100 FUNDAMENTAL PRINCIPLES OF MATHEMATICS. 



addition of pure numbers, and showed that the axioms of arithmetic of 

 the equality of two numbers in respect to a third, the association law 

 of addition, and the commutation law, can only be proved by the agree- 

 ment of the results arithmetically derived with those which can 

 be obtained by counting of exterior numerable objects. That the 

 objects should be numerable, certain conditions must be fulfilled con- 

 cerning whose presence only experience can decide. Since that objects 

 which in any particular respect are alike and can be numbered may be 

 regarded as units of number, the result of their enumeration as a defi- 

 nite number, and the kind of units which compose it as the denomina- 

 tion of the number, the conception with respect to the equality of two 

 groups containing given numbers of objects of the same denomination 

 is given by these numbers. If we designate as quantities objects or 

 attributes of objects which, when compared with similar ones, may be 

 greater, equal, or less, and if we can express these quantities by known 

 numbers, we call these numbers the values of the quantities, and the 

 process by which we find them measurement. Thus we measure a force 

 by the masses and displacements of systems upon which they have 

 been exercised; or in dynamic measurements by the masses and move- 

 ments of systems upon which they are working; or in the static method 

 of measurement by bringing the forces into equilibrium with others 

 already known. It only remains to consider under what circumstances 

 quantities may be expressed by numbers and what is thereby attained 

 in actual knowledge. With this purpose were instituted interesting and 

 valuable considerations on physical equality and the commutation and 

 association law for physical combination. Addition was regarded as a 

 combination of quantities of the same kind, such that the result remained 

 unchanged when the single elements were exchanged, or when the num- 

 bers were replaced by equal quantities of the same kind. In introduc- 

 ing irrational relations Helmholtz placed himself at the standpoint of 

 the physicist, and in later development of the principles of mechanics 

 he retained, as we shall see, the same point of view, showing that in 

 geometry and physics no discontinuous functions are met with for which 

 it is not enough to know with sufficient accuracy the bounds within 

 which the irrational values lie. The mathematicians, however, it must 

 be said, recognize functions of another kind, and the recent investi- 

 gations of Boltzmann seem to point to a physical application of such 

 analytical conceptions. 



"We now come to by far the most difficult part of our task as we 

 attempt to describe the service of Helmholtz to analytical mechanics; 

 for in order to understand the partial reconstruction of the science in 

 consequence of some of his most brilliant researches it will be neces- 

 sary to accurately follow him through his great series of wonderful 

 mathematical-physical investigations and far-reaching physical discov- 

 eries in the great fields of hydrodynamics, aerodynamics, and electricity, 

 which have contributed to the investigation of the axioms of mechanics. 



