1000000030

¡ oS

1\1

the

fl,h~roill

cut

pl.ncs rerpendicular 10 the axis. For

the !rgment of Ihe 'l'hlro:.1 Illade

Ihe rot"ion of lhe

' pace i\i-lIIE,

rll,,~d

the I:xis ¡\E, is 10 lhe fegmcnt of

~he

fl'hc,e ha ving thc

f.me

oxis AC, ,nd maJe by the

fOldtlOn

01'

Ihe fegmcnt of lhe cirele

AII'IGE,

CFq

D1·

BIIt if the

Ill~afure

of this folid he wanted wilh Ids 1, –

hour, by Ihe 341h prop. of

.~r('hil11ed{s

of conoids anJ

Jpheroids, it will be as HE to AC+ EB; fo is Ihe cune

genermd by the romion of Ihe tri, ngle AH\:: round Ihe

,xis A

10 Ihe [egment of Ihe [phercmade by lhe rola–

tion of ll'e fpaee

I\NHE

l'Ound lhe

f.me

axis AE; "hich

could e.Gly be demonfl raled by the method of indlviGblrs.

e o

ROL

A 11

Henee il IS eafy to find the [olid content of Ihe fegmen l

of a Iphere or fphcroid intercepled between

lWO

pm llel

planes, perpendicular to the axis. This agrces as

~ell

IO the oblate os 10 the obloog fphcroid; as is Obl'ioU5.

e o

A R

FtG .30. If a e>1k is to he valued as lhe middle piece

uf ao obloog fpheroid, etu hy the twOplanes

and fC,

right angles lO the.•

~is:

lirll, let the [olIJ content of

rhe halffpheroid AllCED be ",eafured by the preceding

prop. from which Ict the folidity of lhe fegment

DEC

be fubtraéled, aod thm will remaio the fegmeo l

ARCD;

Rod lhis doubled will give lhe eapaeity of the can, re–

~ui rcd.

The (ollowiog melhod is gcnerally made o[e o( (or

{¡odiog lhe folid cootent of lúeh velTds. The double

area of lhe grearel! ci rele, tll'It is, of that whieh is de·

ferihed by the diameter A

al the middle of Ihe can" is

added to the area of the cireie al the eod, that is, of lhe

eirc/e

or FC (for lhey are ufually equal), aod lhe

third pan of this fum is takeo for a meao bafe of the

can,.; whieh therefore multiplied ioto the length of lhe

eafk

OP,

gives lhe eooteol of the velTtI

rC~lIired

Sontelint<s relTds have other

r.gu

res, dilTereol

fr~m

,Ilúfe we hat'e mroliooed ; the eafy mClhoó of meafuriog

\Vhieh Ol<y be learned from lhofe who praéllfe lhis arto

Wh"

h.th

already been dclivered, is fuflieieO! fo r our

purpofe.

PRO P O S [ T I O N U .

IG. 3

aod

J2.

70 filld

hO'lu mueh

ronlo¡"',d;"

• 1)(/[..1

li al

i, i"

/,arl

111.'1")',

'Whofe

a\iJ

porall,'/ lo

Ih(

horiwl.-L. t

AGR/l

ue lhe great eirele in Ihe

0",1-

die of dIe

e.tn"

whofe fegment GIl H is Glled with li–

quor, lhe fegmeol

G .~ H

heing empty: lhe fegmeo l GHH

is k'oown, if Jite tkplh EH be knoll'n. an,; EH a meao

prop0rlionall'etwcco the fegmeots ·of lhe dianU l., 1\ 13

and EL;; \lhichare fouotl by arad or ruler pUl iolO tite

v,lTe! al the onGce. Let lhe baGs of lhe can" at a Ole–

dillm, be fOllnd, whieh fnppofe

be theeirc/e C'K DL ;

aod lel lhe fcgmeot ¡(eL be Gmil" 10 the fegr.leol CMI

(which is eitlter fouod by

lh~

rule of threr, beeauf. as

the cilele ACli ll is 10 the eird, CK DL, lo is the fc,;'

MOl

I tOthe fl';:n lem ¡(el.; tlr

iuund from Ihe

t,bles (lf f"gm'ots m,de by auJilo,,) ; aod Ihe prodllle¡ of

,his f" gOlenl Inlllul'ltrd by the length of lite ,,11, will give

UIC

ItquiJ eon"nt rCOl,lininr, in ,lte CAlle.

O S I T I ON Uf.

jÍlid

II.(

filt"

,'0/11(1.1

a Ifgll/ar O/i.l

ordiJlal'

h/l.-tI

w,"edroo bcing a pyrami,I, lhe fel id eoolcot

i, 10u",1 by the 441h prrp. The hrxardton, or cube,

b,ing a

kll,"

oí I'fllm,

meafured by Ihe 43d prop.

A" o<\,e,I",,, conlill, of tIVO pyramtcls

úl

lhe lilO;e fqu.,e

b.le

"OU uf

Iqll.tI

h(·ights; euofec¡u.otly

tlS

Oleafure

fOllod by lite 4/lh prop. A duJeeaedron eonlills of

t"elve py"mid,

ha \'in~

equ, l e1u,!alerallnd equiaogular

rentJgondl bales; an,l lo or.e of thele beiog meafured (by

,Ite

~41h

p"·p. of ,his) aod Olulttplied by

12,

the pro·

dll~1

wIII be ttl\l.

tO Ihe folid content of lhe Jodeeaed,Oo.

The ieoli..droo eonlifis of 20 equal pyramids I""iog tri–

aogular bal,s ; lite fulid eonlcnt of om: of whieh being

rouotl (I,y Ihe 44'h prop.) aod tnuhiplied by 20, gives

lhe wllt'le lollJ. The bafes aoJ heighls of thefc pyra–

nllds, il yo" 1\'.10110 proee.<d more exaél ly, may be fouoJ

t rigon~metry.

Sce T RlGONOM ETRY .

R O P O S 1 T I ON LIlI.

jÍlld

Ih,

lid

COI:I",I

a.be

h'/lI,(Vtr

irr(gll/".

- Let tito given body be immerfed iota a velTe! of waler,

haviog ,he r,gltl'e of a parJlldopipedco or prtfm, and let

it be ootld

h~w

mueh tbe waler

raifed upoothe imoter–

(¡oo of lhe body. f or il is plaio, tha! lhe fpaee \l ltieh

the w"er fi lls, afler the immerGoo of the body, exeeed,

the fpaee filled before

tlS

immerGoo, by a fpace equal to

the lolid ec,nteol of lhe body, howevee irregular. Bllt

wheo this exeefs is of the figllre of a parallelopipedoo or

prilill , il is eaGly meafured by the 43d p, op. oí lhis,

oi:.

~y mulal,:)'io~

lhe area of the bafe, or moulh of the vef–

rel, iota thed.tfmoee of Ihe elevalionsof lhe water befare

aod afl er immerGoo. Wheoee is (ouod lhe folid eonleol

of tite body

gi.cn

'lo lhe fame lVay tite foliJ eonteot of a part of a body

may be rOllod, by Immerling lhat part ooly io water.

There Is 00 oeeellily to inGIl here00 dimioilhiogoren–

largiog [olid bodics io a

gire~

proportion.

\l/ill be eafy

10 dedllee thefe tltiogs from the 1tlh aod 121h books oí

Euelid.

T~e

(ollowiog rules are fubjoined (or the ready eom–

" putatioo of lhe eOOleolS of "eOds, aoci of aoy fulid! In

" lhe meafures ioufe in Great Blitain.

" I.

To fi nd lhe eonteot of a eylindrie "clTel in Eog–

" lin. wioe gallons, the diameler of lhe bafe aotl altitude

" of the vdlid beiog gil'eo in ioehes and

deeim.ls

of an

" inch.

" Sqllare the ollmber of ioches in lhe diameter of the

" I'en'cl; muhiply lhis fquare by Ihe oumber of ioehes

" in the hcig!n: theo mulliply lhe produél by the deei–

" m,1 fr.lélion .00 )4; and this laft prodllél ntall gi.e

" ,he eOOleOI in wioe gallons and d(eiotals 01' fueh •

"galloo. 1'0 exp"fs the rllle arithmelicall)'; let

" reprefent lhe oumber of inehes ao,1 decimals of ao

" in,h io the diameter of lhe I'ell'cl, aoJ H Ihe ioches

" .ntl decim,ls of ao ineh in lhe hcight of lhe vttTd ;

" lheolhe eOllteot in wioe'gallonsIhall be

DDHX,...' · ,

" or DDH X.coI4'

F.....

Lel.l,. Jiamcler D

51.2

" ioehlS. the hel"ht II

61.) ioehes, lheo the con–

" tlnt IIt.oIl be

·s

6'].;

.C"'.¡~

555.:i. ; ~2

lI'illc·g, llons. Ths Ide follolls Iroll

pr op

.i 3.

:lnd 4S. fur,

the

ftlllller,

Ihe :He;!

,tf

thlo!

" hale 01 the . cfI' l is in J']ltare ioches

DDx

. ;S; ~ ;

" and by Ihe IJl:er, Ihe rOOl,"l uf

ti:.

"I'c/

111

1, ItJ

" ILh:jII,'S