two sides CE and EB, together, greater than two quadrants the angles CBE, ECB are, together, greater than two right angles :

For, let CE and CB, produced, meet in D: wherefore, (Art. 7.) CAED is equal to two quadrants; since, therefore, CE and EB are greater than two quadrants, they are greater than CE and ED; take away CE from both, and there remains BE greater than ED; therefore, (Art. 129.) the angle D is greater than the angle EBD; that is, (Art. 56.) the angle C is greater than EBD; add to both the angle EBC; and the two angles ECB, EBC are greater than the two EBD, EBC; that is, (Art. 42.) greater than two right angles.

Next, let the two sides CA, AB, of the spherical triangle ACB, be, together, less than two quadrants; then, the same construction having been made, as in the preceding case, it may be shewn, by the same mode of reasoning, and by the very same previous propositions, that the two angles ACB and ABC are, together, less than two right angles.

Conversely: first, if the two angles ECB and EBC, of the spherical triangle ECB, be greater than two right angles, they are greater (Art. 42.) than the two EBC and EBD: take away the common angle EBC; and ECB is greater than EBD; that is, (Art. 56.) the angle D is greater than EBD: therefore, (Art. 129.) EB is greater than ED; add to both EC, and the two BE, EC are, together, greater than DE, EC; that is, greater (Art. 7.) than two quadrants.

And, if the two angles ACB and ABC, of the spherical triangle ACB, be, together, less than two right angles, it may be proved, in like manner, that the aggregate of the two sides CA, AB is less than two quadrants.

(133.) COR. The base of a spherical triangle having been produced, the exterior angle will be greater than the interior opposite angle, if the aggregate of the other two sides be less than two quadrants: And, if the aggregate of those other two sides be greater than two quadrants, the exterior angle will be less than the interior opposite angle: and, conversely.

PROP. VIII.

(134.) Theorem. The angles at the base of a spherical-triangle are of the same, or different species, accordingly as an arch of a great circle, drawn from the vertex at right angles to the base, falls within or without the base: and the converse proposition is also true.

Let ABC be a spherical triangle, having AC for its

D

base; and let BD be an arch of a great circle, perpendicular to AC: if BD fall within AC, the angles A and Care of. the same species: but if BD fall without AC, the angles A and C are of different species: and, conversely.

First, let BD fall within AC: then (Art. 127.) in the two-right angled triangles BDA, BDC, the angle A, and the side BD, are of the same species, as also are BD and the angle C: wherefore the angles A and C are both of the same species.

Next, let ABC' be a spherical triangle, and let the perpendicular arch BD fall without AC': and since the angle BAD in the triangle BAD is of the same species (Art. 127.) as the side BD, the angle BAC' (Art. 42. and 126.) is not of the same species as BD: but (Art. 127.) the angle C' is of the same species as BD: therefore, the angles BAC' and BC'A, of the spherical triangle BC'A, are of different species.

Lastly, it is manifest, that if the converse of the proposition be not true, neither can the proposition itself be true but it has been demonstrated to be true: therefore, its converse is also true.

(135.) COR. The three sides, of an acute-angled spherical triangle, aré, each of them, less than a quadrant.

For, it is evident, that in this case, the perpendicular arch, drawn from any one of the angles, will fall within the base: Thus, if all the angles of the triangle ABC be

acute angles, the perpendicular arch BD, drawn from B, will fall within AC, and the two angles DBA, DAB, of the right-angled triangle BDA, will, both of them, be acute: wherefore, (Art. 130.) AB is less than a quadrant: and, in the same manner, may any other side, of the acuteangled triangle ABC, be shewn to be less than a quad

rant.

PROP. IX.

(136.) Theorem. If from the ends of the side of a spherical triangle, there be drawn to a point, within the triangle, two arches of great circles, they shall be, together, less than the other two sides, of the triangle.

The proposition is proved, by the help of Art. 74. exactly in the same manner, as the twenty-first proposition of the first Book of Euclid's. Elements.

PROP. X.

(137.) Theorem. The side of an isosceles spherical triangle, is less or greater than a quadrant, accordingly as the angles, at the base, are acute, or obtuse angles : and, conversely.

First, let TQR be an isosceles spherical triangle having the angles at its base TQR, TRQ acute angles; TQ, or TR, is less than a quadrant.

For, draw (Art. 70.) the arches QP, RP each at right angles to QR, and meeting in P.

Then (Art. 51.) P is the pole of QR; and PQ and PR (Art. 36.) are quadrants: And, it is manifest, since the angles TQR, TRQ are, by the hypothesis, less than

W

P

R

the right angles PQR, PRQ, that the point T is within the triangle PQR. Therefore, (Art. 136.) QT and TR are less than QP and PR: that is, the two equal sides of the triangle are, together, less than two quadrants : therefore, each of these equal sides is less than a quad

rant.

Secondly, if the angles WQR, WRQ, at the base of the isosceles spherical triangle WQR, be obtuse angles, the same construction having been made, as before, it may be shewn, in like manner, that either side of WQR is greater than a quadrant.

Lastly, the converse proposition is necessarily true; otherwise, it is evident, that the proposition itself cannot be true.