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In order that space may be discussed mathematically it must be reduced conventionally to number; this is done through the agency of measure. Then units of measure are devised giving rise to fractions and whole numbers, multiples, and sub-multiples, when it becomes amenable to the operations of mathematics.

Speed exists in the unit of extension whether there be other units or not; speed, therefore, is unrelated or absolute. But the extended unit having motion must also have path, which is a change of position to others and variable by collision with others. It is thus relative. Speed is constant in the ultimate unit, which will be demonstrated in a subsequent chapter; but path is change of position in relation to others, and motion therefore is absolute in speed and relative in path.

There is persistence or indestructibility in the fundamental unit of extension and motion, but this unit changes its relation to other units in position and also in trajectory; the persistence is absolute and constant, the change relative and variable.

Motions are related to one another in direction and also in the positions of trajectories. Directions may differ in innumerable ways and paths may have innumerable deflections and thus trajectories may have innumerable variables. In order that direction and trajectory may be treated mathematically it becomes necessary to devise methods for the measurement of directions which are expressed in degrees and of lengths which are expressed in various measures. By these conventions motions are