{ Let AB, BC, be the two quantities; place them in the fame right line, as AC; and bifect AC in E; and cut off AD equal to BC; then DB is the difference of the two quantities, and EB half their difference; therefore, if to AE, half their fum, EB, half their difference, be added, the fum is equal to AB, the greater quantity; and if from AE, half the fum, ED, half their difference, be taken, gives AD equal to BC, the leffer quantity. Wherefore, &c, THE ELEMENTS OF SPHERICAL TRIGONOMETRY! TH DEFINITION S. } I. HE poles of a sphere are two points in the fuperficies of the sphere that are the extremes of the axis. II. The pole of a circle in a sphere, is a point in the superficies of the sphere from which all right lines, drawn to the circum. ference of the circle, are equal to one another. III. A great circle in a sphere is that whose plain paffes through the center of the sphere, and whofe center is the fame with that of the sphere, or whofe plain bisects the sphere. IV. A fpherical triangle, is a figure comprehended under the arches of three great circles of a fphere. V. A spherical angle is that which is contained under two arches of greater circles in the fuperficies of the fphere. GR RE AT circles in a sphere mutually bifect each other. Let the two great circles be ACB, AFB, they will mutually bifect each other; for their common fection AB is the diameter of both circles. PROP IF from the pole of any circle, to its center, a right line be drawn, it will be perpendicular to the plain of that circle. Let the circle be AFB, and its pole C; from which draw CD to the center, then CD will be perpendicular to the plain of that circle. For, in it draw any diameters EF, GH, and join CG, CH, CE, CF; then, in the triangles CDF, CDE, the two fides. CD, DE, are equal to the two fides CD, DF, and their bafes CF, CE, are equal; therefore the angle CDF is equal to the angle CDE ; therefore CD is perpendicular to the plain of the circle AFB. Wherefore, &c. b COR. I. Hence, if this circle be a great circle, the distance upon the superficies of the sphere betwixt the pole and great' circle is a quadrant, for the plain of it bisects the fphere. II. Great circles, that pass through the pole of fome other circle, make right angles with it; for the right line CD is the d 19. 11. common section of such plains a. a cor. 2. PROP. III. Fa great circle is defcribed about the pole of a sphere, and from that pole two right lines be drawn to the circle, the arch of that circle contained by the two right lines is the measure of the angle at the pole. Let A be the pole of a sphere, and ECF the great circle defcribed about it, and let the right lines AC, AF, be drawn to the great circle; then the arch CF is the measure of the angle at A. For, let D be the center of the sphere, then the angles ADC, ADF, are right angles ; and the angle CDF is the inclination of the plains ACB, AFB, and equal to the spherical triangle b def. 6. CAF, or CBF. II. C 4. II. COR. I. If the arches AC, AF, are quadrants, then A is the pole of the circle paffing through the points C, F; for AD is at right angles to the plain FDC č. II. The vertical angles are equal, for each is equal to the inclination of the circles; alfo, the adjacent angles are equal to two right angles. PROP. |