July 18, 1913] 



SCIENCE 



91 



of either can be associated in a one-to-one way 

 ■with the elements of the other. This section 

 consists of seven chapters dealing respectively 

 ■with elementary properties of cardinals; and 

 1 and 2; cardinals of assigned types; homo- 

 geneous cardinals; ascending cardinals; de- 

 scending cardinals; and cardinals of relational 

 types. Then follo^ws a section treating of ad- 

 dition, multiplication and exponentiation, 

 ■where the logical muse handles such themes as 

 the arithmetical sum of t'wo classes and of t'wo 

 cardinals; double similarity; the arithmetical 

 sum of a class of classes; the arithmetical 

 product of t'wo classes and of t'wo cardinals; 

 next, of a class of classes; multiplicative 

 classes and arithmetical classes; exponentia- 

 tion; greater and less. Thus no less than 186 

 large symbolically compacted pages deal 'with 

 properties common to finite and infinite classes 

 and to the corresponding numbers. Neverthe- 

 less finites and infinites do differ in many im- 

 portant respects, and as many as 116 pages 

 are required to present such differences under 

 such captions as arithmetical substitution and 

 uniform formal numbers; subtraction; induc- 

 tive cardinals; intervals; progressions; Aleph 

 null, Jij„; reflexive classes and cardinals; the 

 axiom of infinity; and typically indefinite in- 

 ductive cardinals. 



As indicating the fundamental character of 

 the " Principia " it is note'worthy that the 

 arithmetic of relations is not begun earlier 

 than page 301 of the second huge volume. In 

 this division the subject of thought is rela- 

 tions including relations between relations. 

 If iJ, and B, are two relations and if F^ and F, 

 are their respective fields (composed of the 

 things between which the relations subsist), it 

 may happen that F, and F, can be so correlated 

 that, if any two terms of F^ have the relation 

 B^, their correlates in 2^^ have the relation B^, 

 and vice versa. If such is the case, B^ and B, 

 are said to be lihe or to be ordinally similar. 

 Likeness of relations is analogous to similar- 

 ity of classes, and, as cardinal number of 

 classes is defined by means of class similarity, 

 so relation-number of relations is defined by 

 means of relation likeness. And 209 pagea 

 are devoted to the fundamentals of relation- 



arithmetic, the chief headings of the treat- 

 ment being ordinal similarity and relation- 

 numbers; internal transformation of a rela- 

 tion; ordinal similarity; definition and ele- 

 mentary properties of relation-numbers; the 

 relation-numbers, Or, 2^ and Ij ; relation-num- 

 bers of assigned types; homogeneous relation- 

 numbers ; addition of relations and the product 

 of two relations ; the sum of two relations ; ad- 

 dition of a term to a relation; the sum of the 

 relations of a field; relations of mutually ex- 

 clusive relations; double likeness; relations of 

 relations of couples; the product of two rela- 

 tions; the multiplication and exponentiation 

 of relations; and so on. 



The last 259 pages of the volume deal with 

 series. A large initial section is concerned 

 with such properties as are common to all 

 series whatsoever. From this exceedingly 

 high and tenuous atmosphere, the reader is 

 conducted to the level of sections, segments, 

 stretches and derivatives of series. The vol- 

 ume closes with 58 pages devoted to converg- 

 ence, and the limits of functions. 



To judge the " Principia," as some are wont 

 to do, as an attempt to furnish methods for 

 developing existing branches of mathematics, 

 is manifestly unfair; for it is no such attempt. 

 It is an attempt to show that the entire body 

 of mathematical doctrine is deducible from a 

 small number of assumed ideas and proposi- 

 tions. As such it is a most important contribu- 

 tion to the theory of the unity of mathematics 

 and of the compendence of knowledge in gen- 

 eral. As a work of constructive criticism it 

 has never been surpassed. To every one and 

 especially to philosophers and men of natural 

 science, it is an amazing revelation of how the 

 familiar terms with which they deal plunge 

 their roots far into the darkness beneath the 

 surface of common sense. It is a noble monu- 

 ment to the critical spirit of science and to the 

 idealism of our time. 



Of the making of many text-books of the cal- 

 culus there is no end. The phenomenon is 

 doubtless due to a variety of causes, literary, 

 economical, scientific and educational. Chief 

 among the causes is the felt desirability of 

 producing text-books of mathematics that will 



