600 APPLIED MATHEMATICS 



It is not my intention to confine the thought merely to the above 

 facts and their resulting consequences; these are not sufficient to 

 carry through the question as to the finite or infinite divisibility 

 of matter. If we are going to think of the atoms of chemistry as 

 made up of electrons, what would hinder us from considering the 

 electrons as particles filled with rarefied, continuous matter? We 

 shall adhere faithfully to the previously developed philosophical 

 principles and shall examine in the most unhampered manner the 

 concepts themselves in order to express them in a consistent and 

 most useful form. 



It appears now, that we are unable to define the infinite in any other 

 way except as the limit of continually increasing magnitudes, at 

 least no one has hitherto been able to set up any other intelligible 

 conception of the infinite. Should we desire a verbal picture of the 

 continuum, we must first think of a large finite number of particles 

 w r hich are endowed with certain properties and study the totality 

 of these particles. Certain properties of this totality may approach 

 a definite limit as the number of particles is increased, and their 

 size decreased. It can be asserted, concerning these properties, that 

 they belong to the continuum, and it is my opinion that this is the 

 only self-consistent definition of a continuum which is endowed 

 with certain properties. 



The question if matter is composed of atoms or is continuous 

 becomes then the question if the observed properties are accurately 

 satisfied by the assumption of an exceedingly great number of 

 such particles or, by increasing number, their limit. We have not 

 indeed answered the old philosophical question, but we are cured of 

 the effort to answer it in a senseless and hopeless manner. The 

 thought-process, that we must investigate the properties of a finite 

 totality and then let the number of members of this totality increase 

 greatly, remains the same in both cases. It is nothing other than 

 the abbreviated expression in algebraic symbols of exactly the same 

 thought when, as often happens, differential equations are made 

 the basis of a mathematical-physical theory. 



The members of the totality which we select as the picture of the 

 material body cannot be thought of as absolutely at rest, for there 

 would then be no motion of any kind, nor can the members be thought 

 of as relatively at rest in one and the same body, for in this case it 

 would be impossible to account for the fluids. No effort has been 

 made to subject them to anything more than to the general laws 

 of mechanics. In order to explain nature we shall therefore select 

 a totality of an exceedingly large number of very minute funda- 

 mental individuals which are constantly in motion, and which are 

 subject to the laws of mechanics. But an objection is raised that 

 will be an appropriate introduction to the final considerations of 



