gents of these arcs at the point A, and it is measured by the arc DB described from the vertex A as a pole.

C

E

For the tangent AE, drawn in the plane of the arc AB, is perpendicular to the radius AC (Prop. IX., B. III.); also, the tangent AF, drawn in the plane of the arc AD, is perpendicular to the same radius AC. Hence the angle EAF is equal to the angle of the planes ACB, ACD (Def. 4, B. VII.), which is the same as that of the arcs AB, AD.

Also, if the arcs AB, AD are each equal to a quadrant, the lines CB, CD will be perpendicular to AC, and the angle BCD will be equal to the angle of the planes ACB, ACD; hence the arc BD measures the angle of the planes, or the angle BAD.

Cor. 1. Angles of spherical triangles may be compared with each other by means of arcs of great circles described from their vertices as poles, and included between their sides; and thus an angle can easily be made equal to a given angle.

Cor. 2. If two arcs of great circles AC, DE cut each other, the vertical angles ABE, DBC are equal; for each is equal to the angle formed by the two planes ABC, DBE. Also, the two adjacent angles ABD, DBC are together equal to two right angles.

A

D'

-E

B

PROPOSITION VIII. THEOREM.

If from the vertices of a given spherical triangle, as poles, arcs of great circles are described, a second triangle is formed, whose vertices are poles of the sides of the given triangle.

Let ABC be a spherical triangle; and from the points A, B, C, as poles, let great circles be described intersecting each other in D, E, and F; then will the points D, E, and F be the poles of the sides of the triangle ABC.

For, because the point A is the pole of the arc EF, the distance from A to E is a quadrant. Also, because the E point C is the pole of the arc DE, the

D

distance from C to E is a quadrant. Hence the point E is at a quadrant's distance from each of the points A and C; it is, therefore, the pole of the arc AC (Prop. V., Cor. 3). In the same manner, it may be proved that D is the pole of the arc BC, and F the pole of the arc AB.

Scholium. The triangle DEF is called the polar triangle of ABC; and so, also, ABC is the polar triangle of DEF.

Several different triangles might be formed by producing the sides DE, EF, DF; but we shall confine ourselves to the central triangle, of which the vertex D is on the same side of BC with the vertex A; E is on the same side of AC with the vertex B; and F is on the same side of AB with the vertex C.

PROPOSITION IX. THEOREM.

The sides of a spherical triangle, are the supplements of the arcs which measure the angles of its polar triangle; and conversely.

Let DEF be a spherical triangle, ABC its polar triangle; then will the side EF be the supplement of the arc which measures the angle A; and the side BC is the supplement of the arc which measures the angle D.

I

E

G

D

H

K

F

Produce the sides AB, AC, if necessary, until they meet EF in G and H. Then, because the point A is the pole of the arc GH, the angle A is measured by the arc GH (Prop. VII.). Also, because E is the pole of the arc AH, the arc EH is a quadrant; and, because F is the pole of AG, the arc FG is a quadrant. Hence EH and GF, or EF and GH, are together equal to a semicircumference. Therefore EF is the supplement of GH, which measures the angle A. So, also, DF is the supplement of the arc which measures the angle B; and DE is the supplement of the arc which measures the angle C.

Conversely. Because the point D is the pole of the arc BC. the angle D is measured by the arc IK. Also, because C is the pole of the arc DE, the arc IC is a quadrant; and, because B is the pole of the arc DF, the arc BK is a quadrant. Hence IC and BK, or IK and BC, are together equal to a semicircumference. Therefore BC is the supplement of IK. which measures the angle D. So, also, AC is the supplement of the arc which measures the angle E; and AB is the supplement of the arc which measures the angle F.

PROPOSITION X. THEOREM

The sum of the angles of a spherical triangle,is greater than two, and less than six right angles.

Let A, B, and C be the angles of a spherical triangle. The arcs which measure the angles A, B, and C, together with the three sides of the polar triangle, are equal to three semicircumferences (Prop. ÎX). But the three sides of the polar triangle are less than two semicircumferences (Prop. İV.); hence the arcs which measure the angles A, B, and C are greater than one semicircumference; and, therefore, the angles A, B, and C are greater than two right angles.

Also, because each angle of a spherical triangle is less than two right angles, the sum of the three angles must be less than six right angles.

Cor. A spherical triangle may have two, or even three, right angles; also two, or even three, obtuse angles. If a triangle have three right angles, each of its sides will be a quadrant, and the triangle is called a quadrantal triangle. The quadrantal triangle is contained eight times in the surface of the sphere.

D

PROPOSITION XI. THEOREM.

If two triangles on equal spheres are mutually equilateral, they are mutually equiangular.

Let ABC, DEF be two triangles on equal spheres, having the sides AB equal to DE, AC to DF, and BC to EF; then will the angles also be equal, each to each.

Let the centers of the spheres be G and H, and draw the radii GA, GB, GC, HD, HÉ, HF. A solid angle may be con ceived as formed at G by the three plane angles AGB, AGC

BGC; and another solid angle at H by the three plane angles DHE, DHF, EHF. Then, because the arcs AB, DE are equal, the angles AGB, DHE, which are measured by these arcs, are equal. For the same reason, the angles AGC, DHF are equal to each other; and, also, BGC equal to EHF

Hence G and H are two solid angles contained by three equal plane angles; therefore the planes of these equal angles are equally inclined to each other (Prop. XIX., B. VII.). That is, the angles of the triangle ABC are equal to those of the triangle DEF, viz., the angle ABC to the angle DEF, BAC to EDF, and ACB to DFE.

Scholium. It should be observed that the two triangles ABC, DEF do not admit of superposition, unless the three sides are similarly situated in both cases. Triangles which are mutually equilateral, but can not be applied to each other so as to coincide, are called symmetrical triangles.

PROPOSITION XII. THEOREM.

If two triangles on equal spheres are mutually equiangular, they are mutually equilateral.

Denote by A and B two spherical triangles which are mutually equiangular, and by P and Q their polar triangles.

Since the sides of P and Q are the supplements of the arcs which measure the angles of A and B (Prop. IX.), P and Q must be mutually equilateral. Also, because P and Q are mutually equilateral, they must be mutually equiangular (Prop. XI.). But the sides of A and B are the supplements of the arcs which measure the angles of P and Q; and, therefore, A and B are mutually equilateral.

If two triangles on equal spheres have two sides, and the in cluded angle of the one, equal to two sides and the included angle of the other, each to each, their third sides will be equal. and their other angles will be equal, each to each.

Let ABC, DEF be two triangles, having the side AB equal to DE, AC equal to DF, and the angle BAC equal to the angle EDF; then will the side BC be equal to EF, the angle ABC to DEF, and ACB to DFE.

A

D

If the equal sides in the two triangles are similarly situated, the triangle ABC may be applied to the triangle DEF in the same manner as in plane triangles (Prop. VI., B. I.); and the two triangles will coincide throughout. Therefore all the parts of the one triangle, will be equal to the corresponding parts of the other triangle.

C

F

E

But if the equal sides in the two triangles are not similarly situated, then construct the triangle DF/E symmet- B rical with DFE, having DF' equal to DF, and EF' equal to EF. The two triangles DEF', DEF, being mutually equilateral, are also mutually equiangular (Prop. XI.). Now the triangle ABC may be applied to the triangle DEF', so as to coincide throughout; and hence all the parts of the one triangle, will be equal to the corresponding parts of the other triangle. Therefore the side BC, being equal to EF', is also equal to EF; the angle ABC, being equal to DEF', is also equal to DEF; and the angle ACB, being equal to DF/E, is also equal to DFE. Therefore, if two triangles, &c.

PROPOSITION XIV. THEOREM.

If two triangles on equal spheres have two angles, and the included side of the one, equal to two angles and the included side of the other, each to each, their third angles will be equal, and their other sides will be equal, each to each.

If the two triangles ABC, DEF have the angle BAC equal to the angle EDF, the angle ABC equal to DEF, and the included side AB equal to DE; the triangle ABC can be placed upon the triangle DEF, or upon its symmetrical triangle DEF', so as to coincide. Hence the remaining parts of the triangle ABC, will be

A

F

D

E

F

equal to the remaining parts of the triangle DEF; that is, the side A will be equal to DF, BC to EF, and the angle ACB to the angle DFE Therefore, if two triangles, &c.