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motion which he put into their final general shape, proved adequate to explain all astronomical phenomena, including Kepler's laws, and has formed the basis of modern physics. Among other things he proved that comets might move in very elongated ellipses, or in parabolas, or in hyperbolas, which are nearly parabolas. The comets which return–-such as Halley's comet–-must, of course, move in

ellipses. But the essential step in the proof of the law of gravitation, and even in the suggestion of its initial conception, was the verification of Kepler's laws connecting the motions of the planets with the theory of conic sections.

From the seventeenth century onwards the abstract theory of the curves has shared in the double renaissance of geometry due to the introduction of coordinate geometry and of projective geometry. In projective geometry

the fundamental ideas cluster round

the consideration of sets (or pencils, as they

are called) of lines passing through a common point (the vertex of the "pencil"). Now (cf. [fig.]19) if $A$, $B$, $C$, $D$, be any four fixed points on a conic section and $P$ be a variable point on the curve, the pencil of lines $PA$, 19 $PB$, $PC$, and $PD$, has a special property, known as the constancy of its cross ratio. It

will suffice here to say that cross ratio is a fundamental idea in projective geometry. For projective geometry this is really the definition of the curves, or some analogous property which is really equivalent to it. It

will be seen how far in the course of ages of study we have drifted away from the old original idea of the sections of a circular cone. We know now that the Greeks had got hold of a minor property of comparatively slight importance; though by some divine good fortune the curves themselves deserved all the attention which was paid to them. This unimportance of the "section" idea is now marked in ordinary mathematical phraseology by dropping the word from their names. As often as not, they are now named merely "conics" instead of "conic sections."

Finally, we come back to the point at

which we left coordinate geometry in the last chapter. We had asked what was the type of loci corresponding to the general algebraic form $ax+by=c$, and had found that it was the class of straight lines in the plane. We had seen that every straight line possesses an equation of this form, and that every equation of this form corresponds to a straight line. We now wish to go on to the next general type of algebraic forms. This is evidently to be obtained by introducing terms involving $x^2$ and $xy$ and $y^2$. Thus the new general form must be written:–- $a{x}^2+2hxy+b{y}^2+2gx+2fy+c=0.$ What does this represent? The answer is

that (when it represents any locus) it always represents a conic section, and, furthermore, that the equation of every conic section can always be put into this shape. The discrimination of the particular sorts of conics as given by this form of equation is very easy. It entirely depends upon the consideration of $ab-h^2$, where $a$, $b$, and $h$, are the "constants" as written above. If $ab-h^2$ is a positive number, the curve is an ellipse; if $ab-h^2=0$, the curve is a parabola: and if $ab-h^2$ is a negative number, the curve is a hyperbola.

For example, put a = b = 1 , h = g = f = 0 , c = − 4 . We then get the equation x 2 + y 2 − 4 = 0 . It is easy to prove that this is the equation of a circle, whose centre is at the origin, and radius is 2 units of length.