346 METHODOLOGY OF SCIENCE 



of arrangement, but they are again the simplest of all. We come 

 to the concept of magnitude only in the theory of time and space. 

 The theory of time has not been developed as a special science; on 

 the contrary, what we have to say about time first appears in me- 

 chanics. Meantime we can present the fundamental concepts, which 

 arise in this connection, with reference to such well-known charac- 

 teristics of time that the lack of a special science of time is no dis- 

 advantage. 



The first and most important characteristic of time (and of space, 

 too) is that it is a continuous manifold; that is, every portion of 

 time chosen can be divided at any place whatever. In the number 

 series this is not the case; it can be divided only between the single 

 numbers. The series one to ten has only nine places of division and 

 no more. A minute, or a second, on the other hand, has an unlimited 

 number of places of division. In other words, there is nothing in the 

 lapse of any time which hinders us from separating or distinguishing 

 in thought at any given instant the time which has elapsed till then 

 from the following time. It is just the same with space, except that 

 time is a simple manifold and space a threefold, continuous manifold. 



Nevertheless, when we measure them, we are accustomed to indicate 

 times and spaces with numbers. If we first examine, for example, the 

 process of measuring a length, it consists in our applying to the dis- 

 tance to be measured a length conceived as unchangeable, the unit 

 of measure, until we have passed over the distance. The number of 

 these applications gives us the measure or magnitude of the distance. 

 The result is that by the indication of arbitrarily chosen points upon 

 the continuous distance, we place upon it an artificial discontinuity 

 which enables us to associate it with the discontinuous number series. 



A still further assumption, however, belongs to the concept of 

 measuring, namely, that the parts of the distance cut off by the unit 

 used as a measure be equal, and it is taken for granted that this 

 requirement will be fulfilled to whatever place the unit of measure 

 is shifted. As may be seen, this is a definition of equality carried 

 further than the former, for one cannot actually replace a part of 

 the distance by another in order to convince one's self that it has 

 not changed. Just as little can one assert or prove that the unit of 

 measure in changing its place in space remains of the same !cr>gth; 

 we can only say that such distances as are determined by the unit of 

 measure in different places are declared or defined as equal. Actually, 

 for our eye, the unit of measure becomes smaller in perspective the 

 farther away from it we find ourselves. 



From this example we see again the great contribution which 

 arbitrariness or free choice has made to all our structure of science. 

 We could develop a geometry in which distances which seem sub- 

 jectively equal to our eye are called equal, and upon this assumption 



