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. i, are projected into the nght lines Pe and Pf passing through P; and the intercepted arcs representing equal arcs on the pire, are therefore e jual ; 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 other circle be described, the G vertex of which is P, the arc ef 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. 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 aro parallel, and the arc NI=PH, K N therefore, <FLI = NKT or nKI=GIP; therefore, the tri P angles IEL and IEG are similar; whence EL : EI :: EI : EG. E LV CDI Again, the angle EMI=KNI =PIQ ; and, therefore, the triangles IEM and IED are similar ; whence SM : EI :: EI : ED. Therefore, EI? = EL XEG = EM X ED '; conse EM + EL quently, EM: EL :: EG : ED; and, by composition 2 EM EL EG+ED EG-ED : 2 that is, CM:EC :: FG: EF, 2 ; 2 Q. E. D. Cor. 1. Hence, if the circle KN be as far from t':e projecting point, as QH is from either of its poles, and if E and 0 bc iis project. d poles, then will EL : EM :: ED: EG :: OD:0G. This follows from the foregoing demonstration, d Cor. 4, Prop. 1. Cor. 2. Hence, also, if F be the centre, and FD the radius of any circle QH, and E and ( the projected poles, then as EF: DF :: DF: FO. For it follows, from Cor. i, that ÉG +ED:EG-ED ::0G+OD:UG-OD 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 morasure 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 DI, aud 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. Cor. 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 be 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 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 dif. ferent 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 É 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 radius CI, equal to the semi-tangent of the distance of 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 CH B в dicular to it; with the radius FG, describe the required circle GI. The same by the Scale. Set off the 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 this, make BG equal to the distance of the circle from its pole, and GF F its tangent set off from G, given F the centre; through G describe the c.rcle GI from the centre F. Cor. Hence, a great circle perpendicular to the primitive, is a right-hne CDE drawn throngh the centre perpendicular to the line of measnres. N.B. When the centre F lies at too great a distance, draw EG io cut AB in H; or lay the semi-tangent of DG from C to h; and, throngi tie thuec points G, H, I, draw a circle (which may be done, if only a small portion is required, by an instrument called a cylographi, invented by the Editor of this work). Prop. 13. To describe an oblique circle, at a given distance from a given pole. Draw the line of measures AB through the point p, if it is given, and draw DE perpendicular to it; also draw EpP; or, i the pointpis not given, B В set off the height of the pole above the primitive from B to P, then from P set off PH=PI equal the distance of the circle from E its pole, and draw EH and EI to intersect AB in F and G. About the diameter FG describe the required circle. The same by the Scale. If the point P is given, apply Cp to the semi tangent, and it will give the distance of the pole from D, the pole of projection opposite to the projecting point. This distance being obtained, it will be easy to find the greatest and nearest distances of the circle from the pole of the primitive opposite to the projccting point; take the semi-tangent of these distances, and set thein off from C to G and F, both the same way if the circle lie all on one side; but each its own way, if on different sides of D. And then FG is the diameter of the circle required. Cor. t. If F be the pole of a great circle as of DLE, draw EFH through the pole F; make HK equal 90° ; draw EK cutting the line of measures iu L, through the three points D, L, E, draw the required great circle. C. p. 2. Hence it will be easy to draw one circle parallel to another. Prop. 14. Through two given points A and B, to draw a g.eat circle. Through ope of the points A, and through the centre draw a line ACG, B 71 and EF perpendicular to it; draw AE and EG perpendicular to AE ; then, through the three points A, B, G, draw G the required circle. Or thas, from E, found as before, draw EH, HCI, and EIG, which gives a third point G, throngh which the circle most E pass. The same by the Scale. Draw ACG, and apply AC 10 the semi tangents; find the degrees; set off the semi-tangent of its supplement from C to G for a third point Or tuus, apply AC to the tangents, and set off the tangent of its comple. ment from C to G; and, through the three points A, B, G, describe the required circle. For, since HEI, or AEG, is a right-angle, A and G represent opposite points of the sphere, and whence all circles, passing through A and G, are great sircles. Prop. 15. About a given pole, and through a given point, to describe the representation of a circle. Let P be the pole, and B the given point; through P and B de IL scribe the great circle AD, by Prop. B 14, the centre of which is E; through the centre C draw CPH ; and, from the centre E, draw EB and BF perpendicular to it. With the centre F, and radius FB, decribe the required circle BHA, P H to P, Prop. 16. To find the poles of any circle FNG. D draw the line of mea H sures AG, and DE K f B Epp, then p is the pole. Or thos, draw EFH ain! EIG, and binect HI in P, and draw EpP, and p is the internal pole. Lastly, draw PCQ, EQq, and q is the external pole. In a great circle DLE, draw ELK, and makc DHAK, or KH=AD, and draw EFH, and F is the pole. Q E The same by the Scale. Apply CF to the semi-tangents, and note the degrees. Take the sum of the degrees, and of the distance of the circle from its pole, if the circle be all one side, but their difference if it encompass the pole of projection, set off the semi-tangent of this sum, or difference, from C to the internal pole p, and the semi-tangent of its supplement Cp, will give the external pole g. Or thus, apply CF and CG to the semi-tangents, set off the semi-tangent of half the sum of the degrees, if the circle lies all one way, or of half the difference if it encompass the pole of projection, from C to the pole p, and the semi-tangent of the supplement, Co gives the external pole q. In a great circle, as DLE, draw the line of measnres AB perpendicular to DE, and set the tangent and co-tangent of half its inclination from the centre C, different ways, to F and f. |