1000000031

T R

N O M E T R

Be (or fine of the fame angle in .he cirele ",hofe radius is

A e .) Now the angle A being 33° 40" Ihe angle e mufl

56°

20'

lherefore

looking in the

lablt

of artiticill

(jn~s,

Oc.

for Ihe fines of Ihe two angles, and in the uble of lo·

g.rilhm. (or Ihe logarilhm o( 86 the givcn leg, we (hall find,

procccding according to the foregoing proportían, that

Ihe required Icg se is 51.29 ; and ,be operotion will fland

as follow,:

1. 9345° A

9 .74380 S, A 33° 40'

1 ..

3°

9 .9202 7

56°

20'

1.758°3 S e

Secondly. making

IIlhe radiu., it is

pl.in

se, the leg

required, will be the tangent o( the given angle A; and fa

the proportion for 6nding

e, wheo

S is mode the

ra–

dius, wlll be:

R : T , A :: AS:Se

i .

as the radius in the table" is

la lhe

tangent of the

angle A in Ihe fame ; fo Ihe lenglh of S A, or radius in Ih.

feheme, to the lenglh o( S e or tangent of A in the (eheme:

therefore looki ng in the tables for ¡he parts given in

lhe

foregoing proponian, ilnd proceeding

Wilh

fhem

according

' o tha, rul<, we (hall fin d

e lO be

57.28

beloro, and

,be operalion wiU be

follow. :

9 .8 23\2

A 33° 40'

1.93450

A B

lJ ·75

802

10.0~OO

Rad. 90°

171802 S e )7.28

Lamy , by m.king Be, Ihe lec required, the.radiu,. il is

plai o Ihal

A R

wilJ be Ihe ..ogent of C, aod the proportion

for finding BC will be as folJows:

-r,

R. : :

A B : S C

í. ~.

Ihe tangeot o( C

56°, 20'

10.17648

is la

ndius

900

10.00000

the length of AB

.Y34So

'o Ih. lenglh

c (

)7.08

1.71802

C ...

I ~.

T ite angles and one of lhe

leg.

given, lO find

the hypolhenufe.

EX 'MPL':

In Ihe tr

ianr.le

ABC,

(ibid.

4')

fuppofe

124.

and the angl.c A 34°

20';

confcq uently {he

angle e

,,0

40" requi«d Ihe hypolhen\1fe A e, in Ihe

fame

pans

Wilh

A B.

(,'(~7JI~/"¡colly :

Tbis care is

confiru(led after lhe

fa,me

manner

witb

the former ; and the hypothenufc. A

fdund , by laking il! I<nglh in yO\1r eompalfes, , nd applying

that

the (ame lioe o. cqual parts (rom which AH was

taken .

11 . B,

cocll/lllion:

Fidi,

mak-ing

the

r::tdius,

(hall have lhe following proportion for

findin~

A C,

viz.

S.e:R :: AB:AC

as the fine of

40'

radiQ5

900

fo is

A13

12 4

150. 2

9.9 1686

JO 00000

2 .0t)~42

• .

17 656

Secondly, making A S the radius, we have this propor ·

tion,

viz.

R :

fec.

A : : A D : A C.

;. (. as

the.

radius

90°

JO.ooooo

to the recant of A

34° 2Ó

10.093 14

roisAS

1'4

' .0934 '

toAe

150 . '

2. 176,6

This n"y . Ifo be done, wilhou, Ihe hdp",f Ihe fec.n,,:

for fince R : fce . :: Co-S .: R ; lherefore, ,he former pro·

portian

will

be.come.

Co-S,

A : R :: A B: A

í .

, .

as the

eo-~ne

of A

34°, . 0'

9 .91686

is to tbe radius

90°

10. 00000

fo i, A

l'.)

2.0934'

to A e

150 ."

. l

7.656

Thirdl)', makin& S e the radius, we have the (ollowiog

proponían,

viz.

T . C : fec . C : : A S : A e.

t .

the taogeDl 'of C

51 ° , 40'

10 1651 8

is to fee. C

51°, 4°'

1°"4872

foisAB

1' 4

• .°9342

to AC

11°2

' . 17656

This Jikewife may be don. wilhou, Ihe help of (ecan" ;

for finee

T, :

Sec.:: S . : R

{herdare the forruer analogy

wall be reduced

tO this, TÚZ.

S, C: R :: A S : A

where no Cecants do appear; and it coincides Wilh that in

,he

r.r11

fuppofilioo of Ihis eafe, fo we {hall DOI repea, Ihe

operation

CASE IlI . The angles and hypolbenufe given, tO find

either

of lhe legs.

EX'MOLE.

In ,he triangle ABe,

(iiid.

nO 4') ruppofe

the hypolhtnufe AC = 146, and Ihe angle A=36°

25' ;

eonfequently Ihe angle C = 53° 31'; requi red ,he leg

A B.

l .

G~Dmerrically:

Dri1w lhe

!tne

at pleafure, and

make the angle

13 A

C equal lO 36° 21 ' ; Ihea t>ke A C

eqn,l

146 (rom , ny Jine o( equal pans ; laOly, from Ihe

poinl e, let (aJl Ihe perpendicular C B. on ,he line A B.

So ,he triangle is

connru~ed,

and A B may be meafured

(roOl Ihe line uf equal parlS .

11. By

colr,'¡ariDII :

Fira, m:lking

A-e

the

radius,

fu ..

1I have the following

propon

ion,

viz.

R : S, C.:: A C: A

i. ~.

As radius

90°

1000000

,he line o( C

53 °

35'

9.9°565

foi sAC

146

2. 16435

1" ·5

• .°7°00

Sccoodly . making AS the radlus, we have Ihe following

analogy:,

VIZ ,

Seco

A: R·: : A

e :

A S.

t .

Ihe f<C'DI

36°

25'

JO.094~S

radiu5

goO

10 ,00000

foi.AC

146

2., 64H

lO AH

1/7' 5

2.°7°00

Thi. m,y alfo be done ",ilhoul Ihe help

01'

fecants; for

fince fec :

R :: H::

Co-S, Ibe (ormer proportipDmay be re–

duccd

this,

vi=.

R :

eo-s,

A: : A C : A S,

wbi..:h

¡,

the

r,ime

""ilh lhe

proportion

in the

Grn

fl1ppofitÍon.

Thirdly, by fuppofing

C Ihe radius, we have the

(01:

Jowing

proponíon,

{,izo

Sec.

e :

T ,

e ::

a :

A D,

as Ihe fccanl of e

53°,

10.226.\1