derived from the two Greek words trigonon a triangle and metria measurement. The fundamental question from which trigonometry arose is this: Given the magnitudes of the angles of a triangle, what can be stated as to the relative magnitudes of the sides. Note that we say "relative magnitudes of the sides," since by the theory of similarity it is only the proportions of the sides which are known. In order to answer this question, certain functions of the magnitudes of an angle, considered as the argument, are introduced. In their origin these functions were got at by considering a right-angled triangle, and the magnitude of the angle was defined by the length of the arc of a circle. In modern elementary books, the fundamental position of the arc of the circle as defining the magnitude of the angle has been pushed somewhat to the background, not to the advantage either of theory or clearness of explanation. It must first be noticed that, in relation to similarity, the circle holds the same fundamental position among curvilinear figures, as does the triangle among rectilinear figures. Any two circles are similar figures; they only differ in scale. The lengths of the circumferences of two circles, such as $A P A^{'}$ and $A_{1} P_{1} A_{1}^{'}$ in the [fig.]26 are in proportion to the lengths of their radii. Furthermore, if the two circles have the same

centre $O$ , as do the two circles in [fig.]26, then the arcs $A P$ and $A_{1} P_{1}$ intercepted by the arms of any angle $A O P$ , are also in proportion to their radii. Hence the ratio of the 26 length of the arc $A P$ to the length of the radius $O P$ , that is $\frac{arc A P}{radius O P}$ is a number which is quite independent of the length $O P$ , and is the same as the fraction $\frac{arc A_{1} P_{1}}{radius O P_{1}}$ . This fraction of "arc divided by radius" is the proper theoretical way to measure the magnitude of

an angle; for it is dependent on no arbitrary unit of length, and on no arbitrary way of dividing up any arbitrarily assumed angle, such as a right-angle. Thus the fraction $\frac{A P}{O A}$ represents the magnitude of the angle $A O P$ . Now draw $P M$ perpendicularly to $O A$ . Then the Greek mathematicians called the line $P M$ the sine of the arc $A P$ , and the line $O M$ the cosine of the arc $A P$ . They were well aware that the importance of the relations of these various lines to each other was dependent on the theory of similarity which we have just expounded. But they did not make their definitions express the properties which arise from this theory. Also they had not in their heads the modern general ideas respecting functions as correlating pairs of variable numbers, nor in fact were they aware of any modern conception of algebra and algebraic analysis. Accordingly, it was natural to them to think merely of the relations between certain lines in a diagram. For us the case is different: we wish to embody our more powerful ideas.

Hence, in modern mathematics, instead of considering the arc $A P$ , we consider the fraction $\frac{A P}{O P}$ , which is a number the same for all lengths of $O P$ ; and, instead of considering the lines $P M$ and $O M$ , we consider

the fractions $\frac{P M}{O P}$ and $\frac{O M}{O P}$ , which again are numbers not dependent on the length of $O P$ , i.e. not dependent on the scale of our diagrams. Then we define the number $\frac{P M}{O P}$ to be the sine of the number $\frac{P A}{O P}$ , and the number $\frac{O M}{O P}$ to be the cosine of the number $\frac{P A}{O P}$ . These fractional forms are clumsy to print; so let us put $u$ for the fraction $\frac{A P}{O P}$ , which represents the magnitude of the angle $A O P$ , and put $v$ for the fraction $\frac{P M}{O M}$ , and $w$ for the fraction $\frac{O M}{O P}$ . Then $u$ , $v$ , $w$ , are numbers, and, since we are talking of any angle $A O P$ , they are variable numbers. But a correlation exists between their magnitudes, so that when $u$ (i.e. the angle $A O P$ ) is given the magnitudes of $v$ and $w$ are definitely determined. Hence $v$ and $w$ are functions of the argument $u$ . We have called $v$ the sine of $u$ , and $w$ the cosine of $u$ . We wish to adapt the general functional notation $y = f (x)$ to these special cases: so in modern mathematics