NATURE 



657 



THURSDAY, OCTOBER 31, 



TUE THEORY OF FUXCTIOXS OF A REAL 



VARIABLE. 



The Theory of Functions of a Real Variable and 



the Theory of Fourier's Series. By Dr. E. W. 



Hobson, F.R.S. Pp. xvi + 772. (Cambridge: Uni- 

 versity Press, 1907.) Price 21s. net. 

 IT is impossible to read Dr. Hobson 's book without 

 reflecting on the marvellous change that has 

 come over Cambridge mathematics in the last twenty 

 vears. Twenty years ago Cambridge mathematics 

 was a thing standing by itself, and with its own 

 virtues and defects. Pure mathematics in Cambridge 

 meant Cayley and a few disciples ; and Cayley (widely 

 as he read) owed little or nothing to anyone but him- 

 self. Certainly he never appreciated the most funda- 

 mental ideas of modern Continental analysis. It is 

 probable that he could not have defined a function or 

 a limit in a way which would have satisfied Weier- 

 strass or Dr. Hobson : it is certain that he would have 

 been as incapable as any Senior Wrangler of proving 

 anv of the less obvious theorems of convergence. The 

 first signs of the absorption of these ideas are to be 

 found, not in Cayley, but in Stokes. 



Now Cambridge has fallen into line. There are no 

 Cayleys, perhaps, but there is quite a flourishing 

 school of pure mathematics, working by what may be 

 called German methods and on German lines, and 

 making up in numbers and soundness for anything 

 that it has lost in distinction. The school of Cayley 

 is dead, and so (what is perhaps even more to be 

 regretted) is the old Cambridge school of applied 

 mathematics : pure mathematics and experiment have 

 combined to kill it, and the Stokes Lecturer in Applied 

 Mathematics writes books like this. We wonder what 

 Clerk Maxwell or even Stokes himself would have 

 thought of it. 



However, all this is not Dr. Hobson 's fault, and 

 we must not blame him if the reflections which it 

 inspires are not altogether pleasant. And we hasten 

 to congratulate him on the completion of what is, 

 without a doubt, a magnificent piece of work. It 

 would be a fine piece of work even if were a mere 

 compilation ; for the subject is one of which there was 

 no systematic account in English, and which no pre- 

 vious English writer had ever reallv mastered. But 

 the book is far from being a compilation, for Dr. 

 Hobson has made the subject his own, and writes 

 with the air of mastery that onlv orig^inal work can 

 giv-e : and even in French, German, or Italian, 

 there is no book which covers anything like the same 

 ground. Dini (whom Dr. Hobson has obviously taken 

 as his model) has held the fifld for a long time, and 

 Dr. Hobson can fairly claim to have superseded him. 



In taking Dini as his model. Dr. Hobson has made 

 the "theoretically general," rather than what Borel 

 has called the "practically general," his goal. No 

 doubt he had to make his choice, but we must confess 

 that he seems to us to have gone too far. Let us 

 consider his treatment of " double limit problems," 

 for example, problems such as those of the differentia- 

 N'-'. 1983, VOL. 76] 



tion or integration of an infinite series or an infinite 

 integral (why will he persist in making the uninitiated 

 scoff by his fondness for the word "improper"?). 

 .Such problems may be approached from two difl'erent 

 points of view. We may ask, " What is absolutely 

 the most general form in which we can state our 

 theorems, when we utilise all the most modern theories 

 of sets of points, Lebesgue integrals, and the like? " 

 This is the point of view of Dini and Dr. Hobson. On 

 the other hand, we may ask, " In what special forms 

 do these problems naturally occur in analysis? What 

 are the i>eally important cases? Can we state our 

 theorems in such a way that wTiters on applied mathe- 

 matics, or other branches of pure mathematics, when 

 they are confronted, as they continually are. with 

 particular problems of this kind in all conscience 

 difficult enough, will be able to turn to us for a 

 solution of their difficulties? " These questions must 

 be continually before us, if we are aiming at Borel's 

 " practically general " completeness, and even an 

 author who has decided to aim at the other ideal will 

 do well to keep them clearly in sight; and wc wish 

 that Dr. Hobson had more often adopted this point 

 of view. He might then have made his book a good 

 deal more useful and attractive for the ordinary 

 worker in the fields of analysis. The latter, as it is, 

 is likely to find himself faced by many theoretical 

 difficulties to which he will not easily find an answer 

 in Dr. Ilobson's pages. However, it is perhaps as 

 well that Dr. Hobson should leave something for 

 someone else to do. 



But it is time that we said a little of the details 

 of the book. It is needless to say that it is beautifully 

 and almost faultlessly printed. It is a pity, though, 

 that the chapters are so long. Long chapters do not 

 make a difficult book easier to read, nor do they make 

 it easier for the author to arrive at the proper logical 

 arrangement of the subject-matter — as appears very 

 clearly in chapters v. and vi., which had much better 

 have been broken up into half a dozen shorter 

 chapters. We should like to have seen a great 

 many more examples. Summaries of the chapters, too, 

 would have been useful; and the author is too sober 

 in his use of different kinds of type. In a word, he 

 shows too great a contempt for the arts of popularity. 



There are seven chapters in all. For the first three, 

 which are of a particularly abstract character, we have 

 practically nothing but praise. The matter is admir- 

 ably selected and admirably arranged, and Dr. 

 Hobson writes with a lucidity and distinction rare 

 indeed among mathematicians. Nothing could be 

 better in its way, for example, than his terse criticism 

 of the " formal " view of mathematics (pp. 9-10). We 

 cannot entirely agree with the conclusions at which 

 he arrives in the course of the critical discussions of 

 chapter iii, but we can appreciate the clear and 

 temperate manner of his criticisms, advanced, as he 

 says, " with some diffidence, on account of the great 

 logical difficulties of the subject," and in the hope 

 that " they mav be of utility as a contribution towards 

 the discussion of questions of great interest which, at 

 the present time, cannot be regarded as having been 

 decisively settled." 



In chapter iv. we begin for the first time to be 



E E 



