PROFESSOR IN BERLIN 377 



a series, and all enumeration is the arrangement of the things 

 enumerated in a time-series ; he regards the composition of 

 fractions of time into magnitudes of time as the primitive type 

 of addition. He further proposes definitions for objects in 

 general; the definition of equality is that if two things are 

 equal to a third thing they are equal to each other : permutation 

 (combination) is the association of different things, in which the 

 order of association is not indifferent ; addition is the combining 

 of homogeneous things, independent of the order of association ; 

 multiplication is (at least in all its applications), as he sets forth 

 in a note, the combining of heterogeneous magnitudes, for which 

 the order of association is indifferent since units of any kind 

 are multiplied by abstract figures, or horizontal by vertical 

 lines, or distances by masses. In the other mathematical 

 operations the order is not an indifferent matter. Lastly, he 

 regards magnitude as an additive combination of homogeneous 

 units or parts : equal magnitudes are composed of pairs of equal 

 parts ; while a sum is the additive combination of magnitudes. 



While therefore objects which are equal in any definite 

 respect, and are enumerable, are termed units of numeration, 

 their number is termed a denominated number, and the special 

 kind of unit which they comprise is the denomination of the 

 number ; the concept of the equality of two groups of denomi- 

 nated numbers of similar denomination is established by their 

 having the same number. If the objects, or the attributes of 

 objects, which when compared with others admit of the dis- 

 tinctions of greater, equal, or less, are termed magnitudes (as 

 to which only empirical knowledge of certain sides of their 

 physical reactions in coincidence and co-operation with others 

 can decide), and if we can express these magnitudes by a certain 

 number, we term this the value of the magnitude, and the 

 method by which we find the given number we call measure- 

 ment. Thus we measure a force either by the masses and 

 motions of the system by which it is exerted, or in dynamic 

 measurement by the masses and the motions of the system on 

 which it acts, or lastly by the static method of measurement of 

 force, by bringing the force into equilibrium with known forces. 

 There remains only one question to be answered : when may 

 we express magnitudes by denominated numbers, and what 

 actual knowledge do we gain by doing so? To answer this, 



