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at that point the vertical angles of all the triangles. Then, by theor. 1, it will be as 360°: As :: A + B -- c — 180°: its area. Therefore, putting 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 — 180°. n : surface of the polygon. But v is manifestly equal to 360° or 180°x2. Therefore,
area of the polygon. Q. E. D.
Cor. 1. If w and d represent the same quantities as in theor. 1, cor. 2, then the surface of the polygon will be - P – (n − 2) 1800
Cor. 2. If R^ = 57.2957795, then will the surface of
the polygon in square degrees be = R*. [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 i. than, (2n - 4) right angles, n denoting the number of angles of the polygon.
Nature and Measure of Solid Angles.
45. 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,
expressed by ad".
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 angles.* 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 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 (theor. 1); and the areas of spherical polygons of n sides, by the excess of the sum of their angles above 2n — 4 right angles (theor. 2); 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 two 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 man
* This disquisition on solid angles was first published in the 3d volume of Dr. Hutton's Course of Mathematics, in the year 1811. At that time I thought the notion of measuring this class of geometrical magnitudes by means of spherical triangles and polygons was, though extremely obvious and natural, perfectly new. I have since found, by consulting Montacla's History of Mathematics, vol. ii. p. 8, that Albert Girard, in his Invention nouvelle en Algèbre, advanced an analogous theory.
While I am adverting to the third volume of Dr. Hutton's Course, I beg to mention, in order to account for any instances of close correspondence which may be found between parts of this volume and of that (though they; I believe, will occur quite as seldom as can well be expected when the same person is treating of the same subjects), that the 2d, 3d, 4th, 5th, 6th, 7th, 8th, 9th, and 11th chapters of that volume were composed by me; the remainder by my excellent friend the author of that Course.
ner 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 180°. 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 4 (= #”) 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. 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 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 3 of the maximum angle. And thus it may be found, i. * solid angle of a right prism, with an equi9ter
triangular base is + max. angle - # ... 1000,
square base is +.......... = # . 1000, }. base is . . . . . . . . . . = Hos. 1000. xagonal is + .......... = 1°, . 1000. heptagonal is ".......... = Tor. 1000. octagonal is . . . . . . . . . . . = r. 1000. nonagonal is . . . . . . . . . . = + , . 1000. decagonal is # . . . . . . . . . . = *... 1000. "undecagonal is . . . . . . . . . . . = or. 1000. duodecagonal is for.......... = 4;. 1000. ** - . - - m — 2 m gonal . is ........ == 1000.
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 = # ............ ... decagonal,
hexagonal = + . . . . . . . . . . . . . . . . duodecagonal, 'm — 4
§m gonal = H. - - - - - - - - - - - - - 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 half 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 . - o tetragonal baseis = $sumofang.in prism of pentag.base,
pentagonal . . . . =# . . . . . . . . . . . . .... hexagonal, hexagonal .... =# . . . . . . .......... heptagonal, m gonal . . . . . . ===}. . . . . . . . . . . . . (m+1)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; #O = half the circumference or 180°; and A . plane angle made by two adjacent faces;