February i, 19 12] 



NATURE 



443 



THEORY OF ALGEBRAIC NUMBERS. 

 Introduction d, la Thdorie des Nombres Algebriques. 



By Prof. J. Sommer. Edition Frangaise revue 



et augment^e. Traduit de TAllemand par Prof. A. 



L^vy. Pp. x + 376. (Paris: A. Hermann et Fils, 



191 1.) Price 15 francs. 

 nPHIS book is a French translation of the work on 

 -»- algebraic numbers by Dr. J. Sommer, professor 

 of the Technical High School of Danzig. It is an 

 elementary exposition designed to be an introduction 

 to the researches of modern German mathematicians, 

 particularly of David Hilbert, whose masterly resume 

 in " Jahrsbericht der deutschen Mathematiker 

 Vereinigung, 1895," is likely to be regarded as a 

 classic. Gauss was the first to enlarge the field of the 

 higher arithmetic by including therein numbers of the 

 form a + b»/ — i. This led to a beautiful theory in the 

 (Ii velopment of which he was followed by Dirichlet, 

 Kummer, Dedekind, and Kronecker, to mention only 

 a few of the most important and successful investiga- 

 tors. The results included extensions and generalisa- 

 tions of many theorems of the higher arithmetic, and 

 in particular of the theorem of quadratic reciprocity. 

 The modern theory of algebraic numbers involves a 

 further extension of the domain of numbers in that 

 every root of an irreducible algebraic equation with 

 rational coeflRcients is said to be an algebraic number. 

 For a given degree n of such an equation the corpus 

 (if such numbers comprises every rational function of 

 such roots. When n is 2 we have the quadratic 

 corpus which involves the irrationality a/w, in being 

 a given integer, not a perfect square, which defines 

 the corpus. This book treats the domain of these 

 numbers with some completeness (pp. 16-183). 



The difficult subject of "ideal numbers," an inven- 

 liiin of Kummer, is introduced lufcidly, and subse- 

 quently well handled. The reader is then brought to 

 the chapter entitled, "Applications of the Theory of 

 the Quadratic Corpus," and will there find the most 

 interesting part of the book. One of these applica- 

 tions is to the world-famous "last theorem" of Fer- 

 mat, which asserts the insolubility in integers of the 

 equation 



when n is an integer superior to 2. The proofs when 



n is 3, 5, II, or 14 resulted from the attacks of Euler, 



Legendre, and Dirichlet, while Kummer, utilising his 



ideal numbers, established the theorem for all prime 



' values of n less than 100. The theorem as asserted 



l-'ermat still awaits proof. The next application is 



i lie theory of quadratic forms in the higher arith- 



I ic of integers. The theory of the extended domain 



Iiown to involve remarkable extensions of theory in 



restricted domains which were initially handled with 



much success by Gauss. The remaining chapters 



nn the "Cubic Corpus," and the " Relative Corpus," 



latter a notion due to Hilbert somewhat analogous 



ilie ideal numbers of Kummer. 



I he field of pure mathematics into which Dr. 



uner's book gives an insight has been somewhat 



;l('-cted in France, and in England it is correct to 



that nothing of any moment has been accom- 



|ilished. German mathematicians have shown it to 



NO. 2205, VOL. 88] 



be a singularly attractive and fruitful if recondite 

 j subject of thought. Now that the field is brought 

 I closer to the view of English mathematicians it may 

 be that they will take more interest in it. 



The work, though on the whole well produced, is 

 marred by many misprints in the early pages. On 

 p. 41 there are six misprints, on p. 42 four, &c. It 

 is a pity that difficulties already considerable should 

 be thus increased. P. A. M. 



THE RESOURCES OF BRITISH WEST AFRICA. 



The Agricultttral and Forest Products of British 

 West Africa. By G. C. Dudgeon. Pp. x+170, 

 with five maps. (Imperial Institute Handbooks.) 

 (London : John Murray, 191 1.) Price 55. net. 



THIS book forms the introductory volume of a 

 series to be known as the " Imperial Institute 

 Series of Handbooks to the Commercial Resources 

 of the Tropics, with' Special Reference to British 

 West Africa." The handbooks will be edited by 

 Prof. W. R. Dunstan, F.R.S., and will be issued 

 under the authority of the Secretary of State for the 

 Colonies. 



Mr. Dudgeon was for several years Inspector of 

 Agriculture in British West Africa, and in that 

 capacity visited annually the five colonies and pro- 

 tectorates which together form that territory. He 

 writes therefore with full knowledge of the conditions 

 under which agriculture and forestry- are carried on 

 there. 



The five dependencies are dealt with in geographicaF 

 order, commencing with Gambia to the west, and 

 finishing with northern Nigeria to the east. By way 

 of introduction to the section on each country, the 

 salient facts regarding area, population, habits and 

 disposition of the natives, and the peculiarities of 

 the climate and soil are set out, followed by a brief 

 account of whatever local attempts are being made tO' 

 improve native methods. The improvement of native 

 agriculture is largely a question of better tillage and 



i the introduction of a proper rotation of crops, and 

 from this point of view Mr. Dudgeon's brief but 

 careful descriptions of the implements in use by the 

 natives are of value. The greater part of each sec- 

 tion is, however, occupied by descriptions of the 

 principal crops, more detailed reference to any par- 

 ticular crop being reserved for the section relating to 

 the dependency in which that crop is predominant; 

 thus, under Gambia there is a good account of ground- 

 nut cultivation, and under the Gold Coast there is. 

 an excellent summary of the present position of the 

 flourishing cocoa industry of that colony. These 

 matters will, however, receive detailed attention ia 

 subsequent books of the series, and the author rightly 

 confines himself in this volume to giving the reader 

 a general idea of the agricultural and forest resources 



I of each country and of the means whereby these may 



I be most surely and safely developed. 



I It is clear from this book that the two chief 



I obstacles encountered in the agricultural development 



i of British West Africa are the laclc of transport 

 facilities and the ignorance and conservatism of the 



, native population. The first of these obstacles is 



