July 15, 1897] 



NATURE 



245 



likely to confuse or excite the mind of the student ; on 

 the contrary, the various contributors insert carefully 

 -chosen appropriate illustrations as the best means of 

 elucidating the difificulties of the abstract theory. With 

 us the spirit of the schoolmaster is too much abroad in 

 •our mathematical writings ; it has even been objected 

 that these illustrations tend to obscure a subject, as it 

 were, with the smoke of its own guns : a musty simile 

 in these days of smokeless gunpowder. 



Thus, for instance, the solution of a quintic equation 

 is presented as required for the determination of the 

 supply of a water-main (p. 13); very vulgar this, our 

 <:ollege professor will say. So, too, the intuitive reason- 

 ing of a graphical procedure with an appropriate diagram 

 has been freely employed to replace the tedious and 

 unconvincing procession of formulas which impede the 

 progress of our own students. A mention of M. Felix 

 Lucas's electrical determination of the roots, real and 

 imaginary, might well have found a place here. 



Each of the eleven chapters of the book is undertaken 

 by a different writer— Chapter i., on the Solution of 

 Equations, by Mansfield Merriman ; and Chapter ii., on 

 Determinants, by Laenas Gifford Weld ; both complete 

 and original in their way. 



The treatment, in Chapter iii., of Projective Geometry, 

 by George Bruce Halsted, is very bright and stimulat- 

 ing ; this is a subject ignored in our own mathematical 

 curriculum. 



The two forms of spelling "centre' and "center," 

 appear on the same page (95) ; the second is, of course, 

 phonetically correct, as the English pronunciation always 

 inverts the liquid and the vowel in the French spelling, 

 here and in all similar words. 



Chapter iv. is on Hyperbolic Functions, by James 

 McMahon. Our scholastics look upon this subject as 

 a temporary fad, which has not come to stay ; how- 

 ever, electricians find them indispensable, and many 

 elegant electrical applications, among others equally 

 important, of mechanical and astronomical interest, such 

 as catenaries, loxodromes, charts, conjugate functions, 

 will be found collected here. 



The long form of these functions, cosh, sinh, tanh, . . . 

 has been retained, with a suggestion that the ugly sounds 

 they suggest should be avoided by pronouncing them 

 /«-cosine, ^-sine, ^-tangent, &c. But the modern con- 

 tinental practice is to abbreviate the symbols to ch, sh, 

 th, pronouncing only the letters c-h^ s-.h, t-h, as with the 

 Elliptic F^unctions ; so also for their inverse functions, 

 ch"i, sh~\ th-', employed here, for their obvious advan- 

 tages in integration. A well-arranged table concludes 

 this chapter ; we miss, however, Bernoulli's numbers in 

 their proper place in the expression of tan .v, th x. . . , 



Prof. Byerley, of Harvard, contributes Chapter v., on 

 Harmonic Functions. When his genial treatise on 

 Fourier's Series and Harmonic Analysis made its ap- 

 pearance, some four years ago, it was welcomed by all 

 physicists as the long-desired manual, which placed this 

 subject before them in an intelligible manner, devoid of 

 artificial obstacles and impediments. Unfortunately the 

 treatise fell into the hands of mathematical critics, who 

 could see little merit in the book, because it passed over 

 in silence the tedious, and useless, arguments concerning 

 the legitimacy of the expansions. If an electrician is to 

 NO. 1446. VOL. 56] 



employ a Fourier Series, he will content himself with the 

 first two or three terms of the series ; just as the calcu- 

 lator of mathematical tables will not, for practical pur- 

 poses, employ more than three, or four terms at most, in 

 Taylor's Series. But where the applicability becomes 

 doubtful, by reason of the neighbourhood of a discon- 

 tinuity, he will assure himself, by a diagram such as those 

 on p. 199, of the limits of the divergence. 



These difficulties concerning the discontinuity of func- 

 tions is very properly relegated to another chapter, number 

 vii., on Functions of Complex Variables, by Thomas S. 

 Fiske, which gives us a very clear account of the most 

 recent manner, of the school of Weierstrass, of approaching 

 such refinements of argument. We are pleased to find 

 the name "one-valued function" instead of "uniform 

 function," which is misleading to the beginner. 



Prof. Woolsey Johnson, of the U.S. Naval Academy, 

 contributes Chapter vii., on Differential Equations. His 

 own formal treatise on the subject is well known and 

 highly popular ; and the present chapter incorporates the 

 essential, or what Maxwell called the "gentlemanly," 

 knowledge of the subject. 



The next two chapters — Chapter viii., on Grassmann's 

 Space Analysis, by Edward W. Hyde, and Chapter ix., 

 on Vector Analysis and Quaternions, by Alexander 

 Macfarlane — seem to us by comparison to be of the 

 nature of luxuries, appealing to the purely analytical 

 spirit ; although even here electrical applications are in- 

 troduced to show how the theories may be usefully 

 applied. 



Chapter x. is a short and useful r^suiiu\ by R. S. 

 Woodward, of the principal parts of Probability and the 

 Method of Least Squares, with which every physical 

 student should now be familiar; and the volume concludes 

 with Chapter xi., on the History of Modern Mathematics, 

 by David Eugene Smith, in which the author is com- 

 pelled to apologise for the incompleteness imposed upon 

 him by the exigencies of room, but which, nevertheless, 

 provides the most important details required for reference. 



The account given by the Editors, in the preface, of 

 the work expected of the average American student, 

 shows that the standard of requirement is much higher 

 than in this country, and not hampered by traditional 

 prejudice. A. G. Gr ken hill. 



OSTEOLOG Y. 

 The Vertebrate Skeleton. By Sidney H. Reynolds, M.A. 

 Pp. xvi -f- 559. (Cambridge: University Press, 1897.) 



THIS most recent addition to the Biological Series 

 of the Cambridge Natural Science Manuals edited 

 by Mr. A. E. Shipley, is an attractive-looking volume, 

 well printed, and with the monotony of the text agreeably 

 broken by a judicious use of small capitals, italics, and 

 clarendon type. The numerous illustrations, which are 

 probably accountable for the high price {\2S. 6d.) of the 

 book, though simple in execution are clear in detail, and, 

 on the whole, chosen with discretion. The majority of 

 the figures have not been published before, and are 

 based on specimens contained in the Cambridge Univer- 

 sity Museum and the Natural History Museum, London. 

 The text is remarkably free from typographical errors, 

 but is frequently bald in style and irritating from 



