356 THE POPULAR SCIENCE MONTHLY. 



covery of substitution, to abandon the old electro-chemical dualism, 

 has seemingly taken a retrograde step in its advance toward science 

 in this sense. The resolution of all changes in the material world into 

 motions of atoms caused by their constant central forces would be the 

 completion of natural science." 



How do these sentences of one of the foremost physicists of the 

 day now present themselves to our view in the light of the preceding 

 discussion ? Atoms are absolute physical constants, or constants of 

 mass ; and I have shown that there are, and can be, no absolute con- 

 stants of mass. And it is evident now, I trust, that there are similarly 

 no constant central forces belonging to, or inherent in, constants of 

 mass as such. Both the constants of mass and the constant forces 

 assumed to belong to them are simply parts of the scaffolding of the 

 intellect, when it seeks to subject the phenomena of the material world 

 to exact mathematical determination. They aie, as I have already 

 intimated, instrumental fictions which are, for the moment, indispen- 

 sable by reason of the inability of the intellect to deal with phenomena 

 otherwise than singly and under a definite aspect. In order to effect a 

 quantitative determination of material action, the mathematician is 

 constrained to insulate the conceptual elements of matter and to treat 

 them as separate and distinct terms. He is constrained to represent 

 as discrete what is continuous, and as absolute what is relative. In 

 this, as he knows, or ought to know, very well, he is doing violence to 

 the fact. But this violence is harmless, provided he does not forget 

 that what appears in abstract insulation in his formulae exists in con- 

 crete and indissoluble union in Nature, and what he exhibits as uncon- 

 ditioned in thought is essentially conditioned in objective reality. The 

 steps to all scientific knowledge consist in a series of representative 

 fictions. When the old Greek sought to determine the properties of 

 the circle, he began by constructing a polygon whose sides he sub- 

 divided until they were supposed to become infinitely small ; and in 

 his view every line of definite extent and form, i. e., every line which 

 could become the subject of mathematical investigation, was composed 

 of an infinite number of infinitely small straight lines. But he speedily 

 found that, while this fiction enabled him to deduce a rule for calcu- 

 lating the area of the circle and otherwise to determine a number of 

 its properties, nevertheless the circle and its rectilinear diameter were 

 fundamentally incommensurable, and the quadrature of the circle was 

 impossible. The modern analyst similarly determines the " locus " of 

 a curve by the relation of small increments of coordinates arbitrarily 

 established ; but he is well aware that the curve itself has nothing to 

 do with this arbitrary representation, and he very emphatically asserts 

 the continuity of the curve by differentiating, or passing to the 

 " limit " of, his increments — at the same time transforming his coor- 

 dinates by changing their origin or their inclination, or even their 

 system, from bilinears to polars, whenever he finds it convenient, with- 



