MATHEMATICAL CONCEPTS OP THE MATERIAL WORLD. 469 



reals (e.g., in Concept I.). The essential relation ol any one concept will be a relation 

 between a definite finite number of terms, for example, between three terms in 

 Concepts I. and II., between four terms in Concept III., and between five terms in 

 Concepts IV. and V.* 



Definition. In the exposition of every concept, the essential relation of that 

 concept is denoted by R. 



The Extraneous Relations. In all the concepts here considered, other relations, 

 here called the extraneous relations, will be required in addition to the time-relation 

 and the essential relation. In Concepts I. and II. and IV. an indefinite (if not 

 infinite) number of extraneous relations are required, determining the positions of 

 particles. In Concepts III., IV. and V. one tetradic extraneous relation is required 

 to determine the "kinetic axes" of reference for the measurement of velocity. 



The time-relation, the essential relation and the extraneous relations form the 

 fundamental relations of any concept in which they occur. 



It will now be necessary to define geometry anew, since the previous definition has 

 essential reference to the dualism of the classical concept. A proposition of geometry 

 is any proposition (l) concerning the essential relation ; (2) involving one, and only 

 one, instant of time ; (3) true for any instant of time. 



In the classical concept everything is sacrificed to simplicity in reference to 

 geometry, probably because it arose when geometry was the only developed science. 

 The result is that, when the properties of matter are dealt with, an appalling number 

 of extraneous relations are necessary. 



Judged by OCCAM'S principle, this class of extraneous relations forms a defect in 

 Concepts I. and II. and IV. Also, in both forms of the classical concept (viz., in 

 Concepts I. and II.) geometry is segregated from the other physical sciences to a 

 greater degree than in the other concepts. 



In the study of any concept there are four logical stages of progress. The first 

 stage consists of the definition of those entities which are capable of definition in 

 terms of the fundamental relations. These definitions are logically independent of 

 any axioms concerning the fundamental relations, though their convenience and 

 importance are chiefly dependent upon such axioms. The second stage consists of the 



* The idea of deriving geometry (at least protective geometry without reference to order) from a single 

 triadic relation was (I believe) first enunciated and investigated by Mr. A. B. KEMPE, F.R.S., in 1890, 

 cf. "On the Relation between the Logical Theory of Classes and the Geometrical Theory of Points," 

 ' Proc. Lond. Math. Soc.,' vol. XXI. It has since been worked out in detail for Euclidean geometry by 

 Dr. 0. VEBLEN, cf. "A System of Axioms for Geometry," 'Trans. Amer. Math. Soc.,' vol. 5, 1904. 

 Also cf. Professor J. ROYCE on " The Relation of the Principles of Logic to the Foundations of Geometry," 

 'Trans. Amer. Math. Soc.,' 1905. Professor ROYCE emphasises the importance of KEMPE'S work and 

 considerably extends it. This memoir (which unfortunately only came into my hands after the completion 

 of the present investigation) anticipates a general line of thought of the present paper in the emphasis laid 

 on the derivation of geometry from a single polyadic relation ; otherwise our papers are concerned with 

 different problems. 



