6=;8 



NA TURE 



[October 31. 1907 



bored in places. The four derivatives of a function 

 lire dull, and no one will over make them seem any- 

 I'lintj else; and a jj^ood deal of Broden's work is much 

 more solid than inspiring,'-. Occasionally we do not 

 quite like Dr. Hobson's choice of words — in parti- 

 cular we may instance his use of " indefinitely great," 

 in such phrases as "has indcfinitelv great values," 

 " the functional value is regarded as indefinitely 

 yrcat," " the lov.-er limit is indefinitely great." Why 

 not, in the last case, simply " there is no lower 

 limit "? Dr. Hobson could reply that he has ex- 

 pressly warned the reader against any such confusion 

 of thought as is sometimes implied in modes of ex- 

 pression such as these; and there is certainly none 

 in his own mind. None the less we wish that he 

 had expressed himself in a different manner. 



In this chapter, let us single out for special praise 

 the sections on double and repeated limits (pp. 303 

 (•/ scq.). We particularly like the author's generalisa- 

 tion of the definition of a repeated limit, which enables 

 him to simplify the statements of a number of 

 theorems. We have already said that we do not alto- 

 gether like the. arrangement of the next two chapters. 

 .Surely it would have been better to introduce the 

 notion of a series at an earlier stage. As it is, some 

 of the theorems concerning integrals are separated 

 from one another in a rather irritating way. But 

 most of the discussions of particular theorems are 

 adniirnble. We may mention especially the treat- 

 ment of the " absolutely convergent improper 

 integral " (pp. 364 et seq.). the sections on the trans- 

 formation of double integrals (pp. 445 et scq.), and the 

 account of Baire's theory of the representation of 

 functions (pp. 522 et seq.). A few criticisms of details 

 suggest themselves. Is it worth while to define 

 " principal values " if nothing more is to be said 

 about them? There is a curious slip on p. 454, 1. 14; 

 obviously the condition stated is not necessary': 

 and it is very odd that Dr. Hobson should define 

 divergence and oscillation in such a way that 

 '~/+3 4+ ■ ■ . is a divergent rather than an oscil- 

 lating series. The last word has not yet been said 

 about Weierstrass's non-difTerentiable function (pp. 

 620 et seq.). What about ^a" cos i''.x, where ab is 

 only a little greater than i ? One would e.xpect the 

 function to have no differential coefficient whenever 

 rt/'i I ; but no one seems to have found out whether 

 this is the case or not. 



Finally, chapter vii. (Fourier's .Series) shows Dr. 

 Hobson quite at his best. The last part, in which he 

 supplies a final touch of rigour to some of Riemann's 

 work, is extremely diflScult, but that was inevitable. 

 The remark at the foot of p. 647 is open to dispute. 

 Was not something very much like the theorem, 

 ascribed to Lerch on p. 727, also proved bv Stokes? 

 On p. 732, 1. 24, for "diminished " read " increased." 



.\ sliort appendix contains some further critical 

 remarks, in addition to chapter iii. Wc wish that 

 there had been space for a summarv of Konig's rather 

 watery theories, and the author's neat and convincing 

 reply in the London Mathematical Society's Pro- 

 ceedings. We must confess to a strong temptation to 

 argue with Dr. Hobson concerning the remarks at the 

 top of p. 765, but the temptation must be resisted. 

 NO. 1983, VOL. 76] 



Dr. Hobson has attempted an appalling task. There 

 is no region of pure mathematics (unless it be the 

 theory of numbers) which is quite so difficult as this ; 

 certainly none of which the literature is so scattered 

 and so difficult to collate, or in which the writing of a 

 big book requires a greater combination of drudgery 

 and critical insight. .Ml things considered, he has 

 succeeded wonderfully. We can think of no one else 

 who would have done half as well. G. H. H. 



LIEBIG .\ND GCSSEFELD. 

 Justus von Liebig und Emil Louis Ferdinand Giisse- 

 jeld. Briefwechsel : 1S62-1866. Herausgegeben von 

 Dr. O. E. Giissefeld. Pp. viii4-72. (Leipzig: 

 Johann Ambrosius Barth, 1907.) Price 3 marks. 



THIS little book has a twofold interest. To the 

 scientific agriculturist it is interesting as 

 elucidating the history of the introduction of the 

 modern methods of agriculture into Germany, and 

 especially of the introduction of the so-called chemical 

 fertilisers, due largely to the teaching and influence of 

 Liebig ; it serves also to throw some sidelights upon 

 the character and habits of Liebig himself, and is 

 therefore of interest to the historian of chemistry. 

 It consists simply of a collection of thirty-eight letters 

 which passed between Liebig and Emil Giissefeld 

 from 1862 to 1866, twenty-two of which are contri- 

 buted by Liebig, and the whole has been arranged for 

 publication, with explanatory notes and annotations, 

 by the pious care of the son of one of tlie correspon- 

 dents. 



Emil Giissefeld was a Hamburg merchant, of the 

 conventional type, dealing mainly in coffee and other 

 colonial products. In a fortunate hour he accepted 

 an agency from an American company for the sale in 

 Germany of guano from Baker Island, in the Pacific 

 Ocean, and thereby laid the foundations of a pros- 

 perous business in phosphatic manures. Emil Giisse- 

 feld indeed stands to Germany in much the same rela- 

 tion that the late Sir John Bennett Lawes stands to this 

 country, and both reaped fame and fortune by the far- 

 sighted enterprise which induced tliem to give prac- 

 tical effect to the theoretical views of Liebig. 

 As a prudent man, Giissefeld, before undertaking the 

 agencv, seems to have consulted Liebig as to the prob- 

 ability that the Baker guano, of the merits of which he 

 was well assured, would find a ready sale among a body 

 of agriculturists who are even more conservative than 

 our own, and Liebig's reply constitutes the first letter 

 in the series. It is in every respect worthy of him — • 

 sound, thoughtful, and considerate, and with that 

 note of cautious optimism which the eminently prac- 

 tical mind of the Hamburg merchant could not fail 

 to appreciate. Liebig, as this correspondence abund- 

 antly testifies, never spared himself when his interest 

 was aroused, and he was ever ready to give of his 

 best, without fee or thought of reward, when the 

 object commended itself to him. In this large-hearted 

 liberality Liebig resembled Davy, who nearly half a 

 century previously had striven in the same self-sacri- 

 ficing way to infuse something of the scientific spirit 

 into the oldest of the arts. Liebig's letters are rich in 

 practical advice, business hints, analytical information 



