November i, 1917] 



NATURE 



103 



(4) Differential Calculus. By Dr. H. B. Phillips. 

 Pp. vi + 162. (New York: John Wiley and 

 Sons, Inc.; London: Chapman and Hall, Ltd., 

 1916.) Price 55. 6d. net. 



(i) 1\/[R' BRUNT gives an account of the 

 ■•■▼J- method of least squares, without 

 entering into elaborate descriptions of instruments 

 or experimental methods. Ihe book is not easy 

 reading, but this is principally because the material 

 is chosen from a rather difficult branch of applied 

 mathematics, and modern work is discussed very 

 thoroughly. The proof of the law of error is 

 based on Hagen's hypothesis regarding errors of 

 observation, and a generalised form of it, due to 

 Prof. Eddington, is given on p. 15. Other chap- 

 ters deal with the case of one unknown, observa- 

 tions of different weight, observations involving 

 several unknowns, conditioned observations, the 

 rejection of observations, and alternatives to the 

 normal law of errors. The three last chapters deal 

 with correlation, harmonic analysis from the point 

 of view of least squares, and the periodogram. 

 This last part is of great interest in connection 

 with the modern work of Sir Ronald Ross and 

 Dr. Brownlee, and a recent paper by Sir Joseph 

 Larmor on what may be called " practical harmonic 

 analysis." There are a great number of valuable 

 references in the book, which is much to be com- 

 mended. 



(2) This volume is the first part of a projected 

 work in thirteen parts, of which a synopsis is given 

 at the end of the book. Judging by the synopsis, 

 the future parts may very possibly be interesting, 

 but, on the whole, this first part cannot be said 

 to be a useful contribution to our knowledge of 

 the fundamental conceptions of mathematics. " It 

 is, we beHeve," say the authors (p. iii), "the first 

 attempt made on any extensive scale to examine 

 critically the fundamental conceptions of mathe- 

 matics as embodied in the current definitions." 

 After this extraordinary statement we are not un- 

 prepared to find that the contributions of Frege, 

 Russell, and Whitehead are judged merely from 



I a very small part of them, and that a part which, 



on account of its popular character, does not make 



any pretensions to finality. Further, the remarks 



1 about Frege and RusseU (pp. 152-53) are quite 



[superficial and valueless, as well as scarcely true 



;Or polite. Apart from this, there are some good 



[features in the book. Thus, the criticism of the 



[usual mathematicians' confusion of sign with 



things signified, and so on (pp. 2-3, 97-110, 180), 



is quite good, though it is unnecessarily lengthy 



and rather superfluous after the weighty and witty 



tremarks of Frege, which Messrs. Richardson and 



ILandis do not mention. Also the remarks on 



Urichlet's "definition" of a function (pp. 182-90) 



lare correct, but much too long-winded. There 



ire many other true things, but also many mis- 



iken ones which do not even strike us as honest 



ittempts to get at the truth. 



(3) Mr. Borchardt's book is a very useful 

 graduated set of one hundred papers of examples 

 >n the usual course of arithmetic from the first 

 )ur rules up to logarithms and compound interest, 



NO. 2505, VOL. 100] 



and including the measurement of areas and 

 volumes. It is always interesting for a student to 

 come across problems which bear some relation 

 to practical life — housekeeping, the profits or 

 otherwise arising from publishing books, the value 

 of the time of a man of business who finds it pays 

 him to take taxicabs. A suggestion that may be 

 offered is that there might be some problems on 

 the calculation of the amount of income-tax re- 

 claimable on a dividend when the dividend is 

 paid "free of tax." This problem is of great 

 practical importance nowadays. Though the 

 book was published quite recently, we have, so far 

 as we can see, problems about income-tax at 

 pre-war figures (c/. pp. 126, 150), but possibly a 

 sign of the times is the problem (p. 154) on British 

 and German rifles. 



(4) Dr. Phillips has the very laudable purpose 

 of making of the differential calculus "only a brief 

 text suitable for a term's work," so as to leave 

 "for the integral calculus, which in many respects 

 is far more important, a greater proportion of time 

 than is ordinarily devoted to it." We should 

 have expected, then, a suggestive and "intui- 

 tional " introduction to the calculus. We find, 

 however, a treatment rather late in the course of 

 rates of change, velocity, and acceleration in 

 straight and curved paths, which seem by far the 

 most stimulating subjects to anyone approaching 

 the calculus for the first time. It is not strictness 

 of logic that banishes what we would call sugges- 

 tiveness and pedants would call " rough-and-ready 

 methods." For example, in the treatment of 

 Rolle's theorem (p. 94) we have two pictures 

 pretending to show what may happen if the first 

 derivative of f{x) is discontinuous. In both pic- 

 tures the points in question are points for which 

 this derivative does not exist Other defects in 

 logic are the way in which "the limit of a func- 

 tion " is treated (pp. 5-6) : it should, we think, be 

 pointed out that the value of a function at a definite 

 point need not necessarily be the limit of neigh- 

 bouring function-values at this point. Also the 

 difficulty about higher differentials when a variable 

 is changed, is not satisfactorily put (p. 30). This 

 book should be stringently revised before it is put 

 into a student's hands, and then its shortness 

 might combine with accuracy towards making it 

 a good text-book. <f> 



OUR BOOKSHELF. 



An Introduction to the Physiology and Psychology 



of Sex. By Dr. S. Herbert. Pp. xii + 136. 



(London: A. and C. Black, Ltd., 1917.) Price 



35. 6d. net. 



Knowlkdge, it has been said, is not virtue, but 



it is often on the way to it ; and we agree with 



Dr. Herbert that the time has come for franker 



and fuller sex-instruction. In regard to sex, it 



cannot be said that ignorance is bliss ; it often 



leads quite gratuitously to vice and to discoloured 



views of one of the great facts of life. As a 



medical man. Dr. Herbert is able to deal with 



difficult subjects in a very matter-of-fact way, and 



