Again we generalize $x^{2} + y^{2} = 1$ into $a x^{2} + b y^{2} = c$ , or still further into $a x^{2} + 2 h x y + b y^{2} = c$ , or, still further, into $a x^{2} + 2 h x y + b y^{2} + 2 g x + 2 f y = c$ .

Here again we are led to variable correlations which are indicated by their various algebraic forms.

Now let us turn to geometry. The name of the science at once recalls to our minds the thought of figures and diagrams exhibiting triangles and rectangles and squares and circles, all in special relations to each other. The study of the simple properties of these figures is the subject matter of elementary geometry, as it is rightly presented to the beginner. Yet a moment's thought will show that this is not the true conception of the subject. It may be right for a child to commence his geometrical reasoning on shapes, like triangles and squares, which he has cut out with scissors. What, however, is a triangle? It is a figure marked out and bounded by three bits of three straight lines.

Now the boundary of spaces by bits of lines is a very complicated idea, and not at all one which gives any hope of exhibiting the simple general conceptions which should form the bones of the subject. We want something more simple and more general. It is this obsession with the wrong initial ideas–-very natural and good ideas for the creation

of first thoughts on the subject–-which was the cause of the comparative sterility of the study of the science during so many centuries. Coordinate geometry, and Descartes its inventor, must have the credit of disclosing the true simple objects for geometrical thought.

In the place of a bit of a straight line, let us think of the whole of a straight line throughout its unending length in both directions. This is the sort of general idea from which to start our geometrical investigations. The Greeks never seem to have found any use for this conception which is now fundamental in all modern geometrical thought. Euclid always contemplates a straight line as drawn between two definite points, and is very careful to mention when it is to be produced beyond this segment. He never thinks of the line as an entity given once for all as a whole. This careful definition and limitation, so as to exclude an infinity not immediately apparent to the senses, was very characteristic of the Greeks in all their many activities. It is enshrined in the difference between Greek architecture and Gothic architecture, and between the Greek religion and the modern religion. The spire on a Gothic cathedral and the importance of the unbounded straight line in modern geometry are both emblematic of the transformation of the modern world.

The straight line, considered as a whole, is accordingly the root idea from which modern geometry starts. But then other sorts of lines occur to us, and we arrive at the conception of the complete curve which at every point of it exhibits some uniform characteristic, just as the straight line exhibits at all points the characteristic of straightness. For example, there is the circle which

at all points exhibits the characteristic of being at a given distance from its centre, and again there is the ellipse, which is an oval

curve, such that the sum of the two distances of any point on it from two fixed points, called

its foci, is constant for all points on the curve. It is evident that a circle is merely a particular case of an ellipse when the two foci are superposed in the same point; for then the sum of the two distances is merely twice the radius of the circle. The ancients knew the properties of the ellipse and the circle and, of course, considered them as wholes. For example, Euclid never starts with mere segments (i.e., bits) of circles, which are then