Let ADBH (fig. 41.) be the inclined circle, P its centre; and let it be projected into adbh; draw the plane ABFCa through the centre C of the sphere, perpendicular to the plane of the given circle and plane of projection, to intersect them in the lines AB, ab; draw GPH, DE perpendicular, and DQ parallel to AB; then, because the line GP, and the plane of projection, are both perpendicular to the plane ABF, GH is parallel to the plane of projection, and therefore to gh.

In the circle ADB, DQ2 = GQH = gqh, and BP2= GP2 =gp. And (16. 2. Supp.) BP: EP or DQ :: bp: ep or dq, and BP2 : DQ2 : : bp2 : dq2; that is, gp2: gqh:: bp2 : dq; and therefore agbh is an ellipsis, whose transverse gh is the diameter of the circle (Cor. 2. to 15. 2. Conics). 2. E. D.

COROLLARY 1.

Since ab is perpendicular to gh, therefore ab is the conjugate axis, and is twice the sine of the angle ABỏ to the radius gp: that is, the conjugate axis is equal to twice the cosine of the inclination, to the radius of the circle.

COROLLARY II.

The transverse axis is equal to twice the co-sine of its distance from its parallel great circle. For gh = GH = 2 AP twice the sine of AK.

COROLLARY III.

The extremities of the conjugate axis are distant from the centre of the primitive, by the sines of the circles nearest and greatest distance from the pole of the primitive. Thus aC is the sine of AN, and bC the sine of BN.

COROLLARY IV.

Hence also it is plain, that the conjugate axis always passes through the centre C of the primitive; and is always in the line of measures of that circle.

SCHOLIUM.

Every circle in the projection represents two equal cireles, parallel and equidistant from the primitive. Every right line represents two semicircles, one towards the eye, the other in the opposite side. Every ellipsis represents two equal circles, but contrarily inclined, as AB, CD; one above the primitive and the other below it.

SECTION II.

The Stereographic Projection of the Sphere.

PROP. I.

Any circle passing through the projecting point, is prajected into a right line.

For lines drawn from the projecting point to any part of the circle will be in its plane; and will therefore meet the plane of projection in the common section of that plane and the plane of the circle, which is a right line (3. 2 Supp.)

A great circle passing through the poles of the primitive is projected into a right line passing through the centre.

PROP. II.

Any point on the sphere is projected into a point, distant from the centre of the primitive the semi-tangent of its distance from the pole opposite to the projecting point.

Let E (fig. 42.) be the point to be projected; A the projecting point; M the opposite pole; BH the plane of projection (seen edgewise); ABEM a great circle perpendicular to the plane of projection. Since AM is the axis of the primitive, AC is at right angles to BH ; therefore GC is the tangent of GM, or the semi-tangent of EM.

COROLLARY I.

Any circle passing through the projecting point is projected into a right line perpendicular to the line of measures, and distant from the centre the semi-tangent of its nearest distance from the pole opposite to the projecting point.

Thus if AE be a circle passing through A, and at right. angles to ABEM, the common section of AE and the plane of projection will be perpendicular to the line of measures (def. 5. and 18. 2 Supp.), and its distance GC is the semitangent of EM.

COROLLARY II.

Any arc EM of a great circle perpendicular to the primitive, is projected into the semi-tangent of it. Thus EM is projected into GC.

COROLLARY III.

A great circle perpendicular to the primitive, is projected into a line of semi-tangents passing through the centre, and produced infinitely.

For MF is projected into its semi-tangent CH, and EM into the semi-tangent CG.

COROLLARY IV.

Any arc EMF of a great circle perpendicular to the primitive, is projected into the sum or difference of the

semi-tangents of its greatest and least distances from the pole opposite the projecting point, according as the extremities lie on different or the same side of that pole.

PROP. III.

Every circle, that passes not through the projecting point, is projected into a circle.

Let the original circle EF (fig. 43.) be parallel to the primitive BD; lines drawn from all points of it to the projecting point A, will form a conical surface, which being cut by the plane BD parallel to the base, the section GH (into which EF is projected) will be a circle. (Simson's Conics, book 1. prop. 23.)

Case 2. Let ABMF (fig. 42.) be a great circle perpendicular to the primitive and to the circle EF, A the projecting point, BH the line of measures to the circle EF; draw FK parallel to BD, then the arc AK AF, and therefore angle AFK or AHG AEF; therefore in the triangles AEF, AGH, the angles at E and H are equal, and the angle A common, therefore the angles at F and G are equal. Therefore the cone of rays AEF (whose base EF is a circle) is cut by sub-contrary section, by the plane of projection BD, and therefore the section GH (which is the projection of the circle EF, will also be a circle. (Simson's Conics, book 1. prop. 24.) 2. E. D.

COROLLARY 1.

When AF is equal to AG, the circle EF is projected into a circle equal to itself.