geometric properties of the figures from the calculatory properties of the formulas, and vice versa. We must inquire how such close and unambiguous relations can exist between things so diverse. The answer is that we have to deal in this instance with a particularly obvious case of association. The manifoldness of numbers is far greater than that of planes or space, for, whereas the latter are determined by but two or three independent measurements, any number of independent numerical series may be made to react upon one another. We therefore arbitrarily limit the manifoldness of the numbers to two or three independent series, and determine (by means of the cosinus law) their mutual relations so that a manifoldness corresponding exactly to that of the space arises to which we are able completely to refer it. We have then two manifoldnesses of identical character; and all properties of arrangement and size of the one are 'depicted' in the other. In this an extremely important scientific procedure is indicated which consists in giving to the experience-content of a given field a formal manifoldness to which we impart the same manifoldness-character as that possessed by the former. Every science thus develops a formula language of its own, perfect in proportion to the accuracy with which the manifoldness-character of the object has been recognized, and the fitness of the formulas selected. Whereas, in arithmetic and algebra this problem has been solved quite perfectly (though by no means absolutely so), chemical formulas, for instance, express only a relatively small part of the characteristics which they ought to express, while in biology and sociology we have hardly progressed beyond the very beginnings of the solution of this problem.
One of these universal manifoldnesses designed to express our experiences is speech. Inasmuch as it was developed in a primitive civilization it is by no means regular and complete enough to fulfil its purpose satisfactorily. On the contrary, it is quite as unsystematic as were the events in the history of the various peoples. The need to express the infinite variety of events in daily life has been filled by allowing word and concept to correspond only within a wide limit of variation. Therefore all research in the sciences which are forced to employ this means of expression (psychology and sociology or philosophy generally) is greatly impeded by the struggle with the indefiniteness and ambiguity of language. An improvement of these conditions is to be attained only by the introduction, as rapidly as the progress of the science warrants it, of symbols to which we refer the manifoldness which experience tells us is peculiar to the concept.
The sciences which have been classed above as a part of energetics occupy an intermediate position. In addition to the concepts of order, number, magnitude, space and time we meet in this branch of knowledge with the new concept of energy, which is applied as universally to
