X

has only one vertex. Apollonius provedCf. Ball, loc. cit., for this account of Apollonius and Pappus. that the ratio of $P{M}^2$ to $AM·M{A}^{\prime }$ $\left(\mathit{\text{i.e.}}{\displaystyle \frac{P{M}^2}{AM·M{A}^{\prime }}}\right)$ remains constant both for the ellipse and the hyperbola (figs. [fig:16]16 and [fig:18]18), and that the ratio

of $P{M}^2$ to $AM$ is constant for the parabola of [fig.]17; and he bases most of his work on this fact. We are evidently advancing towards the desired uniform definition which does not go out of the plane; but have not yet quite attained to uniformity.

In the diagrams [fig:16]16 and [fig:18]18, two points, $S$ and $S^{\prime }$, will be seen marked, and in [diagram]17 one point, $S$. These are the foci of the curves, and are points of the greatest importance. Apollonius knew that for an ellipse the sum of $SP$ and $S^{\prime }P$ (i.e. $SP+S^{\prime }P$) is constant as $P$ moves on the curve, and is equal to $A{A}^{\prime }$. Similarly for a hyperbola the difference $S^{\prime }P-SP$ is constant, and equal to $A{A}^{\prime }$ when $P$ is on one branch, and the difference $S{P}^{\prime }-S^{\prime }{P}^{\prime }$ is constant and equal to $A{A}^{\prime }$ when $P^{\prime }$ is on the other branch. But no corresponding point seemed to exist for the parabola.

Finally $500$ years later the last great Greek geometer, Pappus of Alexandria, discovered

the final secret which completed this line of thought. In the diagrams [fig:16]16 and [fig:18]18 will be seen two lines, $XN$ and $X^{\prime }{N}^{\prime }$, and in [diagram]17 the single line, $XN$. These are the directrices of the curves, two each for the ellipse and the hyperbola, and one for the parabola. Each directrix corresponds to its nearer focus.

The characteristic property of a focus, $S$, and its corresponding directrix, $XN$, for any one of the three types of curve, is that the ratio

$SP$ to $PN$ $\left(\mathit{\text{i.e.}}{\displaystyle \frac{SP}{PN}}\right)$ is constant, where $PN$ is the perpendicular on the directrix from $P$, and $P$ is any point on the curve. Here we have finally found the desired property of the curves which does not require us to leave the plane, and is stated uniformly for all three curves. For ellipses the ratioCf. Note B, noteB.136 is less than $1$, for parabolas it is equal to $1$, and for hyperbolas it is greater than $1$.

When Pappus had finished his investigations,

he must have felt that, apart from minor extensions, the subject was practically exhausted; and if he could have foreseen the history of science for more than a thousand years, it would have confirmed his belief. Yet in truth the really fruitful ideas in connection with this branch of mathematics had not yet been even touched on, and no one had guessed their supremely important applications in nature. No more impressive warning can be given to those who would confine knowledge and research to what is apparently useful, than the reflection that conic sections were studied for eighteen hundred years merely as an abstract science, without a thought of any utility other than to satisfy the craving for knowledge on the part of mathematicians, and that then at the end of this long period of abstract study, they

were found to be the necessary key with which to attain the knowledge of one of the most important laws of nature.

Meanwhile the entirely distinct study of astronomy had been going forward. The