ts some curve in a plane; a curve
moreover that can be drawn, or its properties completely
investigated without drawing, from the equation. Thus algebra is
wedded to geometry, and the investigation of geometric relations by
means of algebraic equations is called analytical geometry, as
opposed to the old Euclidian or synthetic mode of treating the
subject by reasoning consciously directed to the subject by help of
figures.

If there be three variables--_x_, _y_, and _z_,--instead of only
two, an equation among them represents not a curve in a plane but a
surface in space; the three variables corresponding to the three
dimensions of space: length, breadth, and thickness.

An equation with four variables usually requires space of four
dimensions for its geometrical interpretation, and so on.

Thus geometry can not only be reasoned about in a more mechanical
and therefore much easier, manner, but it can be extended into
regions of which we have and can have no direct conception, because
we are deficient in sense organs for accumulating any kind of
experience in connexion with such ideas.

[Illustration: FIG. 54.--The eye diagram. [From Descartes' _Principia_.]
Three external points are shown depicted on the retina: the image being
appreciated by a representation of the brain.]

In physics proper Descartes' tract on optics is of considerable
historical interest. He treats all the subjects he takes up in an able
and original manner.

In Astronomy he is the author of that famous and long upheld theory, the
doctrine of vortices.

He regarded space as a plenum full of an all-pervading fluid. Certain
portions of this fluid were in a state of whirling motion, as in a
whirlpool or eddy of water; and each planet had its own eddy, in which
it was whirled round and round, as a straw is caught and whirled in a
common whirlpool. This idea he works out and elaborates very fully,
applying it to the system of the world, and to the explanation of all
the motions of the planets.

[Illustration: FIG. 55.--Descartes's diagram of vortices, from his
_Principia_.]

This system evidently supplied a void in men's minds, left vacant by the
overthrow of the Ptolemaic system, and it was rapidly accepted. In the
English Universities it held for a long time almost undisputed sway; it
was in this faith that Newton was brought up.

Something was felt to