Cor. 1.-Each of the quadrants contiguous to the pro jecting point is projected into an indefinite straight line, and each of those that are remote, into a radius of the primitive. Cor. 2. -Every small circle which passes through the projecting point is projected into that straight line which is its common section with the primitive CoR. 3.-Every straight line, in the plane of the primi tive, and produced indefinitely, is the projection of some circle on the sphere passing through the project ing point. CoR. 4.-The stereographic projection of any point in the surface of the sphere is distant from the centre of the primitive, by the semi-tangent of that point's distance from the pole opposite to the projecting point. PROPOSITION II. Every circle on the sphere which does not pass through the projecting point, is projected into a circle. If the circle be parallel to the primitive, the proposition is evident. For, a straight line, drawn from the projecting point to any point in the circumference, and made to revolve about the circle, describes the surface of a cone, which is cut by a plane (namely, the primitive), parallel to the base; and therefore the section (the figure into which the circle is projected) is a circle. If the circle MN be not parallel to the primitive BD; let the great circle ABCD, passing through the projecting D Em Con point, cut it at right angles, in the diameter MN, and the primitive in the diameter BD. Through M, in the plane of that great circle, let MF be drawn parallel to BD; let AM, AN, be joined, and meet BD in m, n. Then, because AB, AD, are quadrants, and BD, MF, parallel, the arc AM = AF, for BM = DF; since, if B and F were joined, the alternate angles would be equal; hence the angle AMF or Amn = ANM. Thus, the conic surface, described by the revolution of AM, about the circle MN, is cut by the primitive in a sub-contrary position (Conic Sections); therefore the section mn is, in this case, likewise a circle. Cor. 1.-The centres, and poles of all circles, parallel to the primitive, have their projections in its centre. Cor. 2.-The centre, and poles of every circle, inclined to the primitive, have their projections in the line of measures. Cor. 3.-All projected great circles cut the primitive in two points diametrically opposite; and every circle in the plane of projection, which passes through the extremities of a diameter of the primitive, or through the projections of two points that are diametrically opposite on the sphere, is the projection of some great circle. For the original great circles cut the primitive in two points diametrically opposite. Cor. 4.-A tangent to any circle of the sphere, which does not pass through the projecting point, is projected. into a tangent to that circle's projection; also, the circular projections of tangent circles touch one an other. Cor. 5.-The extremities of the diameter, on the line of measures of any projected circle, are distant from the centre of the primitive, by the semi-tangents of the circle on the sphere's least and greatest distances from the pole opposite to the projecting point. Cor. 6.—The extremities of the diameter, on the line of measures of any projected great circle, are distant from the centre of the primitive, by the tangent and cotangent of half the complement of the great circle's incli nation to the primitive. For BM (second figure) measures the inclination of the circle MN to the primitive BD, and MC is its complement, and angle MAC half its complement. Also, since MAN is a right angle, EAN is the complement of MAC. Also mE is the tangent of MAC, and Én its cotangent. Cor. 7. The radius of any projected circle is equal to half the sum, or half the difference, of the semi-tangents of the circle's least and greatest distances from the pole opposite to the projecting point, according as the circle does or does not encompass the axis of the primitive. PROPOSITION III. An angle, formed by two tangents, at the same point, in the surface of the sphere, is equal to the angle formed by their projections. Let FGI and GH be the two tangents, and A the projecting point; let the plane AGF cut the sphere in the circle AGL, and the primitive in the line BML. Also, let MN be the line of common section, of the plane AGH, with the primitive. Then the angle FGH = LMN. If the plane FGH be parallel to the primitive Al Jilboa BLD, the proposition is manifest. If not, through any point K, in AG produced, let the plane FKH, parallel to the primitive, be extended to meet FGH in the line FH. Then, because the plane AGF meets two N parallel planes, BLD, FKH, the G) u bolon A tout lines of common section, LM, FK, / are parallel ; therefore the angle / AML= AKF. But, since A is the k Fljud diod pole of BLD, the cords, and conse- . # foto Slytis quently the arcs AB, AL, are equal ; and the arc ABG is, the sum of the arcs ÁL, BG. Draw GP parallel to BL; then the are BG = LP; for if B and P were joined, the alternate angles would be equal. Hence, the arc ABG = ALP, and the angle APG = AGP = AML = FKG. But angle APG - AGI (Pl. Ge. III. 32) = FGK. Consequently the angle FGK = FKG, and the side FG = FK. In like manner, HG = HK. Hence the triangles GHE, KHF, are equal in every respect, and the angle FGHE FKH = LMN. it! I r/ BI Cor. 1.-An angle, contained by any two circles of the sphere, is equal to the angle formed by their projec tions. For, the tangents to these circles on the sphere are projected into straight lines, which either coincide with, or are tangents to, their projections on the primitive. Cor. 2.--An angle, contained by any two circles of the sphere, is equal to the angle formed by the radii of their projections, at the point of concourse. When one of the given projected circles is a diameter of the primitive, for its radius a line perpendicular to it must be taken. PROPOSITION IV. The centre of a great circle's projection is distant from the centre of the primitive by the tangent of the great circle's inclination to the primitive, and its radius is the secant of the same. Let A be the projecting point, ABC a great circle passing through it, perpendicular to the proposed great circle, KEL their line of common section, and BED the line of common section of ABCD, and the primitive. Then, because I F ABC is perpendicular B both to the proposed great circle, and to the primitive, it is perpendicular to their line of common section, and consequently BE, EK, are likewise perpendicular to the same. Hence BEK is the angle of inclination of the proposed circle to the primitive. Let AK, AL, be drawn, and meet BG in F, G, the straight line FG is the diameter of the projection. Let it be bisected in H, and let A, H, be joined. Because FAG is a right angle, HA = HF (IV. Cor. 5), and the angle HAF =HFAS FEK +FKE; from these equals, taking the equal angles EAF, FKE, there remains HAE = FEK, the angle of in clination. And, in the right-angled triangle AEH, to the radius AE, EH, is the tangent, and AH the secant of HAE. Cor. 1.-All circles which pass through the points A, C, are the projections of great circles, and have their centres in the line BG. All circles which pass through the points F, G, are the projections of great circles, and have their centres in the line II', perpendicular to BG. For, considering ABC as the primitive, any circle through A, C, is the projection of a great circle (St. Pr. II. Cor. 3). Let AFC be an arc of a circle described on FG as a diameter ; then, because HA =HF, as was proved, the circle cuts the primitive in A, C, the extremities of a diameter. Let IHI' be perpendicular to FG, then, if an arc of a circle having its centre in II' pass through F, it must pass through G. Also the points K, L, of its intersection with the primitive, are diametrically opposite. For FE · EG = AE2 = KEEL; and hence KEL is a diameter, and KFL is the projection of a great circle. Cor. 2.-It appears from this that II' is the locus of the centres of the projections of all great circles that pass through the point F. Cor. 3.-BK is the measure of the inclination of the great circle to the primitive, and CS=2 BK. For HAE= BEK ; therefore CS = 2 BK. PROPOSITION V. The centre of projection of a small circle, perpendicular to the primitive, is distant from the centre of the primitive, the secant of the circle's distance from its nearer pole, and the radius of projection, is the tangent of the same. Let ABCD be a great circle, passing through the projecting point, and perpendicular to the proposed small circle, MON their line of common section, and BOD the line of common section of ABCD, and the primitive. Then BD is the axis, and O the centre of the small circle. Let AM and AN be drawn to meet BD in G, F; FG is the diameter on the line of measures of the small circle's projection. Let it be bisected in H, and EM, MH, joined. Then, because AE: EF=NO:OF, CE: EF=MO:OF; therefore the points C, F, M, are in a straight line, and that straight |