Proposition XVIII 713. 1. On the surface of a sphere, draw an isosceles spherical triangle; draw the arc of a great circle from its vertex to the middle of the opposite side. In the two triangles thus formed, how do the sides of one compare with the sides of the other? Then, how do the angles of one compare with the angles of the other? In the original triangle, how do the angles opposite the equal sides compare with each other? 2. How does the great circle arc from the vertex to the middle of the base of an isosceles spherical triangle divide the vertical angle? What is its direction with reference to the base? Into what kind of triangles does it divide the given triangle? Theorem. In an isosceles spherical triangle, the angles opposite the equal sides are equal. Data: An isosceles spherical triangle, as ABC, in which AB = AC. To prove Proof. LB LC. Draw the arc of a great circle, as AD, from the vertex 4, bisecting the side BC. Then, in ▲ ABD and ACD, AD is common, AB= AC, DB = ᎠᏟ ; that is, the triangles are mutually equilateral; .. § 712, A ABD and ACD are mutually equiangular. Why? Q.E.D. 714. Cor. The arc of a great circle drawn from the vertex of an isosceles spherical triangle to the middle of the base bisects the vertical angle, is perpendicular to the base, and divides the triangle into two symmetrical triangles. Ex. 875. If the sides of a spherical triangle are 50°, 75°, and 110°, what are the angles of its polar triangle ? Ex. 876. If the sides of a spherical triangle are 54°, 89°, and 103°, what is the spherical excess of its polar triangle? Proposition XIX 715. On the surface of a sphere, draw two mutually equiangular triangles; draw also their polars. How do the sides of their polars compare, each to each? Then, how do the angles of the polars compare, each to each? How, then, do the sides of the given triangles compare, each to each? If the equal parts in the given triangles are arranged in the same order in each, how do the triangles compare? If the equal parts are arranged in reverse order, are the triangles equal or equivalent? Theorem. Two mutually equiangular triangles on the same sphere, or on equal spheres, are mutually equilateral, and are either equal or equivalent. Data: Two spherical triangles, as A and B, that are mutually equiangular. To prove A and B mutually equilateral, and either equal or equivalent. Proof. Suppose A 4' to be the polar of ▲ 4, and ▲ B' the polar of Δ Β. Data, AA and B are mutually equiangular; .. § 701, their polar A, 4' and B', are mutually equilateral; .. § 701, AA and B are mutually equilateral. Q.E.D. 716. Cor. I. If two angles of a spherical triangle are equal, the sides opposite these angles are equal, and the triangle is isosceles. 717. Cor. II. If three planes are passed through the center of a sphere, each perpendicular to the other two, they divide the surface of the sphere into eight equal trirectangular triangles. § 695 Proposition XX 718. 1. On the surface of a sphere draw a spherical triangle, two of whose angles are unequal. How do the sides opposite these angles compare? Which one is the greater? 2. Draw a spherical triangle, two of whose sides are unequal. How do the angles opposite these sides compare? Which one is the greater? Theorem. If two angles of a spherical triangle are unequal, the sides opposite are unequal, and the greater side is opposite the greater angle; conversely, if two sides are unequal, the angles opposite are unequal, and the greater angle is opposite the greater side. Data: A spherical triangle, as ABC, in which the angle ACB is greater than the A angle ABC. or To prove Proof. AB > AC. Draw CD, the arc of a great circle, making / BCD = ▲ B. Conversely: Data: A spherical triangle, as ABC, in which the side AB is greater than the side AC. Therefore, both hypotheses, namely, that ACB = B and that ACB is less than B, are untenable. Consequently, ZACB is greater than B. Therefore, etc. Q.E.D. SPHERICAL MEASUREMENTS 719. The portion of the surface of a sphere included between two parallel planes is called a Zone. The perpendicular distance between the planes is the altitude of the zone, and the circumferences of the sections made by the planes are called the bases of the zone. If one of the parallel planes is tangent to the sphere, the zone is called a zone of one base. ABCD is a zone of the sphere. 720. The portion of the surface of a sphere bounded by two semicircumferences of great circles is called a Lune. The angle between the semicircumferences which form its boundaries is called the angle of the lune. ABCD is a lune of which BAD is the angle. 721. Lunes on the same sphere, or on equal spheres, having equal angles may be made to coincide, and are equal. 722. A convenient unit of measure for the surfaces of spherical figures is the spherical degree, which is equal to of the surface of a hemisphere. Like the unit of arcs, it is not a unit of fixed magnitude, but depends upon the size of the sphere upon which the figure is drawn. It may be conceived of as a birectangular spherical triangle whose third angle is an angle of one degree. The distinction between the three different uses of the term degree should be kept clearly in mind; an angular degree is a difference of direction between two lines, and it is the 360th part of the total angular magnitude about a point in a plane (§ 35); an arc degree is a line, which is the 360th part of the circumference of a circle (§ 224); a spherical degree is a surface, which is the 360th part of the surface of a hemisphere, or the 720th part of the surface of a sphere. Proposition XXI 723. Represent an axis and a line oblique to it, but not meeting it; draw lines from the extremities and middle point of this line perpendicular to the axis; from the nearer extremity draw a line parallel to the axis; also a line perpendicular to the given line at its middle point and terminating in the axis. If the given line revolves about the axis, what kind of a surface will it generate? To what is this surface equivalent? (§ 628) By means of the proportion of lines from similar right triangles, express the surface in terms of the projection of the given line on the axis and the circumference of a circle whose radius is the perpendicular from the middle point of the given line. Would this result hold true, if the line should meet the axis or be parallel to it? Theorem. The surface generated by a straight line revolving about an axis in its plane is equivalent to the rectangle formed by the projection of the line on the axis and the circumference whose radius is a perpendicular erected at the middle point of the line and terminated by the axis. Data: Any line, as AB, revolving about an axis, as MN; its projection upon MN, as CD; and EO perpendicular to AB at its middle point and terminating in the axis. To prove surface AB rect. CD · 2 πEO. If AB neither meets nor is parallel to MN it generates the lateral surface of a frustum of a M C E B K cone of revolution whose slant height is AB and axis CD; 20 § 307, and If AB meets axis MN, or is parallel to it, a conical or a cylindrical surface is generated, and the truth of the theorem follows. |