spherical excess, furnishes a correct measure of the surface of that triangle. Cor. 2. If=3.141593, and d the diameter of the sphere, A+B+C-180° then is ad3.. angle. 720° the area of the spherical tri Cor. 3. Since the length of the radius, in any circle, is equal to the length of 57-2957795 degrees, measured on the circumference of that circle? if the spherical excess be multiplied by 57-297795, the product will express the surface of the triangle in square degrees. Cor. 4. When a = 0, then A+B+C=180°: and when as, then A+B+c=540°. Consequently the sum of the three angles of a spherical triangle, is always between 2 and 6 right angles: which is another confirmation of th. 3. Cor. 5. When two of the angles of a spherical triangle are right angles, the surface of the triangle varies with its third angle. And when a spherical triangle has three right angles its surface is one eighth of the surface of the sphere. Remark. Some of the uses of the spherical excess, in the more extensive geodesic operations, will be shown in the following chapter. The mode of finding it, and thence the area when the three angles of a spherical triangle are given, is obvious enough; but it is often requisite to ascertain it by means of other data, as when two sides and the included angle are given, or when all the three sides are given. In the former case, let a and b be the two sides, c the included angle, and cot la cot b+cos c E the spherical excess: then is cotle= sin c When the three sides a, b, c, are given, the spherical excess may be found by the following very elegant theorem, discovered by Simon Lhuillier : tan = √(tan a+b+c a+b-c a-b+c . tan tan a+b+c). The investigation of these theorems would oc cupy more space than can be allotted to them in the present volume. THEOREM VI. In every Spherical Polygon, or surface included by any number of intersecting great circles, the subjoined proportion obtains, viz. As Four Right Angles, at 360°, to the Surface of a Hemisphere; or, as Two Right Angles, or 180°, to a Great Circle of the Sphere; so is the Excess of the Sum VOL. II. 6 of of the Angles above the Product of 180° and Two Less than the number of Angles of the spherical polygon, to its Area. For, if the polygon be supposed to be divided into as many triangles as it has sides, by great circles drawn from all the angles through any point within it, forming at that point the vertical angles of all the triangles. Then, by th. 5, it will be Therefore, putas 360°: s:: A+B+C−1809 : its area. ting P for the sum of all the angles of the polygon, n for their number, and v for the sum of all the vertical angles of its constituent triangles, it will be, by composition, as 360°: s:: P+v 180o n: surface of the polygon. But Therefore, as v is manifestly equal to 360o or 180° x 2. 360° of the polygon. P-(n-2) 180° the area 360o Q. E. D. Cor. 1.If and d represent the same quantities as in theor. 5 cor. 2, then the surface of the polygon will be expressed P-N- - 2) 180o by ad'. 7200 Cor. 2. If Ro--57.2957795, then will the surface of the polygon in square degrees be=Ro. (P (n-2) 180°). Cor. 3. When the surface of the polygon is 0, then P = (n-2) 180°; and when it is a maximum, that is, when it is equal to the surface of the hemisphere, then p⇒(n-2) 180° +360°=n. 180°: Consequently P the sum of all the angles of any spheric polygon, is always less than 2n right angles. but greater than (2n-4) right angles, n denoting the number of angles of the polygon. GENERAL SCHOLIUM. On the nature and Measure of Solid Angles. A Solid Angle is defined by Euclid, that which is made by the meeting of more than two plane angles, which are not in the same plane, in one point. Others define it the angular space comprised between several planes meeting in one point. It may be defined still more generally, the angular space included between several plane surfaces or one or more curved surfaces, meeting in the point which forms the summit of the angle. According to this definition, solid angles bear just the same relation to the surfaces which comprise them, as plane angles do to the lines by which they are included: so that as in the latter, it is not the magnitude of the lines, but their mutual inclination, which determines the angle; just so, in the former it is not the magnitude of the planes, but their mutual inclinations which determine the angles. And hence all those geometers, from the time of Euclid down to the present period, who have confined their attention principally to the magnitude of the plane angles instead of their relative positions, have never been able to develope the properties of this class of geometrical quantities but have affirmed that no solid angle can be said to be the half or the double of another, and have spoken of the bisection and trisection of solid angles, even in the simplest cases, as impossible problems. But all this supposed difficulty vanishes, and the doctrine of solid angles becomes simple, satisfactory, and universal in its application, by *assuming spherical surfaces for their measure; just as circular arcs are assumed for the measures of plane anglest. Imagine, that from the summit of a solid angle (formed by the meeting of three planes) as a centre, any sphere be described, and that those planes are produced till they cut the surface of the sphere; then will the surface of the spherical triangle, included between those planes be a proper measure of the solid angle made by the planes at their common point of meeting; for no change can be conceived in the relative position of those planes, that is in the magnitude of the solid angle, without a corresponding and proportional mutation in the surface of the spherical triangle. If, in like manner, the three or more surfaces which by their meeting constitute another solid angle, be produced till they cut the surface of the same or an equal sphere, whose centre coincides with the summit of the angle; the surface of the spheric triangle or polygon, included between the planes which determine the angle, will be a correct measure of that angle. And Circular arcs are not merely assumed to be the measures of plane angles, they are demonstrated to be so. See Sim Euclid, Prop. 33, Book VI. It ought also to be demonstrated that spherical surfaces are the measures of solid angles. Ed. : tIt may be proper to anticipate here the only objection which can be made to this assumption; which is founded on the principle, that quantities should always be measured by quantities of the same kind. But this, often and positively as it is affirmed, is by no means necessary; nor in many cases is it possible. To measure is to compare mathematically and if by comparing two quantities, whose ratio we know or can ascertain, with two other quantities whose ratio we wish to know, the point in question becomes determined: it signifies not at all whether the magnitudes which constitute one ratio are like or unlike the magnitudes which constitute the other ratio. It is thus that mathematicians, with perfect safety and correctness, make use of space as a measure of velocity, mass as a measure of inertia, mass and velocity conjointly as a measure of force, space as a measure of time, weight as a measure of density, expansion as a measure of heat, a certain function of planetary velocity as a measure of distance from the central body, arcs of the same circle as measures of plane angles; and it is in conformity with this general procedure that we adopt surfaces of the same sphere as measures of solid angles. the the ratio which subsists between the areas of the spheric triangles, polygons, or other surfaces thus formed, will be accurately the ratio which subsists between the solid angles, constituted by the meeting of the several planes or surfaces, at the centre of the sphere Hence, the comparison of solid angles becomes a matter of great ease and simplicity; for, since the areas of spherical triangles are measured by the excess of the sums of their angles each above two right angles (th. 5); and the areas of spherical polygons of n sides, by the excess of the sum of their angles above 2n-4 right angles (th. 6); it follows, that the magnitude of a trilateral solid angle, will be measured by the excess of the sum of the three angles, made respectively by its bounding planes, above 2 right angles; and the magnitudes of solid angles formed by n bounding planes, by the excess of the sum of the angles of inclination of the several planes above 2n -4 right angles. As to solid angles limited by curve surfaces, such as the angles at the vertices of cones; they will manifestly be measured by the spheric surfaces cut off by the prolongation of their bounding surfaces, in the same manner as angles determined by planes are measured by the triangles or polygons, they mark out upon the same, or an equal sphere. In all cases, the maximum limit of solid angles, will be the plane towards which the various planes determining such angles approach, as they diverge further from each other about the same summit: just as a right line is the maximum limit of plane angles, being formed by the two bounding lines when they make an angle of 180o. The maximum limit of solid angles is measured by the surface of a hemisphere, in like manner as the maximum limit of plane angles is measured by the arc of a semicircle. The solid right angle (either angle, for example, of a cube) is (=12) of the maximum solid angle: while the plane right angle is half the maximum plane angle. The analogy between plane and solid angles being thus traced, we may proceed to exemplify this theory by a few instances; assuming 1000 as the numeral measure of the maximum solid angle 4 times 90° solid 360° solid. 1 1. The solid angles of right prisms are compared with great facility. For, of the three angles made by the three planes which, by their meeting, constitute every such solid angle, two are right angles: and the third is the same as the corresponding plane angle of the polygonal base; on which, therefore, the measure of the solid angle depends. Thus, with respect to the right prism with an equilateral triangular base, each solid angle is formed by planes which respectively make angles angles of 90°, 90°, and 60°. Consequently 90° +90°+60°-180° 60°, is the measure of such angle, compared with 360° the maximum angle. It is therefore, one-sixth of the maximum angle. A right prism with a square base, has, in like manner, each solid angle measured by 90°+90° +90° — 180° = 90°, which is of the maximum angle. And thus may be found, that each solid angle of a right prism, with an equilateral = triangular base is is max. angle 1.1000. =} = 1000. hexagonal is 16 T3 · 1000. 20 Hence it may be deduced, that each solid angle of a regular prism, with triangular base, is half each solid angle of a prism with a regular hexagonal base. Each with regular square base of each, with regular octagonal base,. pentagonal = hexagonal m gonal m-4 m decagonal, m gonal base. Hence again we may infer, that the sum of all the solid angles of any prism of triangular base, whether that base be regular or irregular, is holf the sum of the solid angles of a prism of quadrangular base, regular or irregular. And, the sum of the solid angles of any prism of tetragonal base is sum of angles in prism of pentag. base, pentagonal hexagonal m gonal = 2. Let us compare the solid angles of the five regular bodies. In these bodies, if m be the number of sides of each face; n the number of planes which meet at each solid angle ; =half the circumference or 180°; and a the plane angle made by two adjacent faces: then we have sin A |