any desired moment from separating or distinguishing in thought the time that has passed from that which is to follow. Space is of the same nature, except that time is simple, space threefold.
Nevertheless, we are accustomed to describe both time and space by means of numbers whenever we measure them. If we examine into this procedure, for instance in the case of measuring length, we find that it consists in applying a length considered fixed, the measure, as often to the length to be measured as is necessary to cover it. The number of applications gives us the measure or magnitude of the length. It merely amounts to forcing an artificial discontinuity upon a continuous length by marking off arbitrarily chosen points, allowing us to refer it to the discontinuous numerical series.
The equality of the portions of distance set off by the measuring-rod is an essential part of the concept of measuring. We assume this condition fulfilled no matter how the measuring-rod be shifted. As we see, this is a more forced definition of equality than heretofore made, for it is actually quite impossible to substitute a given portion of a distance for another in order to become convinced that the validity of our definition is not impaired, that nothing is changed thereby. It is quite as impossible to prove that the measuring-rod in being shifted in space remains of the same length. We may only affirm that such distances as are determined in various places by means of the measuring-rod are declared or defined as equal. As a matter of fact, the measuring-rod in perspective looks smaller the further it is away from us.
This example demonstrates anew the great arbitrariness with which we shape science. It is conceivable that a geometry might be developed in which the distances are considered equal which subjectively appear to our eye to be so, and we should then be quite as able to develop a consistent system or science. A geometry of this kind would, however, be of too complicated a nature to be advantageous for any objective purpose (e. g., surveying). Therefore we endeavor to develop a science as free as possible from subjective factors. Historically the Ptolemaic astronomy and that of Copernicus present an illustration in point. The former was formed according to subjective appearances in its assumption that the stars revolved about the earth. It proved most complicated when confronted with the problem of expressing these motions mathematically. The latter gave up the subjective point of view of the observer who regarded himself as the center; and, by transferring the center of motion to the sun, produced: an enormous simplification.
A few more words are necessary at this point concerning the application of arithmetic and algebra in geometry. It is well known that under certain assumptions (coordinates) geometric figures may be expressed in algebraic formulæ so that it is possible to deduce the
