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APPENDIX

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SPHERICAL

TRIGONOMETRY.

CONTAINING

NAPIER'S RULES OF THE CIRCULAR PARTS.

The rule of the Circular parts, invented by NAPIER, is of great use in Spherical Trigonometry, by reducing all the theorems employed in the solution of right angled triangles to two. These two are not new propositions, but are merely enunciations, which, by help of a particular arrangement and classification of the parts of a triangle, include all the six propositions, with their corollaries, which have been demonstrated above from the 18th to the 23d inclusive. They are perhaps the happiest example of artificial memory that is known.

DEFINITIONS.
I.

If in a spherical triangle, we set aside the right angle, and consider only the five remaining parts of the triangle, viz. the three sides and the two oblique angles, then the two sides which contain the right angle, and the complements of the other three, namely, of the two angles and the hypotenuse, are called the Circular Parts. Thus, in the triangle ABC right angled at A, the circular parts are AC, AB with the complements of B, BC, and C. These parts are called circular; because, when they are named in the natural order of their succession, they go round the triangle.

II.

When of the five circular parts any one is taken, for the middle part, then of the remaining four, the two which are immediately adjacent to it, on the right and left, are called the adjacent parts; and the other two, each of which is separated from the middle by an adjacent part, are called opposite parts.

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Thus in the right angled triangle ABC, A, being the right angle, AC AB, 90°-B, 90°-BC, 90°-C, are the circular parts, by Def. 1.; and if any one as AC be reckoned the middle part, then AB and 90° -C, which are contiguous to it on different sides, are called adjacent parts; and 90°-B, 90°- BC are the opposite parts. In like manner

B

A

if AB is taken for the middle part, AC and 90°-B are the adjacent parts: 90°-BC, and 90°-C are the opposite. Or if 90° - BC be the middle part, 90°-B, 90°-C are adjacent; AC and AB opposite, &c. This arrangement being made, the rule of the circular part is contained in the following

PROPOSITION.

In a right angled spherical triangle, the rectangle under the radius and the sine of the middle part, is equal to the rectangle under the tangents of the adjacent parts; or to the rectangle under the cosines of the opposite parts.

The truth of the two theorems included in this enunciation may be easily proved, by taking each of the five circular parts in succession for the middle part, when the general proposition will be found to coincide with some one of the analogies in the table already given for the resolution of the cases of right angled spherical triangles. Thus, in the triangle ABC, if the complement of the hypotenuse BC be taken as the middle part, 90°-B, and 90°-C, are the adjacent parts, AB and AC the opposite. Then the general rule gives these two theorems, Rxcos BC=cot Bxcot C; and Rxcos BC = cos ABXcos AC. The former of these coincides with the cor. to the 20th; and the latter with the 22d.

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To apply the foregoing general proposition, to resolve any case of a right angled spherical triangle, consider which of the three quantities named (the two things given and the one required) must be made the middle term, in order that the other two may be equidistant from it, that is, may be both adjacent, or both opposite; then one or other of the two theorems contained in the above enunciation will give the value of the thing required.

Suppose, for example, that AB and BC are given, to find C; it is evident that if AB be made the middle part, BC and C are the oppo

site parts, and therefore Rxsin AB=sin Cxsin BC, for sin C=cos (90° C), and cos (90°-BC) = şin BC, and consequently

sin AB

sin C = sin BC

Again, suppose that BC and C are given to find AC; it is obvious that C is in the middle between the adjacent parts AC and (90°-BC), therefore Rxcos C=tan ACxcot BC, or tan AC =

tan BC; because, as has been shown above,

1

cot BC

COS C

cot BC

=cos C+

=tan BC.

In the same way may all the other cases be resolved. One or two trials will always lead to the knowledge of the part which in any given case is to be assumed as the middle part; and a little practice will make it easy, even without such trials, to judge at once which of them is to be so assumed. It may be useful for the learner to range the names of the five circular parts of the triangle round the circumference of a circle, at equal distances from one another, by which means the middle part will be immediately determined.

Besides the rule of the circular parts, Napier derived from the last of the three theorems ascribed to him above, (schol. 29.) the solutions of all the cases of oblique angled triangles. These solutions are as follows: A, B, C, denoting the three triangles of a spherical triangle, and a, b, c, the sides opposite to them.

I.

Given two sides b, c, and the angle A between them.
To find the angles B and C.

tan (B-C)=cot & Ax

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sin (b—c). (31.) cor. 1.
sin(b+c)
cos (b-c)

tan (B+C)=cot Axcos (b+c) (31.) cor. 1.

To find the third side a,

sin B sin A: sin b: sin a.

:

II.

Given the two sides b, c, and the angle B opposite to one of them. To find C, and the angle opposite to the other side.

sin b sin c :: sin B: sin C.

To find the contained angle A.

cot A-1an (B-C) sin (b+c) (31.) cor. 1.
sin (b−c).

To find the third side a.

Sin B: sin A :: sin b: sin a.

III.

Given two angles A and B, and the side c between them.
To find the other two sides a, b.

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Given two angles A and B, and the side a, opposite to one of them. To find b, the side opposite to the other.

sin A: sin B:: sin a: sin b.

To find c, the side between the given angles.

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The other two cases, when the three sides are given to find the angles, or when the three angles are given to find the sides, are resolved by the 29th, (the first of NAPIER'S Propositions,) in the same way as in the table already given for the case of the oblique angled triangle.

There is a solution of the case of the three sides being given, which it is often very convenient to use, and which is set down here, though the proposition on which it depends has not been demonstrated.

Let a, b, c, be the three given sides, to find the angle A, contained between b, and c.

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In like manner, if the three angles, A, B, C are given to find c, the side between A and B.

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These theorems, on account of the facility with which Logarithms are applied to them, are the most convenient of any for resolving the two cases to which they refer. When A is a very obtuse angle, the second theorem, which gives the value of the cosine of its half, is to be used; otherwise the first theorem, giving the value of the sine of its half is preferable. The same is to be observed with respect to the side c, the reason of which was explained, Plane Trig. Schol.

END OF SPHERICAL TRIGONOMETRY.

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