And in this case it will be, as rad. : sin. C:: sin. a or sin. b; sin. c And, as rad. cos. A or cos. B: tan. a or tan. b tan. c. 2. The preceding values of tan. (A + B) and tan. (A B) are very well fitted for logarithmic computation: it may, notwithstanding, be proper to investigate a theorem which will at once lead to one of the angles, by means of a subsidiary angle. In order to this, we deduce immediately from the second equation in the investigation of prob. 3,

Then, choosing the subsidiary angle, ø, so that tan. tan. a cos. C*,

that is, finding the angle ø, whose tangent is equal to the product tan. a cos. C, (which is equivalent to dividing the original triangle into two right-angled triangles,)the preceding equation will become

Given two angles of a spherical triangle, and the side comprehended between them; to find expressions for the other two sides.

1. Here, a similar analysis to that employed in the preceding problem, being pursued with respect to the equations Iv, in prob. 3, will produce the following formulæ :

* In order to comprehend the principle of the simplifying transformations that spring from the introduction of a subsidiary angle, let it be recollected, that in all such cases the object is to change into a product a binomial of the general form

M sin. + N cos. 0.

Here the contrivance consists in putting as a common factor one of the quantities M or N ; whence, for example, we have

N
M

Thus, putting equal to a tangent, or to a cotangent, lines which are susceptible of receiving all possible values, we shall have

M sin. (0+)

cos.

M cos. (0-4)
sin.

The two latter are evidently fitted for logarithmic computation.

+ The formulæ marked vi and VII, converted into analogies, by making the denominator of

2. If it be wished to obtain a side at once, by means of a subsidiary angle, cot. A cot. c

find so that

cot.c

tan. ; then will cot. a cos. (B

-

cos.

THEOREM VI.

Given two sides of a spherical triangle, and an angle opposite to one of them; to find the other opposite angle.

Suppose the sides given are a, b, and the given angle B: then from theor. 7, sin. a sin. B we have sin. A = ; or, sin, A, a fourth proportional to sin. b, sin. B, sin. b

and sin. a.

PROBLEM VII.

Given two angles of a spherical triangle, and a side opposite to one of them; to find the side opposite to the other.

Suppose the given angles are A, and B, and b the given side; then theor. 7

the second member the first term, the other two factors the second and third terms, and the first member of the equation the fourth term of the proportion, as

cos. (a+b): cos. (a - b):: cot. C: tan. (A + B),
sin. (a+b): sin. (a - b):: cot. §C; tan. (A — B), &c. &c.

are called the Analogies of Napier, being invented by that celebrated geometer. He likewise
invented other rules for spherical trigonometry, known by the name of Napier's Rules for the
circular parts, which will be found under the head Mnemonics in Spherical Trigonometry. Four
other analogies, corresponding to these, have been given by the celebrated astronomer, Gauss, of
Göttingen, which, in subsequent investigations in Spherical Trigonometry, are of great value,
and even for the solution of the cases of spherical triangles to which they are adapted, are often
more convenient than those of Napier. Those two which apply to the solution of Problem IV.
bear the same relation to Thacker's theorems on plane triangles (Course, vol. i. p. 395) that
Napier's do to the corresponding plane problem solved in the usual manner.
Their peculiar
advantage is felt when the base alone is sought, without requiring the other two angles, as they
save three openings of the tables and two other steps of the process. We give them in this
note, with an indication of the method employed in their investigation annexed.
From the formulæ in Problem II. we have, by multiplication,

Also, taking the values of sin. A sin. B and cos. A cos. B from problem III., we shall find in precisely the same manner

sin. (a+b)
sin.e

sin. c. . . . . (3)

which are the third and fourth analogies of Gauss: and from these four, the analogies of Napier might be at once deduced, by division.

sin. b sin. A

gives sin. a sin. A.

; or, sin. a, a fourth proportional to sin. B, sin. b, a

sin. B

THEOREM VIII.

In every right-angled spherical triangle, the cosine of the hypothenuse equal to the product of the cosines of the sides including the right angle. For if A be measured by 40, its cosine becomes nothing, and the first of th equations I becomes cos. a=cos. b. cos. c. Q. E. D.

THEOREM IX.

In every right-angled spherical triangle, the cosine of either oblique angle, i equal to the quotient of the tangent of the adjacent side divided by the tangen of the hypothenuse.

If, in the second of the equations 1, the preceding value of cos. a be substi tuted for it, and for sin. a its value tan. a cos. a tan. a cos. b cos. c; then, re collecting that 1 cos.2c sin.2 c, there will result, tan. a cos. c cos. B = sin. c. whence it follows that,

In any right-angled spherical triangle, the cosine of one of the sides about the right angle, is equal to the quotient of the cosine of the opposite angle divided by the sine of the adjacent angle.

In every right-angled spherical triangle, the tangent of either of the oblique angles, is equal to the quotient of the tangent of the opposite side, divided by the sine of the other side about the right angle.

Whence, because (theor. 8) cos, a = cos. b. cos. c, and since sin. a cos. a tan. a, we have

sin h

sin h

sin h

1

tan h

THEOREM XII.

In every right-angled spherical triangle, the cosine of the hypothenuse, is equal to the quotient of the cotangent of one of the oblique angles, divided by the tangent of the other angle.

For, multiplying together the resulting equations of the preceding theorem, we have

In every right-angled spherical triangle, the sine of the difference between the hypothenuse and base, is equal to the continued product of the sine of the perpendicular, cosine of the base, and tangent of half the angle opposite to the perpendicular; or equal to the continued product of the tangent of the perpendicular, cosine of the hypothenuse, and tangent of half the angle opposite to the perpendicular *.

Here retaining the same notation, since we have sin. a =

tan. c

sin. b , and cos. B = sin. B'

; and if for the tangents there be substituted their values in sines and

tan. a

sin. b

cosines there will arise sin. c cos. a= cos. B cos. c sin. a = cos. B cos. c. Then substituting for sin. a, and sin. c. cos. a, their values in the known formula (equ. v. page 20,) viz.

sin. B

sin. c,

in sin. (ac) = sin. a. cos. c- Cos. a.

and recollecting that

it will become, sin. (a

c) sin. b cos. c tan. B;

which is the first part of the theorem: and, if in this result we introduce, instead of cos. c, its value

COS. a

cos. b

c) =

(theor. 8), it will be transformed into sin. (a tan. b cos. a tan. B; which is the second part of the theorem t. Q. E. D.

Scholium.

In problems 2 and 3, if the circumstances of the question leave any doubt, whether the arcs or the angles sought are greater or less than a quadrant, or than a right angle, the difficulty will be entirely removed by means of the table of mutations of signs of trigonometrical quantities, in different quadrants, marked VII in chap. 3. In the 6th and 7th problems, the question proposed will often

This theorem is due to M. Prony, who published it without demonstration in the Connaissance des Tems for the year 1808, and made use of it in the construction of a chart of the course of the Po.

This theorem leads manifestly to an analogous one with regard to rectilinear triangles, which, if h, b, and p denote the hypothenuse, base, and perpendicular, and B, P, the angles respectively opposite to b, p; may be expressed thus:

be susceptible of two solutions: by means of the subjoined table the student m always tell when this will or will not be the case.

1. When the data a, b, and B, there can be only one solution

when BO (a right angle),

=

2. With the data A, B, and b, the triangle can exist but in one form,

Those formulæ are always to be preferred in calculation in which the functions, as sine, tangent, &c. are given by means of the half arcs or angles, to those where they are given by means of the whole arcs or angles: as in this case no ambiguity can possibly occur. All such expressions, if adapted to logarithmic calculation, are real contributions to practical science; and to the discovery of such, the student should assiduously direct his attention.

Notes. It may here be observed, that all the analogies and formulæ, of spherical trigonometry, in which cosines or cotangents are not concerned, may be applied to plane trigonometry; taking care to use only a side instead of the sine or the tangent of a side; or the sum or difference of the sides instead of the sine or tangent of such sum or difference. The reason of this is obvious: for analogies or theorems raised, not only from the consideration of a triangular figure, but the curvature of the sides also, are of consequence more general; and therefore, though the curvature should be deemed evanescent, by reason of a diminution of the surface, yet what depends on the triangle alone will remain, notwithstanding.

We have now deduced all the rules that are essential in the operations of spherical trigonometry; and explained under what limitations ambiguities may exist. That the student, however, may want nothing further to direct his practice in this branch of science, we shall add three tables, in which the several formulæ, already given, are respectively applied to the solution of all the cases of right and oblique-angled spherical triangles, that can possibly occur.