II. There are real movements.
The mathematician, expressing with greater precision an idea of common sense, defines positionMovement is relative only for the mathematician, real for the physicist. by the distance from points of reference or from axes, and movement by the variation of the distance. Of movement, then, he only retains changes in length; and as the absolute values of the variable distance between a point and an axis, for instance, express either the displacement of the axis with regard to the point or that of the point with regard to the axis, just as we please, he attributes indifferently to the same point repose or motion. If, then, movement is nothing but a change of distance, the same object is in motion or motionless according to the points to which it is referred, and there is no absolute movement.
But things wear a very different aspect when we pass from mathematics to physics, and from the abstract study of motion to a consideration of the concrete changes occurring in the universe. Though we are free to attribute rest or motion to any material point taken by itself, it is none the less true that the aspect of the material universe changes, that the internal configuration of every real system varies, and that here we have no longer the choice between mobility and rest. Movement, whatever its inner nature, becomes an indisputable reality. We may not be able to say what parts of the whole are in motion;
