April 21, 1899.] 



SCIENCE. 



587 



numbers. The arguments cannot have much 

 meaning to the student until the material of 

 Chapter I. has been grasped, and this seems out 

 of the question. 



Before leaving this section we call attention 

 to a curious break. On page 48 complex func- 

 tions of a real variable are differentiated and in- 

 tegrated. This certainly is illogical until such 

 operations have been defined. We are tempted 

 to believe that the beauties of this chapter vifill 

 fall very ilat with the average student. If the 

 geometrical theory of the logarithm is to appeal 

 to him, what is stated here so rapidly should 

 be given with leisure and detail. 



The next 60 pages, Chapters VIII. -XII., 

 deal with infinite series, and so lead us to 

 Weierstrass's conception of analytic functions. 

 This, as is known, depends on infinite series 

 ascending according to integral powers of 

 {x — a). The treatment here is very superior — 

 the authors show a masterly grasp of the sub- 

 ject. A short chapter on the analytic theory of 

 the exponential and logarithmic function now 

 follows. 



Chapters XIV. and XV., pp. 178-209, turn 

 again to the general theory. Singular points 

 are discussed, and Weierstrass's decomposition 

 of a function into prime factors is deduced. 

 Application is made to show that 



sin !!•.•);= 7r 0:11(1 — x'^jn''). » = ], 2, •■■, oo . 



The consideration of the zeros gives at once 



sin n-i^ ,Te'''-''n(l — x^jn'). 



The determination of the integral transcen- 

 dental function G is singularly difficult. It seems 

 a pity that the method invented for Cauchy for 

 the same purpose and which may easily be 

 made rigorous is to-day quite neglected. By 

 this method G is readily found. 



With Chapter XVI., which treats of integra- 

 tion, we arrive at the starting point of the 

 Cauchy-Bieman theory. It seems to us that our 

 authors have not maintained the high ideals 

 here as well as elsewhere. In a passage, pp. 

 11, 12, we read: "But in using geometric 

 intuitions * * * we must emphasize one 

 lesson of experience ; that the intuitional 

 method is not in itself sufficient for the super- 

 structure. It has been found that only by the 

 notion of number * * * can fundamental prob- 



lems be solved. If, however, we are prepared 

 to replace when occasion arises these geometric 

 intuitions * * * then and only then is the use 

 of geometry thoroughly available." It is true 

 that the authors here speak of points, distances 

 and angle only, but these remarks apply with 

 equal cogency, as they will be the first to admit, 

 to all geometric intuitions when used in analy- 

 sis. We are, therefore, surprised to find the ob- 

 scure notion of curve, of its length, of a closed 

 curve, of a region, etc., freely used without 

 any attempt to put them on a number basis. 

 Such statements as that on p. 189, viz : that a 

 circuit divides the entire plane into two regions 

 will certainly embarrass the authors to prove in 

 its generality. Again, on p. 213, we see the 

 authors implicitly define the length of a curve C 

 to be C\dx\. This definition differs from the 

 one given our text-books, viz.: J dxVl + /'{x)'. 

 As our authors propose to use a broader defini- 

 tion than usual, it seems only fair that they state 

 this to the reader. Still a more serious objec- 

 tion is to be urged to their procedure. It re- 

 sults in stating Cauchy's fundamental theorem 

 and other important theorems of this chapter 

 without any restriction regarding the path of 

 integration. This seems to us like talking of 

 infinite series without bothering ourselves about 

 convergence. 



Chapter XVII. brings a brief discussion of 

 Laurent's and Fourier's series. Then follow 

 two excellent chapters on the elliptic functions. 

 These are followed by two chapters or about 30 

 pages devoted to Algebraic functions and Rie- 

 mann surfaces. 



It appears to us that the fictitious number 

 and point oo has been treated too hurriedly. 

 These notions are very important and also 

 diflBcult for the student to master. Our authors 

 have followed the usual custom of disposing of 

 them with a few words here and there. We 

 believe the custom of introducing the number 

 00 is bad. The theory of functions of a com- 

 plex variable is a theory of two very special 

 real functions of two real variables. In the 

 theory of functions of real variables the num- 

 ber 00 does not exist. It seems to us that its 

 introduction can only produce confusion and 

 embarrassment. 



