274 



NATURE 



[December 28, 191 1 



of the subject. In the case of the present book the 

 unevenness. though not entirely absent, is scarcely 

 apparent. It fakes the form of slight inad«!quacy on 

 the mat! ' side, appearing in the brief nnd not 



entirely <iinR account of the measurement of 



seoiiation intonsiiies, and more prominently in the 

 long and otherwise excellent chapter on "The Inter- 

 relations of Mental Function,'* where the important 

 method of correlation now in general use for the 

 measurement of these interrelations is not mentioned, 

 and is only referred to indirectly by the quotation, in 

 one sentence, of a research somewhat out of date 

 and certainly not representative. 



In other respects, Prof. Pillsbury has written an 

 exceptionally useful and effective book, for which one 

 can safely predict a high degree of popularity among 

 students. The earlier chapters are devoted to a very 

 interesting analysis of the general characteristics of 

 consciousness, such as attention, retention, and asso- 

 ciation, and the descriptions of perception, memory, 

 reasoning, &c., are all based upon this earlier account, 

 and form the later chapters of the book. The dis- 

 cussions of attention, memory, and imagination, 

 reasoning and work, fatigue and sleep, are exception- 

 ally good, and sum up concisely a great many of the 

 results of modem experimental work on these topics. 

 Perception is not so well done. It is surely incorrect 

 to say that "perceptions always involve centrally 

 aroused sensations or memories, as well as sensa- 

 tions " (p. 157). Evidence from pathology and animal 

 psychology makes dead against this view. Inherited 

 structure of the nervous system, as Prof. Stout sug- 

 gests, " explains " the function better in such cases, 

 and even in normal human psychology these addi- 

 tional mental images and ideas are largely mythical 

 and unidentifiable by introspection. 



Another small point : in the chapter on cutaneous 

 sensations no mention is made of the distinctness of 

 sensatk>ns of warmth, coolness, and light touch from 

 those of heat, cold, and heavy touch respectively, 

 although this result, based upon the work of Drs. 

 Head, Rivers, and Sherren, is now three years old and 

 well authenticated. A reference to it would not have 

 conflicted with the elementary character of the book. 

 This is one instance among several of the tendency to 

 ignore important work done by English psychologists, 

 which is more pronounced than it might be in some 

 American and Continental writers. 



At the end of each chapter of the book there is a 

 series of "exercises" in experimental introspection, 

 for which one is grateful. They add considerably to 

 its value for class-work. W. B. 



NUMBER AND QUANTITY. 



Grandeurs et Nomhres — Arithmdtique Ginirale. By 



Prof. E. Dumont. Definitions et Propriet<5'S fonda- 



mentales des Grandeurs g^om^triques et de leurs 



Mesures; Nombres Naturels, Qualifies, Complexes, 



Ternions et Quaternions. Pp. xvii + 275. 



(Paris: A. Hermann et Fils, 191 1.) Price 10 francs. 



T N mathematics, as in other affairs, a great move- 



■»- ment happens now and then which is a kind of 



revolution; and whenever this occurs there is sure to 



be a body of stalwart veterans, who refuse to budge 



NO. 2200, VOL. 88] 



from their old position, however untenable or worth- 

 less it may be. 



M. Dumont 's book is an illustration of this familiar 

 fact. So far is he from accepting the modern view 

 of mathematical science that it stirs him to a pas 

 sionate revolt; he invokes the shades of the old 

 masters, from Archimedes down to Hamilton, an'i 

 denounces the logicians as conspiring to m.i' 

 matics a barren pastime, instead of the i: 

 of the natural philosopher. 



In order to justify his protest, he has atu...,,.^^ . 

 give a theory based on the definition of a number n- 

 the ratio of two quantities. As might be expected 

 he constantly begs the question, and makes a variet-. 

 of tacit assumptions, far more complicated than t' 

 really necessary in applying mathematics to ph;. 

 phenomena. For instance, he says (p. 7), "To mu 

 tiply a quantity G by the number A, /A is to appi 

 to G the same treatment which, applied to A, product 

 A,." What is "the Same treatment"? G may L- 

 a length, and A, A, volumes or masses; how can "th- 

 same treatment " be defined without begging the whol' 

 question at issue? A little further on we read tha' 

 "it is not always possible to multiply a quantit;. 

 {grandeur) by a number, as we shall see in the theory 

 of quaternions." Here our Don Quixote betrays som-- 

 sense of discomfort in his antiquated armour ; the reason 

 appears subsequently. Length is defined (p. ii.^ 

 as " une grandeur lin^aire relative et orient^e . . . qu 

 se d^veloppe dans deux sens opposes, k partir d'un^ 

 origine arbitraire, et dans une direction variable. 

 On p. 195 we read, "the ratio of two vectors or c 

 two angles, thus conceived, is called a quaternion c: 

 quaternary number"; after this it is not surprising 

 to find a treatment of quaternions quite needless!} 

 complex, with definitions stated as theorems, and 

 formulse of the most repellent kind ; moreover, we 

 have a separate chapter on "ternions," which are only 

 special cases of quaternions. To make confusion 

 worse confounded, the author writes (a, b denoting 

 vectors) ajh as the equivalent of b-'a, and calls it 

 " the ratio of a and b," w^hile a : b is the equivalent 

 of ab-', and is called " th^ quotient of a by h." 

 Almost immediately before this we read : " Quant k 

 ajh, on ^crira aussi volontiers a/b = b-'a que 

 ajb = ab-'"l 



M. Dumont expressly denounces the views of hi? 

 distinguished countrj-men Jules Tannery and M. 

 Hadamard; they need no better justification than is 

 unconsciously given by this attempt to prove them 

 wrong. At the same time, some of us will partly 

 sympathise with M. Dumont, although entirely dis- 

 agreeing with his doctrine. It would indeed be 

 lamentable if mathematics w-ere to be entirely divorced 

 from its physical applications, and simply cultivated 

 as an intellectual game. Fortunately, there is no 

 reason to fear that this will ever be the case; elec- 

 trical theory atone will continue to attract many o: 

 the ablest mathematicians of the time. And however 

 fully we may admit that arithmetical analysis is in- 

 dependent of measurement, we cannot ignore the fact 

 that measurements must precede any physical theorj- 

 of a mathematical kind. Moreover, the data properly 

 belonging to any physical science are not themselves 



