NATURE 



485 



THURSDAY, MARCH 22, 1900. 



THE CAMBRIDGE CRYSTALLOGRAPHY. 

 A Treatise on Crystallography. By W. J. Lewis, M.A.' 

 Professor of Mineralogy in the University of Cam- 

 bridge. Pp. xii + 612 ; 553 figures. (Cambridge : 

 1 University Press, 1899.) 



IT is now more than sixty years since Prof. Miller, of 

 Cambridge, published his famous " Treatise on Crys- 

 tallography." At that time crystallography was a new 

 science, and studied by few. Since that date it has 

 entered into the educational programme of most uni- 

 versities, and at Cambridge is now (combined with 

 mineralogy) a recognised Tripos subject, pursued by a 

 { considerable number of students. 



Miller's successor, under whose hands the Cambridge 

 School has developed its present activity, now issues a 

 volume the substantial dimensions and weighty contents 

 of which are worthy of a university publication ; this 

 volume and Maskelyne's "Morphology of Crystals" 

 provide English students with a pair of adequate text- 

 books on the geometry of crystals. 



Prof. Lewis preserves in his book all Miller's results 

 and methods ; his treatment of the subject, however, re- 

 sembles that of Maskelyne and other recent authors, in 

 attaching primary importance to the subject of symmetry ; 

 the general relations of crystal symmetry are, in fact, 

 briefly stated in the third chapter ; although the mathe- 

 matical development of these principles is reserved for 

 Chapter ix. Chapters iv. to viii., being devoted to the 

 law of rational indices, the relation of zones, the methods 

 of drawing and projecting crystals, and the anharmonic 

 ratio of four planes, are almost necessarily an exposition 

 of the work of Miller, Mohs and Naumann. 



It is to Chapter ix. that the critical student will first 

 turn for possible novelty of treatment ; here he will find 

 a series of thirteen propositions establishing the nature, 

 order, number and disposition of axes and planes of 

 symmetry ; a footnote on p. 119 gives for the first time 

 the interesting information that the trigonometrical 

 proof now familiar to all students is due to Prof. Story- 

 Maskelyne, and was given by him in lectures in 1869, 

 two years before the publication of Gadolin's classical 

 memoir, in which a similar proof was independently em- 

 ployed. The author calls the reader's attention to the 

 assumption that an axis of symmetry is parallel to a 

 possible edge and perpendicular to a possible face of the 

 crystal, and points out that this cannot be proved for a 

 three-fold axis. The fact is commonly ignored, but does 

 not affect the main object of the argument, which is to 

 show that four-fold and six-fold axes are the only axes of 

 symmetry of degree higher than three which are possible. 

 Euler's theorem is then employed to show how axes of 

 symmetry may be combined, and how two or more such 

 axes involve the presence of others ; and the number 

 possible in a crystal is deduced from the expression for 

 the area of a regular closed polygon on a sphere. At 

 this point complaint may fairly be made of a serious 

 omission, for the whole course of the argument in 

 Chapter ix. prepares the reader to expect that the thirty- 

 two classes of crystals are about to be established, whereas 

 NO. 1586, VOL. 61] 



the following chapters which contain the detailed descrip- 

 tion of the various classes are not pVeceded by any proof 

 that they alone are possible. A link is wanting in the 

 logical sequence, and since the principle of merohedrism 

 is expressly rejected (see p. 259), there remains no prin- 

 ciple of development or classification to correlate the 

 thirty-two classes. 



The author, in his preface, expresses the opinion that 

 the accurate drawing of crystals develops the student's 

 power of solving crystallographic problems, and his 

 book differs from other text-books above all in the 

 attention paid to the construction of diagrams, and in 

 the number of examples by which this subject is illus- 

 trated. An early chapter describes the methods of 

 crystal drawing, including orthographic and clino- 

 graphic projections, and they are constantly illustrated 

 in the subsequent chapters. The greater portion, the 

 systematic section of the book, consists of a detailed 

 discussion of the various classes ; each of these is 

 treated in a very complete manner ; formulae and 

 methods of calculation are established ; numerous pro- 

 positions concerning the elements of symmetry and 

 their mutual relations are proved, many of them new ; 

 crystals of many substances are figured and described, 

 and (a special feature of the book) a number of fully 

 worked examples are given as exercises in computation 

 and drawing ; this affords opportunity for the descrip- 

 tion of several specimens in the University collection. 

 Excellent also in its wealth of detail is the long chapter 

 on twin crystals, which follows the systematic section, 

 and here again each substance described is treated as 

 an exercise in crystallographic determination, calcula- 

 tion and drawing. To gain an idea of the unusually 

 elaborate, as well as practical, manner in which these 

 various problems are treated, let the reader refer, for 

 example, to the geometrical propositions concerning 

 rhombohedral crystals on pp. 365-403, to the four 

 pages relating to Gypsum in Chapter xii., and to the 

 nine pages devoted to the twinning of Cassiterite in 

 Chapter xviii. 



In the systematic treatment of the thirty-two classes, 

 the less symmetrical systems are treated first, an arrange- 

 ment introduced by Groth in a non-mathematical treatise, 

 but one which introduces the most difficult calculations 

 at the outset ; unfortunately also, the somewhat arbitrary 

 sequence adopted in the present book does not bring the 

 most symmetrical (holohedral) class to the end, or even 

 to the same place, in each system. 



It is really difficult to make an elementary treatise 

 on geometrical crystallography a readable book. The 

 principles of symmetry must be established by the aid of 

 the zone law, so that propositions on indices and anhar- 

 monic ratios must precede the description of the crystals 

 and their symmetry, and yet these propositions are 

 scarcely intelligible without some knowledge of the 

 crystals. Prof. Lewis makes no attempt to surmount 

 this difficulty— and, in fact, recommends his reader to 

 travel backwards and forwards] rather than to read 

 consecutively ; but he succeeds in his main object of 

 presenting the essential features of the science to a 

 student who is not required to possess more than ele- 

 mentary mathematical knowledge, and gives him a hand- 



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