330 SCIENCE PROGRESS 



(i) Fundamental Conceptions of Modern Mathematics. Variables and Quan- 

 tities, with a Discussion of the General Conception of Functional Relation. 

 By Robert P. Richardson and Edward H. Landis. [Pp. xxi + 216.] 

 (London and Chicago : The Open Court Publishing Company, 1916. Price 

 $1.25 net.) 

 (2) Numbers, Variables, and Mr. Russell's Philosophy. By Robert P. 

 Richardson and Edward H. Landis. [Pp. 59.] Reprinted from The 

 Monist of July 191 5 . (London and Chicago : The Open Court Publishing 

 Company, 191 5.) 

 On first reading these books, one is not at all favourably impressed by them. It 

 appears remarkable that nearly all the leading mathematicians should so often be 

 in the wrong, while Messrs. Richardson and Landis are always in the right — 

 especially when we observe several small possible flaws, such as split infinitives. 

 We do not know whether split infinitives have ever been absolutely proved to be 

 vicious, but they are a kind of vegetable which grows largely in this particular 

 garden. A greater fault appears to be that in the first book the whole matter is 

 not set out in well-contrived compartments, but the authors ramble on indefinitely 

 from the first page to the last. Indeed the first book is only the first part of the 

 large subject stated at the top of the title-page ; and at the end of the book the 

 authors give us brief accounts of no less than twelve other parts which they 

 project. As we said, it ought in our opinion to have been arranged in a more 

 formal manner. There is art as well as science in the writing even of books of 

 science. Perhaps the final test of such books is the test which Euclid has with- 

 stood for two thousand years. That poor discredited volume still remains with 

 us. By the universal consent of modern mathematicians the opening postulates, 

 definitions, and axioms are far from perfect : but nevertheless the work proceeds in 

 an orderly manner from proposition to proposition, and the result is that the 

 work has been done and finished once and for ever. We cannot say the same of 

 this book. On reading one page we are driven often to hunt up at random the 

 results of previous or future pages, and the final impression is therefore confusing. 

 Still worse, we are personally doubtful whether the authors have really seen to the 

 bottom of the bucket ; and in science it often happens that the most important 

 facts lie there. As Byron remarked, " When Bishop Berkeley said there was 

 no matter, and proved it — 'twas no matter what he said'' — because the Bishop 

 had entirely overlooked the fact that the mind possesses a kind of muscular sense 

 which enables it to distinguish between real and ideal cognitions. Similarly the 

 writer of this review thinks that the authors have failed to draw the very important 

 distinction between operations and the arguments to which they may be applied, 

 and he therefore believes that the whole of the work will ultimately have to be 

 rewritten. The omission would probably have never occurred if the authors had 

 adopted the wise exemplar of Euclid. 



Now let us haste to say that our first impression was by no means altogether 

 sound. The books are both very interesting, and the strictures on the errors of 

 the great mathematicians indicate only the enthusiasm which the authors have 

 brought to their studies ; and we gladly admit that their criticism lies well 

 within the concluding paragraph of their Preface. Indeed many people might 

 think that a rambling naturalists' excursion such as this will gather more specimens 

 than will a formal search ; and many of the authors' criticisms, distinctions, 

 expositions, and labellings appear to be of distinct value for future work in the 

 philosophy of mathematics— subject, however, to the necessity that the final book 

 upon this subject does reach nearer to the bottom of the bucket. For example, 



