IX

Coordinate Geometry

The methods and ideas of coordinate geometry

have already been employed in the previous chapters. It is now time for us to consider them more closely for their own sake; and in doing so we shall strengthen our hold on other ideas to which we have attained. In the present and succeeding chapters we will go back to the idea of the positive and negative real numbers and will ignore the imaginaries which were introduced in the last two chapters.

We have been perpetually using the idea that, by taking two axes, $XO{X}^{\prime }$ and $YO{Y}^{\prime }$, in a plane, any point $P$ in that plane can be determined in position by a pair of positive or negative numbers $x$ and $y$, where (cf. [fig.]13) $x$ is the length $OM$ and $y$ is the length $PM$. This conception, simple as it looks, is the main idea of the great subject of coordinate geometry. Its discovery marks a momentous epoch in the history of mathematical thought. It is due (as has been

already said) to the philosopher Descartes,

and occurred to him as an important mathematical method one morning as he lay in bed. Philosophers, when they have possessed a thorough knowledge of mathematics, have been among those who have enriched the 13 science with some of its best ideas. On the other hand it must be said that, with hardly an exception, all the remarks on mathematics made by those philosophers who have possessed but a slight or hasty and late-acquired knowledge of it are entirely worthless, being either trivial or wrong. The fact is a curious one; since the ultimate ideas of mathematics

seem, after all, to be very simple, almost childishly so, and to lie well within the province of philosophical thought. Probably their very simplicity is the cause of error; we are not used to think about such simple abstract things, and a long training is necessary to secure even a partial immunity from error as soon as we diverge from the beaten track of thought.

The discovery of coordinate geometry, and also that of projective geometry about the same time, illustrate another fact which is being continually verified in the history of knowledge, namely, that some of the greatest discoveries are to be made among the most well-known topics. By the time that the seventeenth century had arrived, geometry had already been studied for over two thousand years, even if we date its rise with the Greeks. Euclid, taught in the University of Alexandria,

being born about 330 ; and he only systematized and extended the work of a long series of predecessors, some of them men of genius. After him generation after generation of mathematicians laboured at the improvement of the subject. Nor did the subject suffer from that fatal bar to progress, namely, that its study was confined to a narrow group of men of similar origin and outlook–-quite the contrary was the case; by the seventeenth century it had passed

through the minds of Egyptians and Greeks, of Arabs and of Germans. And yet, after all this labour devoted to it through so many ages by such diverse minds its most important secrets were yet to be discovered.