PROPOSITION V.

If an oblique circular cone be truncated by a plane at (588) right angles to that plane which passes through the axis and is perpendicular to the base, the section will, in general, be an ellipse.

Let AoOBy be the truncating plane perpendicular to the plane, AMVBN, passing through the axis at right angles to the base; draw oy = y perpendicular to AB, and pass the plane, MNy, parallel to the base, it will be a circle, and the common ordinate, oy = y, will be perpendicular to the diameter, MoN;

(oy)(oy) = (oM)(0N).

N

Fig. 126.

M

But

ON sinoBN and =

sinoMA' OB sinoNB

.., putting OA = OB = a, Oo = x, and b = yx=09

Cor. The section becomes a circle when the angles, (589) which the truncating plane and circular base make with the sides of the cone, are equal and contrary.

For, when the angle oAM = oNB, then ▲ oBN = oMA, and the equation becomes

y2 = a2 - x2, that of a circle.

PROPOSITION VI.

The Stereographic Projection of any circle of the (590) sphere is, in general, itself a circle.

In the stereographic projection every point of the spherical surface is thrown upon an equatorial plane, denominated the primitive circle, by the intersection with this plane of a line drawn through the point and the eye situated in the pole of the primitive. Now, let AB be that diameter of any circle,

STEREOGRAPHIC PROJECTION.

whose extremities lie in the circumference, AP,BOH.P.EH,, passing through the poles, E, O, of the primitive H2H¡H ̧H2, whose centre, o, is the same with the centre of the sphere, and draw AE, BE, intersecting HH, in a, b; ab is the stereographic projection of AB. We have,

measure of

EBA = arcEH„A = †(EH, + H„A)= meas. of ▲ Eab;
ZEBA
Eab,

Fig. 127.
L

=

and (589) the projection of the circle described on AB is itself a circle situated in the primitive H‚Í‚H ̧H2, and having the diameter, ab.

PROPOSITION VII

The distance from the centre of the primitive at which (591) any point will be projected, will be equal to the tangent of half the arc intercepted between the point and the pole of the primitive, the radius of the sphere being taken for unity.

or

For let A be any point on the surface of the sphere; we have

Cor. 1. If m be the centre of the circle, ab, we have,

(591)

dist. of centre om = (oa + ob) = (tan+OA+tan+OB), (592) where it is to be observed that oa, ob, or their equivalents, tan OA, tan OB, change signs on A, B, passing O.

Cor. 2. The radius ma =

(tanОA — tanдOB).

(593)

Cor. 3. The projection of a circle parallel to the primi- (594) tive will be concentric with the primitive.

For, if A,B, be parallel to H,H,, we have

OA2 = OB

om2 = (tan10A, — tan‡OA1⁄2) = 0, ma2 = (tanдOA2+ tanдOA,) = tanдOA2.

Cor. 4. If the pole of the circle be in the primitive, the (595) distance of its projected centre will be the secant of its radius, and the radius of projection the tangent of the same arc.

Cor. 5. A great circle perpendicular to the primitive (596) will be projected in a diameter of the primitive.

The same is obvious from the figure.

Cor. 6. Any great circle will have for the distance of its (597) projected centre, the tangent of the arc measuring its inclination to the primitive, and the radius of projection will be the secant of the same arc.

Let PP, be the diameter of any great circle, PH ̧P ̧H„, and imitate the reasoning in (595.)

Scholium I. Any point situated in the primitive, is its (598) own projection.

Scholium II. If a great circle and the primitive intersect (599) each other in the diameter, H,H,, and a third circle passing through the axis of the primitive in the diameters, P,P., H2H,, the points, H1, H2, may be found by spherical trigonometry, and the point, pr the projection of P, having been determined by (591), there will be three points given through which to pass the circumference, HpH, the projection of H,P„H2.

HH, will be found (586),

if we put

H1H1 = H, H„P„H, = h‚H„P2 = l

PROPOSITION VIII.

30°

N

30°

60°

To make the Stereographic Projection of the Sphere. I. Let the eye be situated at the south pole in order to project the northern hemisphere. Describe the primitive and graduate its circumference according to the number of meridians which it is intended to lay down; these will be diameters passing through the several points of division (596).

OG

S

Fig. 128.

For the parallels of latitude, we have only to count off from the

STEREOGRAPHIC PROJECTION.

287

north point the corresponding degrees, and from the points of division to draw lines to the south point, and the intersections thus formed with the east and west diameter will be in the circumferences of the circles of latitude whose centres will be that of the primitive, (591), (594).

W

TE

II. Let the eye be in the equator. The primitive will pass through the poles, N, S, and the central meridian, NS, and equator, EQ, will be north and south, and east and west lines intersecting in the centre of the primitive, (596). Graduate the circumference as above, and, laying the corner of a wooden square upon a point of division, direct the edge of one arm through the centre. The intersection of the corresponding edge of the other arm with the production of the line, NS, will be the centre, and the distance of this point to the point of division the radius, for the description of a parallel of latitude, (595). The radii just employed set off from the centre upon EQ will give the centres for describing the meridians, which will pass through the points, N, S, (597), (598).

III. When the eye is sit

uated otherwise than as above, consult (598), (599), and figure 130.

Scholium. It will be observed that the middle of the map is comparatively contracted in the stereographic and enlarged in the orthographic projection. To avoid this, the lines, NS, EQ, are sometimes divided into equal parts, and the

map thus constructed is im

properly denominated a globular projection.

Fig. 130.

Fig. 129.

PROPOSITION IX.

To make the Conical Projection.

In this projection, the eye, situated at the centre of the sphere, throws every point of its surface upon that of a tangent or secant cone, the axis of which is coincident with the axis of the sphere. Let a sphere and cone be generated by the revolution of the arc, NnsES, and the straight line, vns, intersecting each other in n, s, about the common axis, vNo„0,OS, O being the centre of the sphere, N and S the poles, E a point of the equator, and no„, so„ perpendiculars upon NS.

1°. For the magnitude of a parallel of latitude, putting

OE=0s = On = ON = OS = radius = 1, arcA= length of an arc of A° in the

parallel through n,

E

arc Ado. through s, arcA;= do. through E;

Fig. 131.

(600)

=

2o. For the radii, vn, vs, with which to describe the sectoral surface, snNn,s,S, the development of the conical surface upon a plane, we have,

3o. To find the number of degrees in the angle, svs2, corresponding to a difference of latitude of A°; with the trigonometrical radius, vr = 1, describing the arc, rr, there results,