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Or, the sides a, b, or c, respectively, may be found by the formulæ in the fifth set of equations, page 184.

OF THE AMBIGUITY OF THE DIFFERENT CASES.

(F) When two sides, and an angle opposite to one of them, are given, to find the rest (see Case 1st, 11d, and 111d), the -values of these required parts are sometimes ambiguous.

(G) Practical Rules for determining whether the quantities sought are acute, obtuse, or ambiguous, are given in the solutions of the different cases.

The two following tables are the same as those given by Legendre at pages 400 and 401 of the 6th edition of his Geometry, and are deduced from Prop. XVII, page 149, and Prop. XVIII, page 150 of this treatise.

TABLE I. Let A, a and b be the given parts. Then,

90° arb two solutions.
Sa-b one solution.

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90°

Ja+b-180° one solution.

a+b180° two solutions.

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a+b180° two solutions.
a+b180° one solution.

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90° fab two solutions.
Lab one solution.

If A=90°, a=b, or a+b=180, the cases will not be ambiguous, but if b-90°, there will be two solutions.

When two angles, and a side opposite to one of them, are given, to find the rest (see Case Ivth, vth, and vith), the values of the required parts are subject to ambiguity; this triangle being supplemental to that wherein two sides and an angle opposite to one of them are given.

TABLE II. Let A, B, and a be the given parts. Then,

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ACB one solution.
LAB two solutions.

2. a 90°, 90° A+B 180° one solution.

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180° two solutions.

180° two solutions. A+B 180° one solution.

3. a 90°, 90°

4. a90°, B90°

AB one solution.

ACB two solutions.

If a 90°, AB, or A+B=180°, there will be but one solution, but if B-90°, there will be two solutions.

(H) We may likewise remark, that since any side or angle of a spherical triangle is less than 180°, the half of any angle, or half the difference between any two sides, or half the difference between any two angles, must be acute.

Hence in the equation, where cot c. cos (a-b)=tang (A+B), cos (a+b) (M. 188) it is plain that cotc and cos (a-b) are both positive (K. 100.), and therefore tang (A+B) and cos (a+b) must be both positive; consequently, half the sum of any two sides of a spherical triangle is of the same species as half the sum of their opposite angles. This rule is applied in the practical solutions of the different cases, and will frequently remove the ambiguity which would otherwise arise, where a quantity sought is to be determined by means of a sine.

CHAP. VI.

I. PRACTICAL RULES FOR THE SOLUTIONS OF ALL THE DIFFERENT CASES OF RIGHT-ANGLED SPHERICAL TRIANGLES, WITH THEIR APPLICATION BY LOGARITHMS.

Every spherical triangle consists of six parts, three sides, and three angles; any three of which, being given, the rest may be found.

In a right-angled spherical triangle, two given parts, besides the right angle, are sufficient to determine the rest.

The questions arising from a variation of the given and required parts are 16, but if distinguished by the data, the number of cases is 6.

THE GIVEN QUANTITIES ARE, EITHER

1. The hypothenuse and an angle.
2. The hypothenuse and one side.
3. A side and its adjacent angle.
4. A side and its opposite angle.
5. The two sides.
6. The two angles.

(I) RULE I.

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Draw a rough figure as in the margin; and let AG, AH; FB, FH; CI, CD; EI, EG, be considered as quadrants, or 90° each; then you have eight right-angled spherical triangles, every two of which will have equal angles at their bases: And the triangles CGF and EDF will have their respective sides and angles either equal to those of ABC, the triangle under consideration, or they will be the complements thereof. (L. 151.)

Then,

In triangles having equal angles at their bases.

B

H

The sines

of their bases have the same ratio to each other, as the tangents

of their perpendiculars.

And,

ABC and AHG.
and these are propor-
tional by inversion.

The sines of the hypothenuses have the same ratio to each other, as the sines of the perpendiculars. (M. 167.) (K) Illustration. 1st. In the triangles 1. Sine AG: sine AC:: sine HG: sine BC. 2. Sine AH: Sine AB::tang HG: tang BC. 2d. In the triangles FGC and FHB. - 3. Sine BH: sine CG::sine FB: sine FC.

5. Sine DC

6. Sine CG

and inversely, &c.

3d. In the triangles CGF and CID.
sine Fc:: sine ID: sine FG.

sine cr::tang FG tang ID.

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and inversely, &c.

4th. In the triangles EDF and EIG.

7. Sine EI sine ED: tang IG: tang DF, and inversely, &c. The student must remember that ABC is the proper triangle in the preceding proportions, and that AG, AH, &c. are each 90°, consequently sine AG, sine AH, &c. are each equal to radius.

BH is the complement of the base AB.
CG is the complement of the hypoth. Ac.
FC is the complement of the perp. BC.
FG is the complement of the angle A.
ED is the complement of the angle c.
HG EF is the measure of the angle A.
ID is the measure of the angle c.

In any of these cases, for sine, or tangent, write cosine, or co-tangent.

IG AC the hypothenuse, and DFBC the perpendicular. Since AB and Bс are perpendicular to each other, either of them may be considered as the base, and to avoid a number of

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different figures it will sometimes be necessary to make a and c change places, as is done in some of the succeeding cases; then, in considering these remarks, where AB occurs read BC, and instead of the angle a read c; and the contrary.

(L) RULE II. BARON NAPIER's rules.

1. In every right-angled spherical triangle there are five parts, exclusive of the right-angle, which is not taken into consideration; and these five parts are the hypothenuse, the two sides or legs, and their opposite angles. Now in every case proposed for solution, there are three of these five parts concerned, that is, two given (together with the right-angle) and a third required.

2. If the three quantities under consideration, viz. the two which are given and that which is required, are joined together, or follow each other in a successive order, without the intervention of a side, or angle, not concerned in the question; the middle one is called the middle part, and the other two the extremes conjunct, because they are joined to the middle part.

3. If only two of the three things under consideration are connected, or joined together, they are invariably called extremes disjunct, that is, not joined to the middle part, and the other term which is not joined with them, is called the middle part.

Then,

4. Radius × sine of the middle part-rectangle, or product, of the tangents of the extremes when they are conjunct.

5. Radius × sine of the middle part=rectangle, or product, of the cosines of the extremes when they are disjunct.

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Observe to write cosine or co-tangent, instead of sine or tangent; and sine instead of cosine, &c. when make use of either of the angles, or the hypothenuse; but not when you use the sides or legs.

6. Having found which is the middle part, and written the terms down as expressed in one or other of the above rules, according as the extremes are conjunct or disjunct, turn the terms into a proportion, thus: Put the required term last, that with which it is connected first, and the remaining two in the middle in any order.

(M) RULE III.

1. Radius × sine of either side sine of the opposite angle x sine of the hypothenuse.

2. Radius × sine of either side-tangent of the other side x cot of its opposite Z.

=

3. Radius x cosine of either of the angles tangent of the adjacent side x cot hypothenuse.

4. Radius x cosine of either of the angles angle x cosine of its adjacent side.

5. Radius x cosine of the hypothenuse cot of the other ▲.

6. Radius x cosine of the hypothenuse cosine of the other side.

sine of the other

cot of one angle x

cosine of one side x

Then put the required term last, that with which it is connected first, and the remaining two in the middle in any order, and you will have a correct proportion.

NOTE. This rule is the same as BARON NAPIER'S (O. 169.)

OF THE DIFFERENT SPECIES OR AFFECTIONS OF RIGHTANGLED SPHERICAL TRIANGLES.

(N) I. When the hypothenuse and an angle are given.

1. The side opposite to the given angle, is of the same species with the given angle.

2. The side adjacent to the given angle, is acute or obtuse, according as the hypothenuse is of the same, or of different species with the given angle.

3. The other angle is acute or obtuse, according as the hypothenuse and the given angle are of the same or of different species.

(0) II. When the hypothenuse and one side are given.

1. The angle opposite to the given side, is of the same species with the given side.

2. The angle adjacent to the given side, is acute or obtuse, according as the hypothenuse is of the same or of different species with the given side.

3. The other side is acute or obtuse, according as the hypothenuse is of the same, or of different species with the given side.

(P) III. When a side and its adjacent angle are given.

1. The other angle is of the same species as the given side. 2. The other side is of the same species as the given angle. 3. The hypothenuse is acute or obtuse, according as the given angle is of the same or of different species with the given side.

(Q) IV. When a side and its opposite angle are given.

The required parts are always ambiguous or doubtful, that is, they may be either greater or less than 90°, and therefore admit of two answers.

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