C Let A be the projecting point, B EF the great circle, GH the projected diameter, D the centre, Iraw DA. The angle EAF being in a semi-circle is a right-angle. In be right-angled triangle GAH, T D G AC is perpendicular to GH; therefore, 2 GAC= AHC and their double, ECB = ADC, and their complements ECI=CAD. Therefore, CD is the tangent of ECI, and the radius AD is its secant. Q. E. D. Cor. 1. If the great R obliqne circle AGBHbe actually described upon the primitive AB, all great circles passing through (), will liave the centre of their projection in the line RS, drawn through the centre D perpendicular to the line of measures IH. For, since all great 1 D circles cut one another at the distance of a semicircle, all circles pass. ing through G must cut at the opposite point H; and therefore their centres must be in the line RDS. Cor. 2. Hence, also, if any obliqne circle GLH be required to make any 'S given angle with another circle BGAH, it will be projected the same way, with regard to GAH considered as a primitive, and RS its line of measures, as the circle BGA is on the primitive BIA, and line of measnres ID; and, therefore, the tangent of the angle AGL, to the radius GD, set off from D to N, gives the centre of GL. For the < NGD will then be equal to AGL, by Cor. ?, Prop. 4; and, therefore, GLH is truly projecteil. N.B. Of all great circles in the projection the primitive is the least; for the radins of any oblique great circle being the secant of the inclination, is greater than the radius of the primitive; as the secant is always greater than the primitive. Therefore, every oblique greene in the projection is greater than the primitive. Prop. 7. The projected extremities of the diameter of any circle (see the first figure of Prop. 6), are in the line of measures, distant from the centre of the primitive circle the semi-tangent of its nearest and greatest distances from the pole of projection opposite to the projocting point. А A; E B For the diameter of the circle EF is projected into GH from the projecting point E IC G C circle cuts the line of measures, within and without the primitive, is distant from the centre of the primitive the tangent and co-tangeot of half the complement of the inclination of the А circle to the primitive. For CG is equal to the tangent of half EB, or of half the complement of IE the inclination, and because the < EAF is a right aogle, CH is the co-langent of GAC, or half EB. Prop. 8. The projected poles of any circle are in the line of measures within and without the primitive, and distant from its centre the tangent and co-tangent of half the inclination to the primitive. For the poles Pp of the circle EF, are projected into D and d, and CD is the tangent of CAD, or half BCP, that is, of half GCI, the inclination of the circle I G R CK parallel to EF. Likewise, Cd is the tangent of CAd, or the co-tangent of CAD. Q. E. D. Cor. 1. Tie pole of the primitive is its cen. tre, and the pole of a right circle is in the primitive. Cor. 2. The projected centre of any circle is always between the projected pole nearest to it on the sphere and the centre of the primitive ; and the projected ceptres of all circles lie between the projected poles. For the middle point of EF, or its centre, is projected into S; and all the points in Pp, in which are all the centres, are projected into Dd. Cor. 3 If P be the projected centre of any circle EFG, any right line EG, FH passing through P will intercept equal arcs EF, GH. For in any circle of the sphere, any two lines passing through the centre intercept G equal arcs, and these are projected into ECP 19 right lines passing through the projected centre P, and therefore EF, GH, repre. rent equal arcs. hH Prop. 9. If EFGH, efgh (see the last figure), represent two equal circles, whereof EFG is as far distant from its pole P, as efg is from the projecting point, any two right lines, eEP and fFP being drawn inrough P, will intercept equal arcs in the representations of these circles ; if P falls within the circles on the same side ; but on the contrary side if without; that is, EF=ef, and GH=gh. For, by the nature of the section of a sphere, any two circles passing H through two given points, or poles, h on the surface of the sphere, will inter- Pee G cept equal arcs of two other circles 9 EC cqui-castant from these poles. Therefore, the circles EFG and efg on the sphere, are equally cut by the planes of any two circles passing through the projecting point and the pol Pon the sphere. But these circles, by Prop. 1, are projected into the nght lines Pe and Pf passing through P; and the intercepted arcs representing equal arcs on the pare, are therefore equal ; that is, EF=ef, and GH=gh. Cor. 1. If a circle is projected into a right line EF, perpendicular to the line of measures EG, and if froni the centre C a circle of P be described passing F through its pole P, and Pf be drawn, then the arc ef=EF; and, if any otber circle be described, the E vertex of which is P, the arc es will always be cqnal to EF. Cor. 2. Hence, also, if from the pole of a great circle there be drawn two right lines, the intercepted arc of the projected great circle, will be equal to the intercepted arch of the primitive. 77 : centre of DG, and E the projected pole, then the pole E will be distant from their centres in pri portion to the radii of the circles, that, is, CE: EF :: CL : DF or FG. For, since NK and ML are parallel, and the arc NI=PH, K N therefore, <FLI =NKI Or NKI=GIP; therefore, the triangles IEL and IEG are similar; whence EL : EI :: EI : EG. E Ľ CDM Again, the angle EMI=KNI =PIQ; and, therefore, the Р triangles IEM and IED are similar ; whence SM : EI :: EI : ED. Therefore, EI? = EL XEG = EM X ED; consequently, EN: EL :: EG : ED; and, by composition EM + EL 2 EM - EL EG+ED EG-ED : that is, CM:EC:: FG: EF. 2 2 2 Q. E. D. Cor. 1. Hence, if the circle KN be as far from te projecting point, as QH is from either of its poles, and if E and 0 bc its project d poles, then will EL : EM :: ED: EG :: OD:OG. This follows from the foregoing demonstration, d Cor. 4, Prop. ". Cor. 2. Hence, also, if F be the centre, and FD) the radius of any circle QH, and E and O the projected poles, then as EF: DF :: DF : FO. For it follows, from Cor. 1, that ÉG+ED:EG-ED::0G+OD:UG-OD : ز For from the centres C, P draw the lines CN, FK; then, since the angles CPN and FPK are equal, and by this proposition, CP: CN:: PP : FK; therefore, the tri. angles PCN, and PFK are similar; and the angle PCNsl the angle ÞPK; therefore, the arcs MN aud GK are similar. a Cor. 4. Hence, also, if through the projected pole P of any circle DBG a right line BPK be drawn, then will the degrees in the arc GK be the mrasite of DB in the projection; and the degrees in DB will be the neasare of GK in the projection. For, by Prop. 9, the arc MN is the measure of DB, and therefore GK, which is similar to MN, will also be the measure of it. Cor. 5. The centres of all projected circles are all beyond the projected poles, with respect to the centre of the primitive, and none of their centres can fall between them. Cur. Fig. 29. Hence it follows (by Cor. 5, and Prop. 8, Cor. 3), that all circles that are not parallel to the primitive, have equal arcs on the sphere represented by unequal arcs on the plane of projection. For, if P he the projected centre, then GH is greater than EF. N.B. It will be easy, by the foregoing propositions, to describe the representation of any circle, and the reverse will easily shew what circle of the sphere any projected circle represents. What follows hereafter, is deduced from the foregoing propositions, and will be easily understood without any other demonstration. If the sphere were to be projected on any plane parallel to the primitive, it is all the same thing; for the cones of rays issuing from the projecting point, are all cut by parallel planes into similar sections, and it only makes the projection greater or less, according to the distance of the plane of projection, whilst they are still similar, and it amounts to no more than projecting from different scales npon the same plane; and, therefore, the projecting the sphere on the plane of a lesser circle, is only projecting it upon the great circle parallel thereto, and continuing all the lines of the scheme to that lesser circle. Prop. 11. To draw a circle parallel to the primitive, at a given distance from the pole. Through the centre B draw two diameters D AB, DE, perpendicular to one another. Take with the compasses the distance of the circle from the pole of the primitive, oppo. site to the projecting point, and set it off A B from D to F; from E draw EF to intersect AB in 1; with the radius Cl, and centre C, describe the circle required Gİ. The same by the plain Scale. With the radins CI, equal to the semi-tangent of the distance of E the circle from the pole of projection, opposite the projecting point, describe the circle IG. Here the radius of the projection CA, is the tangent of 45°, or the semi-tangent of 90°. Prop. 12. To draw a lesser circle perpendicular to the primitive, at a given distance from the pole of the circle. Through the pole B draw the line of measures AB; make BG equal the distance of the circle from its pole, and draw CG and GF perpen- A dicular to it; with the radius FG, CAH B describe the required circle GI. The same by the Scale. Set off the Die secant of the distance of the circle from E its pole from C to F, which gives its centre. With the tangent of that distance for a radius, describe the circle GDI. Or thus, make BG equal to the distance of the circle from its pole, and GF |