January 13, 1905.] 



SCIENCE. 



65 



modified, restricted or completed. It thus be- 

 came necessary to rewrite textbooks on an- 

 alysis or to prepare new ones more in harmony 

 with the new teachings. In this way arose 

 the new edition of Jordan's ' Cours d'Analyse ' 

 and Harnack's edition of Serret's ' Calcul,' 

 as well as the new works of Stolz, ' Allgemeine 

 Arithmetik,' and ' Grundziige ' ; Tannery, 

 ' Theorie des fonctions d'une variable'; Dini, 

 ' Fondamenti per la teorica delle funzioni di 

 variabili reali.' 



In England and America more progressive 

 teachers have felt for some time the need of 

 a modern text-book on the calculus, which is 

 at once rigorous and elementary. The task 

 of writing such a work is not easy. On the 

 one hand, it is necessary to avoid the worth- 

 less and even vicious forms of i-easoning which 

 mar so many elementary treatises and which 

 are simply intolerable to one educated accord- 

 ing to modern standards of rigor. On the 

 other hand, the author must not introduce 

 subtilties of reasoning and logical refinements 

 beyond the needs and comprehension of those 

 who are to use the book. 



The volume under review is an attempt to 

 solve this difficult problem. To our mind the 

 efforts of its author have been abundantly 

 crowned with success. In perusing Dr. Gran- 

 ville's book one feels throughout that the 

 author has in mind the requirements of mod- 

 ern rigor. The demonstrations, it is true, 

 often rest on intuition; but this is necessary 

 in a first course, as all will admit. They are, 

 however, usually correct as far as they go, and 

 free from the defects we have mentioned 

 above. We believe the present volume is 

 eminently a safe book to put in the hands of 

 the beginner. He will get no false notions 

 which afterwards will have to be eradicated, 

 with much difficulty; he will, on the other 

 hand, acquire a considerable acquaintance 

 with the principles of the calculus and a good 

 working knowledge of its methods. 



We make now a number of criticisms and 

 suggestions. 



The definition of limit given in § 29 is not 

 the one given by Cauchy and Weierstrass and 

 now universally accepted. Looked at care- 

 fully, we see it supposes that all variables are 



functions of an auxiliary variable, the time. 

 This leads to unnecessary complications in 

 the definition of the limit of a function in 

 § 32. We believe the strict Weierstrassian 

 definition should be given and used. As an 

 aid to comprehension, the author's notions in 

 these articles might prove useful. In § 34 the 

 notion of a graph is explained; but not with 

 sufficient care, to our mind. How is the reader 

 to know from their graphs that x and log x 

 are continuous functions? The three proper- 

 ties of the exponential function given in this 

 article result from their arithmetical proper- 

 ties and not from their graph, as the author 

 seems to imply. 



The definition of the derivative given in 

 § 41 is not satisfactory ; what the author really 

 defines is the differential coefficient at a point. 

 It is their aggregate that forms the derivative. 



In § 55 the author has avoided an error 

 which is very prevalent. His passage to the 

 limit is, however, not completely justified. He 

 has yet to show that 



,. Aw At) ,. All Ai' 

 lim — • = lini ■ • lim 



The demonstration in § 56 should, it seems 

 to us, be replaced by a simpler one. The au- 

 thor obtains the equation 



and then remarks : if dx/dy + 0, we have 



dx dx 

 Jy 



He should see that there can be no need of 

 making the further assumption, dx/dy + : 

 for if it were, the equation (1) could not exist. 



In § 133 the author introduces a double 

 limit without any explanation. As such 

 limits are used in connection with double in- 

 tegrals, § 231, seq., they should be explained 

 with care. The footnote on page 194 is un- 

 intelligible to us and certainly will give rise 

 to misapprehension. 



The theory of total differentiation does not 

 meet our approval at all. The author has 

 treated the subject from the standpoint that 

 the variables x^, x.,, ■ ■ • x „ are all functions 



