Motion, whether it be that of our own body in controlled muscular activity, or that imaginary motion of the environment which we call giddiness, or a sensibly perceived motion of some part of the environment, that is, a motion which we can compensate by some actual or imaginary change in the position of our own body produced by our own exertion, is an intuitively felt change, and is incapable of intellectual representation. It is not clearly conceived either in ancient or in modern geometry. Euclidean geometry is, as we have seen, based directly on our intuition of bodily exertion, but it is essentially static in treatment. Let it be admitted that we can draw a straight line of any length and in any direction, and so on; then we regard these straight lines, etc., as motionless, abstract things, and we proceed to discuss their relationships. Cartesian geometry, and the methods of the infinitesimal calculus, do not treat of real motion, and the concept, if it is introduced at all, is introduced illegitimately and surreptitiously. Consider what we do when we “plot a curve.” Let the latter be a parabola having the equation _y_ = 1/2 _x_. Now a parabola is defined as “the locus of a point which _moves_, so that its distance from a fixed point is in a constant relation to its distance from a fixed straight line.” How do we construct such a curve? [Illustration: FIG. 5.] We proceed to fix the positions of a series of points in this way: there are two straight lines, _OX_ and _OY_, at right angles to each other, and we measure off certain steps along the line _OX_; these steps are _OX_↓{0·5}, _OX_↓{1}, _OX_↓{1·5}, _OX_↓{2}, and so on, the small numerals indicating the distance of each point (_OX_↓{0·5}, etc.) from the origin _O_. We then draw lines perpendicular to the _X_-axis through these points. We have now to calculate one-half of the square of each of these lengths _OX_↓{0·5}, _OX_↓{1}, etc., and then we mark off these calculated lengths along the perpendicular lines. The point _A_, for instance, is 1/2(0·5)^2 from the point _X_↓{0·5}, _B_ is 1/2(1)^2 from _X_↓{1}, and so on. In this way we obtain a series of points, _A_, _B_, _C_, _D_, _E_, etc., and these are points on the locus of the “moving” point. [Illustration: FIG. 6.]