INTRODUCTION

TRIGONOMETRY is primarily the science concerned with the measurement of plane and spherical triangles, that is, with the determination of three of the parts of such triangles when the numerical values of the other three parts are given. This is done by means of the six trigonometric functions, defined in article 4 following. But these functions enter so intimately into many branches of mathematical and physical science not directly concerned with the measurement of angles, that their analytical properties are of fundamental importance. Analytical trigonometry, that is, the proof and use of various algebraic relations among the trigonometric functions of the same or related angles, is therefore, in modern times, of equal importance with the trigonometry which deals with triangular solutions.

The same functions which enable one to solve triangles constructed in a plane suffice also for the solution of spherical triangles. But the solution of triangles of which the sides are geodetic lines, that is, lines which are the shortest distances between pairs of points on the surface, on a spheroidal surface such as the Earth, requires the use of other functions than those needed for the solution of plane or spherical triangles. This spheroidal trigonometry is very complex, and becomes necessary only in the accurate survey of very large tracts of the Earth's surface. For ordinary purposes of surveying and for the solution of triangles. on the Earth's surface over small areas, plane and spherical trigonometry are sufficient.

The study of trigonometry, as ancillary to astronomy, dates from very early times. Among the Greeks, who,

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however, were more famous as geometers than as investigators in other branches of mathematics, the names of Hipparchus (about 150 B.c.) and of Ptolemy (who lived in the second century of the Christian era), both astronomers, are prominent. Hipparchus left no mathematical writings, but we are told by an ancient writer that he created the science of trigonometry. Ptolemy, making use of the investigations and discoveries of Hipparchus, perfected the form of the science. The theorems of these two astronomers are

still the basis of trigonometry.

Ptolemy calculated a table of chords, which were used in those earliest days of the science, as we now use the sines of angles. The radius of a circle he divided into sixty equal parts. Each of these he divided again into sixty equal parts, called, in the Latin translation of his work the Almagest, "partes minutae primae"; and each of these in turn into sixty, called "partes minutae secundae"; whence have come the names "minutes" and "seconds" for the subdivisions of the angular degree. Ptolemy, however, was not the first to calculate a table of chords, Hipparchus, among others, having done so previously, but he invented theorems by means of which the calculations could be more readily made.

The Hindus, more skillful calculators than the Greeks, acquired the knowledge of the latter and improved upon it, notably in that they calculated tables of the half-chord, or sine, instead of the whole chord of the angle. The Arabs also were acquainted with the Almagest, and with the investigations of the Hindus. It was an Arab, Al Battani or Albategnius, who first calculated a table of what may be called cotangents, by computing the lengths of shadows of a vertical object cast by the sun at different altitudes. Another Arab invented, as a separate function, the tangent, which had previously been used only as an abbreviation of the ratio sine to cosine. Curiously enough this invention was afterwards forgotten until the tangent was re-invented

in the fifteenth century by the German, Johannes Müller, called Regiomontanus, who wrote the first complete European treatise on trigonometry.

When Napier* invented logarithms, in 1614, they were at once adopted in trigonometric calculations, and the first tables of logarithmic sines and tangents were made by Edmund Gunter, an English astronomer (1581-1626). He it was who first used the names cosine, cotangent, and cosecant. During the following century the science of trigonometry progressed slowly, becoming more analytical in form, until, in the hands of Euler (1707-1783), it became essentially what it is at the present day.

With this brief introduction to the history of trigonometry let us now proceed to become acquainted with that homely, perhaps, but most serviceable handmaid to so many of the arts and sciences,

PLANE TRIGONOMETRY

CHAPTER I

THE TRIGONOMETRIC FUNCTIONS OF ANY ANGLE, AND IDENTICAL RELATIONS AMONG THEM

1. Rectangular Coördinates. Two lines, x'x and y'y, drawn in a plane at right angles to each other, as in Fig. 1, form a system of rectangular, Cartesian coördinates. The point O in which the lines intersect is called the origin; the two

lines are called the axes of coördinates. One of these, usually the horizontal line, is called the axis of abscissæ, or the axis of x. The other is called the axis of ordinates, or the axis of y. We shall speak of XOY, YOX', X'OY', and Y'OX as the first, second, third, and fourth quadrants respectively.

2. Angles of any Magnitude. There are many ways in which a system of coördinates is used in mathematics. In trigonometry such a system is used primarily in defining