1000000031

T R '1 G

N O M E T

l. to ,beir di/!'erence

1 . 3 ~l '2

So is the uneeo' of half the fu m

0(1

6 o

tbe angl.. D and C

l .

S 10.273 72

To Ihe langen, of half th';r difFerenee 7°

so'

9. 13806

So by ,heorem 3. we have Ihe angles D aod C thu. :

To halflhe fom of ,he . ngles D aod C

61 °

58'

Add half tbeir differeoce

so'

And ,he fum i, the grea'er angle D

Alfo. from half Ihe lum

Tak. half the di/Fereoce

69°

~8'

58"

so'

Aod Ihere rrmains ,he lelfer angle C

54'

oS'

T ben by Cafe

JI.

have ,~

lollo",iog aoalogy fo r fiad–

iog DC ,h e leg required,

"iz.

S. C : B D : : S, B : D C.

t .

As ,he fioe of C

54°

08'

9'90869

ToBD

133

2. 12385

So i. Ihe Croe of B

56° 03'

99 1883

To DC

136. 2

2. 13399

CAS

VI.

Thret

Crd..

being giyen, 10 find the .ngl...

EXAMPL &: To ,he triangle

B C

(ihid.

•

9.) fuppofe

AB=1 56, AC= 185' 7, aod Be=8 4; ..quired Ihe .ngle.

A, B,

and

Ctomr/rically:

M.ke

AC= 185.7 from ony lioe of e–

qual pa",; and from Ihe

f.me

line taking

56=AB in your

complíres, fix oae foot of

tbero

A. aod with another

fweep aD arch ; then u ke 84=BC in your cOOlpalTes

and

.fixiog one foot in

with

the otber fweep an arch, which

",ill crof. Ibe former in B: 1.(lly. join ,be points S and A,

aod B and C; fo ,he lriaogle wiJl

coollruéled, alld Ihe

aogle. may be m"fured by

lioo of chords.

n .

lJy

calcula/ion :

Le, b Jl , he perpendicular, B

n ,

from ,he ""ex D, opon Ihe bafe AC; which wiJl divide

, he bafe iOlo 'wo fegOlents AD aud DC, ,be !ength. wbere·

may be foo nd by ,heo..

4. ,hus:

As Ihe bafe

A C

185.7

To ,he fu m

the fide. AB aud BC

~40

So i. ,he ddfercoce of Ihe fides

26893

2 ;8031

857 33

To ,he diff. of ,he fep'ments of ,he bafe 93 1.9687 1

And hninl! the fum of the Ccgments.

viz.

the whole bafe,

and

~heir

dlfFcrence,

fino

lhe fegmt:nu thelDfches,

[heorero

3 ..

thus:

T o half ,he fum of ,he fegOlents

A od half ,heir di/fereo<e

And ,be fUIn i. tbe greater fegOl en' AD

139.3

Alfo from h, lf ,he (um of Ihe f'gmen,

92.

T ake balf Ih.i r difference

46.5

T be remainder i. ,he

l.era

fe~m,n,

46.3

Now ,be triangle ASC i. divided. by tbe perp,ndicular

O S , inlo ,wo right·",gled trianeles, ADB .nd DBC ; in

,be fi rn of which are gioen lhe hypolhenufe AB=I 56, and

the bafe AD= 139

' o fi Ad the oblique

an~lcs.

for whicb

we han (Gy

C.fe

of reélangular trigonometry) ,he fol.

lowíng analogy.

víz .

AsAB

T o A O

So is the

radiu.

To ,be co·li,. of ,he an,le A

10 .00000

Alfo ,h. angle C i. found by ,he fanlt

c.fe

A. BC

T o CD

46'3

So is the

radius

90°

tbu, :

) .9

428

66 55

10 .00000

T o ,he co_fine of C

56030'

9.74 130

Hning fOU Dd lhe (\Vo

angle, A

and

luve ..

he third,

13, by

tlking

lhe

fum of the othc:r twO frool ' So,

,hus :

Tbe fum of all ,be Ihree angles i. 180°

T he fu m of

aod e i.

lO'

Tbe angle B i.

96° 50'

AIJ the propo"ions uf.d for ,h. folu,ion, of ,h e fe.." l

cafes

plain

trigooometry,

m3y

be performed

the

feale

and

compars.

On rhe

fcaJe (here

:IIre feveral

Jogarithmic

Jines,

viz.

one

Dumbeu , anothtr of fine,_ IDd one of

tangenu,

ó, .

A nd ,he w.y of working

proportion

,beCe is ,hi.,

tJiz.

ex tend

your compacrc:5 froln

lhe

6tH

rerm of your pro–

portioo, fou nd

0 0

rhe Ccale,

lhe fecoed ; and

with th:lil

extent, fixing

olle foor

in rhe

tbird

((rOl, lbe otbcr wiU

rnch

the Eourth ltrm required.

S P HERIC AL T RIGON OMETRY.

SPHERICAL

T l\ l G O NOM I TRY

is the ¡ rt

whereby,

{,OOl

three

given puu of a

fpherical

tri:llngle. we

d¡[cover

the

rd i ;

and.

Ijke plane

trigonometry,

clther

tight-angled ,

oblique.angl c:d.

But before we

give

the

analogies

foc

the folutioo of ,he feveral caf.. in 'e;,h", i, will be proper

premife the followiog thc:orems.

TH!OREM

In.1I

riehl.aneled fpherieal trianel..,

tbe fine of ,he hypolhenufe : radius: : Crne of a leg: fioe of

i" oppofite angle. Aod the fioe

a 'ee : radiu. :: ' ao·

gen, of the o,h.. leg: tangen' of iu oppoc... angle.

DEMO"URATJO"_ Le,

D ·A F

C (ihid.

fig.

l ') ..-

preCent the eighth

plrt

of a fphere, where

the

qu:tdraotal

planes EDFC, EDBC, are bo,b perpendicular

Ih. quad·

rantal

pl.ne

ADFB; and ,he

quadranl~1

plane ADCC i.

perpendicular 10 the

pl.ne

E D FC; and ,he fp;'erical tri –

angle ABC i. righl.angled at B, whe.. CA is Ih e hypo·

thenure, aod HA,

Be,

are ,he legs.

To Lhe ilrehes G

1-',

13, dra,\V the u ngents H F,

B .

and ,he fi oe.

el on ,he "dii DF D 3; alfo draw

,be line of Ihe arch

AB,

and CK Ihe fi ne of AC.; and

,hcn join IK and OL. Now liF , OS,

CM,

C I. a.. all

perpendicular lO ,h. plane ADFB. And li D, CK, OL,

lie all in tbe fame

pl.ne

AOCC. A lfo FD, IK. S L , lie

. 11

in the fa me plane

A DCC .

T h,refore, ,he righ' _angled

tri.ogle. IiF0, CIl{:, ODL, h.ving Ihe equal angles

HDF, CKI, OLB, are fio,ilar. And e K : OC : : e l : CM;

rhat

¡"

:liS

the (ioe of tbe. hYPolhenufe:

rad. : :

fine of a leg

fine of

i..

oppofil e anele. For CM i. ,he lioe of ,he are

GF, whicb meafures ,he angle CAB. Alfo, LB: DF::

DO :

FH;

,h.. is. as ,be fioe of a leg : radius : : tangen,

of the other

I~g

tangent of

its

oppofitc angle,

D .

Hence il follow•• tha' ,be fines of ,he angles of any 0_

bliqu. fpherical tri.ngle ACD

(ihid.

2.)

are 'o one ano·

thtr,

dirctHy,

as the

fines

the

oppofi1e

fides.

Hence

alfo foll ows, {hu, in

right-angled

Ipher.ical triangles,

ha·

ving Ihe fl\me

perpendicular, the fines of the.

bares

will

to e:tch othcr, iD.,crfdy,

the u ngcnts of

lhe

OI.nglcs ar

Ihe bafes.

TH ,SOR U :t