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GENERAL OBSERVATIONS ON THE SPECIES AND AMBIGUITY OF THE CASES.

(A) The species of the sides and angles may be determined from the equations produced by Baron Napier's Rules, or from the preceding formulae, by attending to the signs of the quantities which compose the equations or formulae.

The sides which contain the right angle are each of the same species as their opposite angles, viz. a is of the same species with A, and b is of the same species with B. (R. 145.)

It may be proper to observe that where a quantity is to be determined by the sines only, and a side or angle opposite to the quantity sought does not enter into the equation, the case will be ambiguous, thus in the XIIth case, where sine b =

sine a . r - -
, the hypothenuse b is ambiguous.

sine A

sine A. sine b
, the sine

Again, in the IId case, where sine a

of a is evidently determinate, because it is of the same species with A which is a given quantity. (B) When an unknown quantity is to be determined by its cosine, tangent, or cotangent, the sign of this value will always determine its species; for, if its proper sign be-H, the arc will be less than 90°; if the proper sign be-, the arc will be greater than 90°. (K. 100.) (C) Again, in Case with, where radix cos b-cos c x cos a, it is obvious that the three sides are each less than 90°, or that two of them are greater than 90°, and the third less; as no other combination can render the sign of cos c x cos a like that of cos b as the equation requires.”

QUADRANTAL TRIANGLES.

(D) Any spherical triangle of which A, B, C, are the angles, and a, b, c, the opposite sides, may be changed into a spherical triangle of which the angles are supplements of the sides a, b, c,

* Legendre's Geometry, 6th Edition, page 381.

and the sides supplements of the angles A, B, C, (U 137.) viz.
if we call A, B, c' the angles of the supplemental triangle, and
a', b’, c' the sides opposite to these angles, we shall have
- A^- 180°–a; B' = 180°-b; C'- 180°–c
a’= 180°– A; bo: 180°–B; co- 180°–c
Hence it is plain, that if a spherical triangle, has a side b equal
to a quadrant, the corresponding angle B' of the supplemental
triangle will be a right angle, and since there are always three
given parts in a triangle, the supplemental triangle will be a
right-angled triangle, having two parts given to find the rest;
consequently, by finding the required parts in the supplemental
right-angled triangle, the different parts of the quadrantal tri-
angle will be known. -
(E) Formulae might have been inserted for solving the dif.
ferent cases of quadrantal triangles, but this would be making
an increase of formulae to very little purpose, since all quadran-
tal triangles are easily turned into right-angled triangles. .

SOLUTIONs of THE DIFFERENT CASEs of OBLIQUE-ANGLED
SPHERICAL TRIANGLES.

CASE I. Given two sides of an oblique-angled spherical triangle, and an angle opposite to one of them, to find the angle opposite to the other.

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CASE II. Given two sides of an oblique-angled spherical triangle, and an angle opposite to one of them, to find the angle contained between these sides. .

SoLUTION. Find the angle opposite to the other given side

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Case III. Given two sides of an oblique-angled spherical triangle, and an angle opposite to one of them, to find the

other side.

. Solution. Find the angle opposite to the other given side

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Also, I. Tang p =

cos C. tanga

II. Tang p

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tang

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CASE IV. Given two angles of an oblique-angled spherical triangle, and a side opposite to one of them, to find the other

opposite side.

SoLUTION. Sinea- -
SIne B

Sine b

sine A. sine b sine A. sine c

sine C

sine B. sine a sine B. sine c

sine A

Sine c

sine C

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CASE W. Given two angles of an oblique-angled spherical triangle, and a side opposite to one of them, to find the side

adjacent to these angles.

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CASE VI. Given two angles of an oblique-angled spherical triangle, and a side opposite to one of them, to find the third angle.

SoLUTION. Find the side opposite to the other given angle by Case IV.

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cos B. sine

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Case VII. Given two sides of an oblique-angled spherical triangle, and the angle contained between them, to find the other angles.

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Here b, c, and the included ZA are given, and formulae will be obtained by a mere change of letters, if a, b, and the Z. C.; or if a, c, and the Z B are given.

CASE VIII. Given two sides of an oblique-angled spherical triangle, and the angle contained between them, to find the

third side.
SoLUTION. Find the other

two angles by Case VII, and

then find the third side by Case V.

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Or, the sides a, b, or crespectively, may be found by the fourth

set of equations, page 184.

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