the common section of the circles : also, first, let the point D, in ADEB, be at the same distance from A, that E is from B; let the point F also, in AFGB, be at the same distance from A, that G is from B: then is D at the same distance from F, that E is from G. For (Art. 7. and 28.) AB is a diameter of AFGB: let, therefore, C be its center : suppose DH and EL to be drawn from the points D and E, in the plane of ADB, perpendicular to AB: wherefore (E. Def. 4. 11.) DH and EL are perpendicular to the plane AFGB: join C, D, and C, F, and C, E and C, G, and D, F and H, F, and E, G and L, G. And since, by the hypothesis, the direct distance of A, D is equal to the direct distance of E, B, the arch AD (E. 28. 3.) is equal to EB: for the same reason, the arch AF is equal to BG: wherefore (E. 27. 3.) the angle ACD is equal to BCE, and the angle ACF to BCG: in the two right-angled triangles, "therefore, DHC, ELC, the side CD is equal to CE, each being a radius of the same circle, and the angle DCH to ECL: therefore (E. 26. 1.) DH is equal to EL, and CH to CL: and, because CH is equal to CL, and the radius CF to the radius CG, and that the angle HCF has been proved to be equal to LCG, therefore (E. 4.-1.) HF is equal to LG; and DH has been shewn to be equal to EL, and the angles at H and L (E. Def. 3. 11.) are right angles : therefore (E. 4. 1.) DF is equal to EG. Next, let D be at the same distance from A that E is from B, and let DF be equal to EG: then, is the distance of A, F equal to the distance of BG. For, the same construction having been made, it may be shewn, as before, that CH is equal to CL, and DH to and that the angles DHF, ELG are right angles : and since, by the supposition, DF is equal to EG; it EL; follows from E. 47. 1. that HF is equal to LG: and the radius CF is equal to CG: therefore the three sides of the triangle CHF are equal to the three sides of CLG, each to each : wherefore (E.8. 1.) the angle HCF is equal to the angle LCG, the arch AF (E. 26.3.) to the arch BG, and also (E. 29. 3.) the direct distance of A, F to the direct distance of B, G. (40.) Cor. It is manifest, from the demonstration, and from E. 28. 3. that if, instead of the direct distances of the several points, the circular arches intercepted between them, be substituted, the proposition is still true. It is evident, also, that the very same proof may be applied to shew, that, if any other circle equal to AFGB, in the same sphere, or in an equal sphere, be cut at right angles by a great circle, and the points E and G be taken at the same distances from either extremity of the common section, as D and F, respectively, are from A, DF will still be equal to EF. PART I. THE ELEMENTS OF Spherical Geometry. SECTION II. ON SPHERICAL ANGLES AND THEIR MEASURES. DEFINITION. (41.) A Spherical Angle* is the inclination of two circular arches to one another, in a sphere's surface, which meet together, but do not belong to the same circle: If, therefore, two straight lines be drawn touching any two circular arches, on a sphere's surface, which include a spherical angle, the one straight line touching the one arch, and the other the other, in their point of concourse, it is evident, that the inclination of these tangents, to one another, is the same as the inclination of the arches to one another. The plane rectilineal angle, therefore, contained by the tangent straight lines, may be taken as a measure of the spherical angle contained by the arches. * The reclination of two such arches from one another, the measure of which might exceed two right angles, is designedly excluded from the definition of a spherical anglè. When an angle is called a Spherical Angle, without any further specification, it is to be understood that the angle is the inclination of two great circles to one another. Thus, if PF and PH be any two circular arches, in the surface of the sphere PHp, which meet in P, and Px be a straight line touching PF in P, and Py a straight line that touches PH in P, the plane rectilineal angle x Py is equal to the spherical angle FPH. (42.) Cor. 1. If two arches of circles, in a sphere's surface, meet together, they make with one another either two right angles, or two angles that are, together, equal to two right angles (Art. 41. and E. 13. 1.) (43.) Cor. 2. If two arches of circles, in a sphere's surface, cut one another, the spherical opposite, or vertical, angles shall be equal (Art. 41. and E. 15. 1.) |