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 be necessary to keep the planets moving on their endless round; the _primum mobile_ of Ptolemy had been stopped; an angel was sometimes assigned to each planet to carry it round, but though a widely diffused belief, this was a fantastic and not a serious scientific one. Descartes's vortices seemed to do exactly what was wanted. It is true they had no connexion with the laws of Kepler. I doubt whether he knew about the laws of Kepler; he had not much opinion of other people's work; he read very little--found it easier to think. (He travelled through Florence once when Galileo was at the height of his renown without calling upon or seeing him.) In so far as the motion of a planet was not circular, it had to be accounted for by the jostling and crowding and distortion of the vortices. Gravitation he explained by a settling down of bodies toward the centre of each vortex; and cohesion by an absence of relative motion tending to separate particles of matter. He "can imagine no stronger cement."