364 SCIENCE PROGRESS 



by its predecessors. In the present state of mathematical 

 recording it is only by repeatedly looking back in this way that 

 any close approximation to completeness can be secured. The 

 valuable Catalogue issued by the Mathematical Association is 

 mentioned with appreciation, and a suggestion is made that it 

 should be brought up to date at intervals of a few years. This 

 paper is a further step towards the very valuable and rigorously 

 full treatment of that part of mathematical history to which 

 Muir has devoted himself. 



Logic and Principles of Mathematics. — P. J. Daniell {Bull. 

 Amer. Math. Soc. 191 7, 23, 446-50) introduces a new concept 

 into the calculus of classes (logical classes or sets of points) 

 which is analogous to the modular difference of two numbers. 

 The " modular difference of two classes " is defined by a direct 

 logical operation on the classes ; some properties are proved, 

 limiting classes of sequences are defined, and some properties 

 of these limits are proved which are of great interest in con- 

 nection with Borel's work in his Theorie des Fonctions des 

 Variables reelles, and, though this is not mentioned, with the 

 much earlier work of Cantor on derivatives of infinite order. 

 However, where the existence of limits has to be proved, DanielPs 

 paper does not seem complete. 



E. V. Huntington and J. R. Kline (Trans. Amer. Math. 

 Soc. 191 7, 18, 301-2$) give various sets of independent postu- 

 lates for " betweenness " which are selected from the twelve 

 postulates due essentially to Pasch in 1882, and show the 

 characteristic properties of the type of system represented by 

 points on a line by which this type is distinguished from that of 

 any other system of elements related by a triadic relation. 



A. Padoa (Rev. de Metaphys. et de Morale, 191 7, 24, 315-25) 

 writes on the consequences of a change of primitive ideas in any 

 deductive theory. 



G. Peano (Boll, delta Mathesis, 191 5, 1 — 1 5) describes the 

 principal definitions by abstraction which are met with in 

 mathematical books : ratios, cardinal numbers, real numbers 

 and vectors. The objection that the sign of equality is defined 

 again by the above definition is not valid, according to the 

 author, because the above sign is not defined, " but the whole 

 equality composed of the two numbers together with this sign." 

 The method used by Russell of replacing such definitions by 

 nominal definitions is called the " theory of classes," and the 



