64 



SCIENCE 



[N. S. Vol. XXXVII. No. 941 



(p + <ip) (f — dv) = K. 



Now V — dv in place oi v -{- dv is just the sort 

 of error we have constantly to warn the fresh- 

 man against. The increment dv may be nega- 

 tive, but should not be written as — dv. The 

 author finds the correct result dp/dv <= — p/v 

 incorrectly from an incorrect equ.ation. On 

 page 67 there is this choice bit : " At this point 

 (such as P) there is a sudden change of direc- 

 tion; it is therefore called a point of inflec- 

 tion." A fine definition ! How could the au- 

 thor have made more errors in so short a 

 sentence ! On page 86 we find : " It must not 

 be supposed, however, that the series obtained 

 by differentiating a convergent series term by 

 term is also convergent. Thus the series 



1 -f a;i -j- a;i-2 -|- a;i-2-3 -j- a;i-2-3-4 + ... 



is convergent for | a; ] < 1, but the series 



1 -\-2x + 6x5 + ■■■, 



obtained by differentiation, is divergent for 

 all values of x." Now if there is any one fact 

 better known or more fundamental than that 

 a power series which converges is differentiable 

 term by term and yields a convergent series, 

 we fail to know what it is. This sort of mis- 

 take can arise only when ignorance is blatant 

 enough to talk about matters of which it is so 

 completely ignorant that it does not even rec- 

 ognize its ignorance. No author can wholly 

 avoid errors, but here they are too many and 

 too gross for any charitable inference. 



But this book is intended for chemists, and 

 in justice it should be judged chiefly upon 

 what it does for them, what it gives them that 

 they need, what it spares them that for them 

 would be superfluous. Here we must ad- 

 mit that we think the work a great success. 

 To the mathematician, the physicist or the 

 electrical engineer the total omission of all 

 reference to the circular functions and their 

 inverses would seem incomprehensible. But 

 the chemist has no need of oscillating func- 

 tions; his phenomena run one way. The re- 

 straint that the author has exhibited in leav- 

 ing entirely aside the trigonometric functions 

 is therefore highly commendable. Again, the 

 author uses differentials in differentiating and 



gives a tolerably full account of partial dif- 

 ferentiation, of the total or exact differential, 

 and of circuit integrals. These matters are of 

 great importance to the chemist. Moreover, 

 though his work is chiefly elementary calcu- 

 lus, it somewhat justifies the more general 

 title Higher Mathematics by the introduction 

 of methods of interpolation, extrapolation, ap- 

 proximation formulas and the like, and it 

 finds place on almost every page to appeal to 

 the chemist by selecting exclusively for its 

 applications problems which actually arise in 

 that subject. 



The titles of the chapters will give an idea 

 of the scope of the text. Functions and limits, 

 rate of change of a function, differentiation 

 of algebraic functions, maximum and mini- 

 mum values of a function, exponential and 

 logarithmic functions, partial differentiation, 

 interpolation and extrapolation, the indefinite 

 integral (two chapters), definite integrals, ap- 

 plication of the definite integral, differential 

 equations (two chapters), and appendices con- 

 taining the theory of quadratic equations, the 

 solutions of systems of linear equations by de- 

 terminants, approximation formulas, and a 

 tabulation of the exponential and natural 

 logarithmic functions. As has been stated, 

 everywhere are found detailed and vital appli- 

 cations to chemistry, to which the list of en- 

 tries in the index bears ample witness. The 

 student who masters the text will do so with 

 the fullest appreciation of its use to him and 

 will attain a knowledge sufficient for most of 

 his needs, albeit if he wishes to read such 

 highly mathematical works as Gibbs's papers 

 he must pursue his studies somewhat further. 

 For the class for whom it is designed the book 

 is far more useful than the ordinary text on 

 calculus. 



Edwix Bidwell Wilson 



Massachusetts Institute of Technology 



Allgemeine Biologie. Vierte umgearheitete 

 und erweitete Auflage. Von Oscar Hert- 

 wiG. Jena, Gustav Fischer. 1912. Pp. 

 787, mit 478, teils farbigen, Abbildungen in 

 Text. 

 The appearance of a fourth edition of this 



