February 23, 1899] 



NA TURE 



387 



to serve as an introduction to a subject of vast extent. 

 It will be obvious from the preceding^ account of its con- 

 tents that the space allotted in this book to algebraic 

 functions is comparatively small. In the present writer's 

 opinion it might with advantage have been considerably 

 increased. Again, it seems a pity that the well-established 

 use of a closed conve.K surface — a sphere, for instance — 

 as a locus in quo for the geometrical representation of a 

 comple.x variable has been omitted. The possibility of 

 its use is indeed implied in one passage (p. 43), but the 

 sphere is not actually used for the purpose of geometrical 

 representation at all. The apparently exceptional nature 

 of the value x = m is undoubtedly at first a stumbling- 

 block to the student, and the use of the sphere as an 

 alternative to the plane would have been a help to him 

 in this respect as well as in others. The e.\cellent and 

 detailed discussion of infinite series should certainly have 

 been supplemented in the proper place by some cor- 

 responding discussion of infinite products. This point 

 is referred to again below. 



In the main, the authors have carried out the pro- 

 gramme they have put before themselves well and 

 thoroughly ; their reasoning is in general rigorous and 

 clearly expressed. Here and there however throughout 

 the book there are signs of what appears to be undue 

 haste in putting the matter together. Sentences not un- 

 frequently occur which it is necessary to read more than 

 once before their meaning is grasped ; and sometimes, in 

 passing from a sentence to the next, one experiences too 

 great a sensation of transition. Moreover, haste appears 

 occasionally to have led to inaccuracy. Two or tliree 

 examples of this may be given. 



The first chapter is intended to give the reader "a 

 distinct image of a number divorced from measurement." 

 On p. 3 occurs the sentence : " We can think of an 

 infinity of objects as interpolated in the natural row, so 

 that each shall bear a distinct rational number, and so 

 that we can assert which of any two comes first." What 

 is meant here by "an infinity of objects" ? No test has 

 been given in the sentences which precede the one 

 quoted by which a finite assemblage of objects can be 

 distinguished from an infinite assemblage ; and without 

 such a test the sentence quoted appears to beg the whole 

 question discussed in the first chapter. 



As a second instance, the opening sentences of Chapter 

 XV. may be quoted. 



"Let fij, a.,, ..., n„, ... be a sequence of positive 

 numbers, less than unity. Then 



(I - a,) (I - o.) > I - Bi - a„ 

 (I - Oi) (I - a.,) (I - Oj) > I - Oi - a.. - 03, 

 and so on. 



" Hence if the series ■ 2a„ has a sum j-, the products 



n(i - «„) form a sequence of numbers which (i) do not 



increase, (2) remain greater than \-s. Hence they have 

 a limit; and the infinite operation n(i -a„) is convergent ; 

 the limit is called the product, and is itself often denoted 

 by (ni -a„)." 



This is the first place in which an infinite product has 



occurred in the book, and what is implied in calling 



such a product convergent has not been explained. The 



statement that "the infinite operation n(i-an) is con- 



NO. 1530, VOL. 59] 



vergent " is therefore meaningless as it stands. More- 

 over, with the usual definition of convergence for an 

 infinite product, the proof as given is inaccurate. For 

 if 2a„ is greater than unity, all that has been proved is 

 that n(i-a„)is less than unity and greater than some 

 definite negative quantity. 



In an illustrative example on p. 232 the 'following 

 passage occurs : 



" By subtraction we have for \.x\=\, x=~\ e.x- 

 cepted. 



Log. 



=(■'-? 



1234 



The rearrangement involved in passing from the second 

 to the third line of this quotation is one which cannot be 

 used with conditionally convergent series, as indeed the 

 authors have most clearly shown in an earlier chapter. 



It is not implied that a few inaccuracies such as the 

 above really impair the value of the book. The authors 

 have certainly made a most useful addition to the 

 gradually increasing number of English text-books of 

 modern type ; and all teachers who have to introduce 

 their pupils to the elements of function-theory will be 

 grateful to them. 



One further remark in conclusion. The reader of a 

 mathematical text-book does not in general expect 

 amusement as well as instruction ; but surely, in such a 

 work as that under notice, the definition of Log .i- by 

 means of a piece of string and a cone which "should not 

 be polished" (p. 47), has its humorous side. 



W. BURNSIDE. 



THE ''IMPROVEMENT" OF FRUITS. 

 Sketch of the Evolution of our Native Fruits. By L. 

 H. Bailey. Pp. xiii -(- 472 ; illustrated. (New York : 

 The Macmillan Company. London: Macmillan and 

 Co., Ltd.). 



THE main purpose of this book is to give illustrations 

 of the progress made in the development of the 

 edible fruits of North .America from their wild progenitors. 

 This is what our fathers would have said ; nowadays we 

 express the same meaning in different words, and, as 

 Prof Bailey writes, we "attempt to expound the progress 

 of evolution in objects which are familiar, and which 

 have not yet been greatly modified by man." The United 

 States offer an exceptionally good field for investigations 

 of this kind. The wild plants are still there, relatively 

 speaking unmodified by man. Cultivation and ex- 

 periment are of recent date as compared with the long 

 ages that have elapsed since " Noah began to be an 

 husbandman" and prehistoric lake-dwellers dropped the 

 seeds of the grape into the mud of Swiss lakes. Through- 

 out Europe and Asia there is but one cultivated species 

 of Vitis recognised, the Vitis vinifera, and from it have 

 sprung the countless host of named varieties which are 

 cultivated in the vineyards, and the smaller, though still 

 considerable, numbers [that are grown in this country 



