June 2, 1898] 



NATURE 



99 



tinction cannot be recognised in systematic bacteriology. 

 Prof. Migula invites the botanist to follow the methods 

 of the medical bacteriologist in the study of bacterial 

 diseases of plants, which he regrets in most cases has 

 been undertaken in a slipshod and careless manner. He 

 gives a number of plant diseases, said to be due to 

 bacteria, to expose the manner in which the subject 

 hitherto has been approached. Anaerobiosis and phos- 

 phorescence, the thio-bacteria and ferruginous bacteria 

 form the subjects of the next few chapters, and then 

 we come to an interesting and concise account of the 

 nitro-bacteria ; the volume is concluded by two short 

 chapters on the influence of heat- and light on bacterial 

 growth. 



We may disagree with the author here or there, 

 but we, and especially the medical bacteriologists, must 

 welcome the appearance of this work. The volume 

 is the result of Prof. Migula's own labours and studies 

 pursued for many years with true German industry, and 

 this enhances its value considerably. It is well written, 

 and the language is not particularly difficult ; the literary 

 references at the end of each chapter are excellent. It is 

 impossible to read the book without regretting that the 

 second volume has not yet appeared. Six plates ac- 

 company the text, but, by an oversight, plates iv. and v. 

 have been placed and numbered in wrong order, and 

 must be transposed. A. A. Kanthack. 



THE PHYSICAL PROPERTIES OF 



CRYSTALS. 



Die fundamental en physikalischen Etgenschaften der 



Krystallc in elementarer Darstellung von Dr 



Woldemar Voigt, o.o. Professor der Physik an der 



Universitat Gottingen. (Leipzig: Veit and Co., 1898.) 



PROF. VOIGT is well known for his researches into 

 the physical properties of crystals. Not only is 

 the mathematical theory of large parts of the subject due 

 to him, but the experiments on which the theory is built 

 were largely made by himself and Prof Riecke, and 

 many of the instruments used were invented or improved 

 by him. His latest contribution to the science is a little 

 book, half-way between a popular exposition and a 

 technical treatise. Last year Prof. Voigt lectured at 

 Gottingen on crystals to teachers in the upper classes of 

 secondary schools, and these lectures form the basis of 

 the book before us. 



Prof. Voigt is a mathematician, and though the mathe- 

 matics is here reduced to a minimum, he assumes a 

 knowledge of the elements which his hearers doubtless 

 possessed. A command not of facts and formuke, but of 

 mathematical and physical ideas and terms is required 

 for a satisfactory study of the book. In particular, 

 some familiarity with the use and transformation of co- 

 ordinates is essential. In England, where the knowledge 

 of elementary mathematics is widely spread, this little 

 volume ought to find many readers, and a good 

 translation is to be desired. 



We have before us no mere text-book, but a book with 



an idea and a plan. Round the idea the facts are 



grouped, and one is carried on naturally from one set of 



properties to another. .'Vfter a preliminary chapter on 



NO. 1492, VOL. 58] 



the symmetry of crystal forms, the leading idea is 

 developed in the second chapter. Prof. \'oigt points 

 out that, in investigating the relation between cause and 

 effect, it is allowable to treat not only effects but causes 

 as states of matter. For instance, electric phenomena 

 produced by heat may be regarded as the relation 

 between the temperature and the electrical state of a 

 body. Temperature is determined by a scalar quantity, 

 and the electrical state of any particle by a vector. This 

 vector is, moreover, a so-called polar vector, i.e. one, 

 like a translation, whose components change sign when 

 the sense of all the coordinate axes is changed, in 

 contradistinction to a so-called axial vector, whose com- 

 ponents retain their signs. Temperature involving no 

 direction, the direction of the vector can only be deter- 

 mined by the crystalline structure, and we should expect 

 such a relation to be possible in acentric crystals pos- 

 sessing one single polar axis of symmetry, such as 

 tourmaline. In fact, the pyro-electric properties of 

 tourmaline have been known for 200 years. 



Besides scalars and vectors there is a third kind 

 of quantity, by which a state of tension or dilata- 

 tion is characterised. It is determined by a magni- 

 tude and a straight line, undetermined in sense. ProC 

 Voigt calls such a quantity a tensor, and three mutually 

 perpendicular tensors a. tensor-iripel, giving in the preface 

 his reasons for the adoption of a new term, and pointing 

 out that in doing so he is merely e.xtending the use of 

 the word in quaternions. By means of these three kinds 

 of quantities and their mutual relations, he is able to 

 classify, in the manner indicated, the different pheno- 

 mena. In every case we have two effects due to the 

 same cause, and the primary effect is taken to represent 

 the cause in its relation to the secondary effect. Each 

 chapter after the second exhibits such a relation. We 

 have an example of the relation between a scalar and a 

 tensor-tripel in that between temperature and deform- 

 ation, between a vector and a tensor-tripel in piezo- 

 electricity, and numerous examples of two vectors ; 

 elasticity is treated as a relation between two tensor- 

 tripels. 



The method gives more than a mere classification, as 

 the example shows. It enables us to say a priori 

 whether a given body, isotropic or crystalline, is capable 

 of exhibiting certain phenomena. In general the pheno- 

 mena which are a priori possible, are il posteriori known 

 to exist. In one case, however, referred to in Chapter iii., 

 a set of phenomena represented by the relation between 

 a scalar and an axial vector, theoretically possible in a 

 large class of crystals, has never been observed, and 

 it remains open to question whether the failure to 

 observe pyromagnetic phenomena is due to an un- 

 known point of theory or to unsuspected difficulties 

 of observation. 



In the chapter on the symmetry of crystals, Prof. Voigt 

 takes three typical forms— Iceland spar (rhombohedron), 

 tourmaline, and quartz — and he derives the two latter 

 from the former by the simple process of joining 

 together two "half rhombohedra." In spite of three 

 excellent figures, the explanation would not be com- 

 prehensible without previous knowledge of the way in 

 which the rhombohedra are to be divided. Even the 

 simplest crystal forms are hard to understand without a 



