MATHEMATICAL CONCEPTS OF THE MATERIAL WORLD. 477 



Definition. The class whose members are the various straight lines is denoted by 

 lin R . The definition in symbols is 



lin R = v {($a, b) . a, b e K ; ( ; ; ; ) . a ^ b . v = R : "o&} Df 



Definition. Any class of points [i.e., members of R ; (;;;)] is called a figure. 

 Definition. The class of lines defined by a figure u is the class of lines defined by 

 any two distinct points of u. Its symbol is ln R '?<. The definition in symbols is 



ln R 'M = v {(RX, y) . x, y e u n R ; ( ; ; ; ) . x ^ y . v = R'iw/} Df 



Definition. The linear figure defined by a figure u is the logical sum of all the 

 lines defined by u (i.e., is all the points on such lines). Thus its symbol is u'ln R 'w. 



Definition. Three points form a triangle, if there is no line which possesses them 

 all. The symbol expressing that a, b, c are points forming a triangle is A H '(abc). 

 The definition in symbols is 



A R '(abc) . = . a, b, c e R : ( ; ; ; ) . -- (QV) . v e lin, { . a, b, cer Df 



Definition. The space defined by the triangle abc is the linear figure defined by 

 the linear figure defined by the three points a, b, c. Its symbol is U H (abc). The 

 definition in symbols is 



U R (abc) = u'ln H 'u c ln B '(i'a u t'b u t'c) Df 



Definition. The class of planes is the class of spaces defined by any three points 

 a, b, c when they form a triangle. Its symbol is ple R . The definition in symbols is 



ple R = v{(^a, b, e) . A R '(abc) . v = U R (abc)} Df 



Definition. The space defined by the four points a, b, c, d is the linear figure 

 defined by the figure which is the logical sum of U R (bcd) and U R (acd) and U R (abd) 

 and H R (abc). Its symbol is U R (abcd). The symbolic definition is 



TI R (abcd) = u'ln n ' {U R (bcd) u H R (acd) u H K (abd) u Tl R (abc)} Df 



The above definitions are sufficient to exhibit the dependence of the various 

 geometrical entities on the essential relation, and also to enable us, as far as geometry 

 is concerned, to pass on to the third stage. Owing to the simplicity of the definitions, 

 the second stage for this concept is of very small importance. 



It will be noticed that none of the definitions contain any reference to length, 

 distance, area, or volume. This is because none of these ideas appear in the axioms, 

 and only such definitions are given here as are necessary for the enunciation of the 

 axioms. According to the well-known* method of projective metrics, the ideas are 

 introduced by definition and require no special axiom. 



* Cf. VEBLEN, loc. at. ; also ' Vorlesungen iiber Geometric,' by CLEBSCH, third part ; also ' The 

 Principles of Mathematics,' by RUSSELL, chap. XLVIII. 



