1000000030

11'1

E T

Wh,'d'm ,

lhe

cafe of trigooo!llwy, BC, tbe di- the clemeots) tO contrive metilotls for this purprl e, ac'

Hance

fou~ht,

wllI be fou0'1.

cordiog tO die occ.fion: fo that there is no oeed of

atl~-

O P O S 1

IO N

XVr.

iog any more of Ihis fOrt. We

{hall

fubjoin hm a m' ·

F,e ,

~O.

ro nlr4/1rr Ih, drjl"nct o[ /wo p/acú

A thod by whidl (he oi:uneter of the earth Illay be luu"d

M.1lJ,

,vhic~

o,,,

ÍI

aCCffiblt,

~)'

Iht gr6pr.ollulrr.-

OUt.

Let there be ereaed at two points A aod C, fuflicienrly

PRO P O S

ION

XIX.

dirhnt, twO vifible ligos ; then (by Ihe t2n, prop. of tbis

PLA,.. CXVL F,c.

r ofind Iht diam"tr

.¡

Il,.

pan)

Id tbe tWO angles BAC, !JCA be t:>.ken by tite

earln[rol/l o/u Obftrvalion,-Let

there

chofeo a high

~r"phometer.

Let the dilfance of thellations A aod C hill AR, oear the Cea-OlOre, and !et tbe obCer·vator vn

be meafured witll a chain. Then the thiro angle B being the top 01 it, with ao' exatl quarlrant divided iuto mi–

koo.,n, aoo Ibe fide AC being likewiCe known; there- nUles and fecoods by traofvcrCe divifions, and filtCd with

fore, by tbe 6rn cafe of trigonomctry, the dillance re-

telcfeope in place of tbe common figlm, meafure ,he

quired, AB, will be fouod.

the aogle ABE cootaioed uoder the righl lioe AB, whkh

PRO

S {

ION

XVU.

goes to tbe ceotre, and the right lioe HE dralVn to the

Flc . 2t.

r o",'af¡rt Ay

Ihe

groph' lntltr, Int di·

Cea, a tangeot to the globe at

let tber.e be drawo

jla/lrt

I,v. plam, /ltilhtr o[ 'W!Jich

aCCfJ!ih/t.-

Crom A perpeodicular 10 BD, the lioe AF meetiog BE

Let

(11'0

Ihtioos C aod D be cbofeo, from each of which io F. Now in Ihe rigbt.angled Iriaogle BAF all

tI~

ao–

the plms may be feeo whofe dillance is fought : let tbe gles are giveo, alfo the fide AB, the Iteight of,the bill .

Rogle5 ACD, ACB, BCD, aod likewiCe the aogles whicb is 10 be found by fome

the foregoiog Dltlbods:

BDC, BDA, CDA, be meafured by tbe grlphomelcr; as exaélly as pollible; and

(by

lrigonometry) the fidco

let Ihe dill.oce of the

Ilatio~s

C aod D be meaCured by a BF aod

are foun.d , But, by eor. 36tb

Euc!. AF

cbaio, or (if it be oeceffary) by the precediag praélice. ;s equal to FE; tberefore

will be koown, More–

Now, io the triangle ACD, there are giveo tWO aogles over, by 361h

3. Elle!.

tbe rellangle under BA aod BO

ACD and

OC; tbereíore, the third CAD is likewife is equal tO tbe fquare of BE, And thenee, by 17lh

giveo ; moreover the fiele CD is giveo ; therefore, by the

6. Euc!,

as AB : BE :: BE :

D. Therefore, ,fioce

6rll caCe of Higooometry, the fide AD will be found . AB aod BE are already ginn, BD will be Cound bJ rrth,

After Ihe fame maooer, in tbe triaogle BCD, from all

Euc!. or by "le rule of three; ood

Cubr~2éting

BA,

the aogles aod ooe lirle CD given, tl:e Gde BD is fouod, there 'l/ill remain AD the diamelar of tbe eanh fought.

Whercfore, io the triangle ADB, from lhe given fioes

L 1 U M.

aod DH, aod rhe aogle ADB contaioed by them,

Many otber metbods might be propofed for meaCuring

the fide AS (the di(laoce Cought) is found by tbe 4tb the diameter of the earth. The mofl ex,él is that pro-

cafe

trigonometry of oblique.angled triaogles.

pofed by Mr Picart of lhe acade/TIy of Ceieoees at Paris.

PRO P O S 1

T ION XVIII,

" Accordiog te Mr Pi<:art, a degree of lbe meridian

C. 22.

rf</uird'by1ht grophol/ul<r and 9ua-

"al the lati'tude of 49° 21 ', was 57,060 Freoch toiCes,

¿ronl, lo II/tajl/r¡ an awJ!iblr hrighl

AB,

plaud jo on a

"eaeh

whicbCOntaiM

fix

Ceet of rhe fa me meaCure'

jlup, Ihat ont con n,ilhtr go n,or

in an horizonlal

"from which it follow!, that, iC Ihe earth

an exad

plant, mr ruede[ronl i/, al 'v< juppofld in Ihtjo/ulian

"fphere, lbe circumfereoce of a great cirele of it will

o/'tht

14/h

P,op.-Let

there be chofen '1oy Í1tuation as "be 123,249,600 Paris fm, and the femidiameter of

C, aod .notber D; where let fome mark be ere@.ed: "the earth r9.6t5,800 feet: but the l'rench malhe–

let tbe angles ACD aod ADC be fouod by Ibe grapbo- "maticiaos, who of late have ·cxamined Mr

Picart~s

meter; then the third aogle DAC Vlill be koown. Let "perations, alfure us, That Ihe ckgree io thar latitude

the fide CD, the dillance of the (la,ions, be Oleafu red "is 57,183 10iCes. They menCured

-a

degree inLap–

with a chaio, aod theoce (by trigoo.) the fide AC will he "lood, in the la!ilUde of 66° 20', and fouod it of

found. Again, io the triaogle ACB, rigbt aogled at B, "570438 toiCes .• By comparing rbefe degrecs, as wcll

naviog fouod by the 9uadraot the aogle ACB, the other .. as by the obCervatioos 00 peodulums, and Ihe thenry

angle CAS is knolVo like"..iCe: but the fide AC io lbe "of gravity, it appears Ihat Ibe earth is ao olilate fphe–

triangle ADC is already known; therefore Ihe heighl "roid; aorl (fuppofing thoCe degrees

be accurmly

req uir~d

AB will be found by tbe 4tb cafe of righ t·aogled "meafured) Ihe axis or diameler that palfes through the

triangles. If theheight of ¡he tower is waoled, the ao· .. poles will be to tite diameter of tbe equator, as

177

glc BC!" wilJ be fouod by the quadrant; wbicb being "1 ¡8i or thecarth will bc 22 miles higher at the equator

takeo froOl me angle ACB already koowo, the angle "than at tbe poles.

degree bas likell'ife ['eeo mN–

ACF \ViII remain: but tbe aosle

FAC

w;tS koowo before ; "fured 1.t tbe cquator, ami found 10 be confidmbly

I~(s

tberefore Ihe reOlaining aogle AFC will be konwo. But "tbao al the I.,titurle of Paris ; whicb

c()nfirm~

r!,c ob.

tbe fine AC lilas alCo knowo before; Ihcrefore, in the "late fi gure of

Ih~

eallh, Bur an accouot 01' this IJI!

triangle AFC, .11 the aoglcs aod ooe of the fides AC be- "meofurat;oo has OOt beco ptlbliOll'o as reto

If Ibe

iog koowo, Af, the beight of the tower l.bove the hill, " e3lth was

ao I:oiformdeoÍlty from Ihe «rrface

will be founJ by trip,onoOletry.

.. the

ce~lre,

!llro, accordiog to Ihe thco,y oCgra l'il,",

L 1 U M.

the OIcridian \Vould be an exaél ellil'fis, aod the axis

werc cafy tO add 0I30y other

Ill~th()ds

of meafl!ring "would be 10 Ihe t1iameler of the c9t1ator as 1,0 10

~eigh:s

and diflaoces ; but, ir what is aboye be tloder- "23r ; aroJ Ihc .!ilI'crence of Ihe fcmi,li'lm",'r oi' the

¡¡uod, it will hc e"fy (. rpetially for ooe th.1 is ..rCld in .. cqtlalor aod f"mi,xis abotlr 17 miles,"

VOL .

JI.

No,

Sí,

7 N