PROBLEM VII.

Through a given point in a given projected great circle, to describe the projection of another great circle, cutting the former at a given angle.

Let ABCD be the primitive, and Z the given angle. 1. When the given circle is the primitive. Let A be the given point; draw the perpendicular diameters AC, BD; make angle EAFZ; and from F, as a centre, with a radius FA, describe the circle AGC; it is the required projection (Pr. I. 3, Cor. 2). When the angle is a right angle, the diameter AC is evidently the required projection.

B F

E G

2. When the given projected circle is a diameter of the primitive.

Let BD be the given projection, and F the given point. Find GH the locus of all the great circles passing through F (Pr. I. Prob. 1); draw IL perpendicular to BD, and FH making an angle LFHZ; from the centre H, with the radius B HF, describe the circle IFK; it is the required projection (Pr. I. 3, Cor. 2).

If the angle be a right angle, G is the centre, and AFC the required projection, for angle LFG a right

L

K

G

D

H

angle. Or, since the required circle is, in this case, perpendicular to BFD, it must pass through its poles A and C. Hence the circle A, F, C, passing

through the three points A, F, C, is

the required projection.

3. When the given circle is inclined to the primitive.

B

Let AFC be the given circle, and F the given point in it. Find EG I the locus of the centres of all the great circles passing through F (Pr. I.

E

F

H

G

Prob. 1). Draw FH a radius of the given circle, and draw FG, making the angle GFHZ; from the centre G, with the radius GC, describe IFE; and it is the required projection (Pr. I. 3, Cor. 2).

When the angle Z is a right angle, draw from F a line perpendicular to FH, and it will cut EG in the centre of the required circle. Or since, in this case, the required projection must pass through the pole of AFC; find its pole, and describe the projection of a great circle passing through this pole and the point F (Pr. I. Prob. 2), and it will be the required circle.

PROBLEM VIII.

Through a given point in the plane of the primitive, to describe the projection of a great circle cutting that of another great circle at a given angle.

Let AKB be the given circle, Z the given angle, and C the given point in the plane of the primitive AMB.

A

Find F the pole of AKB, and about it describe a small circle IGN, at a distance from its pole equal to the measure of angle Z. About the given point C, as a pole, describe a great circle LHM, intersecting the small circle in L and G. About either of these points, as G, for a pole, describe a great circle DCE, and it is the required projection. For the circle DCE must pass through C, since C is at the distance of a quadrant

IN

B

M

G

from G, a point of the circle LGM. Also, the distance between F and G, the poles of AKB and DCE, is the measure of the given angle, and hence the inclination of the circles is equal to that angle.

Schol. 1.-Let an arc of a great circle FCK be described through F and C; then, FK and CH being quadrants, FH=CK. Now, FH must not exceed FN the measure of the angle, otherwise the circle LHM would not meet IGN, and the problem would be impossible. But CK=FH; therefore the distance of the given point from the given circle must not exceed the measure of the angle.

Schol. 2.-If the point C were in the centre of the primitive, the circle LGM would coincide with the primitive. If C were in the circumference of the primitive, the circle LGM would be a diameter perpendicular to that passing through C.

PROBLEM IX.

To describe the projection of a great circle that shall cut the primitive and a given great circle at given angles.

Let ADB be the primitive, AEB the given circle, and X, Y, the given angles, which the required circle makes respectively with these circles.

About F, the pole of the primitive, describe a small circle at a distance equal to the measure of angle X, and about G, the pole of ALB, describe another small circle at a distance equal to the measure of angle Y. Then from either of the points of intersection H, I, as I for a pole, describe the great circle CED, and it is the required circle. For the distances of its pole I from F and G,

B

the poles of the given circles, are equal to the measures of the angles X and Y; and therefore the inclinations of CED to the given circles are equal to these angles.

"

Schol.-When any of the angles exceeds a right angle, the distance of the small circle from its pole is greater than a quadrant. The same small circle will be determined by finding the more remote pole, that is, the projection of the pole nearest to the projecting point, and then describing a small circle about it at a distance equal to the supplement of the measure of the angle. SANA

EXERCISES.

1. If, from one of the points, in which a perpendiculat small circle meets the primitive, a straight line be drawn to any point in the circumference of its projection, and con tinued to meet the diameter of the primitive that is perpen dicular to the line of measures, the point of concourse will be the pole of the projected great circle, which passes through the poles of the small circle, and that point in the circum ference of its projection., lt

2. If two equal circles, one of which is parallel, and the other inclined to the primitive, be projected, the distances of the pole of the inclined projected circle, from the centres of the projections, will be directly proportional to the radii of these projected circles.

3. If two equal circles, one of which is parallel, and the other inclined to the primitive, be projected, straight lines, drawn through the pole of the inclined projected circle, will intercept corresponding arcs on the projection.

SECOND BOOK.

ORTHOGRAPHIC PROJECTION OF THE SPHERE.

In the orthographic projection of the sphere, the projecting point is still supposed to be in the axis of a great circle, assumed as the primitive or plane of projection; but at so great a distance, that a straight line drawn from it to any point of the sphere, may be considered as perpendicular to the plane of the primitive.

The orthographic projection of any point, therefore, is where a perpendicular from that point meets the primitive.

PROPOSITION I.

Every great circle, perpendicular to the primitive, is projected into a diameter of the primitive; and every arc of it, reckoned from the pole of the primitive, is projected into its sine. Let BFD be the primitive, and ABCD a great circle perpendicular to it, passing through its poles A, C; then the diameter BED, which is their line of common section, will be the projection of the B circle ABCD.

For, if from any point as G, in the circle ABC, a perpendicular GH fall upon BD, it will also be

H

E

H

D

perpendicular to the plane of the primitive. Therefore H

G

is the projection of G. Hence the whole circle is projected into BD, and any arc AG into EH GI, its sine.

=

COR. 1.-Every arc of the great circle, reckoned from its intersection with the primitive, is projected into its versed sine.

COR. 2.-The orthographic projection of any point on the surface of the sphere, is, within the primitive, distant from its centre, by the sine of that point's distance from either pole of the primitive.

COR. 3. Every small circle, perpendicular to the primitive, is projected into its line of common section with the primitive, which is also its own diameter; and every arc of the semicircle above the primitive, reckoned from the middle point, is projected into its sine. COR. 4. Every diameter of the primitive is the projection of a great circle, and every other cord the projection of a small circle.

COR. 5. A straight line, perpendicular to the primitive, is projected into a point; a parallel to the primitive, into an equal line; and one inclined to the primitive, into a less line, such that the radius is to the cosine of the inclination, as the inclined line to its projection. COR. 6. A spherical angle, at the pole of the primitive, also any rectilineal angle, whose plane is parallel to the primitive, is projected into an equal angle.

PROPOSITION II.

A circle parallel to the primitive is projected into a circle equal to itself, and concentric with the primitive.

F

Let the small circle FIG be parallel to the plane of the primitive BND. The straight line HE, which joins their centres, is perpendicular to the primitive; therefore E is the projection of H. Let any radius HI and IN perpendicular to the primitive be drawn. Then IN, HE, being parallel, are in the B same plane; therefore IH, NE,

H

E

the lines of common section of the plane IE, with two parallel planes, are parallel, and the figure IHEN is a paral