386 



NATURE 



[February 23, 1899 



Jl 



Prof. Penfield's introduction to the study of crystallo- 

 graphy, in which the subject is admirably treated in sixty- 

 six pages of large print, will prove of not less interest to 

 teachers, as showing how the great fundamentals of the 

 modern presentation of the science can be dealt with in 

 a very moderate space. Questions like those of the pro- 

 jection of crystals, and the calculation of axial ratios from 

 gonioinctric measurements, which are well explained in 

 I'rof. Dana's text-book, are, of course, omitted in the 

 smaller summary of crystallography by Prof Penfield. 



The subject of the optical characters of minerals is one 

 which in the past has always proved to be of peculiar 

 difficulty to students of mineralogy. The important 

 memoir of Mr. Fletcher on " The Optical Indicatrix," has 

 recently led physicists and mineralogists to reconsider 

 the soundness of the postulates on which Fresnel based 

 his theories of the action of crystals on light ; and there 

 can be little doubt that the near future will witness as 

 complete a revolution in the nomenclature and methods 

 of physical optics, as that which we have witnessed in the 

 case of crystallography. 



.^t the present time, however, il seems desirable to 

 adopt the course followed by Prof Dana, and to lay before 

 the student both of the accepted methods of interpreting 

 observed phenomena in connection with the passage of 

 light through crystals of various kinds. We feel little 

 doubt, however, that a future edition of the " Text-Hook 

 of Mineralogy" will break altogether away from the 

 somewhat cumbrous and complicated terminology of 

 Fresnel — hampered as it is by unnecessary assumptions — 

 and that a more simple and rational method of treatment, 

 in harmony with the methods of Mr. Fletcher, will be 

 adopted in its place. The subject of physical optics finds 

 no place in the more elementary work of Prof Penfield. 



The second half of the "Text-15ook of Mineralogy" is 

 a very judicious abridgement of the sixth edition of Dana's 

 excellent "System of Mineralogy,"' the most important 

 features of which were described in this journal at the 

 time the work appeared. It is scarcely necessary to add 

 that the present book has been brought well up to date. 



The concluding portion of Prof Penfield's book is 

 made up of the well-known analytical tables for the deter- 

 mination of minerals. These tables have not only been 

 completely revised, but have now had incorporated in 

 ihem a great number of new species, including not a few 

 which are of very rare occurrence. This increase in the 

 number of species treated of has necessitated a complete 

 rearrangement of the tables. 



Prof Zirkel, in bringing out a new edition — the 

 thirteenth — of Dr. Naumann's well known " Elemente 

 der Mineralogie," has recognised equally with Profs. 

 Dana and Penfield the necessity for a complete change 

 in the mode of treatment of the crystallographic and 

 optical properties of minerals. .As time has not yet per- 

 mitted him to altogether rewrite the introductory portions 

 of this old standard treatise, he has contented himself 

 with issuing a revised edition of the second or systematic 

 portion of the volume. The excellent features of this 

 familiar text-book arc well maintained, and some improve- 

 ments are introduced into it, especially in the clearer and 

 fuller treatment of the mode of occurrence of the different 

 mineral species. We trust that the indefatigable editor, 

 who has so long kept Naumann's Ixjok in the tirst rank 

 NO. 1530. VOL. 59] 



of treatises of the science, may before long be able to- 

 supply us with that complete revision of the groundwork 

 of the subject which he contemplates. 



Jf)HN W. Jl'DIl. 



THE THEORY OF FUNCTIONS. 



Introdiiclion to the Theory of Analytic Functions. By 



J. Harkness, M.A., and F. Morley, Sc.D. Pp. xv 



-t- 336. (London : Macmillan and Co., Ltd., 1898.) 



A NOTICE of the "Treatise on the Theory of 

 Functions," by Profs. Harkness and Morley, 

 appeared in Nature during 1S94 (vol. xlix. p. 477). 

 The object of that work, as of Prof Forsyth's book on 

 the same subject, was to present a complete view of the 

 theory as a whole, and to follow out its various develop- 

 ments as far as space permitted. It would not be correct 

 to regard either of them as written for a student who 

 could be fairly described as a beginner. What Prof. 

 Klein somewhere calls "a certain ripeness of mathe- 

 matical judgment," which is just what a beginner does 

 not possess, would be necessary in a reader who, without 

 previous knowledge of the subject, could study such 

 volumes with profit. 



The new work by Profs. Harkness ami Morley, the title 

 of which is given above, is stated in the preface, and quite 

 justly staled, to be in no sense an abridgment of their 

 earlier and larger treatise. The authors say that their 

 aim in writing it has been purely didactic, and that the 

 book is intended to be an introduction to the subject 

 for a student with no previous knowledge of it. The 

 scope of the book will be best described by giving a 

 short account of what it contains. It commences with 

 an introductory chapter on ordinal numbers. The 

 second chapter explains the representation of a com- 

 plex number by means of an Argand diagram ; and 

 the third and fifth chapters deal at some length with 

 the correspondence established between two planes, 

 distinct or the same, by means of a lineo-linear equation 

 between two variables. The fourth chapter discusses 

 the logarithmic function from a special point of view. 

 Chapter vii. deals with rational algebraic functions. In 

 Chapters vi., viii. and ix. the idea of a limit, the concep- 

 tion of continuity, and the definition of convergence in 

 connection with an infinite series are introduced. The 

 conditions under which an infinite scries has the 

 properties of an ordinary sum are very completely in- 

 vestigated. Then follow five chapters which treat of 

 power-series, and of some of the properties of an analytic 

 function defined by a power-series and its continuations. 

 Chapter xv. considers the representation of an integral 

 function as a product of primary factors, each of which 

 has a single zero. Next come a chapter on the integra- 

 tion of a function of a complex variable, and three 

 chapters treating very briefly of the elliptic functions. 

 Chapters xx. and xxi. deal with some of the properties of 

 algebraic functions and with the construction and use of 

 Kiemann's surfaces in connection with them. The last 

 chapter gives some account of the method of Cauchyand 

 of the theory of the potential. 



Opinions will and must differ as to what should be 

 admitted into and what excluded from a book which is 



