436 



NA TURE 



[April 8, 1922 



somewhat illusory, simplicity; and it is fascinating 

 to investigators, to teachers, and to students alike. 

 It is unlikely that the real theory, more abstract and 

 in many ways more difficult, will ever be so generally 

 attractive. Still, times have changed, very largely 

 through the influence of Prof. Hobson himself. The 

 theory is studied seriously even in England, and 

 ignorance of fundamentals is no longer regarded as 

 proof of physical insight or geometrical intuition. 

 Prof. Hobson has every right to be satisfied with his 

 share in this salutary revolution. 



It must be admitted that there was some excuse 

 for the conservative mathematician of twenty years 

 ago, and his sneers at a theory which he was too lazy 

 to try to understand. The older theory, the theory 

 of 1900, was not only abstract and difficult, but in 

 some ways really unattractive. There was too little 

 simple and positive doctrine, too many intricate and 

 irritating exceptions. Little could be proved, and 

 the theorems which it was possible to prove were 

 difficult to state in a terse and striking form. The 

 theory of content in particular was obviously imperfect. 

 The theory as a whole seemed dried up and infertile ; 

 it is easy to see now how grievously it stood in need of 

 some refreshing storm. 



All this has been changed by the rejuvenating 

 influence of the ideas of Borel and Lebesgue. The 

 storm has broken, and the ground has become fresh 

 and fertile once more. There is, indeed, no other 

 region of pure mathematics that has experienced so 

 drastic a revolution. Prof. Hobson's book is the only 

 English book which contains a systematic statement 

 of the revolutionary doctrine, and it is this, above all 

 else, that gives it its unique position. 



The importance of the new theories of measure 

 and integration is generally admitted, but their effect 

 on the theory of functions is still very widely mis- 

 understood. They are much more general than the 

 older theories, and it is supposed that, being more 

 general, they must be much more complicated and 

 more difficult to understand. The result is that many 

 mathematicians are too frightened to make any 

 serious attempt to comprehend them. This attitude 

 of panic is based on a complete misapprehension. It 

 is not true that the new theories are much more difficult 

 than the old. It is by no means always the most 

 general and the most abstract that is the most difficult 

 to understand. The trouble with the older theory 

 lay not so much in the inherent difficulty of the subject- 

 matter as in the complexity and clumsiness of the 

 results. The modern theory, in acquiring generality, 

 has acquired symmetry, terseness, and to a great 

 extent simplicity as well. It possesses the aesthetic 

 qualities that are characteristic of a first-rate mathe- 

 NO. 2736, VOL. 109] 



matical science. Its theorems can be stated in a 

 concise and arresting form, and make that appeal to 

 the imagination which enables them to be mastered 

 and remembered. It is much easi'er to be a master 

 of the new theories than it was to be a master of the 

 old, and it is also much more necessary. A young 

 mathematician who elects to remain in ignorance of 

 them is certain to regret his laziness or obstinacy in 

 years when it is more difficult to learn. 



It is, then. Prof. Hobson's chapters on measure 

 (chap. 3) and integration (chaps. 6-8) that are un- 

 questionably the most important in the book. His 

 treatment is much more comprehensive and encyclo- 

 paedic than that of any other writer. He has three 

 serious rivals, de la Vallee Poussin, Caratheodory, 

 and Hahn. Hahn may be disregarded for the present, 

 as the second volume of his " Theorie der reellen 

 Funktionen," in which the theory of integration is 

 to be developed, has not yet appeared. The works 

 of de la Vallee Poussin (" Cours d'analyse infinitesi- 

 male," second and third editions, 1909, 1912, 1914 ; 

 " Integrales de Lebesgue, fonctions d'ensembles, classes 

 de Baire," 1916) continue to provide the best intro- 

 duction to the theory. Between Caratheodory and 

 Prof. Hobson it is unnecessary to discriminate, for 

 both are essential for the systematic study of the 

 subject. It is sufficient to say that there is a great 

 deal in this volume which Caratheodory does not 

 touch. 



Chaps. I, on number, and 4, on transfinite numbers 

 and order-types (chap. 3 of the first edition) have 

 not been greatly changed. We must confess that it 

 has always been this part of the book that we like the 

 least. Prof. Hobson often allows himself to use 

 language which suggests the Oxford philosopher rather 

 than the Cambridge mathematician. " The mind " 

 maintains its position in the first sentence of chap, i ; 

 " objects for thought " are " postulated " on p. 29 ; 

 a " fundamental difference of view on a matter of 

 Ontology " is mentioned on p. 249. We have an 

 uneasy feeling th£»,t if one scratched the mathematician 

 one might find the ideaHst, and that all these dis- 

 cussions, and especially those which concern the 

 " principle of Zermelo," ought to be stated in a sharper 

 and clearer form. 



Chaps. 2 and 3 are concerned with sets of points, 

 the theory of content and measure having very wisely 

 been separated from the descriptive theory. The 

 greatest difficulty is to distinguish the theorems for 

 which Zermelo's axiom is required. We could make 

 some criticisms of detail — we found difficulty, for 

 example, in disentangling the proof that the measure 

 of a measurable set satisfies the postulate (3) of p. 

 159, tied up as it is with the corresponding proof for 



