564 GEOMETRY 



certain pairs of points may necessitate its fulfillment for all pairs. 

 If in Euclid's system the postulate of parallels is replaced by the 

 postulate concerning the sum of the angles of a triangle, a well-known 

 example of such a reduction is obtained; for it is sufficient to as- 

 sume the new postulate for a single triangle, the general result being 

 then deducible. As other examples we may mention Peano's reduc- 

 tion of the Euclidean definition of the plane; and the definition of 

 a collineation which demands, instead of the conversion of all straight 

 lines into straight lines, the existence of four simply infinite systems 

 of such straight lines. 1 



These examples illustrate the difficulty, if not the impossibility, 

 of formulating a really fundamental, that is, absolute standard of 

 independence and irreducibility. It is probable that the guiding 

 ideas will be obtained in the discussion of simpler deductive theories, 

 in particular, the systems for numbers and groups. 



Two features are especially prominent in the actual develop- 

 ment of the body of geometry from its fundamental system. First, 

 the consideration of what may be termed the collateral geometries, 

 which arise by replacing one of the original postulates by its opposite, 

 or otherwise varying the system. Such theories serve to show the 

 limitation of that point of view which restricts the term general 

 geometry (pangeometry) to the Euclidean and non-Euclidean geo- 

 metries. The variety of possible abstract geometries is, of course, 

 inexhaustible; this is the central fact brought to light by the ex- 

 hibition of such systems as the non-Archimedean and the non- 

 arguesian. In the second place, much valuable work is being done in 

 discussing the various methods by which the same theorem may 

 be deduced from the postulates, the ideal being to use as few of the 

 postulates as possible. Here again the question of simplicity (simplest 

 proof), though it baffles analysis, forces itself upon the attenti9n. 



Among the minor problems in this field, it is sufficient to consider 

 that concerning the relation of the theory of volume to the axiom of 

 continuity. This axiom need not be used in establishing the theory 

 of areas of polygons ; but after Dehn and others had proved the exist- 

 ence of polyhedra having the same volume though not decomposable 

 into mutually congruent parts (even after the addition of congruent 

 polyhedra), it was stated by Hilbert, and deemed evident generally, 

 that reference to continuity could not be avoided in three dimensions. 

 In a recent announcement 2 of Vahlen's forthcoming Abstrakte 

 Geometric this conclusion is declared unsound. It seems probable, 

 however, that the difference is merely one concerning the interpreta- 

 tion to be given to the term continuity. 



1 Together with certain continuity assumptions. Cf. Bull. Amer. Math. Soc., 

 vol. ix (1903), p. 545. 



2 Jahr. Deut. Math.-Ver., vol. xin (1904), p. 395. 



