The above can be stated in the language of mathematical logic, thereby making the character of the assumptions clearer and the generalization to continuous processes easier. Let be the series of qualities, the series of events in the rhythmic process. Let us imagine the events arranged in rows and columns, so that each row consists of one period and each column consists of all the events having a given quality. We assume a one-many relation , whose domain is the field of and whose converse domain is the field of . When has the relation to , we say " has the quality ." If is any term in the field of , let be the term which has the relation to ; then the next term below a in the same column (i.e. the corresponding in the next period) is the first term in the series which is after and to which has the relation .[Pg 351] The "row of " consists of all 's earlier than and not earlier than . The "column of " consists of all 's to which has the relation . We assume that with its converse domain limited to one row is one-one, so that each row (i.e. each period) is a series which is similar (in the technical sense) to the series . There is no difficulty in adapting the above analysis of periodicity to continuous processes. Instead of an enumerated set of qualities , , ..., we shall have to take some continuous series of qualities, such as the colours of the rainbow, or the notes produced on a violin by running one's finger up and down the string. The number of events compresent with a given event must now be infinite, but must still be less than the whole of one period (ignoring events outside the process concerned). The number of points in one period, or in any finite portion of it, is now infinite, and cannot therefore be used as a measure of distance. Thus in regard to metrical properties there are important differences between continuous and discrete processes. However, I shall not enlarge upon these, as I propose to consider the analysis of "interval" in a later chapter. Hitherto I have been considering processes which may be regarded as taking place in matter, or which, at any rate, do not move with the velocity of light. But light, also, is commonly regarded as consisting of a periodic process. Accepting the wave-theory of light, let us proceed to analyze its periodic character. We shall find that it differs in important respects from that of periodic processes in matter. [Pg 352]