Feb. 13, 1890] 



NATURE 



use Euler's title, an * Introductio in Analysin Infini- 

 torum.' A practice has sprung up of late (encouraged by 

 demands for premature knowledge in certain examina- 

 tions) of hurrying young students into the manipulation 

 of the machinery of the differential and integral calculus 

 before they have grasped the preliminary notions of a 

 limit and of an infinite series, on which all the meaning 

 and all the uses of the infinitesimal calculus are based. 

 Besides being to a large extent an educational sham, this 

 course is a sin against the spirit of mathematical progress. 

 The methods of the differential and integral calculus, 

 which were once an outwork in the progress of pure 

 mathematics, threatened for a time to become its grave. 

 Mathematicians had fallen into a habit of covering their 

 inability to solve many particular problems by a vague 

 wave of the hand towards some generality, like Taylor's 

 theorem, which was supposed to give ' an account of all 

 such things,' subject only to the awkwardness of practical 

 inapplicability. Much has happened to remove this 



danger and to reduce didx and / dx to their proper place 



as servants of the pure mathematician. . . . For the 

 proper understanding of this important branch of modern 

 mathematics \i.e. function-theory], a firm grasp of the 

 doctrine of limits and of the convergence and continuity 

 of an infinite series is of much greater moment than 

 familiarity with the symbols in which these ideas may be 

 clothed. It is hoped that the chapters on inequalities, 

 limits, and convergence of series [chapters xxiv.-xxvi.], 

 will help to give the student all that is required both for 

 entering on the study of the theory of functions and for 

 rapidly acquiring intelligent command of the infinitesimal 

 calculus. In the chapters in question, I have avoided 

 trenching on the ground already occupied by standard 

 treatises : the subjects taken up, although they are all 

 important, are either not treated at all or else treated very 

 perfunctorily in other English text-books." 



No student who masters the present treatise will pass 

 such judgment upon these chapters, or, indeed, upon any 

 part of the work. What the writer aims at, and succeeds 

 in achieving, is thoroughness. 



The first part occupied twenty-two chapters ; the second 

 part occupies chapters xxiii.-xxxvi. 



Following on the lines of our previous notice (/.f .), we 

 give a brief analysis of the chapters : — 23, permutations 

 and combinations (with applications to binomial and 

 multinomial theorems, distributions and derangements, 

 and the theory of substitutions) ; 24-26, see extract above ; 



27, binomial and multinomial theorems for any index : 



28, exponential and logarithmic series (with an account, 

 and applications, of Bernoulli's numbers) ; 29, 30, summa- 

 tion of the fundamental power-series for complex values 

 of the variable, and general theorems regarding the ex- 

 pansion of functions in infinite forms — these are two 

 splendid chapters, which the author says 



" may be regarded as an elementary illustration of the 

 application of the modern theory of functions. They are 

 intended to pave the way for the study of the recent 

 works of Continental mathematicians on the same subject. 

 Incidentally, they contain all that is usually given in 

 English works under the title of analytical trigonometry. 

 If anyone should be scandalized at this traversing of 

 the boundaries of English examination subjects, I must 

 ask him to recollect that the boundaries in question were 

 never traced in accordance with the principles of modern 

 science, and sometimes break the canon of common- 

 sense. . . . The timid way, oscillating between ill-founded 

 trust and unreasonable fear, in which functions of a com- 

 plex variable have been treated in some manuals, is a 

 little discreditable to our intellectual culture, f Some ex- 



pounders of the theory of the exponential function of an 

 imaginary argument, seem even to have forgotten the 

 obvious truism that one can prove no property of a 

 function which has not been defined." 



Chapter 30, moreover, closes with "a careful dis- 

 cussion of the reversion of series and of the expansion in 

 power-series of an algebraic function — subjects which 

 have never been fully treated before in an English text- 

 book, although we have in Frost's curve-tracing an 

 admirable collection of examples of their use" (this is a 

 work often referred to with high commendation in the 

 text). To resume our analysis, chapter 31 is on the 

 summation and transformation of series in general ; 

 32-34 gives a thorough discussion of continued fractions 

 and their applications ; 35 gives numerous general pro- 

 perties of integral numbers ; and 36 is on probability, or 

 the theory of averages. In this last chapter the author 

 has " omitted certain matter of doubtful soundness and 

 of questionable utility ; and filled its place by what I 

 hope will prove a useful exposition of the principles of 

 actuarial calculation." 



The student of the present day knows that " things are 

 not always what they seem," so when he hears that an 

 elementary text-book of algebra occupies more than a 

 thousand octavo printed pages, he is prepared to find 

 that the " elementary" is comparative, and the " algebra " 

 comprises some other subjects, in ordinary parlance, 

 called by other names. He will find the present work 

 most readable, provided he comes to the perusal with 

 the requisite knowledge and ability, and when he has 

 got to the end of the course he will have an excellent 

 foundation for all his after mathematical reading. Prof. 

 Chrystal gives good advice, which we copy. " When you 

 come on a hard or dreary passage, pass it over ; and come 

 back to it after you have seen its importance or found the 

 need for it further on. To facilitate this skimming pro- 

 cess, I have given, after the table of contents, a sugges- 

 tion for the course of a first reading." There are 

 numerous " historical notes," which form a conspicuous 

 and useful feature of the whole work. 



The author uses the expression (see above) " dreary 

 passage " : we have not come across these, but we can 

 certify with regard to the first part, that we have taken it 

 up again and again, and have always found it difficult to 

 rest contented with a brief glance, and the part before us 

 appears, in some respects, to be even more attractive. 



FERMENTATION WITH PURE YEAST. 



The Micro-organisms of Fermentanon, practically 

 considered. By Alfred Jorgensen. Edited from the 

 German by G. Harris Morris, Ph.D., F.C.S., F.I.C., 

 &c. With an Introduction by Horace T. Brown, 

 F.C.S., F.I.C. (London : F. W. Lyon, 1889.) 



DURING the past ten years in which the investigation 

 of micro-organisms and their functions has been 

 so actively pursued there has been a conspicuous absence 

 of any work dealing with the progress made in our know- 

 ledge of those particular forms which are of industrial 

 importance. Thus whilst numerous text-books in various 

 languages have appeared embodying the latest discoveries 

 in the relationship of micro-organisms to disease, the 

 only noteworthy treatise on the technological side of 



