June 30, 1892] 



NATURE 



199 



The examination of water is given iti exienso^ but there 

 is no mention, when discussing the presence of typhoid 

 bacilli in water, of the latest methods for their detection 

 amongst other micro-organisms contained in natural 

 waters. 



The investigation of air for micro-organisms is entirely 

 left out, an omission which renders the book less com- 

 plete than it would otherwise appear to be. 



But there is a great deal of instruction, together with 

 many valuable hints, contained in the comparatively 

 short space of 200 pages ; and whilst, interspersed in the 

 text, wood-cuts serve to supplement some of the descrip- 

 tions of apparatus, it also boasts some very good photo- 

 graphs from original preparations of the Staphylococcus 

 pyogenes citreus, the Streptococcus erystpelatis, the Bacil- 

 lus anthracis with spores, the tuberculosis Bacillus, Koch's 

 comma Spirillum, and others. 



There is also appended a useful list of all the requisite 

 appliances for bacteriological work. 



Grace C. Frankland. 



OUR BOOK SHELF. 



Neue Rechnungsmethoden der Hbheren Mathematik. Von 



Dr. Julius Bergbohm. (Stuttgart : Selbstverlag des Ver- 



fassers, 1892.) 

 Neue Integrationsmethoden auf Grund der Potenzial-^ 



Logarithmal-, und Numeraire chnung. (The same.) 

 The first of these pamphlets contains an account of what 

 the author calls the Immensalrechnung, the Potenzial- 

 rechnung, the Radtkal?-echnung^X^&Logarithmalrechnung, 

 and the Numeralrechnung. In the Itnmensalrechnung z.n 

 attempt is made to provide a calculus of the infinitely 

 great {das Jmmensal\ which shall form a complement to 

 the differential calculus, or calculus of the infinitely small. 

 The Potenzialrechnung contains an account of exponen- 

 tial functions in which the base is an infinitely small or an 

 infinitely great quantity, and the exponent is infinitely 

 small ; and the Radikalrechnung a.n account of the inverse 

 functions that are obtained from these by changing the 

 exponent into its reciprocal. So, too, in the Logarith- 

 malrechnung, logarithmic functions are considered in 

 which the base and the argument are either infinitely 

 small or infinitely great ; and in the Numeralrechnung 

 the inverse functions (antilogarithms or exponential 

 functions) are discussed. The pamphlet is occupied, 

 for the most part, with an exposition of the author's 

 notation, a discussion of certain indeterminate forms, 

 and a calculation of some algebraic functions contain- 

 ing an infinitely small argument, to a first, second, or 

 third approximation. It is hardly possible to compliment 

 the author on his accuracy, seeing that the statement 

 occurs that Lt. log^ris finite when ;r is zero or infinity, the 

 reason given being that Lt. (;rlog^) and Lt. (log xjx) are 

 zero, for these values of x. 



The second pamphlet begins with a resume of some of 

 the results of the first one ; and then proceeds to discuss 

 the application of these results to the evaluation of certain 

 elementary integrals. The author's avowed object is to 

 provide a method for the direct calculation of integrals, 

 comparable with that now employed in differentiation, so 

 that it may no longer be necessary to resort to the indirect 

 methods of integration at present employed. It is impos- 

 sible to deny that the object is a laudable one ; but, to 

 judge from the examples given in this pamphlet, it does 

 not seem likely that the method will be of much use in 

 the case of integrals of any degree of complexity. Dr, 

 Bergbohm promises to supply us in the future with further 

 examples of the application of his methods ; but, until 



NO. 1183, VOL. 46] 



these have appeared, it is hirdly possible to say that stu- 

 dents of mathematics will find these pamphlets repay 

 them for the trouble of reading them and of mastering 

 the author's notation. R. E. A. 



An Elementary Course in Theory of Equations. By Ci 

 H. Chapman, Ph.D. (New York: John Wiley and 

 Sons, 1892.) 



This is really an excellent little book, but is rather mis- 

 named in being called an elementary treatise. The study 

 of the theory of equations, although generally expanded 

 far too considerably, is here dealt wiih in rather the re- 

 verse way, the treatment being somewhat too curt. For 

 anyone beginning this subject the book would be found 

 slightly difficult, but for a student who has already had 

 a little experience in this direction, it should prove 

 a very useful vade mecum, for the author has brought 

 together in a few pages just those portions of the subject 

 that are required in actual practice. The three sections 

 treat respectively of determinants, algebraical equations, 

 and the methods by which the real roots of numerical 

 equations are computed, and they are each accompanied 

 by numerous examples. 



LETTERS TO THE EDITOR. 



{The Editor does not hold himself responsible for opinions ex- 

 pressed by his correspondents. Neither can he undertake 

 to return, or to correspond with the writers of, rejected 

 manuscripts intended for this or any other part of iHATUViZ, 

 No notice is taken of anonymous communications.'^ 



"The Grammar of Science." 



It is very idle as a rule to criticize a critic, especially when 

 he happens, like C. G. K., to be the disciple of a school which 

 the author of the criticized work is gently laughing at through- 

 out his pages. But some of C. G. K.^s remarks might lead 

 your readers to believe that the ' ' Grammar of Science " is 

 nonsense, even when looked at without the spectacles of the 

 Edinburgh physical school, and his review may therefore justly 

 call for a few words of reply. 



Because C. G. K. found himself entirely unable to follow 

 my argument as to the universality of scientific law, he was hardly 

 justified in putting an antecedent before a consequent, and 

 making nonsense of it. The universality of scientific law 

 depends on the similarity of the perceptions and of the reflective 

 faculties in normal civilized man. Why does this similarity 

 exist? asks C. G. K. j and then turns for an answer to an 

 antecedent in the argument — namely, that a condition of this 

 universality is the similarity in those perceptions and reflective 

 faculties. As a matter of fact in the "Grammar" itself, it is 

 pointed out that a society of beings with different perceptions 

 and reflective faculties could hardly survive in the struggle for 

 existence with societies where there was an approach to 

 similarity ; that as soon as the divergence reaches a certain 

 magnitude we lock up the individual as a madman or an idiot, 

 or, in milder cases, bring great social pressure to bear upon him, 

 and mould him to the ordinary standard. 



' ' The laws of Nature are a mental product, yet a certain 

 evolution theory logically based upon them quite eliminates the 

 mental," writes C. G. K. of the "Grammar." Wherehefound 

 this statement I know not, but what the "Grammar" itself 

 states is : that the laws of evolution are themselves a mental 

 product, a description in shorthand of the sense-impressions 

 and stored sense-impresses of the mind at a given instant. They 

 are a mental mode of briefly classifying sense-impressions, and 

 not inherent in something behind sense-impressions themselves. 



C. G. K. then quotes my statements as to Maxwell's descrip- 

 tions of energy and matter. Now what the "Grammar" says 

 is that Maxwell's statements are " extremely valuable as express- 

 ing concisely the nature of certain conceptual processes by aid of 

 which we describe certain phases of our perceptual experience, 

 but as defining matter they carry us no further than the state- 

 ment that matter is that which moves," or indeed than Prof. 

 Tait's statement that "matter is that which occupies space." 

 The whole object of the investigation is to show that mass, but 



