PRINCIPLES FOR THE SOLUTION OF RIGHT-ANGLED SPHERICAL TRIANGLES.

LI. In every right-angled spherical triangle, radius is to the sine of the hypotenuse, as the sine of either of the oblique angles is to the sine of the opposite side.

From O, the centre of the sphere, draw the radii OA, OB, OC; then take OF equal to radius in the tables, and from the point F draw FD perpendicular to OA: the line FD will be perpendicular to the plane OAB, because the angle A being right by hypothesis, the two planes OAB, OAC are thus perpendicular to each other. From the point D, draw DE perpendicular to OB; and join EF: the line EF will also be perpendicular to OB, and thus the angle DEF will measure the inclination of the two planes OBA, OBC, and be equal to the angle B of the triangle ABC.

This being proved, in the triangle DEF, right-angled at D, we have R sin DEF : : EF : DF; now the angle DEF=B; and since OF-R, we have EF-sin EOF=sin BC, DF= sin AC. Hence R sin B:: sin BC: sin AC, or

If we designate by a the hypotenuse or side opposite the right-angle A, by b the side opposite the angle B, by c the opposite the angle C, we shall thus have

R sin a sin B: sin b:: sin C : sin c;

:

a formula, which of itself furnishes two equations among the parts of the right-angled spherical triangles.

LII. In every right-angled spherical triangle, radius is to the cosine of an oblique angle, as the tangent of the hypote nuse is to the tangent of the side adjacent to that angle.

cos DEF :: EF: ED. But we have DEF B, EF=sin BC, OE=cos BC; and in the triangle OED, right-angled at E,

OE tang DOE cos BC tang AB

we have DE

R

; hence R:

R

R: cos B: tang BC: tang AB.

Making, as above, BC=a and AB=c, we shall have R:

cos B: tanga tang c, or cos B=

R tang c tang c cot a

tang a

R

The same principle, applied to the angle C, will give cos C= R tang b tang b cot a

LIII. In every right-angled spherical triangle, radius is to the cosine of a side containing the right angle, as the cosine of the other side is to the cosine of the hypotenuse.

Let ABC (see the preceding figure) be the proposed triangle right-angled at A; we are to show that R: cos AB:: cos AC cos BC.

For, the same construction remaining, the triangle ODF, which is right-angled at D and has the hypotenuse OF=R, will give OD=cos DOF=cos AC: also the triangle ODE, OD cos DOE cos AC cos AB

right-angled at E, will give OE But in the right-angled triangle cos AC cos BC

hence cos BC:

R

R
OEF, we have OE=cos BC;

or what amounts to the same,

R: cos AC:: cos AB: cos BC.

This third principle is expressed by the equation R cos a= cos b cos c; it cannot form a second equation, like the two preceding principles, because a change of b for c in it would produce no alteration.

LIV. By means of these three general principles, three others may be found, which are requisite for the solution of right-angled spherical triangles. They might be demonstrated directly, each by a particular construction; but it seems preferable to deduce them, by way of analysis, from the three which are already proved. We shall now do so.

from which also, by changing the letters, there results,

R: tang B sin c: tang b;

from which also, by changing the letters, there results

R: tang C:: sin b : tang c;

Lastly, these two formulas give

cot B cot CR cos a; or R: cot C:: cot B: cos a. This is the sixth and last principle: it cannot furnish another equation, because the change of B for C in it produces no alteration.

We subjoin a recapitulation of these six principles, whereof four give each two equations:

I. R sin b = sin a sin B, R sin c = sin a sin C.
II. R tang b = tang a cos C, R tang c = tang a cos B.
III. R cos a = cos b cos c,

IV. R cos B sin C cos b, R cos C = sin B cos . c
V. R tang b = sin c tang B, R tang c = sin b tang C.
VI. R cos acot B cot C.

From these are obtained ten equations including all the relations that can exist between three of the five elements B, C, a, b, c; so that two of these quantities with the right angle being given, the third will immediately be discovered in the form of its sine, cosine, tangent or cotangent.

LV. It is to be observed, that when any element is discovered in the form of its sine only, there will be two values for this element, and consequently two triangles that will satisfy the question; because, the same sine which corresponds to an angle or an arc, corresponds likewise to its supplement. This will not take place, when the unknown quantity is determined by means of its cosine, its tangent, or cotangent. In all these cases, the sign will enable us to decide whether the element in question is less or greater than 90°; the element will be less than 90°, if its cosine, tangent, or cotangent has the sign +; it will be greater if one of these quantities has the signOn this point, likewise, some general principles might be established, which would merely be consequences of the six equations demonstrated above.

From the equation R cos a = cos b cos c, for example, it results, that either the three sides of a right-angled spherical triangle are all less than 90°; or that of those three sides, two are greater than 90°, while the third is less. No other combination can render the sign of cos b cos c like that of cos a, as the equation requires.

In like manner, the equation R tang c = sin b tang C, in which sin b is always positive, proves that tang C has always the same sign with tang c. Hence in every right-angled spherical triangle, an oblique angle and the side opposite to it are always of the same species; in other words, are both greater or both less than 90°,

+

SOLUTION OF RIGHT-ANGLED SPHERICAL TRIANGLES.

LVI. A spherical triangle may have three right angles, and then its three sides are each 90°; it may have only two right angles, in which case, the opposite sides are both 90° each, and there remains an angle and its opposite side, both of which are measured by the same number of degrees. These two kinds of triangles can evidently give rise to no problem; we may, therefore, leave them out of view entirely, and limit our attention to such triangles as have only one right angle.

Let A be the right angle, B and C the other two angles which are called oblique; let a be the hypotenuse opposite the angle A; b and c the sides opposite the angles B and C. Two of the five quantities B, C, a, b, c, being given, the solution of the triangle will always be reducible to one of the six following cases.

LVII.

FIRST CASE.

Given the hypotenuse a, and a side b; the two anglés B and C with the third side c may be found by the equa

tions,

The angle C and the side c have in them no uncertainty as to their signs; the angle B must be of the same species with the side b.

SECOND CASE.

LVIII. Given b and c the two sides containing the right angle, the hypotenuse a and the angles B and C may be found by the equations,

There is no ambiguity in any of these values.

THIRD CASE.

LIX. The hypotenuse a and an angle B being given, we shall obtain the two sides b and c, with the other angle C, by the equations,