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The logarithmical co-secants of each of the angles adjacent to the required side, rejecting the indices, and the cosines of the above half sum and remainder; half the sum of these four o is the logarithmical sine of half the side sought. (I. 186.)

OR, RULE II. Take the supplements sf each of the angles, and use the remainders as sides in a new triangle. Find the angles of this triangle, by any of the rules in Case V. the supplements of which will be the sides sought. (U. 137.)

(R) CASE I. Given two sides of an oblique spherical triangle, and an angle opposite to one of them, to find the rest.

In the oblique spherical triangle ABC.
The side Ac-80°. 19'
Given

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By construction. (Plate P. Fig. 15.)

1. With the chord of 60 degrees describe the primitive circle; through the centre P draw CPe, and apr at right angles to it. 2. Set off the side Ac =80°.19% from c to A, by the scale of chords. 3. Through A draw the great circle Abbn, making an angle of 51°.30 with the primitive. (P. 160.) 4. Set off the side BC =63°.50' by a scale of chords, from c to m, and draw the parallel circle mbH.m. (Z. 162.) Through the points b, B, where it euts the oblique circle Abbn, and the point c, draw the great circles Cbe, CBe. 5. Then, Abc or ABC is the triangle required, each having the same data, which shews this example to be ambiguous.

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9.95304 || Because ac--Bc, A + (B acute), and A+ 9-89354 || (B obtuse) are each of the same species 9.993.77 || with respect to 180°, the Z B is ambigu- ~ 9-93427 || ous (Y. 232.) being=59°. 16 or its supplement 120°.44'. '

sine Bc = 63°.50'

: sine ZA=51°30' ::sine Ac – 80°. 19' : sine Z b = 59°, 16'

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In the oblique spherical triangle ABC.
The side Ac- 57°.30'
Given & The side BC = 115°.20/} Required the rest.
U The Z. A - 126°.37'

By construction. 1. With the chord of 60°

(Plate V. Fig. 16.)
describe the primitive circle,

through the centre P draw cre, and apr at right angles to it. 2. Set off the side Ac-57°.30 from c to A, by the scale of

chords.

3. Through A draw the great circle ABn, making an angle

of 126°.37 with the primitive.

4. Set off the side BC = 115°.

c to m, and draw the parallel c

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the point B, where it cuts the oblique circle ABn, and the point

c, draw the great circle cBe.

5. Then ABC is the triangle required; and though it has

exactly the same data as the fo are ambiguous.

To measure the

rmer example, none of the parts

required parts.

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The ZA-51°.30'
Given & The Z. B=599.16' Required the rest.
The side BC=630.50'

"By construction. (Plate V. Fig. 17.) 1. With the chord of 60 degrees describe the primitive circle, through the centre P draw BPe, and DPE at right angles to it. 2. Set one foot of your compasses on 90 degrees, on the hime of semi-tangents, extend the other towards the beginning of the scale, till the degrees between them be equal to the angle B=59°.16%, and apply this extent from E to n (P. 160.); and through the three points Bne draw a great circle. 3. Set off the side BC=63°.50'. taken from a scale of chords, from B to m, and draw the parallel circle mom, cutting the oblique circle, Bne in c. 4. With the tangent of the angle A=51°.30' and P as a centre, describe an arc; and with the secant of the same angle, and c as a centre, cross it in o. 5. With the centre o, and radius oc, draw the great circle acA. Then ABC is the triangle required. ”

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1. With the chord of 60 degrees describe the primitive circle, through the centre P draw BPe, and DPE at right angles to it. 2. Draw the great circle BCe making an angle of 48°30' with the primitive. (P. 160.) 3. Set off the side BC- 115°.20% from B to m, and draw the parallel circle mcm (Z. 162.), cutting the oblique circle BCe in c. : 4. With the tangent of the complement of the angle A = 53°.23%, and centre P, describe an arc ; and with the secant of the same angle and centre C, cross it in o. 5. With the centre o, and radius oc, draw the great circles Acb, acb. Then ABC is the triangle required; and none of the parts are ambiguous.

Required the rest.

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- 990452 || Because ec + (ac acute,) and a + b only,
9.95609 || are of the same species with respect to
- 9'87446 || 180°, the side Ac is acute, and not ain-
- 992603 || biguous. (B. 235.)

sine ZA = 126°.37.
: sine be = 115°.20'
; : sine ZB = 48°.30
; sine Ac-57°.30

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(T) CASE III. Given two sides of an oblique spherical triangle, and the angle contained between them, to find the rest, The side Ac- 80°. 19' Given - The side AB = 120°.47' 5- Required the rest. The Z. A - 51°.30'

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By construction. (Plate V. Fig. 19.)

1. With the chord of 60 degrees describe the primitive circle, through the centre P draw CPe, and a Prat right angles to it. 2. Set off the side Ac-80°.19 by a scale of chords, and through the point A, draw the great circle ABn, making an angle of 51°.30 with the primitive. (P. 160.) 3. Set off the supplement of AB =59°.13' from n to m, and draw the parallel circle mbm (Z. 162.) cutting the oblique circle Abn in B. Through the three points C, B, e, draw a great circle, then ABC is the triangle required." - *

* A perpendicular may be drawn from the vertical angle cupon the base AB, by finding p the pole of the oblique circle Ann, (N. 159.) and drawing a great circle cor, through p and the point C. (W.161.)

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