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H

is, the square of CA is greater than the squares of CG, GD, together, that is, than the square of CD; therefore CA is greater than CD, and the arc CA than the arc CD. In the same manner, since GD is greater than GE, and GE than GF, it is shown that CD A is greater than CE, and CE than

E

G

B

F

CF, and, consequently, the arc CD greater than the arc CE, and the arc CE greater than the arc CF. Also, because AG is the greatest, and GB the least, of all the lines drawn from G, CA is the greatest, and CB the least, of all the lines drawn from C, and therefore the arc CA is the greatest, and CB, its supplement, the least of all the arcs drawn through C.

PROPOSITION XIV.

In a right-angled spherical triangle the sides are of the same affection with the opposite angles; that is, if the sides' be greater or less than quadrants, the opposite angles will be greater or less than right angles.

Let ABC be a spherical triangle right angled at A, any side AB will be of the same affection with the opposite angle ACB.

G

Case 1. Let AB be less than a quadrant. Let AE be a quadrant, and EC an arc of a great circle passing through E, C. Since A is a right angle, and AE a quadrant, E is the pole of the great circle AC, and ECA a right angle; but ECA is greater than BCA, therefore BCA is less than a right angle.

B

E

Case 2. Let AB be greater than a quadrant; make AE equal to a quadrant, and let a great circle pass through C, E. ECA is a right angle as before, and BCA is greater than ECA, that is, greater than a right angle.

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PROPOSITION XV.

If the two sides of a right-angled spherical triangle be of the same affection, the hypotenuse will be less than a quadrant; and if they be of different affection, the hypotenuse will be greater than a quadrant.

Let ABC (last figure) be a right-angled spherical triangle; if the two sides AB, AC, be of the same or of different affection, the hypotenuse BC will be less or greater than a quadrant.

Case 1. Let AB, AC, be each less than a quadrant. Let AE, AG, be quadrants; G will be the pole of AB, and E the pole of AC, and EC a quadrant; but (Sp. Ge. 13) CE is greater than CB, since CB is farther off from CGD than CE. In the same manner, it is shown that CB, in the triangle CBD, where the two sides CD, BD, are each greater than a quadrant, is less than CE, that is, less than a quadrant.

Case 2. Let AC be less, and AB greater than a quadrant; then the hypotenuse BC will be greater than a quadrant; for, let AE be a quadrant, then E is the pole of AC, and EC will be a quadrant. But CB is greater than CE (Sp. Ge. 13), since AC passes through the pole of ABD. COR. 1.Hence, conversely, if the hypotenuse of a rightangled triangle be greater or less than a quadrant, the sides will be of different or the same affection. COR. 2. Since (Sp. Ge. 14) the angles of a right-angled spherical triangle have the same affection with the opposite sides, therefore, according as the hypotenuse is greater or less than a quadrant, the angles will be of different or of the same affection.

PROPOSITION XVI.

In any spherical triangle, if the perpendicular upon the base from the opposite angle fall within the triangle, the angles at the base are of the same affection; and if the perpendicular fall without the triangle, the angles at the base are of different affection.

Let ABC be a spherical triangle, and let the arc CD be drawn from C perpendicular to the base AB.

1. Let CD fall within the triangle; then since ADC, BDC, are right-angled spherical triangles, the angles A, B, must each be of the same affection with CD (Sp. Ge. 14).

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2. Let CD fall without the triangle; then (Sp. Ge. 14) the angle B is of the same affection with CD; and the angle CAD is of the same affection with CD; therefore the angles CAD and B are of the same affection, and the angles CAB and B of different affections.

COR.-Hence, if the angles A and B be of the same affection, the perpendicular will fall within the base; for, if it did not, A and B would be of different affection. And if the angles A and B be of opposite affection, the perpendicular will fall without the triangle; for, if it did not, the angles A and B would be of the same affection, contrary to the supposition.

PROPOSITION XVII.

If to the base of a spherical triangle a perpendicular be drawn from the opposite angle, which either falls within the triangle, or is the nearest of the two that fall without; the least of the segments of the base is adjacent to the least of the sides of the triangle, or to the greatest, according as the sum of the sides is less or greater than a semicircle.

Let ABEF be a great circle of a sphere, Hits pole, and GHD any circle passing through H, which therefore is perpendicular to the circle ABEF. Let A and B be two points in the circle ABEF on opposite sides of the point D, and let D be nearer to A than to B, and let C be any point in the circle GHD, between H and D. Through the points A and C, B and C, let the arcs AC and BC be drawn, and let them be produced till they meet the circle ABEF in the points E and F; then the arcs ACE, BCF, are semicircles. Also ACB, ACF, CFE, ECB, are four spherical triangles

contained by arcs of the same circles, and having the same perpendiculars CD and CG.

B

I. Now, because CE is nearer to the arc CHG than CB is, CE is greater than CB, and therefore CE and CA are greater than CB and CA; wherefore CB E and CA are less than a semicircle; but because AD is, by supposition, o less than DB, AC is also less than CB (Sp. Ge. 13); and therefore in this case, namely, when the pendicular falls within the triangle,

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F

C

H

A

and when the sum of the sides is less than a semicircle, the least segment is adjacent to the least side.

2. Again, in the triangle FCA the two sides FC and CA are less than a semicircle; for, since AC is less than CB, AC and CF are less than BC and CF. Also, AC is less than CF, because it is more remote from CHG than CF is; therefore the least segment of the base AD is in this case also adjacent to the least side.

3. But in the triangle FCE the two sides FC and CE are greater than a semicircle; for, since FC is greater than CA, FC and CE are greater than AC and CE. And because AC is less than CB, EC is greater than CF, and EC is therefore nearer to the perpendicular CHG than CF is, wherefore EG is the least segment of the base, and is adjacent to the greater side.

4. In the triangle ECB the two sides EC, CB, are greater than a semicircle; for, since by supposition CB is greater than CA, EC and CB are greater than EC and CA. Also, EC is greater than CB; wherefore in this case, also, the least segment of the base EG is adjacent to the greatest side of the triangle.

PROPOSITION XVIII.

Any two circles of the sphere, passing through the poles of two great circles, intercept equal arcs upon them.

Let AFB, CFD, be the two great circles intersecting one another in F, and let F be the pole of the great circle ACBD, cutting them in the diameters AEB, CED. The circle

ACBD passes through their poles; let the diameter MN be perpendicular to AB, and PQ to CD; then M, N, are the

M

poles of AFB, and P, Q, the

poles of CFD. Let the small

C

circle PGN pass through the

F

poles P, N, and cut the circle
ACBD in the line of com-
mon section PKLN; the
arcs BH, DG, of the circles A
AFB, CFD, intercepted by
the circle passing through
P, N, are equal.

For, let EH, HK, EG,
GL, PG, NH, be drawn.

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N

In the triangles PEL, NEK, angle P=N, for PE=NE; also NEK, PEL, being right angles, are equal; hence (Pl. Ge. I. 26) PL = NK, and EL: EK. Again, the quadrants PG, NH, are equal, and taking away HG, the are PH NG, and the chord PG NH. Hence, in the triangles PLG, NKH, PL = NK, PG=NH, and angles LPG, KNH, standing on equal arcs NG, PH, are equal; hence, LG KH. Again, in the triangles EKH, ELG, having their sides respectively equal, angle HEK GEL, and hence the arc HB = GD.

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SPHERICAL TRIGONOMETRY.

Spherical Trigonometry treats of those relations between the sides and angles of spherical triangles, by which their numerical values may be computed.

The trigonometrical lines defined in Plane Trigonometry are employed in reference to the sides and angles of spherical triangles.

PROPOSITION I.

In right-angled spherical triangles, the sine of either of the sides about the right angle, is to the radius of the sphere, as the tangent of the remaining side is to the tangent of the angle opposite to that side.

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