Spherical trigonometry

“ The trigonometrical functions are employed in many branches of mathematical and physical science not directly concerned with the measurement of angles, and hence arises the importance of analytical trigonometry. The solution of triangles of which the sides are geodesic lines on a spheroidal surface requires the introduction of other functions than those required for the solution of triangles on a plane or spherical surface, and therefore gives rise to a new branch of science, which is from analogy frequently called spheroidal trigonometry. ”

“ All that is necessary is to calculate by spherical trigonometry the azimuth and distance, with reference to the assumed centre, of all the points of intersection of meridians and parallels within the space which is to be represented in a plane. Then the azimuth is represented unaltered, and any spherical distance z is represented by ρ. Thus we get all the points of intersection transferred to the representation, and it remains merely to draw continuous lines through these points, which lines will be the meridians and parallels in the representation. ”

“ In both elliptic and hyperbolic geometry the spherical geometry, i.e. the relations between the angles formed by lines and planes passing through the same point, is the same as the “spherical trigonometry” in Euclidean geometry. The constant γ, which appears in the formulae both of hyperbolic and elliptic geometry, does not by its variation produce different types of geometry. There is only one type of elliptic geometry and one type of hyperbolic geometry ”

“ If a, b, c, be the sides, and A, B, C, the angles of a spherical triangle, right-angled at A, then . This is frequently called "Geber's Theorem." Radical and bold as were his innovations in spherical trigonometry, in plane trigonometry he followed slavishly the old beaten path of the Greeks. ”

“ Even a general map may be trusted, as long as we keep within ten degrees of its centre. In the case of more considerable distances, however, a globe of suitable size should be consulted, or—and this seems preferable—they should be calculated by the rules of spherical trigonometry. The problem then resolves itself in the solution of a spherical triangle.
In the formulae which follow we suppose l and l′ to represent the latitudes, a and b the co-latitudes (90 − l or 90° − l′) , and t the difference in longitude between them or the meridian distance, whilst D is the distance required. ”

“ Perhaps to the student there is no part of elementary mathematics so repulsive as is spherical trigonometry. Also, everything relating to change of systems of axes, as for instance in the kinematics of a rigid system, where we have constantly to consider one set of rotations with regard to axes fixed in space, and another set with regard to axes fixed in the system, is a matter of troublesome complexity by the usual methods. ”

“ A. D. 100) extant in Arabic; Book I of this work contains propositions about spherical triangles corresponding to the main propositions of Euclid about plane triangles (e.g. congruence theorems and the proposition that in a spherical triangle the three angles are together greater than two right angles) , while Book III contains genuine spherical trigonometry, consisting of ‘Menelaus’s Theorem’ with reference to the sphere and deductions therefrom. ”

“ The crystal is considered to be placed inside a sphere from the centre of which normals are drawn to all the faces of the crystal. The points at which these normals intersect the surface of the sphere are called the poles of the faces, and by these poles the positions of the faces are fixed. The poles of all faces in the same zone on the crystal will lie on a great circle of the sphere, which are therefore called zone-circles. The calculation of the angles between the normals of faces and between zone-circles is then performed by the ordinary methods of spherical trigonometry. ”

“ The Indians did not apply their trigonometrical knowledge to the solution of triangles; for astronomical purposes they solved right-angled plane and spherical triangles by geometry.
The Arabs were acquainted with Ptolemy’s Almagest, and they probably learned from the Indians the use of the sine. ”

“ Since any plane tri angle can be divided into right-angled triangles, the solution of all plane triangles can be reduced to that of right-angled triangles; moreover, according to the theory of similar triangles, the ratios between pairs of sides of a right-angled triangle depend only upon the magnitude of the acute angles of the triangle, and may therefore be regarded as functions of either of these angles. The primary object of trigonometry, therefore, requires a classification and numerical tabulation of these functions of an angular magnitude ”

“ It is rarely that the terms of the fourth order are required. As a rule spheroidal triangles are calculated as spherical (after Legendre) , i.e. like plane triangles with a decrease of each angle of about ε /3; ε must, however, be calculated for each triangle separately with its mean measure of curvature k.
The geodetic line being the shortest that can be drawn on any surface between two given points, we may be conducted to its most important characteristics by the following considerations: let p, q be adjacent points on a curved surface ”

“ We have only to select a number of such triangles and a system of trigonometry ensues, in which base, perpendicular, and hypotenuse of every triangle is a whole measure without fractions, and in which the nomenclature for every angle is clear and simple. ”

“ This method has its roots in the profoundest depths of the human intellect, and hence is of far higher importance than the most ingenious analytical processes which are applicable only to a particular case. True, trigonometry is analytical geometry; as taught in the gymnasia, trigonometry, like stereometry, as both these terms indicate, has to do rather with mensuration, and its use is restricted to a certain class of problems. ”

“ At a first thought it might appear as if these eight movements had nothing in common with trigonometry, but a very fundamental connection may be seen as follows: If the vertex of the angle A is the center of a square and the initial line of A coincides with a line of symmetry of the square, the operations of taking the complement and the supplement of A correspond to movements transforming the square into itself. ”

“ The difficulty experienced by the author of the above remarks is clearly one which can no longer exist, when it is seen that the doctrine of a plurality of worlds is an impossibility. That it is an impossibility is shown by the fact that the Sun, Moon, and Stars are very small bodies, [188] and very near to the earth; this fact is proved by actual non-theoretical measurement; this measurement is made on the principle of plane trigonometry: this principle of plane trigonometry is adopted because the Earth is a Plane; and all the base lines employed in the triangulation are horizontal. ”

“ The figure drawn on the blackboard is the sensible object. The conception of a triangle is the intelligible object formed by abstraction, and universal. In this conception are contained the general notions of a point, a line, an angle; and in these notions are involved several self-evident principles or axioms. From these are demonstrated the various mathematical propositions of trigonometry. It is easy to see that, in the intellectual process of the pupil’s mind, the genesis and development of the act of cognition of mathematical truth is precisely what has been above described. ”

“ Trilinear and Areal Coordinates.—Consider a fixed triangle ABC, and regard its sides as produced without limit. Denote, as in trigonometry, by a, b, c the positive numbers of units of a chosen scale contained in the lengths BC, CA, AB, by A, B, C the angles, and by Δ the area, of the triangle. We might, as in § 6, take CA, CB as axes of x and y, inclined at an angle C. Any point P (x, y) in the plane is at perpendicular distances y sin C and x sin C from CA and CB. Call these β and α respectively. ”

“ A spheroidal triangle differs from a spherical triangle, not only in that the curvatures of the sides are different one from another, but more especially in this that, while in the spherical triangle the normals to the surface at the angular points meet at the centre of the sphere, in the spheroidal triangle the normals at the angles A, B, C meet the axis of revolution of the spheroid in three different points, which we may designate α, β, γ respectively. ”

“ Every new class of surfaces which may be considered would have in this extended sense a trigonometry of its own, which would consist in an investigation of the nature and properties of the functions necessary for the measurement of the sides and angles of triangles bounded by geodesics drawn on such surfaces. ”

“ The unit of circular measure is too large to be convenient for practical purposes, but its use introduces a simplification into the series in analytical trigonometry, owing to the fact that the sine of an angle and the angle itself in this measure, when the magnitude of the angle is indefinitely diminished, are ultimately in a ratio of equality. ”

“ Archimedes also states that the surface of a sphere is two thirds that of the circumscribed cylinder, or the same as the curved surface of this cylinder. This is evident, since the cylindric surface of the cylinder is 2πr × 2r, or 4πr2, and the two bases have an area πr2 + πr2, making the total area 6πr2.
Theorem. The area of a spherical triangle is equal to the area of a lune whose angle is half the triangle's spherical excess. ”

“ It will contain an additional, and at present unknown, constant, which determines the size of the world.
Spherical space is not very easy to imagine. We have to think of the properties of the surface of a sphere—the two-dimensional case—and try to conceive something similar applied to three-dimensional space. Stationing ourselves at a point let us draw a series of spheres of successively greater radii. The surface of a sphere of radius should be proportional to ; but in spherical space the areas of the more distant spheres begin to fall below the proper proportion. ”

“ In one sense, however, the dwellers on the sphere and on the plane have an axiom in common. In each case it will be possible for a figure to be moved about without alteration of its dimensions. A spherical triangle can be moved on the surface of a sphere without distortion just as a plane triangle may be moved in a plane. The sphere-dwellers and the plane-dwellers would be equally able to apply the test of congruence. It is, however, possible to suppose reasoning beings confined to a space in which the translation of a rigid figure is impossible. ”

“ Our measurements of distance in space are found to be subject to certain laws—the laws of geometry. But it has now become impossible to regard the subject of space-geometry as complete in itself. Consider a triangle formed by three points (or events) in the four-dimensional world; if we happen to have drawn our instantaneous strata so that the three points lie in one stratum, then the triangle is a space-triangle and its properties fall within the scope of our classical geometry. ”

“ Without some knowledge of trigonometry, you will not be able fully to understand this process; but, as it is in its nature somewhat curious, and is applied to various other geographical measurements, as well as to the determination of arcs of the meridian, I am desirous that you should understand its general principles. Let us reflect, then, that it must be a matter of the greatest difficulty, to execute with exactness the measurement of a line of any great length in one continued direction on the earth's surface. ”

“ We will consider the tangent as a secant joining two points infinitely near to each other; and then, designating by dy and dx the infinitely small differences of the co-ordinates of those two points, the elementary principles of geometry will immediately give the equation t = dy/dx for the trigonometrical tangent of the angle which is made with the axis of the abscissas by the desired tangent, this being the most simple way of fixing its position in a system of rectilinear co-ordinates. ”

“ Geodetical triangulation is, like many of the other branches of scientific applications, essentially a child of the modern era. It is not older than the seventeenth century; the first application of geodetical triangulation to the measurement of an arc of the earth's meridian having been made in Holland at the beginning of that century. ”

“ For if we stick the paper discs on the globe, so that no disc overlaps another, the surface of the globe will finally become so full that there is no room for another disc. This simply means that the spherical surface of the globe is finite in relation to the paper discs. Further, the spherical surface is a non-Euclidean continuum of two dimensions, that is to say, the laws of disposition for the rigid figures lying in it do not agree with those of the Euclidean plane. ”

“ The first tables of logarithmic sines and tangents were constructed by Edmund Gunter (1581-1626) , professor of astronomy at Gresham College, London; he was also the first to employ the expressions cosine, cotangent, and cosecant for the sine, tangent, and secant of the complement of an arc. A treatise by Albert Girard (1590-1634) , published at The Hague in 1626, contains the theorems which give areas of spherical triangles and polygons, and applications of the properties of the supplementary triangles to the reduction of the number of different cases in the solution of spherical triangles. ”

“ To obviate the inevitable inaccuracies attending this graphic method and as a corroboration of the ship’s position, the changes of latitude and longitude involved in each alteration of course are daily calculated by plane trigonometry, such calculations being materially abbreviated by the use of the Traverse Table, which is a tabulated expression of the solutions of right-angled plane triangles. ”