we suppose the cutting plane to move downwards from a position above the tangent plane (remaining always horizontal), then we shall see the two branches of the first hyperbola approach one another and get sharper and sharper until they meet and become simply two crossing straight lines. These lines will then have their corners rounded off and will be divided in the other direction and open out into the second hyperbola.

This leads us to suppose that a pair of intersecting straight lines is only a particular case of a hyperbola, and that we may consider the hyperbola as derived from the two crossing straight lines by dividing them at their point of intersection and rounding off the

corners.

§ 10. How to form Curves of the Third and Higher Orders.

The method of the preceding paragraph may be extended so as to discover the forms of new curves by putting known curves together. By a mode of expression which sounds paradoxical, yet is found convenient, a straight line is called a curve of the first order, because it can be met by another straight line in only one point; but two straight lines taken together are called a curve of the second order, because they can be met by a straight line in two points. The circle, and its shadows, the ellipse, parabola, and hyperbola, are also called curves of the second order, because they can be met by a straight line in two points, but not in more than two points; and we see that by this process of rounding off the corners and the method of projection we can derive all these curves of the second order from a pair of straight lines.

A similar process enables us to draw curves of the third order. An ellipse and a straight line taken together form a curve of the third order. If now we round off the corners at both the points where they meet we obtain (fig. 28) a curve consisting of an oval and a sinuous portion called a 'snake.' Now just as when we move a plane which cuts a sphere away from the centre, the curve of intersection shrinks up into a

point and then disappears, so we can vary our curve of the third order so as to make the oval which belongs to it shrink up into a point, and then disappear altogether, leaving only the sinuous part, but no variation will get rid of the snake.'

We may, if we like, only round off the corners at one of the intersections of the straight line and the ellipse, and we then have a curve of the third order crossing itself, having a knot or double point (fig. 29); and we can further suppose this loop to shrink up, and the curve will then be found to have a sharp point or

cusp.

It was shown by Newton that all curves of the third order might be derived as shadows from the five forms

which we have just mentioned, viz. the oval and snake, the point and snake, the snake alone, the form with a knot, and the form with a cusp.

In the same way curves of the fourth order may be got by combining together two ellipses. If we suppose

88

FIG. 30.

them to cross each other in four points we may round off all the corners at once and so obtain two different forms, either four ovals all outside one another or an

oval with four dints in it, and another oval inside it (fig. 30).

But the number of forms of curves of the fourth order is so great that it has never yet been completely catalogued; and curves of higher orders are of still more varied shapes.

CHAPTER III.

QUANTITY.

§ 1. The Measurement of Quantities.

WE considered at the beginning of the first chapter, on Number, the process of counting things which are separate from one another, such as letters or men or sheep, and we found it to be a fundamental property of this counting that the result was not affected by the order in which the things to be counted were taken; that one of the things, that is, was as good as another at any stage of the process.

We may also count things which are not separate but all in one piece. For example, we may say that a room is sixteen feet broad. And in order to count the number of feet in the breadth of this room we should probably take a foot rule and measure off first a foot close to the wall, then another beginning where that ended, and so on until we reached the opposite wall. Now when these feet are thus marked off they may, just like any other separate things, be counted in whatever order we please, and the number of them will always be sixteen.

But this is not all the variety in the process of counting which is possible. For suppose that we take a stick whose length is equal to the breadth of the room. Then we may cut out a foot of it wherever we please, and join the ends together. And if we then