[bakeu] the foundations OF GEOMETRY 123 



congruent; also if A B (J ... P contains four non-co-jîlanar points, then 

 the construction of P' is possible in only one way. 



With, this general proposition before us we see that the facts pi 

 congruence are resolvable into " assumptions of congruence " as ele- 

 ments. Or we may express the same idea by saying that the question 

 of the possible existence of two distinct congruent associations of points 

 is shown to be resolvable into the elementary " assumptions of con- 

 gruence" given, above. In consequence, Professor Halsted says that 

 we thus " found the idea of motion upon the congruence assumptions " ; 

 and Sommer of Gottingen, in his review of Hilbert's " Grundlagen der 

 Géométrie," (Bulletin, Am. Math. See, Vol. vi, p. 289), speaks of 

 the definition of motion as based on the assumptions of congruence. 

 It is, however, of importance to note that the words of Hilbert are, 

 "All the facts concerning space which have reference to congruence, 

 that is to say, to displacements in space, are exclusively the consequences 

 of the six assumptions of congruences." Hilbert, therefore, here says 

 in effect that congruence is displacement. When, as above, we state 

 the congruence idea with reference to a system of points, we are stating 

 ii with reference to any rigid body. We, however, are conveying the 

 idea of all its force, though in the simpler form, in the very first of 

 the assumptions of congruence, that respecting two sects. I think Ave 

 may admit that there is a sense, in the purely rational geometry of 

 Hilbert, in which the idea of congruence precedes the idea of motion, 

 for, indeed, Hilbert has so arranged it. We admit that there is a 

 sect congruent to A B in all positions. We conclude, therefore, that 

 space is such as to admit of the transference of A B to any position, 

 i.e., we reach the idea of motion. If, however, we are dealing with 

 a geometry into which experience in the least enters, the existence 

 elsewhere of a sect congruent to A B seems to me to depend on the 

 idea that AB may be transferred anywhere; i.e., the idea of motion 

 precedes the idea of congruence. The matter is of interest since, 

 on the ground that the idea of congruence precedes the idea of motion. 

 Professor Halsted protests against the use of superposition in proving 

 theorems in congruence, as we do, for example, in proving the con- 

 gruence of triangles in our ordinary elementary geometry. He says 

 " to prove the congruence assumptions and theorems with the help of 

 the motion idea, is false and fallacious, since the intuition of rigid 

 motion involves, contains and uses the congruence idea/' 



PAEALIiELS. 



The definition of parallels is that they are coplanar straight lines 

 with no common point. 



