July 7, 1892] 



NA TURE 



219 



about, even when he has reached the end of Book I., 

 which covers the ground of the subject usually called the 

 Dififerential Calculus : there are no illustrations, except 

 for one or two meagre geometrical applications, for the 

 mind to hold on by ; no diagrams, and no examples to 

 test the soundness of the student's knowledge. 



It is true that these collections of examples are 

 decried in certain lofty quarters of the mathematical 

 hierarchy ; but the humbler priests of the science, who 

 are in touch with the noviciate mind of human nature, 

 know their practical value ; and these collections of prob- 

 lems, formerly a feature of our text-books unknown 

 abroad, are now being extensively copied and adopted 

 in other countries. " In scientiis ediscendis prosunt 

 exempla magis quam praecepta " (Newton). 



The Second Book considers Functions of Complex 

 Numbers : we make another fresh start with the opera- 

 tions of Arithmetic, as it is called here ; not that any 

 resemblance can be traced to what generally goes by that" 

 name. In this book the questions of Convergency, of 

 Single- and Multiple-valued Functions, as illustrated by a 

 Riemann surface, and of their Zeros and Infinities, are 

 gone into at great length ; but at the same time the 

 reader will have an impression that the information is 

 given in a very condensed form, and that an attempt has 

 been made to give a brief resume of a subject which 

 requires a large volume to itself. 



This Morbid Pathology of the Mathematical Function, 

 as we may call it, requires a very clear, concise, and 

 cosmopolitan tereminology, which, as Mr. Cathcart points 

 out on p. 148, it does not yet possess ; it is unfortunate 

 that the nomenclature has mostly been formed originally 

 in the agglutinate German language, and in many cases 

 is only very imperfectly translatable. 



This part of the subject, although principally known to 

 us from the researches of later writers, such as Cauchy, 

 Riemann, Dirichlet, and Weierstrass, owes very much to 

 Gauss ; but Gauss deserves to lose the credit of priority, 

 from his baneful habit of bottling up his discoveries, after 

 announcing that he had obtained the solution, so as to 

 warn off all other investigators from his preserves of 

 research. 



The Integral Calculus is developed in Book III. ; here 

 also the treatment,though complete, is very condensed; and 

 but few simple problems and applications are provided to 

 show the use of the subject when the analysis is established. 

 The author never employs the hyperbolic functions, 

 although their use can be traced back to Newton 

 (" Principia," Lib. II., Prop, ix.) ; but in the reductions 

 of the integral of Y{x, v'R) where R is the quadratic 

 a-\-2bx -\- r.i-2,the use of sJR as the argument in conjunc- 

 tion with the circular and hyperbolic functions enables us 

 to present the different results which arise in a more 

 systematic manner than that employed in the present 

 work. A very short sketch is also given of the method 

 of reduction of the integrals when R is of the third or 

 fourth degree ; the elliptic integrals are now introduced, 

 but no mention is made of the tWx'pUc functions, introduced 

 by Abel by the inversion of the elliptic integrals. 



The Fourth Book, which treats of the integrals of 



complex functions and of the general properties of analytic 



functions, is probably the sole presentation of this modern 



and difficult subject in our language. To a mathematician 



NO. I 184, VOL. 46] 



of Mr. Cathcart's development the treatment will appear 

 very concise and elegant, but for our part we miss the 

 footholds afforded by the physical applications of the 

 general theorems of functions ; say to Hydrodynamics, 

 such as those recently published by Prof. W. Burnside in 

 the Proceedings of the London Mathematical Society, on 

 Riemann's Theory and on Automorphic Functions, 

 determined from their discontinuities. 



The book will recommend itself, as we said at the out- 

 set, to the advanced student, who pursues mathematical 

 study as an end to itself, by reason of the strict logical 

 order in which the subjects are presented ; but is this 

 strict logical order the most suitable arrangement for a 

 beginner } 



Herbert Spencer says that " in each branch of instruc- 

 tion we should proceed from the empirical to the rational." 

 In the operatic version of" Manon " the events are pre- 

 sented in chronological order; but in the original "Histoire 

 de Manon Lescaut " the story begins in the middle, so as 

 to excite the reader's curiosity as to the preceding events 

 which led up to the point at which the characters appear 

 on the scene. 



According to Prof. Harnack's preface, the present work 

 may be considered the operatic version of his lectures, 

 while the simple story would appear in the lectures de- 

 livered in the Dresden Polytechnicum to his technical 

 students, who required a knowledge of Analysis chiefly as 

 an instrument for the solution of mechanical problems. 



Mr. Cathcart explains in his Translator's Note the desire 

 he had to make these lectures accessible to the English 

 reader, and records the regret he felt at the news of the 

 death of Prof. Harnack, while engaged on a revision of 

 his notes for a new edition. The thanks of the mathe- 

 matical world are due to Mr. Cathcart for the care and 

 trouble he has taken in this valuable piece of work. 



A. G. Greenhill. 



ALTERA TIONS OF PERSONALITY. 



Les Alterations de la Personnaliti. Par Alfred Binet. 

 Bibliotheque Scientifique Internationale. (Paris : An- 

 cienne Librairie Germer Bailli^re et Cie., 1892.) 



IN what is in ordinary parlance called somnambulism, 

 or sleep-walking, the patient rises in the night, per- 

 forms a number of seemingly intelligent actions directed 

 to some special end, answers questions with regard to such 

 actions with a variable amount of coherence, returns to 

 bed, and generally, but not in all cases, wakes in the 

 morning with no remembrance of that which he has 

 done during the night. Such is somnambulism in its 

 narrower sense. It exhibits the individual in an abnor- 

 mal psychological condition, the actions performed in 

 this abnormal condition being generally unconnected in 

 memory with the normal sequence of events in waking 

 life. The word somnambulism is, however, now used in 

 a wider and at the same time more technical sense, being 

 applied to all cases where the individual, either spon- 

 taneously or through hypnotic suggestion, falls into an 

 abnormal condition distinguishable from the normal con- 

 dition of his or her waking life. It is with the alterations 

 of personality exhibited during the state of somnambulism 

 in this wider sense that M. Binet's volume chiefly deals. 



