In biology I believe he may be considered almost equally great: certainly he spent a great deal of time in dissecting, and he made out a good deal of what is now known of the structure of the body, and of the theory of vision. He eagerly accepted the doctrine of the circulation of the blood, then being taught by Harvey, and was an excellent anatomist.
You doubtless know Professor Huxley's article on Descartes in the Lay Sermons, and you perceive in what high estimation he is there held.
He originated the hypothesis that animals are automata, for which indeed there is much to be said from some points of view; but he unfortunately believed that they were unconscious and non-sentient automata, and this belief led his disciples into acts of abominable cruelty. Professor Huxley lectured on this hypothesis and partially upheld it not many years since. The article is included in his volume called Science and Culture.
Concerning his work in mathematics and physics I can speak with more confidence. He is the author of the Cartesian system of algebraic or analytic geometry, which has been so powerful an engine of research, far easier to wield than the old synthetic geometry. Without it Newton could never have written the Principia, or made his greatest discoveries. He might indeed have invented it for himself, but it would have consumed some of his life to have brought it to the necessary perfection.
The principle of it is the specification of the position of a point in a plane by two numbers, indicating say its distance from two lines of reference in the plane; like the latitude and longitude of a place on the globe. For instance, the two lines of reference might be the bottom edge and the left-hand vertical edge of a wall; then a point on the wall, stated as being for instance 6 feet along and 2 feet up, is precisely determined. These two distances are called co-ordinates; horizontal ones are usually denoted by x, and vertical ones by y.
If, instead of specifying two things, only one statement is made, such as y = 2, it is satisfied by a whole row of points, all the points in a horizontal line 2 feet above the ground. Hence y = 2 may be said to represent that straight line, and is called the equation to that straight line. Similarly x = 6 represents a vertical straight line 6 feet (or inches or some other unit) from the left-hand edge. If it is asserted that x = 6 and y = 2, only one point can be found to satisfy both conditions, viz. the crossing point of the above two straight lines.
Suppose an equation such as x = y to be given. This also is satisfied by a row of points, viz. by all those that are equidistant from bottom and left-hand edges. In other words, x = y represents a straight line slanting upwards at 45°. The equation x = 2y represents another straight line with a different angle of slope, and so on. The equation x2 + y2 = 36 represents a circle of radius 6. The equation 3x2 + 4y2 = 25 represents an ellipse; and in general every algebraic equation that can be written down, provided it involve only two variables, x and y, represents 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.