For finding the ar of any fegment I A 
oa K 
‘Therdly ° y be deduced in the follow- 
(fg. 91. J ofa {phere, a rule m 
ing manner. It appears 
(fee SpuEre,) that-the 
ro ae property of the {phere 
inion propofed, I A K, is equal 
to the difference betwee conical fruftum FC DH, 
and a cylinder of the: me one ftanding 
upon a Bate, w ‘hole radius C A is ha Oo 
e cone bein 
fame bafe and altitude ; and the ee of the cone 
But the content of the 
fruftum F CDH, if the two diameters CD and FH be 
reprefented by A and B, and the ,2618th part of the alti- 
tude 
is, Aa cht to 
will be B ( 
> a paral alle — ipedon whofe siitvies is E and bafe 
+A x B+ B*) . And the content of the ened 
ECD wil be = 5 x Ay or Ex A’; becaufe 
7854. x x D; and the hare: of 
of the bale “multiplied, by its 
‘but the area of the bafe is, from 
x A’, = the = 
5 
a 
height (fee Cy CINDER) $ 
the property of the circle, = 575 54 x 
= D: therefore 3E x A? = 
content of the cylinder. 
content of the fegment I 
B+ “Bi = 
e 
a & 
Qs 
x B-B i 
A _ B ; the former ne of which A?— A x Bis= 
A— re een a (case r CD 
FH=EF+ Bor 2D, becaufe the tri- 
angle EF °C, ne atin edit w Ath COA, is es 
and the latter part wx —~B=A—BxA+B=2D 
x 2A — 2D; A —B being equal CD —- FH =2D, 
and A+ B= EG+FH=EG+EG—2D=2A 
2D. Whence a fum of both parts will be = 2D 
x 3A — 2D; and the content of the fegment itfelf = E 
x20 % 3A =2D = ,2618 Dx2Dx3A— 
ee neers 
ife. To find the ae of the aoe ef a ‘a 
ae add i ait = aie the of the two ends and 
n propor ween them, + and then multiply the 
fe ‘d ‘fam by the perpen 4 ht, and 3d of the prodné 
willbe the folidity. £. be the area of the eens 
end, a a that.of the lefs, a ; a height ; then Atay 
Hence, 1 .u the = 
nd it is 
Itiply the quotient by the height, and the 
4d of the prodnét. 
and t 
be circles, the fruftum will is that .of a cone, and then 
multiply ,2618, viz. 3d of ,7854, by the height, and the 
rodu@t either by tbe quotient arifing from the divifion of. 
arifing from the 
addition of re ae of each diameter and the product of 
e latter or equal to the fphere; and 
by E, willbe = E x A? +A x B+ B? (that 
27854 x ; 
*Confequently, the ieee or re 
K, willbe = E x 3 A? — 
2A°?°—-AxB—B. B 
pene of A —A x Band " p. 206. 
f 
2D 
ce we have the fol- 
n f 
will give the content 
2. If the ends Ww 
m t 
of the diameters’ by the dif.. 
by the fum 
the diameters, or by the fum arifing from the fquare.of the | 
half difference of the diameters added to triple the {quare 
of the half fum. The principles from which ie rules 
are deduced will appear under the‘article Pyra 
© find the = = - fegment of a fp here. “Rule I. 
=e 
FE 
tip 
and a Fs rodu 6 for the folidity. ig. Q2.) - 
- Thus = DE the radius of its bafe, and 4 aa E the 
height; then 5236h x 3rr+hh = the folidity of the’ 
fegm en GF. e 2. From three times - diameter 
of the ie fubt mc twice the height o e fruftum ;" 
multiply the difference by the {quare of the c height and the 
product by ,5236 for the folidity. That is, if d= GH 
the aaa of the fphere, and 4 = G E, the height of the. 
fruftum ; then 5236 4' x 3d—2 ~ 2h = the folidity of 
D F, 
the fum by the faid height, and the produ& again by 
1,5708 for the content. That is, R* + 7° + 5 bx kph 
= the folidity of the fruftum whofe height is}, and _ 
radii of its ends R and zr, p being 351416. For the 
roe of thefe rules, fee Sprrere, or Hutton’s ‘Menfuration, 
To find the content of the evan of a fpheroid ; its’ 
ends being perpendicular to one of the - and one of 
them ar eee the centre. Rule o the 
: the lefs - wite that of the pee 
e fum by de fee of the fruftum ; and $d of the 
fs & will be the content. That is, 2D? 7 4b x ban= 
the fruftufa whofe ends are perpendicular to the fixed axis; 
where D is the Fae of the greater end, d that of the 
kefs, a the altitude, and n =. ,785308. And 2TC hie 
x - an ae! fr bain lca ends are parallel to the fixed. 
ax wher are eure oe ee 
axes “of che. greater end, ofe the 
ere It is 1 evident that — double of ‘the ae 
>= 
ocd 
Q° 
et 
of ‘the zone, or fpheroidal eafk. 
e 2. From three times the {quare of the fenti 
9] I 
3214159, &¢.3 and call the laft produét P: then, 1. If the 
ends be parallel to the fixed axis, fay, ag the revolving axis . 
is to ‘the fixed axis, fo will P be to the content of the fruf-' 
2. Wh 
tum en the ends are aed aah a the fixed 
axidy- fay, as the {quare of the fixed axis i e {quare 
of the revolving a fo is P to the content- of che fui um. 
That is, et ie x parr will be the fruftum thofe 
or Lider, 
ends are eae to the fixed axis: 
ra 
is be ph the fraftum, whofe ends are parallel to, the fixed _ 
3 f bein nS the fixed, and r the revolvirig femj-axis, b. 
ae height, P= 314. For ed can of thefe 
rules, lee ee and Flutton ubi fu 
cylinder is always the fame i in different parts of ‘the fanie, 
3F2 "ot 
