*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 constants 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 are, as I have already intimated, instrumental fictions which are, for the moment, indispensable 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 formulæ exists in concrete and indissoluble union in Nature, and what he exhibits as unconditioned 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 subdivided 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 calculating 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 coördinates by changing their origin or their inclination, or even their system, from bilinears to polars, whenever he finds it convenient, with-