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In the article Mathematics we find what we would naturally look for, namely 

 a "Synopsis of Existing Developments of Mathematics" (p. 882) containing a 

 general account of what articles are to be consulted for information on any 

 particular branch of pure or applied mathematics. But the method adopted in this 

 synopsis is thoroughly unsatisfactory. The classification adopted in the Royal 

 Society's International Catalogue of Scientific Literature is, after some disparaging 

 remarks, such as that "it would be unfair to criticise it [the classification] from an 

 exacting philosophical point of view," laid down as fundamental, the headings are 

 copied out in a column and a half of matter, and those articles of the Encyclopedia 

 which seemed to the copyist, on a casual inspection, to be related closely to those 

 headings are enumerated. The mention of the articles of the Encyclopedia made 

 here can only have been due to a casual inspection, because some of the articles, 

 such as Infinitesimal Calculus and Number, are not mentioned where they 

 palpably ought to be. Any one who has even a slight acquaintance with what these 

 articles contain would see this. Further, there are omissions and inexact references 

 which point to careless proof-reading. But the most serious omission is the omission 

 to draw the editor's attention to the lack of any article in the Encyclopedia 

 which deals with functional equations and operations or the " functional calculus." 

 This subject is one in which a new and great conception was introduced into fairly 

 modern mathematics— namely, that of a calculus of functional operations instead 

 of real or complex numbers — and it could not be urged that this calculus came into 

 importance only after the eleventh edition of the Encyclopedia was planned. 

 The solution of functional equations occupied a fairly large part in British mathe- 

 matical literature during the first half of the nineteenth century, and was mentioned 

 at the end of the late Prof. Cayley's article Function in the ninth edition of the 

 Encyclopedia Britannica ; but afterwards was put into the background by the 

 rapidly growing importance of the theory of functions of a complex variable, and 

 only fairly recently have these researches stepped into the foreground once more. 

 To a mathematician, the mention of the names of Volterra, Frechet, Hilbert, 

 Pincherle, and Hadamard is enough to indicate the subject. 



It is difficult to understand why the author of the article Mathematics could 

 have failed to notice the absence of such an important article, since Frege, whose 

 splendid logical work is mentioned by him, pointed out quite clearly {Function und 

 Begriff, Jena, 1891, p. 31), but in rather different words, the place that a calculus of 

 functions must occupy as an important stage in the building of mathematics. It 

 seems to be a neglect of duty on the part of the author of the article Mathe- 

 matics not to have pointed out to the editor of the Encyclopedia this notable and 

 unfortunate omission, and not to have insisted on the gap being filled. 



The last part of the article Mathematics is devoted to " The History of 

 Mathematics" (pp. 882-3). If, as the author decided, "mathematics" is to be 

 defined as "the science concerned with logical deduction from the premisses of all 

 reasoning," and "the science . . ." here means, as it probably does— since it does 

 with Russell — "the class of propositions . . .," it is difficult to see how "mathe- 

 matics " can have a history at all. The number 2 does not have a history : though 

 it is quite possible that there may be a history of our discovery of the number 2. 

 Thus, in this article, the author commits the elementary blunder of speaking of two 

 things by the same name : a class of logical propositions, and our process of 

 discovery of these propositions. Even if we suppose that necessary explanations 

 of this somewhat obscure procedure are added, the brief and unsatisfactory sketch 

 of history, which begins with a platitude and a false statement : "the history of 

 mathematics is, in the main, the history of its various branches. A short account 



