must be used. Having drawn through q a diameter A B to the primitive circle, and produced it, draw the diameter PS at right angles to it; then join P, Q and make QM, QN each equal to a quadrantal arc. Join P, M and P, N, and produce PN; the intersections m and n will determine the diameter mn, and mPns will be the projected circle.

When the point q is in the circumference of the primitive circle, the projection of the great circle is a diameter of the primitive at right angles to the line joining q and the centre.

PROPOSITION XI.

36. To find the poles of a given projected small circle, and the converse.

also

N

a

P

R

Let APBS be the primitive circle, and namb the given projected small circle. Find s the centre of this circle, and through it draw AB, a diameter of the primitive circle; through C, the centre of the primitive circle, draw PS at right angles to AB. Draw Pn intersecting the primitive in N; also draw Pm and produce the

n

N

с

m

E

A

'n' B

M/

latter till it cuts the primitive in M. Then the line NM will be the diameter of the circle of which namb is the projection; bisect the arc MN in Q, and draw QR through the centre of the primitive; then Q and R will be poles of the given circle. Draw PQ, PR, and produce the latter; then the intersections q and r will be the projected poles of the given circle.

If about a given pole, as q, it were required to describe the projection of a small circle at a distance expressed by a given arc or a given number of degrees, a converse operation must be performed.

Through q and c, the centre of the primitive circle, draw the diameter AB; and draw PS at right angles to it: from P draw the line PQ through q, and make each of the arcs QM and QN, by a scale of chords, equal to the given distance: draw PM and PN, and produce either of those lines if necessary; the intersections m and n will determine the diameter, and the circle namb will be the required projection.

If the given pole were at C, the centre of the primitive, a circle described about c as a centre with a radius equal to the tangent of half the given distance of the circle from its pole would (1 Cor. art. 33.) be the required projected circle.

If the given pole were on the circumference of the primitive circle, as at B; on a diameter of the primitive, passing through B and produced, make CE equal to the secant of the given distance of the circle from its pole; then (2 Cor. art. 33.) E will be the centre, and the tangent of the given distance will be the radius of the required projection.

Or, by a scale of chords make the arc BM' equal to the given distance, and from м' draw a tangent to the primitive circle, meeting the radius CB produced in E; then E will be the centre, and EM' the radius of the required projection.

Or again, having made BM' and BN' each equal to the given distance, make cn' equal to the tangent of half the arc SM', that is, of half the complement of the given distance: then a circle described through м', n', and N', will be that which is required.

PROPOSITION XII.

37. To describe the projection of a great circle of the sphere through two given points on the plane of projection.

A

G F

N

R

Let ABP be the primitive circle, c its centre, and M and N the two given points. Through either of the points, as M, draw a line MR through c, and draw the diameter AP at right angles to MC. Join M, P and draw PR at right angles to MP meeting MC produced in R: then through M, N, R describing a circle; it will be the projection required.

M

C

B

Since the angle MPR

Let the circle intersect the primitive in E and F, and draw CE, CF. is a right angle, and CP is perpendicular to MR, the rectangle MC. CR is (Euc., 35. 3.) equal to CP2; and since CE, CF are each equal to CP, MC.CR CE.CF. Now if EC and CF be not in one straight line, let ECG be a straight line; then (Euc., 35. 3.) MC.CR=EC.CG; therefore EC.CG = EC.CF, and CG would be equal to CF, which is absurd. Therefore ECF is a straight line, and consequently EMNR is the projection of a great circle of the sphere.

PROPOSITION XIII.

38. Through a given point in the circumference of the primitive circle, to describe the projection of a great circle of the sphere making with the plane of projection a given angle.

P

P

Let APBS be the primitive circle in the plane of projection, c its centre, and P the given point. Through P draw the diameter PS, and at right angles to it the diameter AB: then make the angle CPD equal to the given angle. The point D is the centre, and the line DP the radius of the projection (art. 32.): thus the required circle Pm s may be described.

It is obvious that if the given point were, as at P', not in the cir

A

3

m

C

D

S

cumference of the primitive circle, the centre of the required projection might be found thus: describe an arc of a circle with P' as a centre, and with a radius equal to the secant of the angle which the circle makes with the plane of the primitive; and also an arc from c the centre of the primitive, with a radius equal to the tangent of the angle (the radius of the primitive being supposed to be unity): the intersection of these arcs would be the centre of the required projected circle, and the secant of the angle would be its radius.

PROPOSITION XIV.

39. Through any given point to describe the projection of a great circle of the sphere, making a given angle with the projection of a given great circle.

E

M

B

Let MAB be the primitive circle, and MEN the projection of the given great circle, also let P be the given point. Find (art. 35.) H, a pole of MEN, and about H as a pole describe a small circle a b, (art. 36.) at a distance from H equal to the arc, or number of degrees by which the given angle is expressed; or at a distance equal to the supplement of that arc, if the latter were greater than a quadrant. About P as a

A

R

N

H

T

pole, describe the projection QR of a great circle (art. 35.); and if the data be such that the construction is possible, this circle will touch or cut the small circle ab: let it cut the circle in p and q. Then, either from p or q (suppose from p) as a pole describe the projection PT of a great circle, it will pass through P and be the projection which is required.

For let it intersect the circle M E N in E: then, since p is a projected pole of PET, and H is a projected pole of MEN, an arc of a great circle drawn from p to H would measure the distance between the poles of the circles M E N and PET; therefore it would measure the angle M E P, or that at which the circles are inclined to one another. But the circle ab was described about H at a distance equal to the measure of that angle; therefore either the angle MEP or the angle MET is equal to the given angle, and PET is the required projection.

E

P

If through a given point, as P, it were required to describe the projection of a great circle, making with the projection NPS of a great circle perpendicular to the plane of projection any given angle, the construction might be very conveniently effected in the following manner. Through P describe the projection of a great circle at right angles to NS; this will pass through A and B, the poles of N S, at the extremities of the diameter ACB at right angles to NS, and let R be its

B

D

R

X

F

S

centre. Then, considering A P B as a new primitive circle, through R draw the line R X at right angles to NS; and draw PD making the angle R PD equal to the complement of the given angle, that is, equal to the angle which the circle whose projection is required makes with the plane of the circle whose projection is a P B. The point D is the centre, and D P the radius (art. 32.) of the circle EPF, the required projection.

PROPOSITION XV.

M

40. To describe the projection of a great circle making, with two projected great circles, angles which are given. Let MQN be the primitive circle, c its centre, and M P N, QPR, the projections of two given great circles. Find (art. 35.) p a projected pole of M PN, and q a projected pole of QPR: about p as a pole (art. 36.) describe the projection of a circle at a distance equal to the angle which the required projection is to make with the circle M P N, and about q as a

Σ

H

༡

Y

C

T

K

R

N

pole, a circle at a distance equal to the angle which the

required projection is to make with Q PR. Then, if the data be such that the construction of the problem is possible, the circles about p and q will either touch in some point, or cut each other in two points; let them cut each other in H and K, and about either H or K (suppose H) as a pole describe (art. 35.) the projection x Y of a great circle. This will be the circle required, and the angles at s and T will be equal to the given angles.

Р

For, since p is a pole of M P N and H a pole of XS Y, the arc of a great circle which measures the distance between and H will measure the angle PSX or P S T, between the circles MPN and XSY: and for a like reason the arc of a great circle between q and H will measure the angle P T S or PTY. But the distances between p and H, q and H are by construction equal to arcs which measure those angles; therefore the angles at s and T are equal to those which were given.

PROPOSITION XVI.

41. To measure an arc of a projected great circle.

S

If the given arc, as A B, be anywhere on the circumference of the primitive circle, it may be measured by a scale of chords on which the chord of 60° is equal to the radius of the primitive. And if the given arc, as CD, be on the projection of a circle at right angles to P the plane of projection, and one extremity of the arc be at

the centre or pole of the primitive, it may be measured on a scale of tangents; the

m

M

A

m'

P

B

D

number of degrees found on the scale being doubled, because CD (art. 33. 1 Cor.) is equal to the tangent of half the arc which it represents.

If the arc, as M N, be on the circumference of a projected circle which does not coincide with the primitive, it may be measured in the following manner. Find (art. 35.) P a projected pole of S M N A, and draw the lines P M m and PNn, cutting the primitive circle in m and n; the arc m n, measured by a scale of chords, is the value of the arc of which M N is the projection. For PM m and PN n being straight lines, are the projections of two circles which pass through the projecting point, or the pole of the primitive; and since they pass through P, a projected pole of S M N A, the circles which they represent must pass through a pole of the circle