NA rURE 



Ml 



THURSDAY, MARCH 22, 1894. 



THEORY OF FUNCTIONS. 

 A Treatise on the Theory of Functions. By James 

 Harkness, M.A., Associate Professor of Mathematics, 

 Brj'n Mawr College, Pa., and Frank Morley, M. A., Pro- 

 fessor of Pure Mathematics in Haverford College, Pa. 

 (London and New York : Macmillan and Co., 1893.) 



IF evidence were wanted of the recent progress of the 

 study of pure mathematics on English and American 

 soil, none better could be furnished than the appearance 

 on the two sides of the Atlantic, within a short interval, 

 of two important works on the theory of functions of a 

 complex variable. But a few years ago this great 

 modern branch of mathematics was so little known to 

 English-speaking mathematicians that scarcely a trace 

 of its influence could be traced in their writings, and the 

 majority of our text-books were disfigured by incomplete- 

 ness, and not seldom by positive error arising from ignor- 

 ance of its principles. Now the English reader has at his 

 disposal two extensive works dealing with the funda- 

 mental principles of the theory from all the more 

 important points of view ; and also a very useful aid in 

 Cathcart's valuable translation of Harnack's "Elements 

 of the Differential and Integral Calculus." Probably 

 nothing could serve better as an exorcist of the spirit of 

 formalism which has oppressed the English school of 

 mathematicians so heavily, in spite of all the great 

 things that its leaders have done for the science, than the 

 study of the theory of functions. In no other mathe- 

 matical discipline is the fundamental unity of logic kept 

 so constantly before the student ; nowhere else in mathe- 

 matics is it so clearly made evident that the manifold 

 array of symbols is the clothing, and not the soul 

 of mathematical thought ; and nowhere else can we 

 perceive so fully that progress is to be looked for 

 mainly in strengthening our hold upon elementary con- 

 ceptions, in continual refinement of definition and con- 

 tinual increase of stringency in inference, together with 

 the necessary complement of this, viz. a continual 

 widening of our power of imagining logical possibilities.^ 

 A single illustration of these general remarks may be 

 cited here, viz. the important part now played in mathe- 

 matics by the classification of the possible singularities 

 of a function. Although as yet this classification has 

 hardly proceeded beyond the first stage of distinguish- 

 ing between what Weierstrass has called essential and 

 non-essential singularities, yet the exceeding fruitfulness 

 of the idea is very manifest in every part, not only of the 

 theory itself, but of its applications. In this connection 

 we may remark that anyone who is sceptical as to the 

 value of function-theory, should compare the treatment 

 of the theory of elliptic functions as given in chapter 

 vii. of the treatise now before us, with the older method 

 of dealing with the same subject. He will there find the 

 theorems which used to be for many of us a mere 

 savagery of riotous mathematical formulae, sitting now 



1 It is in this particular that the peculiar originality of Cauchy, Riemann, 

 and Weierstrass, the three great leaders in the theory of functions, has been 

 so conspicuous. 



NO. 1273, VOL. 49] 



clothed in their right minds — the cultured dependents 

 of a few leading ideas. 



Our first impulse, after dipping here and there into 

 the work of Messrs. Harkness and Morley, and recog- 

 nising its substantial character, was to regret that so 

 much learning and ability had been wasted in a field 

 already covered by the admirable treatise of Forsyth. A 

 more careful reading convinced us that this feeling was 

 a mistake. The subject is wide enough to allow of two 

 independent treatises ; and the two works are indepen- 

 dent so far as two mathematical works, each partly 

 historical, dealing with the same subject, can be. Like 

 Forsyth, Harkness and Morley are full of valuable refer- 

 ences, not only to the great writers and the great memoirs 

 on the subject, but also to the minor writers and to 

 memoirs dealing with points of detail. So much is this 

 the case, that we doubt whether in the matter of history 

 and references the continental student has anything to 

 equal, and certainly he has nothing to surpass, what the 

 English student now possesses in Forsyth, combined 

 with Harkness and Morley. 



The more recent work does not, it is true, rival Forsyth 

 in style and width of view. It is constructed more 

 nearly on the model of a continental treatise, not reach- 

 ing the airy elegance of a French work, but happily 

 avoiding the intolerable prolixity and dulness of too 

 many continental books, where a parade of generality 

 not unfrequently engenders obscurity, or covers a poverty 

 of fruitful ideas. It is inseparable from the nature of the 

 subject that the unskilled reader should at times find 

 passages that seem obscure. In such cases he will find 

 it of great advantage to turn from Forsyth to Harkness 

 and Morley, or from Harkness and Morley to Forsyth. 

 The greater detail in some of the demonstrations in 

 certain parts of the subject which characterises the 

 treatise before us will often be a help to the reader who 

 has run aground in Forsyth. A mere remark which 

 constitutes a full demonstration to a mind properly pre- 

 pared or naturally sufficiently nimble to receive it, often 

 proves an enigma to another mind not so well "dis- 

 posed,' or, what is worse, is taken after the manner of 

 the patient who, instead of taking his doctor's medicine, 

 swallowed the prescription. If we might advise the 

 beginner, we should say, first read Forsyth rapidly, 

 possibly superficially with judicious omission, in order to 

 get a good idea of the nature and aims of the theory ; 

 then proceed to work carefully through Harkness and 

 Morley ; and, finally, again read Forsyth carefully ; so 

 that the last impressions should be of the " poetry of 

 the subject." 



Chapter i. of Harkness and Morley's work is a very 

 elegant and valuable geometric introduction to the sub- 

 ject, containing, besides the usual matter, a number of 

 excellent graphical illustrations of the theory of invariants 

 by means of Argand's diagram. Chapter ii. gives an 

 account of the more recent refinements in the theory of 

 functions of a real variable, in so far as such are neces- 

 sary for the purpose in hand. In chapter iii. the theory 

 of infinite series is dealt with in sufficient detail, and the 

 reader is thus rapidly introduced to Weierstrass's theory 

 of the analytic function, its continuation, its singular 

 points and lacunary spaces. Chapter iv. deals specially 

 with the algebraic function, its zeros, poles, and branch 



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