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right angle, EAN is the complement of MAC. Also mE is the tangent of MAC, and En its cotangent.

COR. 7.-The radius of any projected circle is equal to half the sum, or half the difference, of the semi-tangents of the circle's least and greatest distances from the pole opposite to the projecting point, according as the circle does or does not encompass the axis of the primitive.

PROPOSITION III.

An angle, formed by two tangents, at the same point, in the surface of the sphere, is equal to the angle formed by their projections.

LMN. If the plane

Let FGI and GH be the two tangents, and A the projecting point; let the plane AGF cut the sphere in the circle AGL, and the primitive in the line BML. Also, let MN be the line of common section, of the plane AGH, with the primitive. Then the angle FGH FGH be parallel to the primitive BLD, the proposition is manifest. If not, through any point K, in AG produced, let the plane FKH, parallel to the primitive, be extended to meet FGH in the line FH. Then, because the plane AGF meets two parallel planes, BLD, FKH, the lines of common section, LM, FK, are parallel; therefore the angle AML AKF. But, since A is the K pole of BLD, the cords, and conse

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quently the arcs AB, AL, are equal; and the arc ABG is the sum of the arcs AL, BG. Draw GP parallel to BL; then the arc BG LP; for if B and P were joined, the alternate angles would be equal. Hence, the arc ABG= ALP, and the angle APG AGP AML FKG. But angle APG AGI (Pl. Ge. III. 32) FGK. Conse quently the angle FGK FKG, and the side FGFK. In like manner, HG HK. Hence the triangles GHE KHF, are equal in every respect, and the angle FGH FKHLMN.

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COR. 1.—An angle, contained by any two circles of the sphere, is equal to the angle formed by their projections.

For, the tangents to these circles on the sphere are projected into straight lines, which either coincide with, or are tangents to, their projections on the primitive.

COR. 2.-An angle, contained by any two circles of the sphere, is equal to the angle formed by the radii of their projections, at the point of concourse.

When one of the given projected circles is a diameter of the primitive, for its radius a line perpendicular to it must be taken.

PROPOSITION IV.

The centre of a great circle's projection is distant from the centre of the primitive by the tangent of the great circle's inclination to the primitive, and its radius is the secant of the same.

Let A be the projecting point, ABC a great circle passing through it, perpendicular to the proposed great circle, KEL their line of com

mon section, and BED
the line of common sec-
tion of ABCD, and the
primitive. Then, because
ABC is perpendicular B
both to the proposed great
circle, and to the primi-
tive, it is perpendicular
to their line of common

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section, and consequently BE, EK, are likewise perpendicular to the same. Hence BEK is the angle of inclination of the proposed circle to the primitive. Let AK, AL, be drawn, and meet BG in F, G; the straight line FG is the diameter of the projection. Let it be bisected in II, and let A, H, be joined. Because FAG is a right angle, HA HF (IV. Cor. 5), and the angle HAF HFA= FEK+FKE; from these equals, taking the equal angles. EAF, FKE, there remains HAE FEK, the angle of in

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clination. And, in the right-angled triangle AEH, to the radius AE, EH, is the tangent, and AH the secant of HAE. COR. 1.-All circles which pass through the points A, C, are the projections of great circles, and have their centres in the line BG. All circles which pass through the points F, G, are the projections of great circles, and have their centres in the line II', perpendicular to BG. For, considering ABC as the primitive, any circle through A, C, is the projection of a great circle (St. Pr. II. Cor. 3). Let AFC be an arc of a circle described on FG as a diameter; then, because HA = HF, as was proved, the circle cuts the primitive in A, C, the extremities of a diameter. Let IHI' be perpendicular to FG, then, if an arc of a circle having its centre in II' pass through F, it must pass through G. Also the points K, L, of its intersection with the primitive, are diametrically opposite. For FE EG = AE2 KE EL; and hence KEL is a diameter, and KFL is the projection of a great circle.

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COR. 2.-It appears from this that II' is the locus of the centres of the projections of all great circles that pass through the point F.

COR. 3.-BK is the measure of the inclination of the great circle to the primitive, and CS=2BK. For HAE BEK; therefore CS2 BK.

PROPOSITION V.

The centre of projection of a small circle, perpendicular to the primitive, is distant from the centre of the primitive, the secant of the circle's distance from its nearer pole, and the radius of projection, is the tangent of the same.

Let ABCD be a great circle, passing through the projecting point, and perpendicular to the proposed small circle, MON their line of common section, and BOD the line of common section of ABCD, and the primitive. Then BD is the axis, and O the centre of the small circle. Let AM and AN be drawn to meet BD in G, F; FG is the diameter on the line of measures of the small circle's projection. Let it be bisected in H, and EM, MH, joined. Then, because AE: EF = NO: OF, CE:EF=MO: OF; therefore the points C, F, M, are in a straight line, and that straight

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COR. 1.-EMH is a right angle, and EM, MH, are tangents to the two circles MFN and AMN.

COR. 2.-Any radius HI is a tangent to the great circle through the points B, I, D.

For (St. Pr. 3) the two circles being perpendicular, so are their projections, and hence also their radii at the point of intersection I.

PROPOSITION VI.

The projections of the poles of any circle, inclined to the primitive, are in the line of measures distant from the centre of the primitive, the tangent and cotangent of half its inclination.

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Because ABCD is perpendicular to the plane of the great circle KL (as in Prop. 4), it passes through its poles (which are also the poles of all its parallel small circles); let these be and let Ap, Aq, meet BD in P, Q, their projections. Then the quad

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rants pK, CB, be

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ing equal, and CK common to both,

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PC will be equal to

BK, which measures the inclination of the great circle (or its parallel small circles) to the primitive. Now, EP to the radius AE, is the tangent of PC, and EQ the tangent of 4C, or cotangent of PC.

COR. 1.-The projection of that pole which is adjacent to the projecting point, is without the primitive, and the projection of the other within.

COR. 2.--The distances of either projected pole from the centres of the primitive and projected great circle, are directly proportional to the radii of these circles.

For AP bisects the angle EAH (Pr. I. 4, Cor. 3), and consequently AQ bisects the external angle EAR. Hence EP: PHEA: AH, and EQ: QH-EA: AH (Pl. Ge. VI. 3).

Schol. The projection of a circle perpendicular to the primitive is a diameter of it, and its poles are in the extremities of another diameter perpendicular to the former.

PROPOSITION VII.

If, from either pole of a projected great circle, two straight lines be drawn to meet the primitive and the projection, they will intercept corresponding arcs of these circles.

From the pole P, of the projected great circle MLN, let there be drawn any two straight lines PL, PQ, meeting the primitive in R, S, and the projection in L, Q; then shall the arc LQ be the projection

of an arc equal to RS.

S

For SS', RR', are the projections of two circles (Pr. I. 1, Cor. 2), each of which passes through a pole of the primitive, and a pole of the great circle, and which there- R fore intercept equal arcs upon them (Sp. Ge. 18). Now, RS

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is one of the intercepted arcs, and the other is projected into LQ. Hence LQ, RS, are corresponding arcs.

COR.-Hence, if, from the point where the projections of two great circles intersect one another, two straight lines be drawn through their adjacent poles, these will intercept on the primitive an arc, which is the measure of their inclination.

For the point of intersection, being common to both circles, is at the distance of a quadrant from their poles; it is therefore the pole of an arc of a great circle passing through these

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