riangle cannot be determined from a knowledge of the angles alone; while that of a spherical triangle always can.

5. The sides of a spherical triangle are all arcs of great circles, which, by their intersection on the surface of the sphere, constitute that triangle.

6. The angle which is contained between the arcs of two great circles, intersecting each other on the surface of the sphere, is called a spherical angle; and its measure is the same as the measure of the plane angle which is formed by two lines issuing from the same point of, and perpendicular to, the common section of the planes which determine the containing sides; that is to say, it is the same as the angle made by those planes. Or, it is equal to the plane angle formed by the tangents to those arcs at their point of intersection.

7. Hence it follows, that the surface of a spherical triangle BAC, and the three planes which determine it, form a kind of triangular pyramid, BCGA, of which the vertex & is at the centre of the sphere, the base ABC a portion of the spherical surface, and the faces AGC, AGB, BGC, sectors of the great circles whose intersections determine the side of the triangle.

G

BM

P

P

Def. 6. A line perpendicular to the plane of a great circle, passing through the centre of the sphere, and terminated by two points, diametrically opposite, at its surface, is called the axis of such circle; and the extremities of the axis, or the points where it meets the surface, are called the poles, of that circle. Thus, PGP' is the axis, and P, P', are the poles, of the great circle CND.

If we conceive any number of less circles, each parallel to the said great circle, this axis will be perpendicular to them likewise; and the points P, P', will be their poles also.

8. Hence, each pole of a great circle is 90° distant from every point in its circumference; and all the arcs drawn from either pole of a little circle to its circumference, are equal to each other.

9. It likewise follows, that all the arcs of great circles drawn through the poles of another great circle, are perpendicular to it for, since they are great circles by the sup position, they all pass through the centre of the sphere, and consequently through the axis of the said circle. The same thing may be affirmed with regard to small circles.

10. Hence, in order to find the poles of any circle, it is merely necessary to describe, upon the surface of the sphere,

two great circles perpendicular to the plane of the former? the points where these circles intersect each other will be the poles required.

11. It may be inferred also, from the preceding, that if it were proposed to draw, from any point assumed on the surface of the sphere, an arc of a circle which may measure the shortest distance from that point, to the circumference of any given circle; this arc must be so described, that its prolongation may pass through the poles of the given circle. And conversely, if an arc pass through the poles of a given circle, it will measure the shortest distance from any assumed point to the circumference of that circle.

12. Hence again, if upon the sides, AC and BC, (produced if necessary) of a spherical triangle BCA, we take the arcs CN, c, each equal 90°, and through the radii GN, GM (figure to art. 7) draw the plane NGM, it is manifest that the point c will be the pole of the circle coinciding with the plane NGM : so that, as the lines GM, GN, are both perpendicular to the common section GC, of the planes AGC, BGC, they measure, by their inclination, the angle of these planes; or the arc NM measures that angle, and consequently the spherical angle

BCA.

13. It is also evident that every arc of a little circle, described from the pole c as centre, and containing the same number of degrees as the arc MN, is equally proper for mea. suring the angle BCA; though it is customary to use only arcs of great circles for this purpose.

14. Lastly, we infer, that if a spherical angle be a right angle, the area of the great circles which form it, will pass mutually through the poles of each other: and that, if the planes of two great circles contain each the axis of the other, or pass through the poles of each other, the ungle which they include is a right angle.

These obvious truths being premised and comprehended, the student may pass to the consideration of the following theorems.

THEOREM I.

Any two sides of a spherical triangle are together greater than the third.

This proposition is a necessary consequence of the truth, that the shortest distance between any two points, measured on the surface of the sphere, is the arc of a great circle passing through these points.

THEOREM II.

The sum of the three sides of any spherical triangle is less than 360 degrees.

For, let the sides AC, BC, (fig. to art. 7) containing any angle A, be produced till they meet again in D: then will the arcs DAC, DBC, be each 180", because all great circles cut each other into two equal parts: consequently DAC + DBC = 360°.. But (theorem 1) DA and DB are together greater than the third side AB of the triangle DAB; and therefore, since CA + CBDA + DB = 360°, the sum CA + CB + AB is less than 360°. Q. E. D.

THEOREM III.

The sum of the three angles of any spherical triangle is always greater than two right angles, but less than six,

G

Aa

For, let ABC be a spherical triangle, G the centre of the sphere, and let the chords of the arcs AP, BC, AC, be drawn : these chords constitute a rectilinear triangle, the sum of whose three angles is equal to two right angles. But the angle at в made by the chords, AB, BC, is less than the angle aвc, formed by the two tangents Ba, BC, or less than the angle of inclination of the two planes GBC, GBA, which (art. 6) is the spherical angle at B; consequently the spherical angle at в is greater than the angle at в made by the chords AB, CB. In like manner, the spherical angles at A and c, are greater than the respective angles made by the chords meeting at those points. Consequently, the sum of the three angles of the spherical triangle ABC, is greater than the sum of the three angles of the rectilinear triangle made by the chords AB, BC, AC, that is, greater than two right angles. Q. E. 1° D.

2. The angle of inclination of no two of the planes can be so great as two right angles; because, in that case, the two planes would become but one continued plane, and the arcs, instead of being arcs of distinct circles, would be joint arcs of one and the same circle. Therefore, each of the three sphe. rical angles must be less than two right angles; and consequently their sum less than six right angles. Q. e. 2od.

Cor. 1. Hence it follows, that a spherical triangle may have all its angles either right or obtuse; and therefore the knowledge of any two angles is not sufficient for the deter mination of the third.

Cor. 2. If the three angles of a spherical triangle be right

or obtuse, the three sides are likewise each equal to, or greater than 90° and, if each of the angles be acute, each of the sides is also less than 90°; and conversely.

Scholium. From the preceding theorem the student may clearly perceive what is the essential difference between plane and spherical triangles, and how absurd it would be to apply the rules of plane trigonometry to the solution of cases in spherical trigonometry. Yet, though the difference between the two kinds of triangles be really so great, still there are various properties which are common to both, and which may be demonstrated exactly in the same manner. Thus, for example, it might be demonstrated here, (as well as with regard to plane triangles in the elements of Geometry, vol. 1) that two spherical triangles are equal to each other, 1st. When the three sides of the one are respectively equal to the three sides of the other. 2dly. When each of them has an equal angle contained between equal sides: and, 3dly. When they have each two equal angles at the extremities of equal bases. It might also be shown, that a spherical triangle is equilateral, isosceles, or scalene, according as it hath three equal, two equal, or three unequal angles and again, that the greatest side is always opposite to the greatest angle, and the least side to the least angle. But the brevity that our plan requires compels us merely to mention these particulars. It may be added, however, that a spherical triangle may be at once right-angled and equilateral; which can never be the case with a plane triangle.

THEOREM IV.

If from the angles of a spherical triangle, as poles, there be described, on the surface of the sphere, three arcs of great circles, which by their intersections form another spherical triangle; each side of this new triangle will be the sup. plement to the measure of the angle which is at its pole, and the measure of each of its angles the supplement to that side of the primitive triangle to which it is opposite.

From B, A, and c, as poles, let the arcs DF, DE, FE, be described, and by their intersections form another spherical triangle DEF; either side, as DE, of this triangle, is the supplement of the measure of the angle A at its pole; and either angle, as D, has for its measure the supplement of the side AB.

D

N

M

B

E

Let the sides AB, AC, BC, of the primitive triangle, be produced till they meet those of the triangle DEF, in the points 1, L, M, N, G, K : then, since the point A is the pole of the arc DILE, the distance of the points A and E (measured on an arc of a great circle) will be 90°; also, since c is the pole of the arc EF, the points c and E will be 90° distant: conse. quently (art. 8) the point E is the pole of the arc AC. In like manner it may be shown, that F is the pole of BC, and

D that of AB.

This being premised, we shall have DL 90°, and re=90°; whence DL + IE = DL+EL+IL DE + IL = 180°. Therefore DE = 180°. IL: that is, since IL is the measure of the angle BAC, the arc DE is the supplement of that measure. Thus also may it be demonstrated that er is equal the supplement to MN, the measure of the angle BCA, and that DF is equal the supplement to GK, the measure of the angle ABC which constitutes the first part of the proposi. tion.

2dly. The respective measures of the angles of the triangle DEF are supplemental to the opposite sides of the triangle ABC. For, since the arcs AL and BG are each 90°, therefore is ALBG GL + AB = 180°; whence GL 180°- AB; that is, the measure of the angle D is equal to the supplement to AB. So likewise may it be shown that AC, BC, are equal to the supplements to the measures of the respectively opposite angles E and F. Consequently, the measures of the angles of the triangle DEF are supplemental to the several opposite sides of the triangle ABC. Q. E. D.

Cor. 1. Hence these two triangles are called supplemental or polar triangles.

Cor. 2. Since the three sides DE, EF, DF, are supplements to the measures of the three angles A, B, C ; it results that DEEF DF + A+B+c=3 × 180° = 540. But (th. 2), de + ef + df < 360°: consequently A + B +c> 180°. Thus the first part of theorem 3 is very compendiously demonstrated.

Cor. 3. This theorem suggests mutations that are some. times of use in computation. Thus, if three angles of a spherical triangle are given, to find the sides: the student may subtract each of the angles from 180, and the three remainders will be the three sides of a new triangle; the angles of this new triangle being found, if their measures be each taken from 180°, the three remainders will be the respective sides of the primitive triangle, whose angles were given.

Scholium. The invention of the preceding theorem is due to Philip Langsberg. Vide, Simon Stevin, liv. 3, de la Cos. nographie, prop. 31, and Alb. Girard in loc. It is often how.