2. To find the Leg A C.

The Analogy, R:s BA :: t B: t AC.

Set the S of 90° on F, to the S of 36o 15' on L; then against the T of 42° 34' on F, is the T of 289 30 on L.

3. To find the Angle C.

The Analogy, R: csBA:: SB: cs C.

Set the S of 90° on F, to the S of 53° 45' on L; then againft the S of 428 34 on F, is the S of 33° 031 on L.

Cafe 4, Ambiguous.

Given {

-

One of the Legs, as AC 28 30 and an oppofite Angle B42 34

1. To find the Hypothenufe B C.

The Analogy, s B: s AC:: Rs BC.

Set the S of 429 34' on F, to the S of 28° 30' on L; then against the S of 90° on F, is the S of 44° 52′ on L.

2. To find the Leg A B.

The Analogy, t BtAC: R: s B A.

Set the T of 429 34 on F, to the T of 28° 30' on L; then against the S of 90° on F, is the S of 369 15' on L.

3. To find the Angle C.

The Analogy, csCA: R::cs B: s C.

Set the S of 61° 30' on F, to the S of 90° on L; then against the S of 47° 26' on F, is the S of 56° 57°

on L.

1. To find the Hypothenufe BC.

The Analogy, R:cs AC: :cs AB: cs B C.

Set the S of 90o on F, to the S of 61° 30′on L; then against the S of 53° 45' on F, is the S of 45° 08′ on L.

2. To find the Angle B.

s

The Analogy, BAR:: tC A: t B.

Set the S of 36° 15′ on L, to the S of 90° on F; then against the T of 28° 30' on L, is the T of 42° 34' on F

3. To find the Angle C.

The Analogy, s CA: R::tBA: t C.

Set the S of 28° 30′ on L, to the S of 90° on F; then against the T of 36° 15' on L, is the T of 56° 57′ on F.

See the Remark at the End of this Chapter.

Cafe

Set the 7 of 56° 57' on F, to the T of 47° 26' on L; then against the S of 90° on L, is the S of 45o 08' on F.

2. To find the Leg A B.

The Analogy, s B: cs C:: R:cs BA.

Set the S of 42° 34′ on F, to the S of 33° 03'on L; then against the S of 90 on F, is the S of 53° 45′ on L.

3. To find the Leg A C.

The Analogy, s C: cs B :: R:cs AC.

Set the S of 569 57 on F, to the S of 47° 26' on L; then against the S of 90o on F, is the S of 61° 30' on L.

Remark.

In working any Analogy, by the Examples above, 'tis plain there is no Difficulty by the Sliding-rule, when the Analogy contains Tangents, both Greater or both Lefs than 45°; but when one Tangent is Greater, and the other Leffer, than 45°, then either the Line of Tangents, both on the Rule and the Slider, must be continued out beyond 45°, to work the Analogy by once fetting the Inftrument, or if (as is most com

mon

mon) the Tangents are numbered back again from 45 towards the Beginning, the Inftrument must be twice fet to give an answer.

Thus in Cafe 5 above, to find the Angle C, we have this Analogy; AC: R::tBA: 1C. Now here, because the Tangent of BA≈ 36° 15′ is lets, and the Tangent of C = 56° 57' is Greater than 45o or Radius; therefore if the Lines of Tangents be not continued, as before faid, the Analogy muft be worked at twice, as in the following manner.

1. Set the Sine of 28° 30' on F, to the Sine of 909 on L; then against the Tangent of 45° on F, is the Tangent of 64° 30' on L.

2. Then fet the Tangent of 36° 15' on L, to the Tangent of 64° 30' on F; and against the Tangent of 45 on L, is the Tangent of 56° 57' on F; which is the Quantity of the Angle C as required.

But if the Analogy be altered, and for the greater Tangent be taken its Complement, as taught heretofore; then it will ftand thus, tBA: R:: SAC :ct C; then fet the T of 36° 15' on L, to the T of 45 on F; then against the S of 28° 30' on L is the

And this, in fuch a Cafe, is much the better way, and more direct and ingenious, than that before by two Operations,

VOL II..

T CHAP.

CHA P. XIV.

The fixth Method of folving Right-angled Spherical Triangles, by Gunter's Scale and Compafles.

T

HE Nature and Conftruction of this Inftrument, you have alfo largely described in its Ufe in Plain Trigonometry, Vol. 1; to which I refer you. And as it is the moft commonly known of any other Trigonometrical Instrument, I fhall not need to fay any thing further of it here, but immediately proceed to fhew its Ufe, in folving all the Cafes of Rightangled Spherical Triangles, in the Manner following.

Cafe 1.

Given {

The Hypothenusfe

BC

44 5:2

and an Oblique Angle, as C = 56 57

1. To find the Leg A B.

The Analogy, R: s C::s BC:s AB.

Set one Foot of the Compaffes in the Sine of 90°, and extend the other to the Sine of 56° 57'; then that Extent will reach from the Sine of 44° 52', to the Sine of 368 15′ = A B.

2. To find the Leg A C.

The Analogy, R: t BC :: cs C: t AC.

Set