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2. To find the Leg A C. The Analogy, R:SBA :: B:t AC. Set the $ of 900 on F, to the S of 360 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:0SBA :: :C. Set the S of 900 on F, to the S of 53° 45' on L; then against the S of 428 341 on F, is the S of 332031

on L.

Given {

Case 4, Ambiguous.

[ One of the Legs, as - AC = 28 30 and an opposite Angle

B = 42 34

1. To find the Hypothenuse B C. The Analogy, s B:s AC :: R:s BC. Set the S of 428 34' on F, to the S of 289 30 on L; then against the S of 902 on F, is the Š of 4452' on L.

2. To find the Leg A B. The Analogy, i B :t AC :: R: s B A. Set the T of 42° 34' on F, to the T of 28° 30' on L ; then against the S of gol on F, is the S of 369 15' on L.

3. To find the Angle C. The Analogy, csCA:R::csB: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.

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1. To find the Hypotbenuse B C. The Analogy, R:CSAC ::05AB :cs B C.

Set the S of 90° 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. The Analogy, s B A:R ::+CA: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 428 34'

on F.

3. To find the Angle C. The Analogy, s CA:R::+BA:1C. Set the S of 280 30' on L, to the S of 909 on F; then against the T of 36° 15' on L, is the T of 56° 57' on F.

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See the Remark at the End of this Chapter,

C.zje

Cafe 6.

Given

{

The Angle
and the Angle

B = 42 34
С 56 57

1. To find the Hypothenuse B C. The Analogy, tC:ctB::R:0 BC. Set the T 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 45° 08' on F.

2. To find the Leg A B. The Analogy, s B:05C ::R:cs B A. 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 532 45'

on L.

3. To find the Leg A C. The Analogy, sС:05 B :: R:cs A C. Set the S of 560 57 on F, to the S of 47° 26' on L; then against the S of 902 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 Less than 45° ; but when one Tangent is Greater, and the other Lesser, than 45o, then either the Line of Tangents, both on the Rule and the Slider, muft be continued out beyond 45°, to work the Analogy by once setting the Instrument, or if (as is most com

mon

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

Thus in Case 5 above, to find the Angle C, we have this Analogy ; s AC: R::t BA :1C. Now here, because the Tangent of B A = 36° 15' is lets, and the Tangent of C = 56° 57' is Greater than 45° or Radius ; therefore if the Lines of Tangents be not continued, as before said, the Analogy must be worked at twice, as in the following manner.

1. Set the Sine of 289 30' on F, to the Sine of 909 on L ; then against the Tangent of 459 on F, is the Tangent of 64° 30' on L.

2. Then set the Tangent of 36° 15' on L, to the Tangent of 649 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 stand thus, tBA:R :: SAC :ctC; then set the T of 362 15' on L, to the T of 45° on F; then against the S of 280 301 on L is the

o 1 Tof 33 03

on F.

56 57

}

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

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CH A P. XIV.

The sixth Method of solving Right-angled

Spherical Triangles, by Gunter's Scale and Compasses.

T

HE Nature and Construction of this Inftrument, you have also largely described in its

Use in Plain Trigonometry, Vol. 1 ; to which I refer you. And as it is the most commonly known of any other Trigonometrical Instrument, I shall not need to say any thing further of it here, but immediately proceed to shew its Ufe, in solving all the Cases of Rightangled Spherical Triangles, in the Manner following.

Cafe 1.

Given {

The Hypothenuse B C 44 52 and an Oblique Angle, as C = 56 57

1. To find the Leg A B. The Analogy, RisC:: SBC: s AB. Set one foot of the Compasses in the Sine of goo, 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, Rit BC ::65C : 1 AC.

:

Set

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