(E. 33. 6.) that the sides PB, BD, DP are the measures of the plane angles PCB, BCD, and DCP, respectively, by which angles the solid angle at Cis contained but (E. 21. 11.) the three plane angles are together less than four right angles; and (E. 20. 11.) any two of them are greater than the third: wherefore, also, any two sides of the spherical triangle are greater than the third, and the three sides, together, are less than the measure of four right angles, that is, they are less than the circumference of a great circle of the sphere.

(75.) COR. In the same manner, it may be shewn, that the aggregate of the sides of a spherical polygon, is less than the circumference of a great circle of the sphere, if the polygon be bounded by arches of great circles, each of which is less than the semi-circumference of a great circle.

(76.) DEF. If, about each of the angular points of a spherical triangle, as a pole, a great circle, in the sphere, be described; the spherical triangle included by the circumferences of the circles, so described, is called the Polar Triangle of the given spherical triangle.

Thus, if ABC be a spherical triangle, and DE, EF, and FD, be the arches of great circles, described from A B and C as their respective poles, the spherical triangle DEF is called the Polar Triangle of ABC*.

(77.) COR. 1. The angular points of the polar triangle are also (Art. 27.) the poles of the great circles, the arches of which include the given triangle; and, therefore, if any sides of the given triangle, as AB, be produced to meet the sides of the polar triangle, in G and L, the side so produced, GL, will measure (Art. 54.) the opposite angle E, of the polar triangle; and if two sides, as AB and AC, of the given triangle be produced, the segment GH which they cut off from the side DE of the polar triangle, will likewise measure the angle A, of the given triangle, that is opposite to DE.

It is also manifest, from Art. 50. that an arch of a great circle, drawn from any angle of either triangle, as from the angle A, of the triangle ABC, at right angles to the opposite side BC, of that triangle, will likewise cut the opposite side DE, of the other triangle, perpendicularly, and will pass through the opposite angle F of the other triangle and conversely.

* For the sake of simplicity in the figure, each of the sides of the triangle ABC is here supposed to be less than a quadrant, and each of its angles to be less than a right angle: in which case, the triangle, ABC lies wholly within its polar triangle. But if any of the sides of the triangle ABC be greater than a quadrant, the sides of the two triangles will intersect one another. The same reasoning, however, which is applied in the one case, is equally applicable in the other.

(78.) COR. 2. In either of the two triangles, the measure, GH, of any angle, A, of the one, together with the side, DE, of the other, that is opposite to that angle, is equal to the semi-circumference of a great circle of the sphere.

For (Art. 36. and 7.) DH* and GE are quadrants, and are, therefore, (Art. 35.) together equal to the semicircumference of a great circle: but DH together with GE is manifestly equal to DG, and HE, and twice GH, taken together; that is, to DE and GH together: therefore, DE and GH are, together equal to the semi-circumference of a great circle. And, in the same manner, may the measure of any angle of the triangle DEF, together with the side of ABC, opposite to that angle, be shewn to be equal to the semi-circumference of a great circle of the sphere.

Hence, the aggregate of any two sides, in either triangle, together with the aggregate of the measures of the two angles of the other triangle, that are opposite to them, is equal to the circumference of a great circle : also, the difference of any two sides, of the one triangle, is equal to the difference of the measures of the angles, of the other triangle, that are opposite to them.

And, if an arch of a great circle be drawn from any angle of either triangle, as from the angle A of ABC, at right angles to the opposite side BC of that triangle, the difference of the segments, into which it divides the

* See the figure in Art. 76.

opposite side DE, of the other triangle, is equal to the difference of the measures of the segments, into which it divides the angle A.

(79.) COR. 3. If a spherical polygon be given, which is bounded by arches of great circles, each less than a semi-circumference, and about each of its angular points, as a pole, a great circle be described, another spherical polygon will thus be formed: and it may be shewn, in the same manner as before, that in either of the two figures, the measure of any angle, in the one, together with the side opposite to it of the other, is equal to the semi-circumference of a great circle of the sphere.

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PROP. II.

(80.) Theorem. The three angles of any given spherical triangle are together greater than two right angles.

For, (Art. 78.) the measures of the three angles of the given triangle, together with the three sides of its polar triangle, are equal to three semi-circumferences of a great circle: but (Art. 74.) the three sides of the polar triangle are less than two such semi-circumferences: wherefore the measures of the angles of the given triangle are, together, greater than the remaining semi-circumference: and the angles themselves are, consequently, together, greater than two right angles.

(81.) COR. 1. Hence, and from Art. 42, it is evident, that the exterior angle, of a spherical triangle, is

PROP. X.

(64.) Problem. An arch of a circle, in the surface of a given sphere, being given, to find the poles of the circle.

Find (Art. 63.) two straight lines equal to the distances between any point in the given arch and its two poles: then, from the extremities of the given arch, as poles, at distances equal to the two straight lines, so found, describe (Art. 59.) two circles in the sphere: the intersections of the circumferences of the circles, so described, are the poles of the circle, to which the given arch belongs.

For, the poles (Art. 31.) are in each of the circumferences, and therefore they are in the intersections of the circumferences.

4.4 (65.) COR. A given arch of any circle, in a sphere, may be produced, in the sphere's surface (Art. 64. and 59.)

PROP. XI.

(66.) Problem. Two points in a sphere's surface, being given, to describe a circle, in the sphere, which shall pass through them, and shall have the direct distance between its pole and its circumference, equal to a given straight line.

* In the following articles, when two points on a sphere's surface are directed to be joined, it is intended that they should be joined by arches of great circles, unless the contrary be specified.