3. Rules for the Twelve Cases of Oblique Angled Spherical Triangles.

Fig. 26, 27.

Given,

Case.

| Sought.

Rule.

1.

Two sides AB, AC, and the included

angle A.

One of the Draw the perpendicular CD from C the unknown
other angles angle, not required, on AB.
B. R:cos A : : tan AC : tan AD, less or greater than 90°,
as A and AC are of the same or of different affection.

BD=AB - AD.
sin BD: sin AD: : tan A : tan B, of the same or of

different affection, as AD is less or greater than AB. The third cos AD: cos BD: : cos AC : cos BC, less or greater side BC. than 90°, as A and B D are of the same or of different

affection.

2.

3.

The side

BC.

Two angles A and ACB,

and AC, the side between them.

Draw the perpendicular CD, from C the end of the given side next to BC, the side sought. R : cos AC: : tun A : cot ACD, less or greater than 90°, as AC and A are of the same or of different affection.

BCD=ACB, ACD. cos BCD: cos ACD :: tan AC : tan BC, less or greater

than 90°, as A and BCD are of the same or of different affection.

4.

The third
angle B.

sin ACD; sin BCD: : cos A : cos B, of the same or of

different affection with A, as ACD is less or greater than ACB.

Two sides AC and BC, and an angle A, opposite to one of

them BC.

5. The angle B, sin BC : sin AC : : sin A : sin B. The affection of B opposite to the is doubtful; unless it can be determined by this rule, other given that according as AC+BC is greater or less than 180°,

side AC A+B is also greater or less than 180°. 6. The included R: cos AC: :tan A : cot ACD, less or greater than 90°, angle ACB. as AC and A are of the same or of different affection.

tan BC: tan AC :: cos ACD: cos BCD greater than

90°, if one or all of the 3 terms ACD, AC, BC, are

greater than 90°; otherwise less than 900, ACD + BCD=ACB, doubtful. If ACD+BCD ex

ceed 180°, take their difference; if BCD is greater

than ACD, take their sum, for the angle ACB. 7. The third R : cos A : : tan AC : tan AD, less or greater than 90°, side AB. as A and AC are of the same or of different affection.

cos AC : cos BC :: cos AD: cos BD, greater than 90°,

if one or all of the 3 terms AD, AC, BC, are greater than 90°; otherwise less than 90°. AD+BD=AB, doubtful. If AD+BD exceed 180°, take their difference; if BD is greater than AD, take their sum for AB.

Given.

Case.

Sought.

Rule.

The side BC sin B : sin A : : sin AC : sin BC, doubtful; unless it opposite to the can be determined by this rule, that according as other given A+B is greater or less than 180°, AC+BC is also angle A.

greater or less than 180°. The side AB R : cos A : : tan AC : tan AD, less or greater than 90°, adjacent to the as A and AC are of the same or of different affection, given angles tan B : tan A : : sin AD : sin BD, doubtful. AB= A, B.

AD+BD. If AD+BD be greater than 180°, take their difference ; if B D is greater than AD, take their sum for AB.

10.

The third angle ACB.

R: cos AC: : tan A : cot ACD, less or greater than

90o, as AC and A are of the same or of different affec

tion. cos A : cos B : : sin ACD : sin BCD, doubtful. ACB

=ACD+BCD. If ACD+BCD exceed 180', take their difference; if BCD be greater than ACD, take their sum for ACB.

4. Rules for the first Ten Cases of Oblique Angled Spherical Triangles,

expressed in general terins.

1.

=tan I;

One of the Find an arc x, so that
other angles. rud Xcos given angle

cot side opposite angle sought
and let y=difference between x and the other given side,

tan given angle X sin x
then will

=tan angle sought.

sin y

The third

side.

Find an arc x, so that
rad Xcos given angle

-=tan x; and
cot one of the given sides
let y=difference between x and the other given side,

cos first side Xcos y
then will

=cos side sought. COST

Two angles and the side between

them.

4. One of the Find an arc r, so that
other sides. radXcos given side

Scotc; and
cot angle opposite side sought
let y=difference between x and the other given angle,

cos xXtan given side then will

Elan side sought.

5. The angle

opposite to sin side op. to ang. soughtX sin giv, ang:—sin ang. sought the other gi sin side opposite to given angle

ven side.

Two sides and an angle opposite to one of them.

6. The angle Find an arc x, so that
included be-radXcos side adjacent to given angle

=cot ;
tween the

cot given angle given sides:

and another y, so that tan same side Xcos x

=cos Y ;
tan other side

then will x+y=angle sought.
7.
The

third Find an arc x, so that
side.
rad X cos given angle

=lar ;
cot given side adjacent given angle

and another y, so that
cos side opposite given angle Xcos x

sy;
cos other given side

then will x+y=side sought.
8. The side op-
posite to the sin ang. op. to side soughtxsin giv. side

=sin side sought. other given sin angle opposite to given side

angle.

two

Two angles and a side opposite to one of them.

=sin yi

9. The side ad Find an arc , so that
jacent to the rad Xcos given angle adjacent to given side

=lan I; given

cot given side
angles.

and another y, so that
tan same angle x sin a

tan other angle

then will x+y=side sought.
10. The third Find an arc x, so that
angle. radXcos given side

cot
cot angle adjacent to it

and another y, so that
cos angle opposite to given side X sin x

cos other given angle
then will x+y=angle sought.

=sin y;

James Walker, Printer, 6. James's Court, Lawnmarket,

Edinburgh.