January 2, 1914] 



SCIENCE 



23 



In the matter of references the author has 

 been particularly unfortunate. In a book barely 

 entering upon the threshold of the theory, a 

 scarcity of references would have been entirely 

 justifiable. But to give hundreds of references to 

 a certain report on the subject (excellent although 

 it be) and to completely ignore the literature and 

 not even mention the names of the discoverers of 

 the theorems is against all scientific traditions. 

 What Professor Skinner says I said is : 

 Again, the reviewer, deploring the omission of 

 references, says: But to give [as above]. 



My second sentence above (not quoted by 

 Professor Skinner) shows that I did not de- 

 plore the omission of references. This sen- 

 tence together with the one actually quoted by 

 him show that what I deplored was misplaced 

 references. As should be well known, Kummer 

 created a highly complex theory of ideal num- 

 bers for the case of fields built upon roots of 

 unity; then Dedekind created a simpler theory 

 of ideals in complete generality and developed 

 the subject at great length; then Hurwitz 

 made a simplification which yields a brief and 

 attractive exposition of the theory; also 

 Dirichlet, Kronecker, Hilbert, Minkowski, 

 Hensel and others have contributed to the 

 development of the subject in various direc- 

 tions. A very large proportion of the theorems 

 stated on pages 218-451, the part dealing with 

 quadratic fields other than Gauss's important 

 case, should have been attributed to Dedekind, 

 provided a reference was to be given. But in 

 these 234 pages, I find only four references to 

 Dedekind, once to an alternative proof, once 

 to a symbol, once to a simple lemma, and 

 finally to a wholly subsidiary theorem. There 

 are two references to Minkowski and one to 

 each Woronoj (on cubic number fields), Hur- 

 "witz, Sommer, H. J. S. Smith and to Chystal's 

 algebra. The references to the main theorems 

 are to that excellent report by Hilbert, re- 

 cently translated into French. As against the 

 four wholly minor references to the originator 

 of the general theory, Dedekind, there are 45 

 references to Hilbert's report (Professor 

 Skinner's count of 38 for the entire book is 

 misleading as he neglected references given in 

 the body of the text). With a single excep- 

 tion, these 5 references are to passages in Hil- 



bert's report in which Hilbert expressly attri- 

 buted the results to other writers; had the 

 author reached the higher parts of the theory, 

 he would have needed many references to Hil- 

 bert's own important contributions. On my 

 own part the impression that there were hun- 

 dreds of such references was wrong; but that 

 exaggeration is really beside the mark. The 

 references are largely misplaced and that is 

 evidently all I was emphasizing in the passage 

 quoted above. I do not begin preparation for 

 writing a book review by counting references, 

 and I do not care a straw whether or not Pro- 

 fessor Skinner's count of 158 as the total num- 

 ber of footnote references is correct; in any 

 event only about 44 of these relate to the part 

 under discussion. In the above extract from 

 my review I expressly limited myself to the 

 subject of the report and hence to algebraic 

 numbers; consequently it is not a fair com- 

 ment on that extract to speak of the large 

 number and nature of the references in the 

 introductory part on rational numbers. In all 

 probability these references would have been 

 like those discussed above had the report 

 treated also rational numbers. 



4. Professor Skinner states that my review 

 was freely interspersed with exclamation. 

 points. As a matter of actual fact only two 

 exclamation points appear in my two-column 

 review. One is in 



The author desires to bring out a closer relation 

 between rational numbers and quadratic numbers. 

 This he accomplishes by complicating the elements 

 of rational numbers with 'the unnecessary ma- 

 chinery of quadratic numbers! We find on page 

 91 Wilson's theorem stated in the form 



V^--- ric + 1 ss (mod p), k^4>{p), 



where J"!, •••,rfc form a complete set of residues 

 modulo p, a prime. 



According to the Index, this is the first 

 statement in the text of Wilson's theorem, 

 which has been known since 17Y0 under the 

 familiar form that 1 • 2 ■ 3 ■ • • (p — 1) + 1 is 

 divisible by the prime p. After the compli- 

 cated theorem is stated, proved, generalized 

 and illustrated by several examples, the usual 

 form is finally given. The second exclamar 

 tion point was used in discussing a three- 



