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By construction. (Plate W. Fig. 13.) 1. With the chord of 60 degrees describe the primitive circle; and through the centre P draw apd, and cre at right angles to it. 2. Set one foot of your compasses on 90 degrees on the line of semi-tangents, extend the other towards the beginning of the scale, till the degrees between them be equal to the angle c=42°.12', and apply this extent from D to A. 3. Through the three points CAe draw a great circle, then AC will represent the quadrantal side. 4. With the tangent of the complement of the angle B, and centre c, describe an arc; with the secant of the complement of the same angle, and centre A, cross it in of with o as a centre, and radius oA, describe the great circle BAb. Then ABC is the triangle required. To measure the required parts. The sides AB and Bc (C. 163.) will be 48° and 64°.35’, And the angle A=54°.43". (G. 164.) Note. The right-angled spherical triangle abc (Plate V. fig. 14.) into which the quadrantal triangle ABC is transformed, may be constructed and measured exactly in the same manner as Case IV. of right-angled spherical triangles was constructed and measured. See Plate V. fig. 9. (P) CASE II. Given a quadrantal side, and the other two sides, to find the rest. The quadrantal side Ac = 90°t Required the Given - The side AB = 115°. 9" angles A, B, The side Bo = 113°.18' and c.

+ This example was formed from Example 2, Case VI. of right-angled spherics. The z b was raade equal to the supplement of the hypothenuse, the two sides A* and Bc were made equal to the angles of the right-angled triangle; and hence the legs of the right-angled triangle become angles in the quadrantal triangle. In a similar manner the other examples were made.

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Here the quadrantal side Ac must C represent the right-angle b : AB the Z c : and BC the Za; ac the supplement of the Z B, ab the Z c, and be the Z.A.

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(R) Case IV. Given a quadrantal side, one of the other sides, and the angle comprehended between them, to find the 2'est. *

The quadrantal side Ac- 90° Required Bc, Given & The side AB = 115°. 9' and the anThe angle CAB = 115°.55' gles B and C. Answer. Bc3=113°.18', the angle B = 101°.40', c=117°.34'. (S) CASE W. Given a quadrantal side, its adjacent angle, and a side opposite to that angle, to find the rest.

The quadrantal side Ac-90° Required Bc, Given { Its adjacent angle c-42°,12. and the anAnd the oppositeside AB=48°.00 gles A and B.

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(T) CASE VI. Given the quadrantal side, one of the other sides and an angle opposite to the quadrantal side, to find the rest. The quadrantal side Ac- 90° \ Required Bc, Given - The side AB = 115°. 9' and the anThe angle B = 101°.40' gles A and C, Answer. Be = 113°.18', the angle A = 115°.55', and c = 1 17°.34'.

CHAP. VIII.

III. PRACTICAL RULEs Fort solvin G ALL THE CASE8 OF oBLIQUE-ANGLED spHERICAL TRIANGLEs, witH A PERPENDICULAR ; AND THEIR APPLICATION BY LoGARITHMS.

RULE I,

(U) The rule Prop. xxv. (X. 173.) will solve ten cases; see also the rules P. 178.

It ULE II.

(W) When the three sides are given, to find the angles.

Any one of the three sides may be called the base. Then,
The tangent of half the base,
Is to the tangent of half the sum of the sides,
As the tangent of half the difference between the sides,

Is to the tangent of the distance of a perpendicular from the

middle of the base. (S. 179.) According as this distance is less or greater than half the

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base, the perpendicular falls within or without the triangle. When the sum of the two sides is less than 180°, the perpendicular falls nearest to the less side, when greater than 180° it falls nearest to the greater side; consequently the greater segment is joined to the greater side in the former case, and to the less side in the latter. The sum of half the base, and the fourth term found by the above proportion, gives the greater segment, their difference gives the less. The triangle being thus divided into two right-angled triangles, the remaining parts must be found by the proper rules.

- RULE III. (X) When the three angles are given, to find the sides. The co-tangent of half the sum of the angles at the base, Is to the tangent of half their difference; As the tangent of half the verticle angle, Is to the tangent of the excess of the greater of the two vertical angles (formed by a perpendicular), above half the aforesaid vertical angle. (W. 180.) If the sum of the base angles be less than 180°, the perpendicular and the less segment are nearest the greater base angle, if greater than 180° they are nearest the less base angle. The sum and difference of this fourth term, and half the vertical angle, gives the greater and less vertical angle formed by the perpendicular. The triangle being thus divided into two rightangled triangles, the remaining parts must be found by the preceding rules. (Y) CASE I. Given two sides and an angle opposite to one of them, to find an angle opposite to the other. The side Ac-80°. 19' Given The side BC=63°.50' X-Required the Z B. The Z. A=51°.30' DETERMINATION OF THE SPECIES. A perpendicular in this case is unnecessary. If Ac-H Bo, A+(B acute), and A+(B obtuse), be each of the same species with respect to 180°, B is ambiguous: — But if only two of these sums be of the same species, that value of B must universally be taken which agrees with the sum of the sides, in all such cases B is not ambiguous.

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BY FoRMUL.At II. page 200. Log sine B-(log sine A+log sine b)-log sine a-59°. 16'. This example is ambiguous, vide Table I. page 207.

EXAMPLE II.

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The Z. A - 126°.37' Answer. The ZB-48°.30', not ambiguous. (Z) CASE II. Given two sides and an angle opposite to one of them, to find the angle contained between these sides.

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DETERMINATION OF THE SPECIES.

1. If Ac and the Z. A be of the same species, the Z. Acd is acute. The perpendicular CD is of the same species as the Z.A. If Bc and DC be of the - i same species, the Z BCD is acute. * B

2. If the Z. BCD be less than the ZACD, and their sum less than 180°, then the ZACB is ambiguous ; but if their sum be not less than 180°, their difference is the true value of the Z ACB, not ambiguous.

If the Z. BCD be not less than the Z. AcD, and at the same time their sum be less than 180°, this sum is the true value of the Z ACB, not ambiguous.

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+ In using the Formulae, the three angles of the triangle are represented by A, B, c, and their opposite sides by a, b, c, The perpendicular is not regarded.

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