a spherical triangle may have all its angles right angles or all obtuse angles.

(43.) The foregoing geometrical properties comprise all that we require, for the foundation of the analytical theory of spherical Trigonometry: we need not, therefore, enumerate any more. We shall, however, in conclusion, endeavour to establish the fact that the arc of a great circle joining two points is the shortest line that can be drawn on the sphere from the one to the other.

The following proof of this property is by Legendre.

Let ANB be the arc of the great circle which joins the points A and B; and without this line, if possible, let M be a point in the shortest path, between A and B. Through the point M

draw MA, MB, arcs of great circles; and take BN = MB.

Then, by (38), the arc ANB is shorter than AM + MB; take BN = BM, respectively from both;

there will remain AN <AM.

M

B

N

Now, the distance of B from M, whether it be the same with the arc BM or with any other line, is equal to the distance of B from N; for by making the plane of the great circle BM revolve about the diameter, which passes through B, the point M may be brought into the position of the point N; and the shortest line between M and B, whatever it may be, will then be identical with that between N and B: hence the two paths from A to B, one passing through M, the other through N, have an equal part in each, the part from M to B equal to the part from N to B. The first path is the shorter by hypothesis; hence the distance from A to M must be shorter than the distance from A to N; which is absurd, the arc AM being proved greater than AN; hence no point of the shortest line from A to B can be out of the arc ANB; hence this arc is itself the shortest distance between its two extremities.

73

СНАРТER II.

INVESTIGATION OF FORMULAS, AND RULES FOR THE SOLUTION OF SPHERICAL TRIANGLES.

(44.) Let ABC be a triangle traced on the surface of a sphere of which the centre is O, and the radius equal to the linear unit. The angles of this triangle we shall represent by the letters at their vertices, A, B, C, and the sides opposite to them by the small

letters a, b, c; so that having drawn the two tangents AD, AE, to meet the radii OB, OC, produced through the other extremities of the arcs AB, AC, we shall have

Draw DE, then in the two triangles ODE, ADE, we have (17)

DE2 OE2+ OD2 — 2 OE⚫OD cos. a

DE2 = AE2 + AD2 — 2 AE·AD cos. A;

recollecting that (p. 68) the plane angle DAE measures the spherical angle A.

Substituting in these equations the values given by (1), they become

cos.acos. b cos. c + sin. b sin. c cos. A;

which is a general expression for the cosine of any side in terms of the other two sides, and their included angle. If we had taken the side b instead of a, the other two would have been a, c, and their included angle B; and if we had taken the side c the other two would have been u, b, and their included angle C; we have, therefore, the three following symmetrical equations, viz.

cos.acos. b cos. c + sin. b sin. c cos. A

cos. bcos. a cos. c + sin. a sin. c cos. B

(A);

cos. c cos. a cos. b + sin. a sin. b cos. C

and these equations embody the whole theory of spherical trigonometry and are sufficient to supply rules for the solution of every case.

(45.) Some interesting geometrical properties flow also from these equations.

1. Suppose two sides b, c, of the triangle are equal, that is, let it be isosceles, then it will follow from the two last of these equations that, like as in the isosceles plane triangle, the angles opposite the equal sides will be equal. For taking the difference of these two equations on the supposition that b=c; we have

0 sin. a sin. b cos. B― sin. a sin. b cos. C;

and, consequently, BC,

2. If a=b=c, then it is in a similar manner proved that A=B = C, that is, every equilateral spherical triangle is equiangular.

3. The arc which bisects the vertical angle A of a spherical isosceles triangle also bisects the base a. For let p represent this bisecting arc, and m, m', the parts into which it divides the base, then the two spherical triangles thus formed give, by the above equations,

cos.mcos.b cos. p + sin. b sin. p cos. A

cos. m' cos. a cos. p + sin. a sin. p cos. A;

therefore, since by hypothesis ab we have mm', that is, the arc

bisecting the vertical angle also bisects the base, and the student will find no difficulty in further showing that this same arc is also perpendicular to the base.

4. If two sides and the included angle in one triangle are equal to two sides, and the included angle in another, the third side of the one must be equal to the third side of the other. This is obvious from the first of (A), which shows that cos. a, and therefore a, becomes fixed when the other two sides b, c, and their included A, is fixed; moreover, the remaining angles of the one triangle are equal to the remaining angles of the other; for by the second and third of (A), cos. B, cos. C, and, therefore, B, C, become fixed when a, b, and c, are fixed.

5. If the three sides of one triangle are severally equal to the three sides of another, the three angles of the one are also severally equal to those of the other, the equal angles being opposite to the equal sides. For with fixed values for a, b, c, the formulas (A) give fixed values for cos. A, cos. B, cos. C, and, therefore, for A, B, C. We may, in like manner, infer the equality of the sides from that of the angles, but perhaps the inference is a little more obvious from the equations (B), p. 81, following.

In these deductions the student will observe that we have abstained from saying that the triangles are equal in all respects as in the analogous theorems of plane geometry; because two spherical triangles may exist, of which the several parts of the one may be equal to the several parts of the other, and yet not admit of coincidence, as plane triangles would under like conditions. Thus, if two plane triangles, of which the sides in the one are equal to those in the other, be joined together by a corresponding side of each, and if we turn one of the triangles about this common side either above or below the plane on which they are situated till it comes to that plane again, we know that we shall thus obtain a perfect coincidence between the two; but if the sides of the triangles thus joined are the chords of two spherical triangles, these triangles will, as we have seen, have all their parts equal, each to each, because, the chords being equal, the arcs must be equal, and yet it is very plain that the corresponding parts of the two triangles cannot be brought into coincidence as in plane triangles, and only in the particular case in which the two triangles are isosceles can they coincide, by being laid the one over the other. We cannot therefore say, as in plane triangles,

that two triangles, whose corresponding parts are equal, have equal surfaces, without distinct proof. This proof will be given in Part IV.

We shall add here but one more inference from the fundamental equations (A).

6. By the first of (A) if the sides b, c, are fixed, cos. a will necessarily diminish as cos. A diminishes; that is, a will increase as A increases: hence if two triangles have two sides in the one equal to two sides in the other, but the included angle in the first greater than the included ungle in the second, then the third side of the first triangle must be greater than the third side of the second.

Let us now proceed with the analytical discussion.

The three general equations above involve all the six parts of a triangle, the sides, and the angles; and in order to solve them, fewer than three of these parts will be insufficient; but, knowing any three, the others may be determined from them by the usual algebraical process of elimination; yet, as in the general formulas for the solution of plane triangles, so here, the results thus obtained would require considerable modification in certain cases to fit them for logarithmic computation, and on this account it is better to deduce particular formulas by a less direct process. Thus, in order to ascertain the relation between the sides and opposite angles of a spherical triangle, we proceed as follows. (46.) From the equation (A)

Now the second side of this equation is plainly of such a form, that, however we interchange the quantities a, b, c, the value of the expres