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Prop. 17. To draw a great circle at any given inclination above the primitive; or, making any given angle with it at a given point.
Draw the line of measures AB, and DCE perpendicular to it; make EK=2HD equal twice the complement of the inclination of
VL В Р the circles; or DK = 2AH = twice the inclination; and draw EKF, then F is the centre of EGD, the circle required.
Or thus, draw DE and AB perpendicular to it, and let D be the point Riven, make AH equal to the inclinatiou, and draw EGH and HCN, and ÉNO to cut AB in O. Then bisect GO iu F, for the centre of the required circle.
The same by the Scale. Set off the tangent of the inclination on the line of measures from C to F, iben F is the centre. Set off the semi-tangent of the complement from Clo G, then GF or DF is the radius.
Or the secant of the inclination being set off from G or D to F, will give the centre.
Prop. 18. Through a given point P, to draw a great circle, to make a given angle with the primitive.
Through the point given P, and the centre C,
draw the line AB and DE per. pendicular to it; set off the given angle from A to H and from H to K, and draw BGK; with the radius CG, and centre A А
IC LP C, describe the circle GIF; and, with the radius BG, and centre P, cross that circle in F; then, with the radius FP, and centre F, describe the circle LPM, required.
The same by the Scale. With the tangent of the given angle, and one foot in C, describe the arc FG with the secant of the given angle, and une foot in the given point P. cross that arc at P; from the centre F describe a circle through the point P.'
Prop. 19. To draw a great circle to make a given angle with a given oblique circle FPR, at a given point P in that circle. Through the centre C, and the given
FG point P, draw the right-line DE, and AB perpendicular to it; draw APG, D and make BM=2DG ; and draw AM to cut DE in I; draw IQ perpendicular to DE, then IQ is the line. wherein the centres of all circles are found, which pass through the point P. Find N, the centre of the given circle FPR, and make the angle NPL equal to the given angle, tha L is the centre of the circle HPK, as
The same by the Soole. Through P and C draw DE; apply CP to the semi-tangents, and set off the tangent of its complement from C to I, or the secant from P to I; on DI erect the perpendicular IQ: fod the centre N of FPR, and make the angle NPL equal the angle given, and L is the centre.
Cor. If one circle is to be drawn perpendicular to another, it must be drawn through its poles.
Frop. 20. To draw a great circle through a given point P, to make a given angle with a given great circle DE. About the given point P, as a pole,
D by Prop. 13, Cor. 1, describe the great circle FG; find I, the pole of
K the given circle DE, and by Prop. 16;
P aboutthe pole I, by Prop. 13, describe a the small circle HKL, at a distance equal to the given angle, to intersect
Б FG in H; about the pole H de
H scribe, by Prop. 13, the great circle APB as required.
Prop. 21. To draw a great circle to cut two given great circles abd, elf, at given angles.
Find the poles ser of the two given circles, by
f Prop. 1ő, about which draw two parallels phk
28 and pnk, at the distances respectively equal to the angles given, by Prop. 13, the point of inter
2 section p, is the pole of the required circle moq.
Cor. Hence, to draw a right circle, to make with an oblique circle abd, any given angle.
Draw a parallel hk at a distance from the pole of the oblique circle equal to the given angle. Its intersection f with the primitive, gives the pole of the right circle gct, as required.
Prop. 22. To lay any number of degrees on a great circle, or to measure any arc of it.
CV bus Let AFI be the primitive; find
CA the internal pole P of the given circle DEH, by Prop. 16, lay the degrees on the primitive from A to F, and draw PA, PF, intercepting the
A DCPS part DE required.
Or, to measure De, draw PEF and PDA, and AF is its measure, which, applied to thu line of chords, shows how many degrees it contains.
Or thus, find the external pole p of the given circle, set off the given degrees from I to K, and draw pl, pk, intercepting the part DE required; or, to measure DE, through D and É draw pl, pk, then KI is the measure of DE.
Or thns, through the internal pole P draw the lines DPG and EPL, setting off the given degrees from G to L on the circle GL; then DE is the arc required. Or, if it be required to measure DE, the degrees in the arc GL is the mcasure of DE,
Or thus, set off the given degrees from G to H on the circle GL, and, from the external pole P draw pg, pH, intercepting DE, the arc required; or, to measure DE, draw pDG, PEH, then the degrees on GH are equal to DE.
By the Seule for Right Circles. See fig. Prop. 22. Let CA be the right circle, take the number of degrees from the semi-tangents, and set them off from C to D for the arc CD.
Or, if the given degrees are to be set off from A, then take the degrees off from the semi tangents from 90° towards the beginning, and set them off from A to D, and if CD was to be measured, apply it to the beginning of the semi. tangents; and, to measure AD, apply it from 90° backwards, and the degrees intercepted gives its measure.
N.B. The primitive is measured by the line of chords, or else it is actually divided into degrees.
Prop. 23. To set off any number of degrees on a lesser circle, or to measure any arc of it.
Fig. 1. Let the lesser circle be DEH; find its internal pole P, by Prop. 16, describe the circle AFK parallel to the primitive, by Prop. 11, and as far from the projecting point, as the given circle DE is from its internal pole P, set off the given degrees from A to F, and draw PA and PF intersecting the given circle in D and E; then DE is the arc required. Or, to measure DE, draw PDA and PEF, and AF exhibits the degrees on DE.
Or thus, find the external pole p, of the given circle, by Prop. 16, describe the lesser circle AFK as far from the projecting point, as DE the given circle is from its pole p, by Prop 11 ; set off the degrees from I to K, and draw PDI and pEK, tben De represents the given number of degrees ; or to measure DE, draw pDI, and pĖK, aud KI is the measure of DĚ.
Or thos, let O be the centre of the given circle DEH; through the internal pole P, draw the lines DPG and EPL, divide the quadrant GQ into ninety equal degrees, and if the degrees be set off from G to L, then DE will repro sent these degrees. Or the degrees in GL will measure DE,
Or thus, divide the quadrant GR into ninety equal parts, or degrees, and set off the given degrees from G to H, and draw pDÅ, and pEH, from the external pole p; then DE will represent the given degrees.
Or, through D and E, draw pDG, and rEH, then the number of equal de grees on GH is the measure of DE.
Scholium. Any circle parallel to the primitive is divided or measured, by drawing lines from the centre to the divisions of the primitive.
Prop. 24. To measure an angle:
By Cor. 1, Prop. 13, about the angular point as a pole, describe a great circle, and note where it intersects the legs of the angle; through these points of intersection, and the angular point, draw two rigbt lines to cut the primitive; the arc of the primitive, intercepted between them, is the measure of the angle. This needs no example.
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 lives 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 angular 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 circles 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-cangent 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 projected 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 langent Cd.
Cor. 2. Any point, as D, 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.
For 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, tben all these circles will be represented by lines parallel to one another, and perpendicular to the line of measures, or represeniation 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 plans 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 CI 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 4from the pole, it is projected into a circle equal to a great circle of the sphere, and may, 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 frons 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. tance.
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 A FE being cut by the plane, f 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 s, 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 bę lens 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 of 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 en byperbola, the vertices of which 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, havings for its vertex.
For the plane dp cutting the cone FAē, is parallel to the side AE. Cor. 4. If I be the ceutre, and Kkl the focus of the ellipsis, hyperbola, or puabola ; theo HK equal Ad-As for the ellipsis, and HR equal
nE+ for the hyperbola, and drawing fx perpendicular on AE, n equal for the parabola, wbich are the representatiods of the circles DF, PG, FRA repectively.