2. When the angle is at the circumference of the primitive.

3. When the angle is in the plane of the primitive.

D

And the supplement of

UW is the measure of the angle NED, or AEF.

PROBLEM

PROBLEM IX.

To draw a small circle parallel to a given great circle, at any given distance.

I. When the small circle required is parallel to the primitive.

2. When the small circle required is parallel to a right circle.

Let the distance be

45° from the right circle BD.

Set the given distancè from B to F; reduce F to G; and through F, G, draw the circle FGH, which will be parallel to the given right circle BD, at the distance of 45°.

A

H

D

NOTE. The point K, which is the centre of the circle FGH, may be found by drawing the tangent FK from the point F. For, the centre is where it intersects the diameter AY produced.

3. When

3. When the small circle required is parallel to an oblique circle.

centre O, and radius OX, or OY, draw XBY for the circle required..

PROBLEM X.

To measure any arc of a projected small circle.

1. When the given small circle is parallel to the primitive.

COR. Any arc of the primitive circle, as DC, is reducible, in this manner, to the similar arc AB of a given parallel circle.,

2. When

2.. When the given small circle is parallel to a right circle.

to TU, and TU is the measure of the arc XY.

COR. By inverting the process any number of degrees may be set on a small circle, parallel to a right circle, from any given point.

3. When the given small circle is parallel to an oblique circle.

Let XY be the given

arc of XYZ, parallel to ABC.

Through the poles P, p, and the points X, Y, the extremities of the given arc, draw the circles PXP, PYP, perpendicular to the given oblique circle ABC. Then OQ is equal to XY, and to LM. And LM is the measure of the arc XY.

COR. By inverting the process any number of degrees may be set on a small circle, parallel to an oblique circle.

PROJECTIONS OF THE SPHERE.

A PROJECTION OF THE SPHERE is a perspective representation of the circles of the sphere. Each of the following figures, or general projections on the planes of great circles, contains a hemisphere. In the stereographic projection,, the hemisphere, opposite to the eye, falls within the primitive, to which hemisphere this kind of projection is generally limited. If it be applied to any part of the hemisphere next to the eye, the representation extends beyond the primitive.

PROBLEM I

To project the sphere stereographically on the plane of the equator.

1. Draw the primitive circle WNES, of a convenient bigness, to represent the plane of the equator; and draw. the diameters WE, NS, at right angles to each other, which here represent the equinoctial and solstitial colures.

2. Divide each of the quadrants NE, ES, SW, WN, in to nine equal parts; by which means the equator will be divided into 36 parts, each 10. Then through each of these divisions draw lines to P, the pole. Each of these 10° may be divided more minutely.

3. Lay a rule over N and a, and it will cross PW at x; over Nb, and it will cross PW at y; and over Nc, and it will cross at z, &c. With radii Px, Py, Pz, &c. and centre P, through the points x, y, z, &c. draw concentric circles, which will be parallels of latitude 10° distant from each other. The tropic and polar circle may be projected by setting 23° 28′, and 66° 32′, from W toward S, reducing them, to WP, and describing circles through the points of intersection in WP, concentric with the parallels of latitude.

4. If the projection be intended for the northern hemisphere, the tropic is that of cancer, and the northern half of the ecliptic is to be projected through WE. In the projection of the southern hemisphere is the tropic of capricorn, and the southern half of the ecliptic touching it.

Stereographic