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to the supplements to the measures of the respectively opposite angles E and F. Consequently, the measures of the angles of the triangle DEF are supplemental to the several opposite sides of the triangle ABC. Q. E. D.
Cor. 1. Hence these two triangles are called supplemental or polar triangles.
Cor. 2. Since the three sides DE, EF, DF, are supplements to the measures of the three angles A, B, C; it results that DEEFDF + A+B+c=3 x 180°=540°. But (th. 2), DE + EF + DF < 360°: consequently A + B + c > 180°. Thus the first part of theorem 3 is very compendiously demonstrated.
Cor. 3. This theorem suggests mutations that are some times of use in computation. Thus, if three angles of a spherical triangle are given, to find the sides: the student may subtract each of the angles from 180°, and the three remainders will be the three sides of a new triangle; the angles of this new triangle being found, if their measures be each taken from 180°, the three remainders will be the respective sides of the primitive triangle, whose angles were given.
Scholium. The invention of the preceding theorem is due to Philip Langsberg. Vide, Simon Stevin, liv. 3, de la Cosmographie, prop. 31 and Alb. Girard in loc. It is often however treated very loosely by authors on trigonometry: some of them speaking of sides as the supplements of angles, and scarcely any of them remarking which of the several triangles formed by the intersection of the arcs DE, EF, DF, is the one in question. Besides the triangle DEF, three others may be formed by the intersection of the semi
circles, and if the whole circles be consisidered, there will be seven other triangles formed. But the proposition only obtains with regard to the central triangle (of each hemisphere), which is distinguished from the three others in this, that the two angles A and F are situated on the same
side of BC, the two в and E on the same side of Ac, and the two C and D on the same side of AB.
In Every Spherical Triangle, the following 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 three angles of the triangle above Two Right Angles, to the Area of the triangle.
Let ABC be the spherical triangle. Complete one of its sides as BC into the circle BCEF, which may be supposed to bound the upper hemisphere. Prolong also, at both ends, the two sides AB, AC, until they form semicircles estimated from each angle, that is, until BAE= ABD = CAF ACD
180°. Then will CBF 180°=BFE; and consequently the triangle AEF, on the anterior hemisphere, will be equal to the triangle BCD on the opposite hemisphere. Putting m, m', to represent the surface of these triangles, p for that of the triangle BAF, q for that of CAE, and a for that of the proposed triangle ABC. Then á and m' together (or their equal a and m together) make up the surface of a spheric lune comprehended between the two semicircles ACD, ABD, inclined in the angle A; a and p together make up the lune included between the semicircles CAF, CBF, making the angle c: a and q together make up the spheric lune included between the semicircles BCE, BAE, making the angle B. And the surface of each of these lunes, is to that of the hemisphere, as the angle made by the comprehending semicircles, to two right angles. Therefore, putting is for the surface of the hemisphere, we have
A:: is: a + m.
180° : B ::
is: a + 9.
180°: c :: is: a + p.
Whence, 180°: A+B+C::s: 3a+m+p+q=2a ++s; and consequently, by division of proportion,
as 180°: A + B + C - 180° :: is: 2a + 4s - s = 2a;
or, 180°: A + B + C-180° :: 48: a = s. Q. E. n*.
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 r3.141593, and d the diameter of the sphere, then is æd2. = the area of the spherical
*This determination of the area of a spherical triangle is due to Albert Girard (who died about 1633). 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 the next chapter of this volume.
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 57-2957795, 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 = 4s, 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 E the spherical excess: then is cot E, 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 :
cot a cot b + cos c sin c
tan = √(tan ++. tan +.tan -. tan=a+b+c).
4 The investigation of these theorem would occupy more space than can be allotted to them in the present volume.
In every Spherical Polygon, or surface included by any num. ber 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 th. 5, it will be
#<‹ $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,