(p) (4) becomes R sin. CD = sin. PC sin.P By observing that sin. CD = cos. AC = cos.b. And that tan.PD = cot.DO = cot.A, etc.; and by running equations (n), (m), (o), and (p), back into the triangle, ABC, we shall have, (5) R cos.b = cot.A cot.C (7) R cos.A = cos.a sin.C (8) R cos.b == cos.a cos.c By observing equation (6), we find that the second member refers The same relation holds in respect to sides adjacent to the angle A. to the angle C, and gives, Making the same observations on (7), we infer, (10) R cos. C = cos.c sin.A. OBSERVATION 1. Several of these equations can be deduced geometrically without the least difficulty. For example, take the figure to Proposition 1. The parallels in the plane, DBA, give, That is, DB : DH=DE : DG. R: cos.a=cos.c: cos.b. A result identical with equation (8), and in words it 18 expressed thus: Radius is to cosine of one side, as the cosine of the other side is to the cosine of the hypotenuse. OBSERVATION 2. The equations numbered from (1) to (10) cover every possible case that can occur in rightangled spherical trigonometry; but the combinations are too various to be remembered, and readily applied to practical use. We can remedy this inconvenience, by taking the complement of the hypotenuse, and the complements of the two oblique angles, in place of the arcs themselves. Thus, b is the hypotenuse, and let b' be its complement. Then, b+b=90°; or, b = 90°-b'; and, sin.b = cos.b', cos.b = sin.b'; tan.b=cot.b'. In the same manner, if A' is the complement to A, Then, sin. A = cos.A'; cos. A = sin.A'; and, tan. A = cot.A'; and similarly, sin. C=cos.C"; cos.C=sin.C"; and tan.C=cot. C". Substituting these values for b, A, and C, in the foregoing ten equations (a and e remaining the same), we have, NAPIER'S CIRCULAR (11) Rsin.c = tan.a tan. A' PARTS. Omitting the consideration of the right angle, there are five parts. Each part taken as a middle part, is connected to its adjacent parts by one equation, and to its extreme parts by another equation; therefore, ten equations are required for the combinations of all the parts. These equations are very remarkable, because the first members are all composed of radius into some sine, and the second members are all composed of the product of two tangents, or two cosines. To condense these equations into words, for the pur-. pose of assisting the memory, we will refer any one of them directly to the right-angled triangle, ABC, in the last figure. When the right angle is left out of the question, a right-angled triangle consists of five parts - three sides, and two angles. Let any one of these parts be called a middle part; then two other parts will lie adjacent to this part, and two opposite to it, that is, separated from it by two other parts. For instance, take equation (11), and call c the middle part; then A' and a will be adjacent parts, and C" and b' opposite parts. Again, take a as a middle part; then c and C will be adjacent parts, and A' and b' will be opposite parts; and thus we may go round the triangle. Take any equation from (11) to (20), and consider the middle part in the first member of the equation, and we shall find that it corresponds to one of the following invariable and comprehensive rules: 1. The radius into the sine of the middle part is equal to the product of the tangents of the adjacent parts. 2. The radius into the sine of the middle part is equal to the product of the cosines of the opposite parts. These rules are known as Napier's Rules, because they were first given by that distinguished mathematician, who was also the inventor of logarithms. In the application of these equations, the accent may be omitted if tan. be changed to cotan., sin. to cosin., etc. Thus, if equation (13) were to be employed, it would be written, in the first instance, Rsin.a = cos.b' cos.A', to insure conformity to the rule; then, we would change it into Rsin.a = sin.b sin.A. REMARK.- We caution the pupil to be very particular to take the complements of the hypotenuse, and the complements of the oblique angles. SECTION III. OBLIQUE-ANGLED SPHERICAL TRIGONOMETRY. THE preceding investigations have had reference to right-angled spherical trigonometry only, but the application of these principles covers oblique-angled trigonometry also; for, every oblique-angled spherical triangle may be considered as made up of the sum or difference of two right-angled spherical triangles. With this explanatory remark, we give PROPOSITION I. In all spherical triangles, the sines of the sides are to each other, as the sines of the angles opposite to them. This was proved in relation to right-angled triangles in Prop. 3, Sec. II, and we now apply the principle to oblique-angled triangles. Let ABC be the triangle, and let Then, by Prop. 3, Sec. II, we have, Also, sin.CB: R = sin. CD: sin. B. C D C B By multiplying these two proportions together, term by term, and omitting the common factor R, in the first couplet, and the common factor, sin. CD, in the second, we have sin.CB: sin. AC= sin.A : sin.B. PROPOSITION II. In any spherical triangle, if an arc of a great circle be let fall from any angle perpendicular to the opposite side as a base, or to the base produced, the cosines of the other two sides will be to each other as the cosines of the segments of the base. By the application of equation 8, (Sec. II), to the last figure, we have, Similarly, R cos.AC = cos.AD cos.DC R cos.BC = cos.DC Cos.BD Dividing one of these equations by the other, omitting common factors in numerators and denominators, we If from any angle of a spherical triangle, a perpendicular be let fall on the base, or on the base produced, the tangents of the segments of the base will be reciprocally proportional to the cotangents of the segments of the angle. By the application of Equation 2, (Sec. II), to the last figure, we have, R sin.CD = tan. AD cot. ACD. |