290 



NA TURE 



[January 30, 1896 



to the teacher a method of escape from a very awkward 

 difficuhy. Prof. Maskelyne, for whose book on Crystal- 

 lography the method was devised, has the intention of 

 presenting the method in text-book form, and of making 

 it accessible and useful to the English student ; un- 

 fortunately the many distractions of a busy public life 

 have led to prolonged delay in the completion and issue 

 ■of that part of his crystallographic treatise. Prof. Groth, 

 however, has been quick to show that the method is one 

 which is practically useful both to the teacher and the 

 student ; in the course of years the idea of optical 

 ■elasticity will disappear from crystallographic text-books, 

 and an unnecessary obstacle will vanish therewith from 

 the path of the optical student. One word on a trivial 

 matter, that of nomenclature : Prof. Groth has translated 

 the term Indicatrix into Index-flache, on the ground that 

 the word Indicatrix is not acceptable in a Teutonic 

 language. To an Englishman it is not intelligible on 

 literary grounds why Index should be within and Indi- 

 catrix be without the limits of what is acceptable in 

 Germany ; nor were any objections to the term Indi- 

 catrix made by the German Professors (Konig and 

 Ambronn), by whom the Tract itself was translated into 

 German and rendered more accessible to the German 

 student. In any case the constant repetition of the word 

 ■" surface " in the proofs of the various geometrical pro- 

 positions would make " Index-surface" unacceptable for 

 •common use in English, and it would have been a simpli- 

 fication of nomenclature if the term Indicatrix {i.e. the 

 indicating surface) could have been adopted bodily in 

 the other European languages. 



The second great change is the total exclusion of the 

 Naumann symbols from the text. There was much to 

 be said for the use of the Naumann symbols in the 

 indication of a limited number of crystallographic forms, 

 but they are too complex in others, and are thus not 

 ^generally useful : further, in the specification of par- 

 ticular faces and in crystal calculations, the Millerian 

 symbol is infinitely superior, and is quite indispensable 

 for the more advanced student. Looking to the slight- 

 ness of the advantages of the Naumann symbol even in 

 the most favourable cases, it has seemed better to Prof. 

 •Groth to now completely dispense with its consideration ; 

 this saves much valuable time to the Professor in his 

 lectures, much space in the text-book, much unavailing 

 toil for the student. 



The third great change has involved a complete altera- 

 tion of treatment of the geometrical characters of 

 <:rystals. A century ago the Father of Crystallography, 

 Rome de I'lsle, introduced the idea of a primitive form, 

 and showed that all the crystal forms of the same kind 

 of substance are such as could be derived from a single 

 primitive form by similar alterations of all those parts 

 which are geometrically similar to each other ; the 

 octahedron and the tetrahedron, for instance, were 

 different kinds of primitive form, and were regarded as 

 mutually independent. Later on, the idea of a primitive 

 form was discarded by crystallographers, and that of 

 crystalline axes, systems and symmetry, was introduced. 

 This necessitated the recognition of symmetry as being 

 either complete or partial in each crystalline system ; and 

 the forms presenting incomplete symmetry were treated 

 as resulting from the suppression of half or three-fourths 

 NO. 1370, VOL. 53] 



of the faces of forms having a complete symmetry 

 proper to the system. Certain elements oif sym- 

 metry were regarded as being then in abeyance. One 

 educational disadvantage of this method is that the 

 student is almost led to imagine that the axes and planes 

 of symmetry have a physical existence anterior to that of 

 the matter itself, and to think that the inert matter is 

 compelled to arrange itself in a particular way through 

 the action of these pre-existing axes and planes. Or 

 again, he imagines that the tetrahedron was at one time 

 an octahedron, and that it only arrived at the tetrahedral 

 form through the excessive growth of four alternating 

 faces of an eight-sided crystal. In the present edition 

 Prof. Groth has followed a method suggested amongst 

 others by Fedorow, now the Professor of Mineralogy at 

 Moscow, and one which presents some analogy to that of 

 Rome de ITsle ; in that, for instance, it treats the 

 octahedral and tetrahedral symmetries as quite in- 

 dependent of each other. It has now been known for 

 some decades that thirty-two, and only thirty-two, 

 classes of symmetry are consistent with that law of 

 whole numbers which had been discovered by Haiiy to 

 control the positions of crystal faces. This limitation of 

 the classes of crystal symmetry was first established by 

 Hessel in 1829, but, owing to some extent to the repellent 

 form in which the reasoning was presented, did not then 

 attract the attention of other crystallographers. It was 

 re-discovered, however, by Axel Gadolin, and made 

 known by him (1866-7) in a very lucid memoir ; since 

 that time the law has been universally recognised. 



The thirty-two classes of symmetry can be grouped in 

 various ways ; and one method is that of the old so-called 

 " systems of crystallisation." In the new mode of treat- 

 ment, the six systems are merely conventional and have 

 no structural importance ; it is the thirty-two classes, not 

 the six systems, which are fundamental, and the classes 

 themselves are regarded as independent of each other and 

 co-equal, in importance. 



In his exposition of geometrical crystallography, Prof. 

 Groth starts from the simplest type of form, that in which 

 there is no symmetrical repetition at all, and thence 

 gradually advances to the most complex type, in which 

 the existence of a single face may involve the co-existence 

 of forty-seven others. This method, so long as it does 

 not involve trigonometrical calculation, presents no 

 greater difficulty than the usual mode of treatment ; 

 indeed. Prof. Groth asserts from practical experience that 

 the student in this way makes quicker progress in the 

 acquisition of knowledge. Each of the thirty-two classes 

 being regarded as independent of the rest, the idea that 

 a form may be hemimorphous, hemihedral, tetarto- 

 hedral or holohedral, is completely eliminated from the 

 science. In one respect the method actually adopted 

 seems to the writer to fail of the desired simplicity ; in 

 the case of an ordinary crystal belonging to the Anorthic 

 system, a crystal of albite for example, every face is ac- 

 companied by one which is parallel and opposite, and the 

 crystal is usually regarded as having a centre, but no 

 planes, of symmetry. For some reason not clear to the 

 writer, Groth, following Federow, rejects the idea of a 

 centre of symmetry, and regards such a crystal as being 

 one in which there is no centre of symmetry at all, and 

 every plane of the crystal, actual or crystallographically 



