713 / 1042
G
F.
o
1\1
E
T R
Y.
~rea
of
Ih~
hafe(fuun,1by rrop.
33 ,)
ue multiplirJ inlO
~
uf pans of Ihe e, lioder,
~nd Ih~t
:he whoi< fphere is ; of
Ihe heieht, the produd w,!1 g"'e the ful,,1
~untrnl
of :he lhe "ouble eylioder eirclIlllfcrtbrcl ahout it.
cone; for by the 10lh 12
~.ud.
a e(lne is lhe tlllro pnrt
1'hls is thal eeleurateJ 391h p"'p. l. "ook of Arehi .
of
01
eylincler
Ih.llhdS tlle fame bafe anO
hei~hl
mede, of lhe fphere aod eylioder; io whieh he deler.
PRO
r
O S 1
l'
ION XLVII .
ollllcsthe proportion of lhe eylillder 10 Ibe fphm ioferi.
F' G. 24 . 25 .
r o filld Ih, ¡olid C"III"1I1 or a
fruflum bcd 10 ue lhl t of
1
to
2 •
•r
4
con< cuf
kv
a p/all'
par.JI,/ f. Ih,
r/f//:'
o[ 11"
e
'0 ROL L AR
r.
~a¡(.-Flffl,
lel Ihe heighlof the clltire eone be found ,
Heoee il foll o\Vs, thal lhe fph ere is cqual to a eooe
and Ihenee (by Ihe preeenlOg prop.) it, foltd eOOlenl; from whofe height is equal to Ihe fenlldiameter of
tI~
fphere,
whieh fublr<tllhc fuhd eonlent of lhe eone eUI olf
~I
Ihe having for ilS bafe a eirele eq ual to lhe fopc,fieilS of Ihe
IOp, Ihere will remdin Ihe folio eonten! of Ihe
[rujllflll
of fphere, or to fuor great eireles of tI,e fphete, or to a eirele
Ihe eooe.
"hofe r.dius is equal to the diammr of the fphere, (by
How Ihe eonlenl of the enlire eooe may be foond , ap' prop. 41. of this.) Aod indeed a fphete dilr", very
pea" lhus: Lel AHCD be
lhefrujlulIl
of Ihe eooe (ei IlIde from the fúm of an intioite oumber of eones that
Iher right or f..lenous, as in Ihe figores 2. and ;.) let
h.ve. lheir bafes io the furfaee of Ihe fphere. and their
Ihe eooe ECD be foppofed 10 be eomrleated; lel AG eommoo vertex in
I~e
centre of the fpll"e; fo tha¡ the
be drdwO paralld to DE, aod
1<1
AH .nd Ef' be perpeo· fuperfieies of Ihó fphcre, (of whofe diOlenfioo fce prop. 4
J.
dieular 00 CD ; it will be (by ,d
6.
Euc!.) "CC:CA: : of Ihis) Olultiplied ioto Ihe third pm of the femidiamcter,
CD:CE; but (byart. 71. ofpar!.
l.)
as CA:AH::CE: giv," thefolid conteotoflhe fphm.
EF; eonfequendy (by 22d
5.
Euel.)
as
CG :AH::CI):
P RO P O S I T ION XLIX.
f'f';
thal is.
as
Ihe exeef, of Ihe diaOleter uf Ihe lelfer
F,
o.
28.
ro find Ihr jo/id w¡lenl
,[.
follor
of
Iht
bafe is to the height of the
frtljlum,
fo is the diameter of
¡phm.-A
fpherieal fellor AJ3C (as appears by the eor.
the gre. tcr
b.fe10 the height of Ihe eOlire eone.
of the prceeding prop.) is very lude diITereol from an io.
e o
ROL L AR
r.
fin ite numm of eones, haviog their bafes in the fu perfi.
F, o.
26.
Sorne eafks ",hofe flaves are remarkably beod- . eies of ,he fphere DEC, aod
I~eir
eommOD .ertex in tbe
ed about the middle, and firait IOwards Ihe eods. may eeolre. Wherefore the fpherieal fuperfieies BEC beiog
be
ta~en
for tINO portioos of eODes, wilhout aoy eooGder· foond (by prop. 42. of this), and multiplied ioto the third
able mor. T hus ABEF is a
¡rujlunl
of a righl eooe, part of AB the radiu, of the fpbere, !he pfodua will
to whofe bafe EF, 00 the other fioe, thae is aoolher
6·
give the folid eooleot of the fellor ABC.
milar
¡"ujl~m
of a eone joiDed, EOCf'. The venices of
e o
ROL LAR
r.
Ihefe eones, if they be foppofed to be eomplcted, will be
It i, ,,'ident how to find the folidity of a fpherical (eg.
fouod at G aod
11.
\Vheoee, (uy the preceding prop.) meot lefs thao a hemifphere, by fubtra(\ing Ihe cone ABe
the folid eoolent of fuch velfds n\ay be found .
from the feaor already found
BUI 'if the fpherieal feg.
P RO
P
O S
1
T ION XLVII I.
ment be grealer thaD a Ilemifphere, Ihe eooe eomfpaod.
F'G. 27. A eylinder cireumfcribed about a fphere , ing mull be addeo ID Ihe feaor, 10 make the fegment.
,hat is, ha.iog its bafe equal 10 a great eirele of the
PRO P O S
1
T I ON
L.
fphm, aod ils heigut eqoal tO the diameter of Ihe (phere,
F'G . 29.
r ofind Ihrjo/idill o( Ih, Jphroid, and
of
is 10 Ihe (phere as
3
to 2.
ilJ fogmrnfJ cul
bJ·
p!,In'! /,rrpwdic.la.r lo Ihr axil.-
Let ABEC be Ihe quadraot of a cirele, and ARDC
In prop. 44. of thlS, It IS fhewo, that every whel e
the eircumferibed fqoare; anJ likewife the trian;:le ADC; EH: EG :: CF : CO; bui eireles are as Ihe fquates
by the revoluli9n of Ihe figure about the right lioe AC, deferibed opon Iheir rays, lhat is, Ihe eirele of Ihe radio
as aKis, a hemifphere will be geoermd by Ihc quadr. ot, liS EH is 10 Ihe cirele of Ihe radius EG, as Cf
q
10 COy.
a
eylinder of Ihe fame bafe aod height by Ihe fquare, and Aod fioee it is fo every where, all Ihe ei,eles dcfcribed
a crne by the triangle . Let thefe Ihree be eot any
)1ow
with Ihe refpellive rays EH, (Ihat is, Ihe fpheroid made
by Ihe plane Hf', par.lld 10 tite bafe AH; Ihe fellion in by Ihe.rolaliunof Ihe femi .cllipfes AfB a,oood the
~xis
the eylioder will be a cirele whofe radius is FH, io the AB) lVill be to alJ the cirel" d&ribed by Ihe refpellive
hemifphcre a eirele of Ihe radius E.f, and in the eone a
r,¡dii
EG,.
~¡hat
i" Ihe (phete
~efcnbld
by Ihe rOlatioo
eirele of the radios GF.
of Ihe fem'Cllde A
DH
00 the axIS AB) as f'Cq to
CDq ;
By (an .
69.
of part l.)
EAq,
or
HFq=EF
q
aod thal is, as the fphcrllld 10 the fphm 00 Ihe fame ax is, fo
FAq
uken logelher, (but Af'r-FGq. heeaofe AC=l'D)
¡
i, the fqnm of lite olher axis of Ihe geoeratiog dlipfc
Ihmfore Ihe eirele of Ihe radills
HF
i, equal to a rirele· ID the fqu are of the axIS of the fphete .
oflhe radiu, Ef' Ingclher wit" a
ei rcl~
of Iheradw. GF:
And .lhis ho/ns, whether the
fpl!~roi~
be fouod by
a
and finee Ihis is lIue every where, alllhe eireles
logel!~er
rel'olullOO arcunOIhe grealcr or lctler axIS.
defcribed by Ihe refpellive
r.dti
¡.¡
r
(th.1 is, Ihe eyllO'
.
e o
ROL L A
JI
r
1.
.
der) are eqoallo all Ihe eireleo defenbed by the refpee-
Heoee
11
apí':ars, ,hal Ihe half
oF
Ihe /phcro,d, Form.
li"
radir
EF and FG (thal is, to the hemifl'here aod Ihe ed hy Il.e ro:."on uf lite fpaee AHI'C around Ihe axis
cOne lokentogether) ; hut, (hy the ,olh 12. Eue!. ) the AC, ,s dCl1ble bf
I~C
cone.geo:raled by Ihe Iriaoglc
i\
~'C
Ihe eooe gcnermo by Ihe triaogle DAC is ooe third part .bol1l Ihe (;,me ax,s
1
wlllch
IS
Ihe 32d prop.
of
Arehi.
of the eyltnder generalcd by ¡he fq uare BC.
Whe~ee
il medes of eunmus ,nn fphe,oitls.
fl!lIo~!,
thallh. heruiFl'htr. geoew cd by Ihe rOl
'llon.of.
(.o
~I
O.L LAR
r
2.
me quadrant
AllE.e
I! cqu¡l
10
lbe rCnlJini" c
I~U
!lurd
Ilenee,
It~evllfe,
IS
cI·"lrnllhe meaCure of frgmcnrs of
the