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39. As it is, obviously, difficult to retain in recollection the necessary rules and formulae in spherical trigomometry, attempts have been made by different mathematicians, to assist the student by contrivances akin to those which occur in repositories of artificial memory.
Napier, to whom this department of science is so much indebted, at the end of his Mirifici Canonis Constructio, obscurely suggested a simple and comprehensive method, characterised by the name of Napier's
_Rules for the Circular Parts; which apply, 1st, to right
angled spherical triangles; 2dly, by means of the polar triangle, to quadrantal triangles; and, 3dly, by means of a perpendicular from the vertical angle, to oblique angled spherical triangles. These rules were developed much more perspicuously by Gellibrand; and have, since his time, been explained by almost every writer on spherical trigonometry.
40. In a spherical triangle ABC, right angled at A, we have, as was shown in arts. 25–31, of this chapter,
These equations include only five of the six parts of the triangle, the sixth part, viz. the angle A being constant. Now, Napier’s circular parts are, the complements of the angles B and C, the complement of the hypothenuse a, and the other two sides 6 and c : that is, 90° — a, 90° — B, 90° — C, b, and c, are the five circular parts. Any one of these five may be called a middle part ; then the two parts next adjacent, one on the right, the other on the left, [not including the right angle] are called adjacent parts; and the next two, each separated from the middle part by an adjacent part, are called opposite parts. This premised, the rules are,
1. The rectangle of the radius and the sine of the
middle part = rectangle of the tangents of the adjacent arts. p 2. The rectangle of the radius and the sine of the middle part = rectangle of the cosines of the opposite arts. p These rules may be comprehended in one expression, extremely easy to remember, simply remarking that the vowels in the contractions sin, tail, cos, are respectively the same as those in the first syllable of middle, adjacent, opposite : for then, regarding unity as radius, we shall have sin mid = rect tan adja = rect cos op.
A rule which, by taking a, b, c, &c. successively for the . middle part, will be found to comprise the five expressions given above.
41. By the circular parts of an oblique spherical triangle are meant its three sides, and the supplements of its three angles. Any of these six being assumed as a middle part, the opposite parts are those two that are
of the same denomination with it; that is, if the middle part is one of the sides, the opposite parts are the other two ; and if the middle part is the supplement of one of the angles, the opposite parts are the supplements of the other two.
42. Mr. Walter Fisher has also given, in the Transactions of the Royal Society of Edinburgh, rules of easy recollection which will serve for the solution of all the cases of plane and spherical triangles. They are included in four theorems, which may be applied to plane triangles by taking, instead of the sine or tangent of a side, the side itself.
Theor. 1. Given two parts, and an opposite one.
S. A : S. O :: S - a - S . O,
Theor. 4. Given the three sides or angles of an oblique angled triangle.
(A + a) + 1 (A + a) - l. 2 * .—a- . —H·-· so. g. Here M denotes the middle part of the triangle, and must always be assumed between two given parts. It is either a side or the supplement of an angle; and is sometimes given, sometimes not. A and a are the two parts adjacent to the middle, and of a different denomination from it. o and o denote the two parts opposite to the adjacent parts, and of the same denomination with the middle art. P l is the last, or most distant part, and of a different denomination from the middle part. That these four theorems may be called to mind with the greater facility, the following four words formed by
abbreviating the terms of the respective analogies should be committed to memory, viz. Sao, satom, tao, sarsalm.
On the Areas of Spherical Triangles and Polygons, and the Measures of § Angles.
Theorem I. “
43. In every spherical triangle, the following proportion obtains, viz. as four right angles, (or 360°), to the surface of a hemisphere; or, as two right angles (or 180°), to a great circle of the sphere; so is the excess of the three angles of the triangle, above two right angles, to the area of the triangle.
Let ABC be the spherical triangle. Complete one of its sides, as Bc, into the circle BceF, which may be supposed to bound the upper hemisphere. Prolong also, at both ends, the two sides AB, Ae, until they form semicircles estimated from each angle, that is, until BAE = ABD = CAF = AcD = 180°. Then will cBF = 180° = BFE ; and consequently the triangle AEF, on the anterior hemisphere, will be equal to the triangle BCD on the opposite hemisphere. Putting'm, m', to represent the surface of these triangles, p for that of the triangle BAF, f for that of cAE, and a for that of the proposed triangle ABc. Then a and m' together (or their equal a and m together) make up the surface of a spheric lune comprehended between the two semicircles AcD, ABD, inclined in the angle A: a and p together make up the lune included between the semicircles car, CBF, making the angle c : a and q together make up the spheric lune
included between the semicircles Bce, BAE, making the
angle B. And the surface of each of these lunes, is to