;56 



NA TURE 



[August 9, 1894 



crystallographic type belongs to such a regular arrangement, 

 one type of symmetry, that presenied by dioptase, is missini; ; 

 and it seems that, in this ca^e at least, the meroiymmetry can 

 only be accounted for by ths meroiymmetry of the particle, or 

 something equivalent to it, if the definition of re^jularlty su;;- 

 gested by Sohncke is to be accepted. It was recognisei by 

 Sohncke that each of his point-systems can be regarded as a 

 composite Bravais system, one of the latter being repeated in 

 various positions corresponding with the symmetry of the paral- 

 lelepiped itself. 



More recently, Schbnflies has made a more general hypo- 

 thesis still — namely, that in each substance, whether its crystals 

 be completely or partially symmetrical in facial development, 

 the particles are not of a single kind, hut of two kinds, related 

 to each other in form in much the same way as a right-hand 

 glove and a left-hand glove. With this hypothesis he finds that 

 all the thirty-two known types are accounted for without any 

 specialisation of the characters of the particle, and that no other 

 type of symmetry is mathematically possible. 



It now only remained to discover that Prof. Hessel had 

 already arrived at the thirty-two types of crystallographic sym- 

 metry by mathematical reasoning more than sixty years aijo ; 

 his work, being far in advance of his time, appears to have 

 attracted no attention, and the memoir remained unnoticed 

 till more than half a century afier its publication. 



Starting from Sohncke's definition of a regular point-system, 

 and proceeding, though independently, by a method which 

 closely resembles that of the regular partitioning of space by 

 Sch'>nl1ies, Mr. William Barlow has given in a paper just 

 issued a general definition applicable to all homogeneous 

 structures whatever, and ha*; shown that every such homo- 

 geneous structure falls into one or other of thirty-two types of 

 symmetr)', coinciding exactly with the thirty-two types of 

 crystal-symmetry. He points out that each of ihose homo- 

 geneous structures which possess planes of symmetry or centres 

 of symmetry does so by reason of its having an additional 

 properly beyond mere homogeneity, namely, that if we dis- 

 regard mere orientation, it is identical with its own image in a 

 mirror. Mr. Barlow (uither discovers that every one of the 

 Sohnckian point-systems can be geometrically constructed by 

 finite repetition of some one of a certain ten of them. 



Lord Kelvin, who, with characteristic versatility, has lately 

 enlightened us with his researches on Molecular Tactics, has 

 quite recently attacked another problem of the same group, and 

 has sought to discover the most general form of cell which shall 

 be such that each cell encloses a single point of a Bravais 

 system, while all the cells resemble the parallelepipeds, of 

 which we have already spoken, in being equal, similar, simi- 

 larly orientated, and in completely filling up space, lie finds 

 that in the general case the cell can have at most fourteen walls, 

 *hich may be themselves either plane or curved, and may 

 meet in edges either plane or curved. Having regard, 

 however, to the limited time at our disposal, we may 

 hesitate before following Lord Kelvin into his curious and 

 many-walled cells. 



The deduction of the thirty-two types ol symmetry by mathe- 

 matical reasoning was also made independently by both Gadolin 

 and Viktor von Lang thirty years ago from the law of ration- 

 ality of indices ; while Fedorow pomts out that the method of 

 dedu;tion recorded in the recent German treatise of Schonflies 

 is remarkably similar to the one independently published by 

 himself in Russia. Both Curie and Mmnigerode have also 

 lately given comparatively brief solutions of the problem. 



Nor must I omit to mention to you the elaborate memoir 

 dealing with the symmetry of parallelepipedal point-systems 

 which wa« written by the lite Prof. Henry Stephen Smith, 

 whose too early death this University has so much re.ison to 

 deplore. To the outer world he was perhap< best known as one 

 of the moit perfect mathematicians of the age, but those who 

 hod the good fortune to find themselves among his pupils will 

 always Ireasnrc up in their memory r.ather the kindly courtesy, 

 the warm sym.aihy of the man, than the genius, however 

 ansccndeni, of the maihemaiicion. 



To sum up this part of ihe subject — it is now c.itabli-hed that 

 a definition of the regularity of a point system can be so frame<l 

 that thirt)-lwo, and only thiriylwo, types of symmetry arc 

 mathematically po<>ible in a regular system, and that these arc 

 identical with the type* of symmciry that have been actually 

 obterved in crystals, or are inferred by analogy to be crystal- 

 lographically possible. 



NJ. J 293, VOL 50] 



It remains for subsequent investigators to determine what the 

 points of the .sysem re.illy correspond to in the crystal ; .accord- 

 ing 10 Schonflies, the physicist and the chemist can be allowed 

 in each crystal absolute control within a definite elementary 

 region of space, and the cryslallographer is only entitled to 

 demand that the features of this region are repeated throughout 

 space according to one or other of the thirty-two types of sym- 

 metry already referred to ; or, what appears to be the same thing, 

 the crysiallographer requires mere homegeneity of structure. 



Simplicily of Indicts. — We have seen that the jilanes con- 

 taining points of a regular point-system have rational indices. 

 But there still remains unaccounted for the remarkable fact that 

 the indices of the natural limiting faces, and also of the cleav- 

 age-planes of a crystal are not merely whole numbers, but are 

 in general extremely simple whole numbers. Bravais and his 

 followers have -sought to .account for this by the hypothesis 

 that both the natural limiting planes and the cleavage-planes 

 are those planes of a point-system which are most densely 

 sprinkled with points of the system. Curie and Liveing, in- 

 dependently of each other, have been led to the same result 

 from considerations relative to capillary constants. Sohncke, 

 however, pointing out that there are many cases— for instance, 

 calcite — where an excellent cleavage-plane is rarely a limiting, 

 plane, suggests that his generalised point-system is more 

 satisfactory than a Bravais system in that not only the density of 

 the sprinkling must be had regard to, but also the tangenti.al 

 cohesion of the particles in the plane, and that in his system 

 these may be independent of each other ; while Wullf remarks 

 that Sohncke's arrangement is identical with th.t'. of Bravais for 

 the anorthic system, where the same objection holds, and he 

 denies the legitimacy of the reasoning by which the hyoothesis 

 of a relation between the density of the sprinkling of points on- 

 a plane and the likelihood of the natural occurrence of the 

 plane as a limiting face is supported. 



Compltxily of Indices. — Doubtless, however, crystal faces 

 are observed of which the symbols involve indices far e.xceeding 

 6 in magnitude — so complex, in fact, that one is tempted to 

 doubt the rigidity of the experimental proof that indices are 

 necessarily rational. Often, though the numbers 'are high, their 

 ratios difler by only small amounts from simple ones. A most 

 patient and detailed study of such faces was made for danburite 

 by the late Dr. Max Schuster of Vienna, and the results were 

 brought by him some years ago to the notice of this Section. 

 From careful examination of similar faces in the case of quartz, 

 Molengraaf has been led to conclude that it is extremely pro- 

 bable that such faces are of secondary origin and have been the 

 result of etching ; they would in such case correspond, not to 

 original limiting planes, but to directions in which the crystal 

 yields most readily to solvent or decomposing influences. 



O/'lical Ckaracters. — Passing from the purely geometrical' 

 characters of crystals to the optical, we may in the first place 

 remark that the relationship between crystalline form and cir- 

 cular polarisation discovered by llerschel in the c.ise of quarti,. 

 has been generalised since the issue of Whewell's Report. We 

 now know that many crystallised substances belonging to 

 different systems give circular polarisation, and that all of them 

 are merosymmelrical in facial development or structure ; further, 

 they belong to types of symmetry which have a common feature, 

 though this is only a necessary, not a sulficieni, condition. 



The importance of the discovery of the dispersions of the 

 mean lines has already been referred to. 



We may recall attention to the fact noticed by Reusch that when 

 cleavage-plates of biaxal mica are crossed in pairs and the |>airs 

 arc piled one upon another in similar positions, the optical figure 

 yielded by the combination appro.aches nearer and nearer to that 

 of aun.axal crystal the thinner the plates and the more numerous 

 the pairs : in the same w.ay, by means of triplets of plates, 

 each plaic being turned through one-third of a coinplelc revolu- 

 tion lioin the position of the preceding one, it is found possible 

 to closely imitate the optical figure ol a right-handed or a left- 

 handed circularly polarising crystal. 



And it h.as been observed that repeated combinations of 

 dilTcrcntly orientated pans actually occur in cryst.als. Large 

 crystals of potassium ferrocyanide, for example, are really cocn- 

 pobite, and the diflercnt parts are differently orientated ; on the 

 one hand, a thick slice may give an optical figure which is 

 uniaxal ; on the other hand, a thin slice shows two optic axe 

 inclined to each other at a considerable angle. 



It has been suggested that the circular polarisation of quarl/ 

 and other crystals is due to a spiral molecular arrangement 



