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APPENDIX

то

SPHERICAL TRIGONOMETRY,

CONTAINING

NAPIER'S RULES OF THE CIRCULAR PARTS.

HE rule of the Circular Parts, invented by NAPIER, is TH To rule of great ufe in Spherical Trigonometry, by reducing all the theorems employed in the solution of right angled triangles to two. These two are not new propofitions, but are merely enunciations, which, by help of a particular arrangement and claffification of the parts of a triangle, include all the fix propofitions, with their corollaries, which have been demonftrated above, from the 18th to the 23d inclufive. They are perhaps the happiest example of artificial memory that is known.

DEFINITIONS.

I.

If in a spherical triangle, we set afide the right angle, and confider only the five remaining parts of the triangle, viz.

Bb 3

the

the three fides and the two oblique angles, then the two fides which contain the right angle, and the complements of the other three, namely, of the two angles and the bypotenufe, are called the Circular Parts.

Thus, in the triangle ABC right angled as 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 fucceffion, 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 feparated from the middle by an adjacent part, are called oppofite parts.

Thus, in the right angled triangle ABC, A being the right angle, AC, AB, 90°-B, 90°-BC, 90° C, are the cir90°C, cular parts, by Def. 1.; and if any one as AC be reckoned

B

A.

90°

the middle part, then AB and 90° - C, which are contiguous to it on different fides, are called adjacent parts; and 90°-B, BC are the oppofite parts. In like manner, if AB is taken for the middle part, AC and 90°-B are the adjacent parts; 90° - BC, and 90°C are the oppofite. Or if 90°-BC be the middle part, 90°-B, 90°-C are adjacent; AC and AB oppofite, &c.

This arrangement being made, the rule of the circular parts is contained in the following

PRO.

PROPOSITION.

N a right angled fpherical triangle, the rectangle under the radius and the fine of the middle part, is equal to the rectangle under the tangents of the adjacent parts; or to the rectangle under the co-fines of the oppofite parts.

The truth of the two theorems included in this enunciation may be eafily proved, by taking each of the five circular parts in fucceffion for the middle part, when the general propofition will be found to coincide with fome one of the analogies in the table already given for the refolution of the cafes of right angled spherical triangles. Thus, in the triangle ABC, if the complement of the hypotenufe BC be taken as the middle part, 90°-B, and 90°C, are the adjacent parts, AB and AC the oppofite. Then the general rule gives these two theorems, Rx cof BC cot B x cot C; and Rx cof BC= cos AB x cof AC. The former of thefe coincides with the cor. to the 20th; and the latter with the* 22d.

To apply the foregoing general propofition, to refolve any cafe of a right angled fpherical triangle, confider 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 equidiftant from it, that is, may be both adjacent, or both oppofite; then one or other of the two theorems contained in the above enunciation will give the value of the thing required.

Suppofe, 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 oppofite parts, and therefore R x fin AB= fin Cx fin BC, for fin C cof (90°-C), and cof (90°-—BC)= fin AB

fin BC, and confequently fin C =

fin BC

Again, fuppofe that BC and C are given to find AC. It is obvious that C is in the middle between the adjacent parts

Bb 4

AC

AC and (90°-BC), therefore R x cof C➡ tan AC x cot BC,

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In the fame way may all the other cafes be refolved. One or two trials will always lead to the knowledge of the part which in any given cafe is to be affumed as the middle part; and a little practice will make it easy, even without fuch trials, to judge at once which of them is to be fo affumed. 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.

Befides the rule of the circular parts, Napier derived from the three theorems afcribed to him above, (fchol. 29), the folutions of all the cafes of oblique angled triangles. These folutions are as follows: A, B, C, denoting the three angles of a spherical triangle, and a, b, c the fides oppofite to them.

I.

Given two fides b, c, and the angle A between them.

To find the angles B and C.

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