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Q. E. D. * Cor. 1. Hence the excess of the three angles of any spherical triangle above two right angles, termed technically the spherical excess, furnishes a correct measure of the surface of that triangle. Cor. 2. If a = 3.141593, and d the diameter of the A + B + c – 1809
7209 = the area of the
sphere, then is ad”.
spherical triangle. 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 ercess be multiplied by 57.2957795, the product will express the surface of the triangle in square degrees. Cor. 4. When a = 0, then A + B + c = 180°: and when a = }s, then A + B + c = 540°. Consequentl the sum of the three angles of a spherical triangle, is always between 2 and 6 right angles: which is another confirmation of art. 19, p. 83.
* This determination of the area of a spherical triang'e is due to Albert Girard (who died about 1683). But the demonstration now commonly given of the rule was first published by Dr. Wallis. It was considered as a mere speculative truth, until General Roy, in 1787, employed it very judiciously in the great Trigonometrical Survey, to correct the errors of spherical angles. See Phil. Trans. vol. 80. and chap. xii. of this volume.
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. The mode of finding the spherical excess, 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 E the spherical excess: then is cot E = worro. 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:
The investigation of these theorems would occupy more space than can be allotted to them in the present volume.
44. In every spherical polygon, or surface included by any number of intersecting great circles, the subjoined proportion obtains, viz. as four right angles, or 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 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 theor. 1, it will be as 360°: #s :: 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 a and d represent the same quantities as in theor. 1, cor. 2, then the surface of the polygon will be
P – (n − 2) 180°
expressed by ado. – #
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 1, 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.
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 de
* 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 Montucla’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. .
scribed, 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