1000000030

I ON X;;:XIX.

r.,

.19

'To mm)ure

¡h,

jiqupciu

ora

q/I/I'¡cr.·-ne·

caut", cylinderd'lr'e, , ,,eryh,,1c 1 rom . priTm,'I'hofe oppofirc

pl.ne

! (ur bales) are ordin'le figures of ao ,nfiolle numo

ber uf lides, il ' pp' ''' lh.u Ihe fuperfieies of a e),linder,

",il!>om lhe bal.s.

equa llO ao inl:n'le oumberofparalle·

l~gral11s;

IheeoOlOlon,ltilurleof

,11

\lhiehi. lhef,mewilh

Ihe hcighl of lhe cylioder. and lh. bales of lhemall oif·

fer I'ery liule from Ihe periphery 01' lhe cireJe which is

lhe bafe 01' lhe eylinder. Therefore l.his periphery mul·

liplied inlo lhe eommon heighl, gives the fuperficies of

lhe eyllOder, exelud,ng Ihe bafes; whieh are 10 be mea·

f"red feparardy by lhe help of lhe 330 prop.

Tllis·p.ropolilion eoneerning the mealu re of lhe furface

of Ihe cyl,oder (exeJuoing its bafis) is evident frolll this,

that IVhen il is eooeeived to be fpread OUI, il beeoOlcs a

pmllclog"Ol. whofe

fe is Ihe periphery of Ihe eireJe

01' Ihe

b.te

of Ihe ellinder Ilrelcheo iOlo a righl lioe,

aod whofe heighl is Ihe fame with lhe he'ghl of Ihe ey·

lindero

PRO

I T ION

XL.

E T

II . Y.

7°5

ED.

ArehiOledes

demo~nl'ms

(;?

,nd

íD. 1

fI,A.c.

ra/, har

Ihe fu ,

f.ce

of IlliS [er.m,,,,, cxeluJiog Ihe ei r.

bafe, i, equal tOIhe area

01'

a eirele wh"fe radilll

is Ihe

ri~hl

line.

dralVn fr om lhe

vcrtcx

or Ihe f('\{.

meollOIhe penphcry of Ihe eirele DE. Tb<refurc,

(Ir¡

rhe 33d prop.)

11 15

eafily mcafured.

e o

R O L L

I .

Heoce Ihal rart of Ihe fllrf,iCe of a fl'here Ihal liuh

belw:en rlVo parallcl planes is e:'[ily mc:dured, by

fu~ '

lraél, ng lhe [urface 01 Ihe

I<lfa

legOleollroOl rhe {ud"e

of Ih. grealer regmen!.

e o

ROL LA R T

llence

I~kewife

ir [olloIVs, Ihal lite furf.iCe of

eyli n.

d<'l', dercrlbed abum

[phere (e::e1uo,ng lhe. bafis) is e.

quallo d,e[urface of lhe fphere, aod lhe parts of lhe

one 10 lhe parts of lhe olher, inlereepled bel"'een plaCe!

parallcl lOIhe bafis of Ihe eylinder.

Salid

F~!i"I'e¡

Ilnd tbcil'

MClljU/'nt!OII,

cQ/llprc.

bendillg tikelU({e ,be Prillciple!

gllugil/rt

VfiJe/¡

alllígure!.

FIG. 20.

To /IIeafu re ¡he furface

r(~ht

cone.–

The furface of

righl cune is very linle dilfereol from

As io Ihe form er part of Ibis !rm ife we look ao ioch

rhe furfact of a righl p)'ramid, haviog ao ordioate poly. for Ihe fm, lIel! meafure io lengdl. aoo ao ioch fquare for.

goo for ilS bafe of ao inhnite oumber of fides; Ihe furo !he flOallcfl fu perGcial meafure; [o OOIV, in Irealiog of

face of whieh (exeludiog Ihe baf,) is equallo Ibe [um of Ihe meo[ur'lioo of [ol,ds, \Ve lake a eubical inch for Ihe

Ihe triaogles . The [nm of Ihe bafes of 'lhefe tri'lngles [,naJ lell [olid mea[ure. Of Ihd'e 109 make a Seuls

piol~

is equal 10 Ihe periph"y of Ihe cirele of Ihe 'bafe, aod olha liquid meafures depclld On Ihis, as is geoeraJly

rheeommoo heighl of Ihe Iriaogles is Ihe fideof lh.eeone koowo.

wherefore Ihe fum of lhe[e tri.ngles is equal to

lo dry meafures, Ihe firlol, by /la:ule, eonlains

I ~{rhe produll of lhe [um of rhe bares

(i.

Ihe periphery of pims; aod 00 IhlS depeod Ihe olher dry meafures: Ihere·

rhe bafe of Ihe eooe) mulliplied iOIO Ihe halr of Ihe fore, if Ihe eomemof

a~y

foliJ be g,veo io eubical ioch·.

common heigh!, or il is equal

the produ{t of. Ihe pe. es, il wiJl be ea[y 10 reduce Ihe fame 10 Ihe eommeo li·.

riphery of Ihe bafe.

quid or dry mea[ures, aoo eonverfely 10 reduce Ihefe 10

If the area of Ihe bare is likewife wanled, il ;s to be folid ioehes. The liquid and dry mea[ures, in ufe a·

fouod feparately by rhe 33d prop. If Ihe furface of a mong olher oalions, are known fronl Iheir IVrilers.

cooe is fuppo[ed 10 be [pread pUl 00 a plaoe, it will be.

.. As 10 lhe Englilh liquid mea fUi es, byaél of p3r1,a.

come a [e{tor of a eirele, who[e radius is Ihe (ide of rhe "meol 1706, any round verTel, commonly cal!ed a cy.

eone; aod Ihe are lermioaring Ihe [télor is made from Ihe "Iinder, haviog ao evenbOIlOO!, beirg feven ioches io di·

peripheryof Ihe bafeo

Whence. by corol. 11d prop. of "ameter Ihroughoul, aod fix ioehes deep from Ihe IOp

Ihis, ils dimeofioo may be found.

" of Ihe iofide 10 Ihe bOIlOO1, (\\ llÍch venid will bc.fouoJ.

e o

ROL LAR

hy compui' lioo

cOnl,in

T~~J

cubie.1inches),

Heoee il \ViII be eafy 10 mea[ure Ihe furflce of a

fruf

"or ally

I'rfrd

cootaioing 23 1 cubical ioches, aod 00

IUIiI

of a cooe Cllt by aplane parallello Ihe bafe.

" more, is deemed 10 be a lawrul wioe·gallen. Ao

PRO

O S I T ION XLI.

" Eoglilh pinl Iherefore COOlalOS

28t

cubleal inches ;

FI G.21.

Tomeofu," ¡he furface af agivtnfpA",.

"IIVO piols make a quart ; four qnans a gaJlon; 18

-Lel Ihere he a [phere, whofeceotre is A, and lel Ihe .. gaJloos a roundlel; Ih, ee roundlels aod ao half, or

area of ils coovex[urface be required. i\rehillledes de· "(,3 grillons, m,ke a hogO,ead ;

lh~

half uf a hogOle:ld

mooílrates

ü ¡

prop.

l .

book of rhe [phere and cylio· "is a barrel: ooe hog(h,a,¡ ano a Ihird, or

g,lJoos,

der) ,h' l ils rurface is equal ro Ihe arca of rour greal eir· "make a ruocheoo; one runehoon and a haJ(, or IIVO

rles of Ihe ¡'phere; lhal is, lel lhe area of Ihe grear cir· "hogn,eaJ!, u,' 126 g.lIolls, make a pipe or hUl!; Ihe

el:

be muhiplied by

and Ihe proc!u<'lwill gil'e Ihe area "Ih"d parl 01' a pipr, or 42 g,lIo"" n",ke a lierce ;

of Ihe [phere; or, (by Ihe

20,h ~.

and 2d 12of Euel.) "IIVO ripes , or Ihroc punch, uns, or four hoglh(,ds.

l~e

area of ,he [pherc giveo is equal 10 Ihe arca of a "make a IOn of wine. Thougb ,h. Eoglifl, lI'ine ",l.

cirele whofe "di"s is Ihe rlghl lioe

Be,

Ihe di,meler of " 1011

's no.' r,x,d 31 '3 1 c"b,c.I,"ehcs, Ihe lIandanl

,oe [phere. 1hererore h.ving me.rureo (by 31d prop ) " kept in Cui"i",1I bcior, me [u, eJ, befvf< m, ny r er ·.

rhe e"e1r. oefenhen " ilh ,he r.J,us Be,tllis Il'ill give lhe "[ons

01'

d,l/inél,oo,

J\1.y

2í'h 16SS, il Iras (our.d lO.

(Jrfaee

lhe (j,h"".

" eonl."n only

lu,h ,nches.

P O

1 T IO N XLII.

" lo dtel:ngltfl, bw mealore, a galloo coolaios

~Sl

rl/; .22 .

T. ":eo/u,, ¡f, fllrfa" of a )eg"enl of

cublC.tI

inel", ; éor.r'qucndy

,sI

<uLi,,1 ioches make

" jph",.-Let

Ihere he a regOleO! eUI off by

I~C

pl.nc

"a 1"01, 111'0 pints m.,ke a

n'31',

rucr '1".r<5 makc "

V'O L.

JI.

}; °5 6.

7 1'

""n,