TRIGON 0 M spherical, relates to tri angles, or figures which are reducible to triangles, whose sides are segments of circles. Thus if we describe a triangle on any spherical body, say a globe, it is evident that all the sides must be com posed of curved lines ; and it is the same in the case of a series of circles, or of or bits, intersecting each other. When two equal circles intersect, they will give a parabolic spindle ; more or lens acute, according as the centres of the two cir cles may be more or less distant. When three circles mutually intersect, there will be formed a great variety of spherical triangles, of which the areas and the pro perties could not be ascertained by plane trigonometry, but come under consider ation as parts of spherical surfaces. The following definitions should be clearly un derstood; they are simple in the extreme, but highly important : 1st. The poles of a sphere are two points in the superticies of the sphere, that are the extreme of the axis. 2d. The pole of a circle in a sphere is a point in the superficies of the sphere, from which all right lines that are drawn to the circumference of the circle are equal to one another. 3d. A great circle in a sphere, is that whose plane passes through the centre of the sphere ; and whose centre is the same as that of the sphere. 4th. A spherical triangle is a figure comprehended under the arcs of three great circles in a sphere. 5th. A 'spherical angle is that which, in the su perficies of the sphere, is contained under two arcs of great circles ; and this angle is equal to the inclinations of the planes of the said circles. It is particularly to be held in mind, that although we can, upon any actual sphere, describe triangles at pleasure, which may nearly embrace the whole circumference, yet that such can not be laid down, so as to be represented on paper ; for every side of a spherical triangle is less than a semi-circle.
With respect to spherical triangles, the learner may generally entertain a correct opinion of their value, if he considers that every arc or segment of a circle may have a chord drawn from one to the other ex tremity; and that the triangle which can be contained within such arc or segment, taking the chord for a hypothenuse, will determine how much of that circle has been cut of, and is included between the extremes of the segment. It is utterly impossible to produce any two measura ble segments taken from two different circles, which, having chords of equal length, will contain the same angle._ A
semicircle, having the diameter for its chord, Will give a right angle ; for if to any point within that semicircle two lines be drawn, from the ends of the chord res pectively,- their union at such assumed point will form a right angle. In pro portion as the chord is less than a diame ter, so must the segment be a less part of the whole circle, and the angle contained therein will be more acute. Spherical triangles may be acute, right-angled, or obtuse, the same as on plane-trigonome try. In all right-angled spherical trian gles, the sign of the hypothenuse : radius:: site of a leg : sine of its Opposite angle. And the sine of the leg: radius :: tangent of the other leg : tangent of its opposite angle. In any right-angled spherical tri angle, A B C (fig. 25,) It will-be as radius is to the co-sine of one leg, so is the co sine of the other leg to the co-sine of the hypothenuse. Hence, if two right-angled spherical triangles, A B C, C B 1) (fig. 26,) have the same perpendicular, BC, the co sines of their by pothenuses will be to each othz-r directly as the co-sines of their ba ses. In any spherical triangle it will be, as radius is to the sine of either angle, so is the co-sine of the adjacent leg to the co-sine of the opposite angle. Hence, in right-angled spherical triangles, having the same perpendicular, the co-sines of the angles at the base will be to each other, uirectly, as the sines of the vertical angles. In any right-angled spherical triangle it will be, as radius is to the co sine of the hypothenuse, so is the tangent of either angle to the co-tangent of the other angle. As the sum of the sines of two arches is to their difference, so is the tangent of half the sum of those arches to the tangent of half their differ ence : and as the sum of their co-sines is to their difference, so is the co-tangent of half the Sinn of the arches to the tan gent of half the difference of the same arches. In any spherical triangle, A B C (fig. 27,) it will be, as the co-tangent of half the sum of the angles at the base is to the tangent of half their difference, so is the tangent of half the verticle angle to the tangent of the angle which the per pendicular C D makes with the line C F, bisecting the vertical angle.