lines CA and CB which it meets in that p.ane; so that (Euc. Suppl. 2, Def. 4) BCA is the angle between the planes ACD and BCD, which angle is the same as the spherical angle ADB: and as AB is the measure of the angle ACB, it is also the measure of the angle ADB; as was to be proved.

SPHERICAL TRIANGLES.

PROP. XIX.

If about the vertices of a triangle, as poles, arcs of great circles be drawn, they will form a triangle whose verticles are poles of the great circles passing through the sides of the first triangle.

Let ABC be a spherical triangle, and about A, B, C, as poles, let arcs of great circles EF, FD, DE, be drawn, forming the triangle DEF; the vertices D, E, F, are respectively poles of the great circles passing through BC, CA, AB.

E

D

B

For since B is the pole of the great circle DF, the arc of a great circle drawn from B to D is a quadrant (Prop. 12); and since C is the pole of DE, the arc drawn from C to D is also a quadrant. Now when the arcs of two great circles drawn from a point in the surface of a sphere to the circumference of another great circle are quadrants, the point is the pole of this circle; therefore D is the pole of the great circle passing through BC. In like manner may it be shown that E is the pole of the circle AC, and F the pole of AB.

Def. Such triangles as those above mentioned are called polar triangles, each being said to be polar to the other.

PROP. XX.

The sides of a spherical triangle are supplements to the measures of the angles of its polar triangle.

B

D

Let ABC, DEF, be polar triangles, A, B, C, being poles of EF, FD, DE, respectively, and D, E, F, poles of BC, CA, AB; the sides of the tri- K angle ABC are supplements to the measures of the angles D, E, F, and the sides of the triangle DEF are likewise supplements to the measures of the angles A, B, C, of the triangle ABC.

E

L

C

F

G

H

Let the arcs AB, AC, produced if necessary, meet EF in G and H; and let the arc BC meet DE and DF, in K and L.

Then, as A is the pole of the great circle EGHF, AG and AH are quadrants (Prop. 12); and GH is the measure of the spherical angle at A (Prop. 18); also GF and HE are quadrants, because E and F are poles of AC and AB respectively; wherefore the arcs GF and HE together are equal to the semi-circumference of a great circle: but these two arcs are together equal to EF and GH; hence EF is the supplement of GH, that is, the supplement to the measure of the angle BAC. In like manner it may be shown that FD, DE are supplements to the measures of the angles ABC, ACB, respectively.

Again, because D is the pole of the great circle passing through BC, KL is the measure of the spherical angle EDF; and since B and C are the poles of DF and DE respectively, BL and CK are quadrants, and are together equal to the semi-circumference of a great circle: but their sum is the same as that of BC and KL; wherefore BC is the supplement of KL, the arc which measures the spherical angle at D. In like manner may it be shown that the other sides AB, AC, of the triangle ABC, are respectively supplements to the measures of the angles F and E of the triangle DEF.

Schol. From this property of polar triangles, they are often called supplemental triangles.

PROP. XXI.

Any two sides of a spherical triangle are together greater than the third side.

Let ABD be a spherical triangle ; any two of its sides, as AD and DB, are together greater than the third side AB.

Let C be the center of the sphere: the planes of the great circles passing through the three sides of the trian

gle form a solid angle at C; and of the plane angles containing this solid angle, any two, as ACD and, BCD, are together greater than the third ACB (Euc. › Suppl. 2, 20); wherefore the ares AD and BD which measure the first two angles, are together greater than AB which measures the third.

PROP. XXII.

The three sides of a spherical triangle are together less than the circumference of a great circle.

Let ABD be a spherical triangle; the sum. of the sides AB, BD, and DA, is less than the circumference of a great circle.

For if C be the center of the sphere, the arcs AB, BD, DA, are respectively the measures of the plane an

A

D

B

gles ACB, BCD, DCA; which are angles containing a solid angle at C, and therefore are together less than four right angles (Euc. Suppl. 2, 21). Hence the arcs which measure these angles, namely, AB, BD, DA, are together less than four quadrants or the circumference of a great circle.

Cor. In like manner it may be proved that the sides of any spherical polygon are together less than the circumference of a great circle.

PROP. XXIII.

Each of the angles of a spherical triangle is less: than two right angles, and the sum of the three angles is greater than two right angles.

Let ABC be a spherical triangle; each of its angles A, B, and C, is less than two right angles, and the sum of the three is greater than twó right angles.

A

B

D

Produce the side AB, so as to make an angle CBD adjacent to ABC; these angles are together equal to two right angles (Cor. Prop. 17); therefore ABC, one of them, is less than two right angles: in like manner it may be shown that A and C, the other two angles of the triangle, are each less than two right angles.

Again, the sides of the triangle that is polar to ABC, are supplements to the measures of the angles A, B, C; wherefore those sides together with the measures of these angles are equal to three semi-circumferences of great circles; but the sides of the polar triangle are together less than the circumference, or two semi-circumferences of a great circle (Prop. 22); wherefore the measures of the angles A, B, C, are together greater than one semi-circumference or two quadrants, and the angles accordingly greater than two right angles.

PROP. XXIV.

If two sides of a spherical triangle are equal, the angles opposite them are also equal.

Let ABD be a spherical triangle of which the sides AD and BD are equal; then will the angle DBA be equal to the angle DAB.

If AD and BD are quadrants, the angles DBA and DAB are equal, each of them being a right c

angle (Props. 14, 12); but if not,

D

A

from C the center of the sphere, draw the radii CA, CB, CD; in the plane CAB, draw AF and BF, tangents to the arc AB; and in the plane CAD draw AE a tangent