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Or thus, by Prop. 16, find the two poles of the containing sides, the nearest If it be an acute angle, otherwise the farthest, and through the angular point and the said poles, draw right lines to the primitive, then the intercepted arc of the primitive is the angle required. As if the angle AEL ( fig. 2) was required. Let C and F be the poles of EA and EL; from the avgular point E, draw ECD and EFH; then the arc of the primitive DH, is the measure of the angle AEL,
N.B. Because, that in the Stereographic projection of the sphere, all oircles are-projected either into circles, or right lines, which are easily described ; therefore, this sort of projection is preferred before all others. Also those planes are preferred to others in order to project upon, where most circles are projected into right lines, they being easier to describe and measure than circles are ; such are the projections on the planes of the meridian and solsti. uial colure.
SECTION III. The Gnomonical Projection of the Sphere. Prop. 1. Every great circle, as BAD, is projected into a right line, perpendicular to the line of measures, and distant from the centre the co-tangent of its inclination, or the tangent of its nearest distance from the pole of projection
Let CBDE (fig. 3) be perpendicular both to the given circle BAD and plane of projection, and then the intersection CF will be the line of measures. Now, since the plane of the circle BD, and the plane of projection are both perpendicular to BCDE, their common intersection will also be perpendicular to BCDF, and, consequently, to the line of measures CF; and, since the projecting point A is in the plane of the circle, all the points of it will be projected into that section ; that is, into a right line passing through d, and perpendicular to Cd; and Cd is the tangent of CD, or co-tangent of Cda. Q. E. D.
Cor. 1. A great circle, perpendicular to the plane of projection, is pro jected into a right line passing through the centre of projection; and any arc is projected into its corresponding tangent. Thus the arc CD is projected into the tangent Cd. Cor. 2. Any point, as V, or the pole
of any circle, is projected into a point d, distant from the pole of projection C the tangent of that distance.
Cor. 3. If two great circles be perpendicular to each other, and one of them passes through the pole of projection, they will be projected into two right lines perpendicular to each other.
Por the representation of that circle which passes through the pole of projection, is the line of measures of the other circle.
Cor. 4. And hence, if a great circle be perpendicular to several other great cir. cles, and its representation pass through the centre of projection, then all these circles will be represented by lines parallel to one another, and perpendicular to the line of measures, or representation of that first circle.
Prop. 2. If two great circles (fig. 3) intersect in the pole of projection, their representation will make an angle at the centre of the plane of projection equal to the angle inade by the said circles on the sphere.
Por, since both these circles are perpendicular to the plane of projection, the angle made by their intersections with this plane, is the same as the angle made by the circles. Q. E. D.
Prop. 3. Any lesser circle, paraliel to the plane of projection, is projected into a circle, the centre of which is the pole of projection ; and the radius the tangent of the circle's distance from the pole of projection.
Let the circle PI (fig. 3) be parallel to the plane GF, then the equal arcs PC and ČI are projected into the equal tangents GC and CH ; and therefore C, the point of contact and pole of the circle PI, and of the projection, is the centre of the representation GH. Q. E. D.
Cor. If a circle be parallel to the plane of projection, and 44° from the pole, it is projected into a circle equal to a great circle of the sphere, and maya therefore, be looked upon as the primitive circle in this projection, and its radius the radius of projection.
Prop. 4. Every lesser circle, not parallel to the plane of projection, is projected into a conic section, the major axis of which is in the line of measures, and nearest vertex is distant fron, the centre of the plane, by the tangent of its nearest distance from the pole of projection, and the other vertex is distant by the tangent of its farthest dis.
Let BE (fig. 4) be parallel to the line of measures dp, they any circle is the base of a cone, the vertex of which is at A, and, therefore, that cone being produced, will be cut by the plane of projection in some conic section; thus the circle, the diameter of which is DF, will be cut by the plane in an ellipsis, whose major axis is df ; and Cd is the tangent of CAD, and Cf of CF. In like manner, the cone AFE being cut by the plane, s will be the nearest vertex, and the other point into which E is projected is at an intinite distance. Also the cone AFG, the base of which is the circle FG, being cut by the plane f, is the nearest vertex ; and, GA being produced, gives d the other vertex. Q. E, D.
Cor. 1. If the distance of the farthest point of the circle be less than 90° from the pole of projection, it will be projected into an ellipsis.
Thus DF is projected into d, and DC being less than 90°, the section df is an ellipsis, the vertices of which are at d and f; for the plane dj cuts both sides of the cone dA fA.
Cor. 2. If the farthest point be more than 90° from the pole of projection, it will be projected into an hyperbola. Thas the circle PG is projected into un hyperbola, the vertices of wbich are f and d, and major axis fu.
For the plane dy cuts only the side Af of the code. Cor. 3. And in the circle EF, where the farthest point E is 90° from C, it will be projected into a parabola, having / for its vertex.
For the plane dp cutting the cone FAE, is parallel to the side AE.
parabola ; then HK equal Adm. As for the ellipsis, and HR equal
netes for the hyperbola, and drawing for perpendicular op AE, n equal for the parabola, wbich are the representatiods of the circles DF, PG, PL, respectively.
Prop. 5. Let the plane TW (fig. 5) be perpendicular to the piane of projection TV, and BCD a great circle of the sphere in the plane TW, and let the great circle BED be projected into the right line bek. Draw CQS 1 lk, and Cm || to it, and equal to CA, and make QS= Qm; then any angle Qst=Qt.
Suppose the hypothenuse AQ to be drawn, then, since the plane ACQ is perpendicular to the plane Tv, and BQ is I to the intersection CQ, therefore 6Q is perpendicular to the plane ACQ, and, consequently, IQ ist to the hypothenuse AQ; but AQ=Qm=Qs, and Qs is also I to IQ. Therefore, all angles made at S cut the line bQ in the same point as the angles made at A ; but, by the angles at A, the circle BED is projected into the line 6Q. Therefore the angles at s, are the measures of the parts of the projected circle bQ, and S is the dividing centre thereof. Q. E. D.
Cor. Any great circle (Qb is projected into a line of tangents to the radius SQ.
For Qt is the tangent of the angle QSt to the radius QS or Qm.
Cor. 2. If the circle bС pass throngh the centre of the projection, then A, the projecting point, is the dividing centre thereof, and Co is the langeut of its corresponding arc CB, to CA the radius of projection.
Prop. 6. Let the parallel circle GEH (fig. 5) be as far from the pole of projection C, as the circle FKI is from its pole P, and let the distance of the poles C and P be bisected by the radius AO; and draw bAD perpendicular to AO; then any right line bek drawn through l, will cut off the arcs hl=Fn, and ge=kf, supposing f the other vertex, in the representation of these equal circles in the plane of projection.
For, let G, E, R, L, H, N, R, K, I, be respectively projected into the points g, e, r, l, h, n, r, k, f. Then, since in the sphere the are BF=DH, and the arc BG=DI, and the great circle BEKD makes the angles at B and D equal, and is projected into a right line as bl; therefore the triangular figures BFN and DHL are similar and equal; and, likewise, BGE and DIK are similar and equal, and LH is equal to NF, and KI equal to EG; whence it is evident, that their projections th=nF, and kf=ge. Q. E. D.
Prop. 7. If hlg and Fnk (fig. 6) be the projections of two equal circles, one of which is as far from its pole P, as the other from its pole C, which is the centre of projection; and, if the distance of the projected poles Cp be divided in o, so that the degrees in Co, op, be equal, and the perpendicular oS be erected to the line of measures gh, then the lines pn, Cl, drawn from the poles Cp, through any point Q, in the line os, will cut off the arc Fn=2QCp.
For, drawing the great circle GPI, in a plane perpendicular to the plane of projection, the great circle AO perpendicular to CP is projected into oS, by Prop. 1, Cor. 3. Now let Q be the projection of 4, and since pQ and CQ are lines, they will represent the great circles Pq, Cq.
But the spherical triangle PoC is an isocelis triangle, therefore the angles at P and C are equal. But, because P is the pole of FI, the