NATURE 



289 



THURSDAY, JANUARY 26, 1893. 



MODERN ADVANCED ANALYSIS. 

 Theory of Numbers. By G. R. Mathews, M.A. Parti. 



(Cambridge : Deighton, Bell and Co., 1892.) 

 ''T^HE book under review is a great contrast in many 

 J- ways to the " Thdorie des Nombres" of M. 

 Edouard Lucas, the first volume of which has recently 

 appeared under the a^gis of Messrs. Gauthier-V'illars. 

 The latter, reminding the reader much of the same author's 

 " Recreations Mathdmatiques," exhales human interest 

 from well-nigh every page. The former is on severe 

 philosophical lines, and may be greeted as the first work 

 of the kind in the English language. That this should 

 be a fact is somewhat remarkable. When the late Prof. 

 H. J. S. Smith died prematurely many years ago he left 

 his fellow-countrymen a very valuable legacy. Fortunately 

 he had been commissioned by the British Association to 

 frame a report on the then present state of the Theory 

 of Numbers, a subject with which he was pre-eminently 

 familiar, and in which his own original researches had 

 won for him a great and world-wide renown. The pages 

 of the reports for the years 1864-66 inclusive yield as a 

 consequence a delightful account of modern research in 

 this recondite subject. It is, however, much more than a 

 recital of victories achieved by many able men in many 

 special fields. Prof. Smith's fertile genius enabled him to 

 marshal the leading facts of the theory, and to impress 

 apon them his own personality in a manner that was 

 scarcely within the reach of any other man. He con- 

 trived to impart a glamour to those abstract depths of 

 the subject to which few mathematicians have sufficient 

 faith and energy to penetrate. Since that day the 

 scientific world has been yearly expecting his collected 

 papers. There is no doubt that their appearance will 

 greatly stimulate interest and research in Higher 

 Arithmetic. The reports of the British Association are 

 not sufficiently accessible. Doubtless the papers will 

 soon emerge from the hands of those upon whom has 

 devolved the responsibility of their production. In the 

 meantime we welcome Part I. of the present work. 



The theory of numbers is the oldest of the mathemati. 

 cal sciences, and may be regarded as their sire. Just as 

 applied mathematics is based on pure, so pure mathe- 

 matics rests on the theory of numbers. Every investigator 

 finds that sooner or later his researches become a question 

 of pure number. Continuous and discontinuous quantity 

 are indissolubly allied. The theory of series, the theory 

 of invariants, the theory of elliptic functions throw light 

 upon and receive light from higher arithmetic. Algebra 

 in its most general sense is everywhere pervaded by 

 numbers. It may safely be affirmed that there is nothing 

 more beautiful or fascinating in the wide range of 

 mathematics than the interchange of theorem between 

 arithmetic and algebra. A proposition in arithmetic is 

 written out as a theorem in continuous quantity or con- 

 versely an algebraic identity is represented by a statement 

 concerning discontinuous quantity. In this country the 

 more recent advances in this attractive method are in 

 large measure due to the labours of Sylvester and J. W. L. 

 Glaishei. In a " Constructive Theory of Partitions," 

 NO. I 2 13, VOL. 47] 



published some half-dozen years ago in the American 

 Journal of Mathematics, Sylvester showed some beautiful 

 progressions from arithmetic to algebra, and was followed 

 in the same line by Franklin, Ely, and others, whilst in 

 the pages of the Quarterly Journal of Mathematics and 

 Messenger of Mathematics Glaisher has applied elliptic 

 function formulas to arithmetical theory. The famous 

 theorem which asserts that every number can be com- 

 posed by four or fewer square numbers, was due to an 

 application by H. J. S. Smith of elliptic functions to 

 arithmetic. These interesting matters are not alluded to 

 in this first volume. 



Chapter I. discusses the divisibility of numbers and the 

 elementary theory of congruences. Euler's function (^(«), 

 which denotes the number of positive integers, unity in- 

 cluded, which are prime to and not greater than n, is not 

 treated as fully as might be desired. Gauss's theorem 



<p{d) + <p{d!) + (/»(^') +... = « 



where d, d', d'', . . . are all the divisions of n (unity and 

 n included) is given, but not some interesting theorems 

 connected with permutations, of which this is a particular 

 case. Sylvester has written much about the same function, 

 which he calls the "totient" of «. M. Ed. Lucas em- 

 ploys the term "indicateur" in the same sense, and 

 believing that there is a great convenience in having a 

 special name for the function, we regret that Mr. 

 Mathews has not taken a course which would have 

 familiarized students with Sylvester's nomenclature, and 

 have enabled them to feel at home with much that has 

 been written by him and others in this part of the theory 

 of numbers. 



The author states that this chapter is substantially a 

 paraphrase of the first three sections of the " Disquisi- 

 tiones Arithmeticae," the classical work of Gauss ; we are 

 inclined to think that advantage would have been gained 

 if the paraphrase had not been quite so close. The next 

 succeeding chapters are occupied with " Quadratic 

 Congruences " and the theory of "Binary Quadratic 

 Forms." 



The account given is fairly complete. There are so 

 many proofs of Legendre's celebrated " Law of Quadratic 

 Reciprocity" that it must have been difficult to make a 

 selection. A wise choice has, we think, been made of 

 Gauss's third proof as modified by Dirichlet and Eisen- 

 stein ; the latter's geometrical contribution to the proof 

 taken from the twenty-seventh volume of Crelle is, in par- 

 ticular, of great elegance. Gauss's first proof is also given, 

 as well as references to several others. In the difficult 

 subject of Binary Quadratic Forms, the author keeps 

 well in view the close analogy with the algebraic theory 

 of forms ; so many additional restrictions present them- 

 selves that a large number of definitions are requisite at 

 the outset, and this circumstance is apt to repel a student 

 who approaches the theory for the first time. The 

 definitions, in fact, constitute the alphabet of the science 

 which must be mastered before progress can be expected 

 in the appreciation of the wonderful beauties which are 

 inherent in it. In this subject, more almost than in any 

 other, the initial drudgery must not be shirked, and it 

 may be said in favour of the present work that clearness 

 of definition and conciseness of statement help the learner 

 much to get quickly over the wearisome preliminaries. 



O 



