PROBLEM II.

To project a right circle, or one, that is perpendicular to the plane of projection.

Through the centre C of the primitive draw the diameter AB, and take the distance from its parallel great circle, and set it from A to E, and from B to D, and draw ED, the right circle required.

E

A

D

B

the projection. Also any arc, as Cc, is projected into Oo, equal to ca, the right sine of that arc; and the complemental arc cB is projected into oB, the versed sine of the same arc CB.

7. A circle parallel to the plane of the projection is projected into a circle equal to itself, having its centre the same with the centre of the projection, and its radius equal to the cosine of its distance from the plane. And a circle, oblique to the plane of the projection, is projected int an ellipse, whose greater axis is equal to the diameter of the circle, and its less axis equal to double the cosine of the obliquity of the circle, to a radius equal to half the greater axis.

and let fall the perpendiculars EF, DG; bisect FG in H, and erect the perpendicular KHI, making KH = HI = half ED; then describe an ellipse, whose transverse is IK, and conjugate FG; and that will represent the given circle.

To measure an arc of a parallel circle; or to set any number of degrees on it.

With the radius of the parallel, and centre C, describe a circle Gg, draw CGB and Cgb; then Bb will measure the given arc Gg; or Gg will contain the given number of degrees, set from B to b.

B

PROBLEM VI.

To measure any part of a right circle

In the right circle ED, let EA AD; and let AB be the part to be measured. On the

=

diameter ED, describe the semicircle END, draw AN, BO, LP, perpendicular to ED. Then ON is the measure of BA, and NP of AL; and ON, or NP, may be measured as in Prob. V.

E

N

P

D.

COR. If the right circle pass through the centre, it is only necessary to raise perpendiculars on it, which will cut the primitive, as required.

PROBLEM VII.

To set any number of degrees on a right circle.

[See Figure under last Problem.]

On ED, the given right circle, describe the semicircle END; then, by Prob. V. set off NP = the given degrees, and draw PL perpendicular to ED; then AL contains the degrees required.

PROBLEM VIII.

To measure an arc of an ellipse; or to set any number of degrees on it.

About AR, the transverse axis of the ellipse, describe a circle ABR; erect the perpendiculars BED, KFI, on AR; then BK is the measure of EF, or EF is the representation of the ark BK. And BK is to be measured, or any degrees set on it, as in Prob. V.

BK

EF

A CD IR

STEREOGRAPHIC PROJECTION.

PROBLEM I.

To find the poles of any projected great circle.

1. The poles of the primitive circle.

They are in the centre C.*

.C

*The following are laws of the stereographic projection.

1. In this projection a right circle, or one perpendicular to the plane of projection, and passing through the eye, is projected into a line of half tangents.

2. The projection of all other circles, not passing through the projecting point, whether parallel or oblique, is into circles.

on the

and E. Suppose now, that
AGB is a circle
sphere to be projected, whose
pole most remote from the eye
is P; and the visual rays
from the circle AGB, meet-
ing in E, form the cone
AGBE, of which the triangle
AEB is a section through the
vertex E, and the diameter of
the base is AB; then will the
figure agbf, which is the pro-
jection of the circle AGB,
be itself a circle. Hence, the
middle of the projected diam-
eter is the centre of the pro-
jected circle, whether it be a
great circle or a small one.
Also the poles and centres of

all circles, parallel to the

RA

plane of projection, fall in the centre of the projection; and all oblique great circles cut the primitive circle in two points di ametrically opposite.