THE NEW YORK 

 ACADEMY OF SCIENCES. 



Rogers — The Logical Basis of Mathematics. 193 



functions, and gives the most satisfactory account of incommensurables. 

 Geometiy may be approached from a different side (as in the worlds 

 mentioned below) ; but when the Manifold forming the subject-matter of any 

 Geometry contains an infinite number of points, the theory of transfinite 

 numbers is involved. Geometry of n dimensions in its most complex form 

 is a special application of the theory of series to the case where each member 

 of the series is itself a serially arranged class. 



[The following list of references, though by no means exhaustive, is fairly 

 representative. More complete references will be found in the English 

 works referred to and in Peano's Forinulaire. In Peano's work a complex 

 system of logical symbolism is used, which appears to be almost inevitable 

 for precise exposition. The other works referred to use, with some trivial 

 exceptions, the ordinary symbolism of Mathematics. 



A. On the subject generally : — 



B. EusSELL, The Principles of Mathematics. Vol. i. (Cambridge, 

 1903). 



B. On the Logic of Number (Integral, rational, irrational, and trans- 

 finite) : — 



Peano, Formulaire cle MatMmatiques. (Paris, 1901, and previous 

 Editions.) 



G. Cantoe, Beitrdge zur Begrunclung der transfiniten Mengenlehre, 

 Math. Ann. XLVI, (1895). XLix. (1897). (A more precise 

 •mathematical exposition of the foundations of the philosophical 

 theory of infinite number expounded in his Mannigfaltig- 

 heitslehre.) 



Dedekind, Stctigkeit unci irrationale Zahlen (1872). 



YouxG AND Young's Theory of Sets of Points (1906), and Hobson's 

 Theory of Functions of a Real Variable (1907), contain full 

 expositions and applications of Cantor's theories. 



C. On the Axioms of Geometry : — 



Hubert, Gncndlagen der Geometric (1899). (Eng. trans, by 



TOWNSEND.) 



A. N. Whitehead, The Axioms of Projective Geometry (1906)]. 



H.I.A. tTvOC, VOL. XXVII., SECT. A. [28] 



