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Case 4, Ambiguous.

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One Leg, as

AC = 28 03 Given { and an Opposite Angle B = 42 34

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1. To find the Hypothenuse B C. The Analogy, s B : SAC :: R:s B C. Make the 11 S of 28° 30', a = S of 42° 34' ; so shall the = R give the || S of 44° 52', the Quantity of BC.

2. To find the Leg B A. The Analogy, t B : TAC :: R:s B A. Make the || T of 28° 30', a = T of 42° 34' ; then will the = R be the || S of 36° 15', for the Leg B A.

3. To find the Angle C. The Analogy, csCA:R::cs B : sc. Make the 11 S of 47° 26', a = S of 61° 30'; so will the = R give the 11 S of 56° 57', the Quantity of C.

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1. To find the Hypothenuse B C. The Analogy, R:csCA::cs BA:08 BC. Make the || S of 61° 30', a = Radius ; so will the = S of 53° 45' be the || S of 45° 08', the Complement of BC. VOL. II.

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2. To

2. To find the Angle B. The Analogy, SBA:R::t CA:t B. Make the || T of 28° 30', a = S of 36° 15' ; fo will the = R be the || T of 42° 34', the Measure of B.

3. To find the Angle C. The Analogy, sCA:R::BA:1C. Make the || T of 362 15', a = S of 28° 30' ; then will the = R be the || T of 56° 57', the Measure of the Angle C.

Cafe 6.

Given

The Angle
and the Angle

B = 42 34 с 56 57

1. To find the Hypothenuse B C. The Analogy, 1 C:ct B :: R:cs B C. For the sector, thus, tB:R::ctc:csBC.

Make the || T of 33° 03', a = T of 42° 34' ; fo will = R be the || S of 45° 08', the Complement of the Leg B C.

2. To find the Leg B A. The Analogy, :csC ::R:cs B A. Make the || S of 33° 03', a = S of 428 34'; then will the = R be the || S of 53° 45', the Complement of the Leg B A.

3. To find the other Leg A C. The Analogy, sc:cs B :: R:05 AC. Make || S of 47° 26', a = S of 56° 57 ; so will the = R be the || S of 618 30', the Complement of the Leg AC.

CH A P. XIII.

The fifth Method of folving Right-angled

Spherical Triangles, by the sliding
Rule.

T

HE Nature and Use of the Sliding Rule also, has been already fufficiently explained in

Vol. 1. of Plain Trigonometry ; however as its Use is here somewhat different ; I shall just give Examples of all the real Cales of a Right-angled Spherical Triangle this way ; in order to understand the Method of operating by this most convenient Instrument, I shall again acquaint the Reader, that, as it consists of two Parts, the Fixed and the Moveable or Sliding Part, and each of these hath a Line of Sines and Tangents, so placed as that the Sines on the Slider move by the Sines on the fixed Part, and the same of the Tangents ; I shall (in the Sequel) mark the Line of Sines and Tangents on the Rule (or fixed Part) by the Letter F, and those on the Slider by the Letter I ; this being premised, and the Reader presupposed to understand the Nature of VOL. II.

S 2

the

the Sines on the Rule ; I shall proceed to the Solution of the Six Cafes as before, with all their Varieties, by this Infirument.

Cafe 1.

8 The Hypothenuse

BC = 44 52 Given {

and an Oblique Angle, as C = 56 57

1. To find the Leg B A. The Analogy, R:sC::sBC:s B A. Set the Sine of 90° on F, to the Sine of 56° 57' on L; then against the Sine of 44° 52' on F, is the Sine of 36° 15' on L.

2. To find the other Leg A C. The Analogy, R:tBC ::csC:t AC. Set the Sine of 90° on F, to the Sine of 33° 03? on L; then against the Tangent of 44 52? on F, is the Tangent of 289 301 on L.

3. To find the Angle B. The Analogy, R:cs BC ::7C:ct B. Set the Sine of 902 on F, to the Sine of 45° 08' on L; then against the Tangent of 56 57' on L, is the Tangent of 47° 26' on F. But the Tangent and Cotangent are in the same Point on the Line ; so that in the same Place you also see 42° 34' = B required.

Case 2.

Given {

The Hypothenuse

B. C = 44 52 and one of the Legs, as B A = 36 15

1. To find the other Leg A C.
The Analogy, CSB A:R::csBC:CS AC.

Set the Sine of 53° 45' on F, to the Sine of 90° on L; then against the Sine of 45° 08' on F, is the Sine of 619 301 01 L.

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2. To find the Angle B. The Analogy, tBC:R:: t B A:0$ B. Set the T of 440 521 on F, to the T of 368 15' on L; then against the S of 90? on F, is the S of 479 261 on L.

3. To find the Angle C. The Analogy, sBC:R::SBA:s C. Set the S of 44° 52' on F, to the S of gol on L ; then against the S of 369 15' on F, is the Sine of 564 57' on L.

Case 3.

BA= 36 15

One Leg, as
Given { and the adjacent Angle B = 42 34

1. To find the Hypothenuse B C.
The Analogy, R:csB :: Ct AB : ct BC.

Set the S of 909 on F, to the S of 478 26 on L; then against the T of 538 45' on L, is the T of

44 52

45 08

{

on F.

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