
scribed in the quadrant will generate a series of cylin- 
ders inscribed in the hemisphere ; and the rectangles 
CR, GS, KT, ME will generate a series of ybawies 
which will compose a cylinder having the same base 
and altitude as the hemisphere. 
The triangles CFE, CGO are manifestly similar, and 
CF=FE; therefore CG=GO: In like manner, it may 
be proved that CK=KP and that CM=MQ. 
Join CH, and because CGH is a right angled triangle, 
a circle described with CH as a radius will be equal to 
two circles described with CG and GH as radii (2 Cor. 
8. 5. Part 1.) but CG=GO, and CH=GR, therefore 
a circle described with GR as a radius will be equal to 
two circles described with GO and GH as radii ; hence 
again it follows, that the cylinder generated by the rec- 
tangle CR will be equal to both the cylinders generated 
by the rectangles CO and'CH, for they have all the 
same altitude, and the base of the first is eqral to the 
sum of the bases of the other two. It may be de- 
monstrated in the same manner that the cylinder ge- 
nerated by the rectangle GS is equal to the sum of the 
cylinders, generated by the rectangles GP and GL, 
and the same of all the rest ; therefore the sum of the 
cylinders, generated by the rectangles CR, GS, KT, 
ME is equal to the two series of cylinders, one gene- 
rated by the rectangles CO, GP, KQ, ME, and the 
other generated by the rectangles CH, GL, KN ; that 
is, a cylinder having the same base and altitude as the 
hemisphere, is pare: to the sum of the two séries of cy- 
linders, one described about the cone, and the other 
described in the hemisphere, 
Pror. ‘VI. Turor. 
Every sphere is two thirds of the circumscribing ey- 
linder. 
GEOMETRY. 
Let a figure be constructed exactly as in last propo- Of ¢ 
sition; and to abridge, let C denote the cone, ¢ the oun 
series of cylinders described about it, H the hemisphere, 
h the cylinders described in it, and K the ey inde ha- 
ving the same base and altitude as the hemisphere, or Fig- 1 
cone: Moreover, put d for the difference between the % 
cone and its circumscribed cylinders, and d’ for the 
difference between the hemisphere and its inscribed 
cylinders ; then we have : wal 
C+d=c, ndH=h+d, 
and adding equals to epee ; 
C+H+d>e+h +d’. 
Butc +h=K (5.); therefore,C + H4d=K+4a’, 
and C+ H + d—d’=K, alsoC H=K4+ d'—d. 
Hence it appears that the difference between C +H 
and K is equal to the difference between d and d’. 
Now d is less than the cylinder generated by the rota- 
= 4 the rectangle ME (Cor. to 4.), and d’ is 
ess than the cylinder generated by the rectangle C 
which is equal * ME, therefore the difference ars 
d and d’ must be less than the same rectangle ; hence 
the difference between C+H and K is less than the cy- 
linder generated by the revolution of the ME, 
or is less than a cylinder having the same base as the 
cone, and the line FM for its altitude. From this we 
may infer, that C-- H is exactly equal to K ; for if there 
can be any difference, let it be a cylinder having the 
same base as the cone, and its altitude equal to FV ; 
then FM must be greater then FV ; but the number of 
s into which FC is divided may be so great that 
*M may be less than FV ; therefore C4+-H cannot be 
unequal to K ; and since C+ H=K, and C= +K (3.), 
therefore H=3K; that is, the hemisphere is two- 
thirds of its circumscribing cylinder ; and taking the 
doubles of these; the whole sphere is two-thirds of its 
circumscribing cylinder. 
An Index to shew the Propositions in the foregoing Treatise, which correspond to the principal Theorems in the vi 
six, and the eleventh and twelfth Books of Euclid’s Elements. ir # 
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