232 POPULAR SCIENCE MO NTH LY 



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 measur- 

 ing-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, how- 

 ever, be of too complicated a nature to be advantageous for any ob- 

 jective 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 mo- 

 tions 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 ap- 

 plication of arithmetic and algebra in geometry. It is well known that 

 under certain assumptions (coordinates) geometric figures may be 

 expressed in algebraic formulae so that it is possible to deduce the 



