:68 



NATURE 



[January 19, 191 i 



sciences, and history, play a much greater part than 

 in antiquity. Result : mysticism and agnosticism. 

 Prof. Dorner combats both. The physical sciences 

 themselves point the way to metaphysical principles ; 

 the problem of philosophy is not merely epistemology 

 or the making of a world-conception out of the dis- 

 parate elements of knowledge and experience, but is 

 rather the search for a unified metaphysic by which 

 the fundamentals of the world and of the spiritual life 

 may equally be grounded in an Absolute Being. 



AUERICA^ TEXT-BOOKS OF MATHEMATICS. 



(i) College Algebra. By Prof. H. L. Reitz and A. R. 

 Crathorne. Pp. xiii-l-261. (New York: H. Holt 

 and Co. ; London : G. Bell and Sons, 1909.) Price 



(2) Trigonometry. By Prof. A. G. Hall and F. G. 

 Frink, Pp. x+ 146 + 93. (New York : H. Holt and 

 Co. ; London : G. Bell and Sons, 1909.) Price 

 75. 6d. 



{3) First Course in Calculus. By Prof. E. J. Town- 

 send and Prof. G. A. Goodenough. Pp. xii + 466. 

 (New York : H. Holt and Co. ; London : G. Bell 

 and Sons, 1908.) Price 12s. 



THESE books are the first three of a series which 

 is intended in the first place for students taking 

 a university course in engineering, and also, to a cer- 

 tain extent, for mathematical students. It will be 

 noticed that each book has two authors, who have 

 been selected to represent the interests of readers of 

 both classes. 



(i) and (2). The chief novelty in these books is to 

 be found in the variety of examples, selected from 

 very different subjects. Thus, as an example on 

 evaluating algebraic expressions ("Algebra," p. 24), 

 the student is asked to verify in a few cases a formula 

 for the day of the week, which (after an obvious 

 simplification) can be written ^ — 



2+J> + 2g + [% (^+l)] + s + [is]-2r + [\r]=t {mod. 7) 



where t is the day of the week (Sunday being i and 

 Saturdav 7), and the date is the pth day of the 5th 

 month in the year loor + s. The reader interested in 

 such matters may find it instructive to reconstruct this 

 formula, of which the most interesting feature is the 

 part depending on q; it will be found that starting 

 from March (and ignoring February) the lengths of 

 the months recur after intervals of five months, and 

 this is the basis of the formula. 



The problems proposed in the trigonometry are 

 chosen so as to illustrate the practical difficulties of 

 surveying so far as possible. Great stress is laid on 

 the advantage of making a form for numerical cal- 

 culations, before starting to use the tables at all. One 

 useful consequence is that, in the typical examples 

 worked out, the logarithms to be added are placed in 

 vertical columns, as would be done in practical work; 

 writers of text-Books are very apt (in order to save 

 space) to arrange such logarithms horizontally. The 



1 The notation is that of the theory of number* : that is, f-f] denotes the 

 integral part of .r, and y=z means that y- z is divisible by 7. Note that 

 January and February are regaided as belonging to the /rez'ious year, with 

 the values 5?= 13, 14. 



NO. 2 15 I, VOL. 85] 



result is that imitative readers are liable to arrange 



their work in the same way, with disastrous results. 



The last ninety-three pages in the trigonometry 



contain a good set of five-figure tables. The table 



of logarithmic functions, however, makes no special 



provision for finding the log sin and log tan of small 



angles ; a very simple rule applies to four-figure or 



five-figure tables (with a difference of i' in angle), 



namely — 



log sin O = log[ sin a 4- (log fl-log a) 



and this (or some similar rule) ought to be given in 

 all tables which do not provide a special table for the 

 first few degrees. The table of squares is interesting, 

 as it gives the exact squares from i to looo*, without 

 occupying more space than an ordinary four-figure 

 table ; this is effected by following the arrangement of 

 Crelle's multiplication tables, where every number in 

 the same horizontal line is terminated by the same 

 two digits. Both in the algebra and trigonometry 

 certain of the best-known power-series are given and 

 used for numerical calculations; but the authors of 

 the algebra are content to refer to the calculus (No. 3) 

 for proofs, while in the trigonometry some proofs are 

 provided, which would not be accepted nowadays. It 

 might be better definitely to cut out all such proofs 

 from books on trigonometry ; in modern teachinjg the 

 elements of the calculus are certainly regarded as 

 easier (and more generally useful) than the "calculus- 

 dodging " of the old-fashioned books. 



(3) Compared with recent English books having 

 similar titles this book contains fuller treatment of 

 the applications of the calculus to applied mathe- 

 matics; for instance, centroids, moments of inertia, 

 resultant fluid pressure, are considered at some 

 length, as exercises on integration. 



As in the other books of the series, a large variety 

 of illustrative examples will be found; thus the expo- 

 nential function is connected with the chemical 

 problem of inversion of cane-sugar. The theory of 

 maxima is illustrated by the efficiency of a rough 

 screw, the speed of signalling in a cable, and the 

 h.p. transmitted by a hemp-rope. 



In dealing with the Taylor's series derived from a 

 given function, care is taken to point out that the 

 series may converge without being equal to the 

 function ; this is a point quite commonly overlooked 

 in the theory, and possibly an example would have 

 helped to emphasise it.^ 



As might be anticipated from the character of the 

 series, a good deal of stress is laid on methods for 

 approximate integration, such as Simpson's rules and 

 other similar methods, and several examples are given 

 of their appHcation to irregular solids such as rails. 

 It seems strange, however, that the exact form 

 of Simpson's rule is not mentioned, for finding the 

 volume and centroid of a railway embankment (or the 

 slice of an ellipsoid) in terms of the areas of the 

 ends and the area of the central section. 



The use of infinite series for finding an integral 

 (-t)"«" 



i Thus, Pringsheim's function 2- 



l(i+.r2«-^") 



has the Taylor's series 



e-''-jr2e^''^+x*e ""...: but if a>i, although both series converge for 

 all tea! values of x, they are unequal except for x = o. For instaiice, if 

 a = 2, Jtr=2, it will be found that the first series is less than o'log, while the 

 second is greater than 0-133; on the other hand if a = i, x = k, both series 

 are equal to 0*434 (nearly). 



