rithms to the base 10 are now used in all but the simplest numerical calculations which it is needful to make in the exact sciences; their value arises solely from the fact that addition and subtraction are easier operations than multiplication and division.

§ 11. The Cartesian Method of Determining Position.

(y) In order to determine the position of a point P1 in space of two dimensions, we may draw the line B A B', joining the given points A B and another line CAC' at right angles to this through a. These will divide the plane into four equal portions termed quadrants. Let P1 м be a line drawn from the point P, (the position of

which relative to A we wish to determine), parallel to CA and meeting B'A B in M. Then we may state the following rule to get from A to P1: Take a step A M from A on the line B'A B, and then a step to the left at right angles to this equal to M P1. Now a step like A м may be taken either forwards along A B or backwards along A B'. Precisely as before (see p. 100) we

shall take +AM to mean a step forwards along A B, and -AM to mean a step A M' backwards along A B′ through the same distance a M. Let us use the letter i to denote the operation, which we have represented by (π/2) on p. 151. Thus applied to unit step it will signify Step forwards in the direction of the previous step and from its finish unit distance, and then rotate this unit distance through a right angle counter-clockwise about the finish of the previous step. The operator i placed before a step, thus i. MP1, will then be interpreted as follows: Step from м in the direction AB a distance equal to the length M P1, and then rotate this step м P, about м counter-clockwise through a right angle. We are thus able to express symbolically the position of P, relative to A, or the step A P1, by the relation

AP, A Mi. M P1.

If we had to get to a point P, in the quadrant B A C′, instead of to P1, we should have, instead of stepping forwards from м, to step backwards a distance M P1, and then rotate this through a right angle counter-clockwise. The step backwards would be denoted by inserting a sign as a reversing operation (see p. 39), and we should have

2

Next let us see how we should get to a point like På in the quadrant C A B', where P2 is at a perpendicular distance P, M' from AB'. First, we must take a step, A M', backwards; this is denoted by -A M'; secondly, we must step forwards from м' a distance M' P2; since this step is forwards, it will be towards A; thirdly, by applying the operation i to this step, we rotate it about

M' counter-clockwise through a right angle, and so reach P. Hence

Finally, if we wish to reach P, in the quadrant B'A C', we must step backwards A M', and then still further backwards a step M' P3, and lastly rotate this step counter-clockwise through a right angle.

will be expressed by

A PA Mi. M′ P2.

This

Now let us suppose P1, P2, P3, P49 to be the four corners of a rectangular figure whose centre is at A and whose sides are parallel to B A B' and CA C'. Let the number of units in a м be x, and the number in M P1 be y, then we may represent the four steps which determine the positions of the P's relative to A as follows:

Here x and y are mere numbers, but, when we represent these numbers by steps on a line, the y-numbers are to be taken on a certain line at right angles to that line on which the x-numbers are taken. Thus the moment we represent our x and y numbers by lengths, they give us a means of determining position.

The quantities x and y might thus be used to determine the position of a point, if we supposed them to carry with them proper signs. Our general rule would then be to step forwards from A along A B a distance x, and then from the end of x a distance forwards equal to y; rotate this step y about the end of a counterclockwise through a right angle, and the finish of y will then be the point determined by the quantities a, y.

If a or y be negative, the corresponding forwards must be read: Step forwards a negative quantity, that is, step backwards. Thus :

P1, or position in the quadrant B A C is determined by x, y.

P2

P3 P4

CA B'

B'A C'

CAB

-x, Y. -x,-Y.

x, -y.

The quantities x and y are termed the Cartesian coordinates of the point P, this method of determining the

position of a point having been first used by Descartes. BAB and CA C' are termed the co-ordinate axes of x and y respectively, while a is called the origin of coordinates. For example, let the Cartesian co-ordinates of a point be (-3, 2). How shall we get at it from the origin A? If P be the point, we have A P = −3+i.2. Hence we must step backwards 3 units; from this point step forwards 2 and rotate this step 2 about the extremity of the step 3 through a right angle counterclockwise; we shall then be at the required point.

If P be determined by its Cartesian co-ordinates a and y, we might find a succession of points, P, by always

taking a step y related in a certain invariable fashion to any step a which has been previously made.

Such a succession of points P, obtained by giving æ every possible value, will form a line or curve, and the relation between x and y is termed its Cartesian equation.

As an instance of this, suppose that for every step , we take a step y equal to the double of it. Then we shall have for our relation y = 2x, and our instructions

A

FIG. 73.

to reach any point P of the series are x+i. 2x. Suppose the quadrant B A C divided into a number of little squares by lines parallel to the axes, and let us take the sides of these squares to be of unit length. Then if we take in succession x=1, 2, 3, &c., we can easily mark off our steps. Thus: 1 along A B and then 2 to the left; 2 along A B and 4 to the left; 3 along AB and then 6 to the left; 4 along A B and then 8 to the left; 5 along AB and then 10 to the left, and so on. It will be obvious (by p. 106) that our points all lie upon a