NA TURE 



12 I 



THURSDAY, JUNE 7, 1894. 



HAG EN'S SYNOPSIS. OF HIGHER MATHE- 

 MA TICS. 



Synopsis der Iloheren Mathemalik. Von Johann (J. 

 Hagen, S.J., Director der Sternwarte des Georgetown 

 College, Washington, D.C. Erster Band: Arith- 

 metische und algebraische Analyse. (Berlin ; Felix 

 L. Dames, iSgi.) 



THE author's object has been to give a bird's-eye 

 view, or synopsis, of the whole range of higher 

 mathematics ; and this handsome volume of 39S pages 

 is a first instalment. The work is not intended as a 

 treatise, or to be merely a book of reference to which 

 the mathematician may turn for his formulae. It has a 

 much more ambitious scope, and aims at presenting a 

 general view of all branches of mathematics, methodically 

 arranged and separated into a great number of sections, 

 each of which contains a notice of the history of 

 the subject to which it relates, followed by a series 

 of numbered paragraphs giving the principal formulas, 

 with full references to the books and writings from 

 which they are taken, and to which the reader must 

 have recourse for further information. 



The branches of mathematics treated of in the present 

 volume may be classed under the four heads of Theory 

 of Numbers, Theory of .Series, Theory of Functions, 

 and Theory of Equations. In this classification, however, 

 an extended meaning must be given to these titles, for 

 the functional branch includes determinants, invariants, 

 and groups. Altogether there are twelve subject head- 

 ings divided into 102 sections, each of which is further 

 subdivided into separate articles when required. As 

 an example of the mode of arrangement, we may take 

 the Partition of Numbers. We first find a general 

 sketch of the algebraical methods of Euler, Cayley, and 

 Sylvester, with many of Euler's most interesting results ; 

 then we pass to partitions into figurate numbers and to 

 quadratic forms, both treated in a similar manner. 



It is evident that any near approach to absolute com- 

 pleteness could not be attained in such a comprehensive 

 u ndertaking. No single person could read and digest the 

 whole of mathematics as it exists in our day, and arrange 

 and systematise it in a series of volumes. It might even 

 be regarded as open to question \vh ether so bold an enter- 

 prise could meet with any measure of success. But no one 

 can look at this volume without admitting that the attempt 

 has been well justified, and that, whatever its im- 

 perfections, we are indebted to the author for a most 

 interesting and valuable work. 



The critical reader naturally turns first to the sub- 

 jects — or, rather, the portions of subjects — with which 

 he is himself best acquainted, and it is not surprising 

 if he should here find omissions ; but, even in this 

 extreme case, the sections in question can scarcely be 

 read without advantage as well as interest. The true 

 test of the utility of the work is afforded by an inspec- 

 tion of the sections relating to subjects which lie adjacent 

 to, but not upon, the direct line of the reader's own 

 studies ; here he cannot fail to be impressed by the new 

 matter which he will find set out before him. 



NO. 128*1, \'OL. 50] 



The history, theorems, and references are grouped to- 

 gether in an attractive manner ; a mathematician could 

 not turn over the pages, even in the most casual manner, 

 without being tempted to stop here and there and pore 

 over some of the paragraphs. The historical introduction 

 is always remarkably clear, and the formulae are 

 sufficiently explained to render them intelligible as they 

 stand. Although the book is to some extent a cyclopasdia, 

 it is not unduly concise, nor is any attempt made to save 

 space by the introduction of special abbreviations in the 

 explanations or references. 



As an illustration of the contents of the sections, we 

 may take the paragraphs which relate to the number of 

 prime numbers. We first find references to the proofs 

 of the theorems that the number of primes is unlimited, 

 and that every arithmetical progression, whose first term 

 and difference have no common factor, must contain a 

 prime. The next paragraph gives an account of Gauss's, 

 Encke's, and Legendre's approximate formulae for the 

 numbers of primes between given limits, with references. 

 Then we come to 2. resuiiu' oi Tchebicheff's memoir of 

 1851, with .Sylvester's additions (1881), followed by a 

 similar statement of Riemann's results (1859) and a refer- 

 ence to Meissel's methods of calculating the exact num- 

 ber of primes up to a given limit (1871). As another 

 illustration, we may take the section relating to the har- 

 monic series. First we find references to works or 

 memoirs where special cases of harmonic series are 

 treated at some length ; then we come to the general 

 summation by means of the semi-convergent series with 

 Eulerian numbers as coefficients ; and the section 

 closes with an account of the history of Euler's 

 constant. From this description it will be seen that 

 the work, covering as it does all higher mathematics, 

 is unique in its character. No other writer has attempted 

 to deal systematically with any large field of mathe- 

 matical research so fully and completely. 



It seems to us that Mr. Hagen has very skilfully com- 

 bined statements of results with references. It is difficult 

 to avoid being too dit'fuse when formute have to be 

 selected from an elaborate memoir ; and it is difficult to 

 render a mere body of references attractive. But in both 

 these respects the author has been successful. The 

 references are always accompanied by enough explana- 

 tory matter to render them interesting ; in fact, unlike 

 most mathematical quartos, every page of the book is 

 "readable" in the ordinary sense of the word. The 

 su"bdivision of the subjects into so many sections, though 

 convenient for the user, must have added considerably 

 to the labour of preparation, and increased the difficulty 

 of arranging the references so as to avoid repetition. 



.•\ list of sixty-six treatises and twenty-one periodicals, 

 which arc referred to in the volume, is given at the end. 

 This list, long as it is, might have been considerably 

 extended, had more complete libraries been accessible to 

 the author. As it is, the works consulted form a most 

 excellent nucleus, which may be supplemented at some 

 future time by the author or a successor. Had many 

 more been included, we think the author's attempt must 

 have failed, no matter what ability and perseverance 

 he might have brought to his task. It is to be 

 remembered that for such a compilation it is necessary 

 to study the memoirs with some care in order to decide 



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