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#<‹ $6 ̃ ̃+48 2; &+B+C−180°: its area. Therefore, putting
♪~(~—2)180° : is.
surface of the polygon. 180° x 2. Therefore,
the area of
and d represent the same quantities as in ha, then the surface of the polygon will bẹ ex
If R° 57′2957795, then will the surface of the on in square degrees be = R ̊. (P−(n−2)180°).
3. When the surface of the polygon is 0, then P = 2) 180°; and when it is a maximum, that is, when it is al to the surface of the hemisphere, then P= (n− 2) 180′ + S60° = n. 180°: Consequently P, the sum of all the angles 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.
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 comprized 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 comprize 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 riod, who have confined their attention principally to the magthe present pe
nitude 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.
* It 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 quan tities 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,
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 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 (42) of the maximum solid angle: while the plane right angle is half the maximum planę 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 corre sponding 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 repectively 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 4 of the maximum angle. And thus it may be found, that each solid angle of a right prism, with an equilateral triangular base is max. angle = .1000.
square base is
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,
hexagonal i'm gonal
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
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= 1
This theorem gives, for the plane angle formed by every two contiguous faces of the tetraedron, 70°31′42′′; of the hexaedron, 90°; of the octaëdron, 109°28'18"; of the dodecaedron, 116°33′ 54′′; of the icosaedron, 138°11′23′′. But, in these polyedræ, the number of faces meeting about each solid angle,
3, 3, 4, 3, 5 respectively. Consequently the several solid angles will be determined by the subjoined proportions:
360°: 3.70°31′42′′-180°:: 1000: 87.73611 Tetraëdron. 360° : 3.90° - 180°:: 1000: 250· Hexaëdron.
360° : 4.109°28′18′′ – 360° :: 1000: 216·35185 Octaëdron. 360° : 3.116°33′54′′-180° :: 1000: 471.395 Dodecaedron. 360°: 5.138°11′23′′ – 540° :: 1000: 419.30169 Icosaedron.
3. The solid angles at the vertices of cones, will be determined by means of the spheric segments cut off at the bases of those cones; that is, if right cones, instead of having plane bases, had bases formed of the segments of equal spheres, whose centres were the vertices of the cones, the surfaces of those segments would be measures of the solid angles at the respective vertices. Now, the surfaces of spheric segments, are to the surface of the hemisphere, as their altitudes, to the radius of the sphere; and therefore the solid angles at the vertices of right coness will be to the maximum solid angle, as the excess of the slant side above the axis of the cone, to the slant side of the cone. Thus, if we wish to ascertain the solid angles at the vertices of the equilateral and the rightangled cones; the axis of the former is 3, of the latter, Hence,
2, the slant side of each being unity.
1:1-1/3: 1000: 133.97464, equilateral cone,
4. From what has been said, the mode of determining the solid angles at the vertices of pyramids will be sufficiently obvious. If the pyramids be regular ones, if N be the number of faces meeting about the vertical angle in one, and A the angle of inclination of each two of its plane faces; if Ê be the number of planes meeting about the vertex of the other, and a the angle of inclination of each two of its faces: then will the vertical angle of the former, be to the vertical angle of the latter pyramid, as NA− (N − 2) 180°, to na− (n − 2) 180°.
If a cube be cut by diagonal planes, into 6 equal pyramids with square bases, their vertices all meeting at the centre of the circumscribing sphere; then each of the solid angles, made by the four planes meeting at each vertex, will be of the maximum solid angle; and each of the solid angles at the bases of the pyramids, will be of the maximum solid angle. Therefore, each solid angle at the base of such pyramid, is one-fourth of the solid angle at its vertex: and, if the angle at the vertex be bisected, as described below, either of the solid angles arising from the bisection, will be double of either solid angle at the base. Hence also, and from the first subdivision