PROPOSITION I. THEOREM. In every spherical triangle, any side is less than the sum of the two other sides. Let O be the centre of the sphere, and ACB a spheri cal triangle: then will any side be less than the sum of the two other sides. For, draw the radii OA, OB, OC. Conceive the planes AOB, AOC, COB, to be drawn; these planes bound a polyedral angle whose vertex is at the centre 0; and the plane angles AOB, AOC, COB, are measured by AB, AC, BC, the sides of the spherical triangle. But each of the three plane angles which bound a polyedral B C angle is less than the sum of the two other angles (B. VI., P. 19); hence, any side of a spherical triangle is less than the sum of the two other sides. PROPOSITION II. THEOREM. The sum of all the sides of any spherical polygon is less than the circumference of a great circle. 1 Let ABCDE be any spherical polygon, and O the cen tre of the sphere. Conceive O to be the vertex of a polyedral angle bounded by the plane angles AOB, BOC, COD, &c. Now, the sum of the plane angles which bound a polyedral angle is less than four right angles (B. VI., P. 20); hence, the sum of the sides of any D E B spherical polygon is less than the circumference. Cor. The sum of the three sides of any spherical triangle is less than the circumference; for, the triangle is a polygon of three sides. PROPOSITION III. THEOREM. The poles of a great circle of a sphere are the extremities of that diameter of the sphere which is perpendicular to the circle; and these extremities are also the poles of all small circles parallel to it. Let ED be perpendicular to the great circle AMB; then will E and D be its poles; and they will also be the poles of every parallel small circle FNG. are each equally distant from all the points of the circumference AMB; hence, they are the poles of that circumference (D. 7). Again, the radius DC, perpendicular to the plane AMB, is perpendicular to the parallel FNG; hence, it passes through O, the centre of the circle FNG (B. VIII., P. 7, c. 4); hence, if the chords DF, DN, DG, be drawn, these oblique lines will cut off equal distances measured from 0; hence, they will be equal (B. VI., P. 5). But, the chords being equal, the arcs are equal; hence, the point D is the pole of the small circle FNG; and for like reasons, the point E is the other pole. Cor. If through the pole D and any point M, in the arc of a great circle AMB, an arc of another great circle MD be drawn, the arc MD is a quarter of the circumference, and is called a quadrant. This quadrant makes a right angle with the arc AM. For, the line DC being perpendicular to the plane AMC, every plane DME, passing through the line DC is perpendicular to the plane AMC (B VI., P. 16); hence, the angle of these planes, or the angle AMD is a right angle. Cor. 2. Conversely: If the distance of the point D from each of the points A and M, in the circumference of a great circle, is equal to a quadrant, the point D is the pole of the arc AM. F B M M For, let be the centre of the sphere, and draw the radii CD, CA, CM. Since the angles ACD, MCD, are right angles, the line CD is perpendicular to the two straight lines CA, CM; hence, it is perpendicular to their plane (B. VI., P. 4): hence, the point D is the pole of the arc AM. Scholium. The properties of these poles enable us to describe arcs of a circle on the surface of a sphere, with the same facility as on a plane surface. It is evident, for instance, that by turning the arc DF, or any other line extending to the same distance, round the point D, the extremity F will describe the small circle FNG; and by turning the quadrant DFA round the point D, its extrem ity A will describe the arc of a great circle AMB. PROPOSITION IV. THEOREM. The angle formed by two arcs of great circles, is equal to the angle formed by the tangents of these arcs at their point of intersection. The angle is measured by the arc of a great circle described from the vertex as a pole, and limited by the sides, produced if necessary. Let the angle BAC be formed by the two arcs AB, AC; then will it be equal to the angle FAG formed by the tangents AF, AG, and be measured by the arc DE of a great circle, described about A as a pole. For, the tangent AF, drawn in the plane of the arc AB, is perpendicular to the radius AO; and the tangent AG, drawn in the plane of the arc AC, is perpendicular to the same radius 40. Hence, the angle FAG is equal to the angle contained by the planes ABDH, ACEH (B. VI., D. 4); which is that of the arcs AB, AC, and is called the angle ВАС. Again, if the arcs AD and AE are both quadrants, the lines OD, OE, are perpendicular to OA, and the angle DOE is equal to the angle of the planes ABDH, ACEH; hence, the arc DE is the measure of the angle contained by these planes, or of the angle CAB. Cor. 1. The angles of spherical triangles may be com pared together, by means of the arcs of great circles described from their vertices as poles and included between their sides: hence, it is easy to make an angle of this kind equal to a given angle. Cor. 2. Vertical angles, such as ACO and BCN are equal; for either of them is still the angle formed by the two planes ACB, OCN. It is further evident, that, when two arcs ACB, OCN, intersect, the two adjacent angles ACO, OCB, taken together, are equal to two right angles. B PROPOSITION V. THEOREM. If from the vertices of the three angles of a spherical triangle, as poles, arcs be described forming a spherical triangle; then, the vertices of the angles of this second triangle, will be respectively poles of the sides of the first. From the vertices A, B, C, as poles, let the arcs EF, FD, ED, be described, forming on the surface of the sphere, the triangle DFE; then will the vertices D, E, and F, be respectively poles of the sides BC, AC, AB. pole of the arc AC (P. 3, c. 2). It may be shown by similar reasoning, that D is the pole of the arc BC, and F that of the arc AB. Scholium. Hence, the triangle ABC may be described by means of DEF, as DEF is described by means of ABC. Triangles so described, are called polar triangles, or supplemental triangles. PROPOSITION VI. THEOREM. The same supposition continuing as in the last Proposition, each angle in one of the triangles, will be measured by a semicircumference, minus the side lying opposite to it in the other triangle. For, produce the sides AB, AC, if necessary, till they meet EF, in G and H. The point A being the pole of the arc GH, the angle A is measured by that arc (P. 4). But, since E is the pole of AH, the arc EH is a quadrant; and since Fis the pole of AG, FG is a quadrant: hence, EH+GF is equal to a semicircumference. But, EH+GF=EF+GH; hence the arc GH, which mea |