August 7, 19 13] 



NATURE 



579 



MATHEMATICAL TEXT-BOOKS. 

 (i) A School Algebra. By F. O. Lane and J. A. C. 

 Lane. Pp. viii + 333. (London: Edward 

 Arnold, n.d.) Price 35. 6d. 



(2) A Treatise on Hydromechanics. Part ii. 

 Hydrodynamics. By A. S. Ramsey. Pp. xiii + 

 360. (London: G. Bell and Sons., Ltd., 1913.) 



(3) Les Appareils d' Integration. By H. de Morin. 

 Pp. 208. (Paris: Gauthier-Villars, 1913.) Price 

 5 francs. 



1. 1 1 Einfuhrung in die hohere Mathematik jitr 

 Naturforscher und lerste. By Dr. J. Salpeter. 

 Pp. xii + 336. (Jena: Gustav Fischer, 1913.) 

 Price 12 marks. 



(5) Elements of the Precision of Measurements and 

 Graphical Methods. By Prof. H. M. Goodwin. 

 Pp. 104. (London: Hill Publishing Co., Ltd.; 

 New York : McGraw-Hill Book Co., 1913.) 



(6) Matrices and Dcterminoids. By Prof. C. E. 

 Cullis. Vol. i. Pp. xii + 430. (Cambridge 

 University Press, 1913.) Price 215. net. 



.(1) TV J ESSRS. LANE'S "Algebra " looks as if 

 i_Vl_ it would prove a useful school-book. 

 In dealing with the binomial and exponential series 

 the authors state certain properties, with the ex- 

 plicit warning that they are not proving them. 

 This is as it should be; but the chapter on ex- 

 ponentials and logarithms is not so clear as it 

 might be; in particular, Arts. 135-7 would be 

 better if arranged in the reverse order. In the 

 earlier pages we have the old fallacious and mean- 

 ingless statement : " to multiply a number a by a 

 second number b, we do to a what is done to the 

 unit to obtain b." It would be much better to give 

 the rule of signs as a rule pure and simple, and 

 then to show by cases of (a — b)(c — d) that it does 

 actually work out in practice. There are hundreds 

 of examples — some, alas, of a highly artificial 

 character; for instance, "If the hypotenuse of a 

 right-angled triangle is x, and the other sides are 

 y and z units of length, show that 



l/log, + J /+l/log. r _ 2 .;' = 2," 



or, again : 



" Multiply 3 iJxy 3 — .ry+2 \Vlr~ 1 by s'.rr- 2 s' .1 y 3 ." 



The most interesting chapters in the book are 

 those on simultaneous equations, which are illus- 

 trated by appropriate graphs. 



(2) Mr. Ramsey's "Hydrodynamics" is a 

 treatise specially suited for university candidates, 

 and as such may be highly praised for its clearness, 

 elegance, and helpfulness. The chapter on discon- 

 tinuous motion is exceptionally good, the cases 

 •discussed being worked out in unusual detail. The 

 book includes a chapter on vibrations of strings, 

 and one on sound waves ; there is a large number 

 NO. 2284, VOL. 91] 



of excellent examples, with their sources indicated; 

 and sufficient references are L;iven to original 

 memoirs. As a text-book for capable students, 

 Mr. Ramsey's work will be very hard to improve 

 upon, and is certain to have a favourable reception. 



(3) Mr. de Morin describes various kinds of 

 planimeters, integrometers, integraphs, harmonic 

 analysers, and compound integrators. We have 

 summaries of the mathematical theories involved, 

 diagrams of the mechanisms, and pictures of the 

 different machines that have actually been con- 

 structed. No written account can be equivalent to 

 inspection and use of the machines themselves ; 

 allowing for this fact, the author seems to have 

 done all that could be expected. By the way, we 

 wonder what the author of " Erewhon " would 

 have said if he had been shown machines for doing 

 sums, and predicting tides, and calculating 

 moments of inertia. 



(4) Dr. Salpeter's work is chiefly interesting as 

 an example of a course in higher mathematics for 

 medical men and men of science drawn up by an 

 author acquainted more or less with modern pure 

 mathematics. As might be expected, he appeals 

 mainly to intuition ; but he gives a whole intro- 

 ductory chapter to the notion of a limit, he proves 

 d 2 z/dxdy = d 2 z/cydx, when it can be proved, by 

 the mean value theorem, and gives in an appendix 

 some examples of discontinuous functions. For 

 some reason, not apparent, there is a chapter on 

 the second law of thermodynamics ; in other re- 

 spects the scope of the book is not of an unfamiliar 

 kind ; we have differential and integral calculus 

 treated separately, then ordinary differential equa- 

 tions of the second order, and then some easy 

 cases of definite integration. Naturally, many of 

 the examples are chosen to illustrate physical or 

 chemical formulae. 



(5) Dr. Goodwin's work is based on a course- 

 given by him for years past in the Massachusetts 

 Institute of Technology. The sort of problem he 

 deals with is such as : "Calculating g from w 2 l/t 2 , 

 how closely should I, t be measured, so that the 

 resulting value of g may be true within o'i per 

 cent.?" He quotes, without proof, certain results 

 of the theory of errors ; in other respects the dis- 

 cussion is quite elementary, and includes a section 

 on graphical methods. We can quite believe that 

 a course of this kind has been of great value to 

 the Massachusetts students ; whether an actual 

 course is given or not, laboratory students must 

 become familiar with the connection between the 

 probable values of their data and the probable 

 value of their result. Dr. Goodwin gives a set of 

 seventy-nine unsolved questions, which ^achers 

 of physics would find very useful exercises. 



(6) Prof. Cullis associates with any rectangular 



