w for the fraction 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

we write " $\sin$ " for " $f$ " when we want to

indicate the special function of "sine," and " $\cos$ " for " $f$ " when we want to indicate the special function of "cosine." Thus, with the above meanings for $u$ , $v$ , $w$ , we get $v = \sin u, and w = \cos u,$ where the brackets surrounding the $x$ in $f (x)$ are omitted for the special functions. The meaning of these functions $\sin$ and $\cos$ as correlating the pairs of numbers $u$ and $v$ , and $u$ and $w$ is, that the functional relations are to be found by constructing (cf. [fig.]26) an angle $A O P$ , whose measure " $A P$ divided by $O P$ " is equal to $u$ , and that then $v$ is the number given by " $P M$ divided by $O P$ " and $w$ is the number given by " $O M$ divided by $O P$ ."

It is evident that without some further definitions we shall get into difficulties when the number $u$ is taken too large. For then the arc $A P$ may be greater than one-quarter of the circumference of the circle, and the point $M$ (cf. figs. [fig:26]26 and [fig:27]27) may fall between $O$ and $A^{'}$ and not between $O$ and $A$ . Also $P$ may be below the line $A O A^{'}$ and not above it as in [fig.]26. In order to get over this difficulty we have recourse to the ideas and conventions of coordinate geometry in making our complete definitions of the sine and cosine. Let one arm $O A$ of the angle be the axis $O X$ , and produce the axis backwards to obtain its negative part $O X^{'}$ . Draw the

other axis $Y O Y^{'}$ perpendicular to it. Let any point $P$ at a distance $r$ from $O$ have coordinates $x$ and $y$ . These coordinates are both positive in the first "quadrant" of the plan, e.g. the coordinates $x$ and $y$ of $P$ 27 in [fig.]27. In the other quadrants, either one or both of the coordinates are negative, for example, $x^{'}$ and $y$ for $P^{'}$ , and $x^{'}$ and $y^{'}$ for $P^{″}$ , and $x$ and $y^{'}$ for $P^{‴}$ in [fig.]27, where $x^{'}$ and $y^{'}$ are both negative numbers. The positive angle $P O A$ is the arc $A P$ divided by $r$ , its sine is $\frac{y}{r}$ and its cosine is $\frac{x}{r}$ ; the positive

angle $A O P^{'}$ is the arc $A B P^{'}$ divided by $r$ , its sine is $\frac{y}{r}$ and cosine $\frac{x^{'}}{r}$ ; the positive angle $A O P^{″}$ is the arc $A B A^{'} P^{″}$ divided by $r$ , its sine is $\frac{y^{'}}{r}$ and its cosine is $\frac{x^{'}}{r}$ ; the positive angle $A O P^{‴}$ is the arc $A B A^{'} B^{'} P^{‴}$ divided by $r$ , its sine is $\frac{y^{'}}{r}$ and its cosine is $\frac{x}{r}$ .

But even now we have not gone far enough. For suppose we choose $u$ to be a number greater than the ratio of the whole circumference of the circle to its radius. Owing to the similarity of all circles this ratio is the same for all circles. It is always denoted in mathematics by the symbol $2 π$ , where $π$ is the Greek form of the letter p and its name in the Greek alphabet is "pi." It can be proved that $π$ is an incommensurable number, and that therefore its value cannot be expressed by any fraction, or by any terminating or recurring decimal. Its value to a few decimal places is $3.14159$ ; for many purposes a sufficiently accurate approximate value is $\frac{22}{7}$ . Mathematicians can easily calculate $π$ to any degree of accuracy required, just as $\sqrt{2}$ can be so calculated. Its value has been actually given to $707$ places of