August 9, 1894] 



NATURE 



355 



advantages ; and in his treatise published in 1839 — a treatise 

 which is a masterpiece of mathematical terseness and simple 

 elegance — he gave the methods of crystallographic calculation 

 which render the advantages of the symbol particularly mani- 

 fest. It may be here remarked that in that treatise the rationality 

 of the anharmonic ratios of any four tautozonal planes of a crystal 

 was first made known, and the properly was largely used in 

 the simplification of the methods of calculation : the fact that 

 the fraction was of the knd uhich had betn already termed an 

 anharmonic ratio, however, had escaped the attention of the 

 author. 



But the change of a method of notation, like a change in the 

 system of weights and measure?, involves such serious practical 

 difficulties that many jears passed away before the Millerian 

 symbol received abroad the consideiation which it deserved. 

 Now, at last, no conlinenlal text-book of mineralogy tails to 

 introduce the Millerian indices, even if the symbols of Levy or 

 of \aumann are given in addition ; and it is evident that 

 within a few more years the mineralogist will be completely 

 relieved from the tiresome necessity of translating each crys- 

 talline symbol into another form to make it intt-lligible to him, 

 and the student will be able to make a more advantageous use 

 of the time which has been hitherto devoted to acquiring a 

 mastery over a second and unnecessary form of crystallographic 

 tiotation. For this result credit is largely due to Prof. Grolh, 

 -of Munich, whi^se adootion of the Millerian symbol in the 

 Zeilichrift fiir Kiyslallografhic has done much to bring home 

 its advantages to the foreign worker. It is to be hoped that 

 Prof. Grolh will earn the further gratitude of students by 

 encouraging the adoption of the true Millerian symbol in the 

 still outstanding case of the Rhonihohedral System. 



Rjlionality of Indices ami the Law of Zones. — It may here 

 te pointed out thai, although the importance of zones for the 

 simplification of crystallographic calculation had been recog- 

 -nised by Weiss, it was only later that Neumann proved that 

 the fact that all possible crystal faces can be derived by means 

 of the intersection of zones is a necessary consequence of the 

 rationality of the indices ; thai, indeed, the law of zones is 

 mathematically identical with the law of rationality. To the 

 same able phy-icist and mathematician we owe the develop- 

 ment of the method of stereographic projection now in common 

 use by cryslallographers for the representation of the poles 0' 

 •crystal faces. 



Symmetry. — We have said th.at the recognition of si.s systems 

 of crystallisation was a result of consideration of the lengths 

 and mutual inclinations of certain lines called axes. Now, it 

 'had long ago been remarked that any one face of a crystal is 

 accompanied by certain others similarly related to the geometric- 

 ally similar parts of what may be regarded as a fundamental 

 figure : such a group of concurrent faces is called a simple 

 form. It came to be recognised, too, that all the faces of such 

 a form can be geometrically derived from any one of them by 

 ■repetition, according to certain laws of symmetry, and that the 

 same laws of symmetry are binding for every simple fiirm or 

 ■combination of forms exhibited by crystals of Ihe same sub- 

 stance. Hence it came to be perceived, though very slowly, 

 that the essential differences of the systems of crystallisation 

 ore not mere differences of lengths and mutual inclinations of 

 lines of reference, but are really differences of symmetry. Ever 

 since his appointment to the professorship of Mineralogy in this 

 University, now thirly-eii,'ht years ago, Mr. Maskelyne has been 

 persistent in directing atteniion to the importance of symmetry, 

 •and such importance now receives universal recognition. 



Thirty-t'oo Types of Symmetry in Crystals. — Hut in each 

 system of crystallisation it becomes necessary to recognise both 

 completely and partially symmetrical types. In the latter, the 

 symmetry is in abeyance relative to various planes or lines which 

 in other crystals of the same system are active as planes or 

 axes of symmetry. But this abeyance of symmetry is itself 

 found to be subject to a law, for .all planes or axes of symmetry 

 which are geometrically similar are either simultaneously active 

 or simultaneously in abey.ance. By means of this law relating 

 to partial symmetry, it has been inferred that altogether thirty- 

 two types of symmetry are possible in the six crystalline 

 ■systems. 



The possible existence of these thirty-two types of symmetry 

 of crystals is thus an induction from observation : the question 

 naturally arises as to why only these thirty-two exist, or are in- 

 ferred by analogy to be possible. Axes of symmetry are ob- 

 served, round which faces of crystals are symmetrically repeated 



NO. 1293, VOL. 50] 



by twos or threes or fours or sixes ; why is it that in crystals no 

 axis of symmetry is ever met with round which the faces are 

 symmetrically repeated by fives or sevens ? A few words as 

 to how this most important problem has been attacked and 

 solved may be of interest. 



We know that the characters of a crystal relative to any line 

 in it vary with the direction of the line, but are the same for all 

 lines parallel to each other. Such a property will result, if we 

 imagine with Bravais that in a crystal elementary particles are 

 arranged at equal distances from each other along every line, 

 and are similarly arranged in all those lines which are parallel 

 to each other ; the distances separating particles being, how- 

 ever, in general diflferent for lines which are inclined to each 

 other. Such an arrangement of particles is termed parallelepi- 

 pedal : space may be imagined to be completely filled with 

 equal and similarly disposed parallelepipeds, and an elementary' 

 particle to be placed at every corner or quoin of each. Further, 

 each particle is regarded, not as being spherical, but as having 

 different characters on its different side ; and the particles must 

 be similarly orientated — that is, have similarly sides in similar 

 positions. 



Now, it will be seen on an examination of a model or figure 

 that with such an arrangement any plane containing three par- 

 ticles will contain an infinite number, all arranged at the 

 corners of parallelograms. Further, .any such plane will clearly 

 have whole numbers for the indices which fix its position, for 

 along any line the distance between two particles is by hypo- 

 thesis a whole multiple of ihe common distance between any 

 two adjacent ones in the same line. Thus the first great 

 crystallographic law — the law of the rationality of the indices — 

 is an immediate consequence. 



In the next place, it was found that the possible modes of 

 symmetry of arrangement of the particles of such a system 

 depend on the form of the paralleliped, and that any possible 

 arrangement of the particles must present a symmetry which is 

 identical with one or other of the six completely symmetrical 

 types already referred to. And calculation shows that any 

 other mode of grouping — a repetition by fives or sevens, for 

 example — round an axis of symmetry, would involve the pre- 

 sence of planes having irrational indices ; and this according to 

 the first law is impossible. 



The abeyance of symmetry, however, met with in the par- 

 tially symmetrical types required the aid of an auxiliary hypo- 

 thesis — namely, that the abeyance of symmetry belongs to the 

 particle itself, and not to the arrangement of the particles. 



But the parallelepipedal arrangement imagined by Bravais is 

 unnecessarily special. Our actual observations of physical 

 characters relate not to single lines of particles, but to groups of 

 parallel lines of particles : the identity of character observed in 

 parallel directions is thus not necessarily due to actual identity 

 of each line with its neighbour, but may be due to statistical 

 equality, an equality of avenages. If, for example, a plane 

 were divided into regular hexagons, and a particle were placed 

 at each corner of each of these figures, the physical properties 

 of the system of panicles would be the same along all lines 

 parallel to each other as far as experiment could decide, and yet 

 the arrangement of the particles in the plane, though possibly 

 crystalline, is not that of a Bravais system. In any straight line 

 passing along the sides of a series ol the hexagons, the particles 

 will not be equidistant Irom each other : they are in equidistant 

 pairs, and ihe two nearest particles of adjacent pairs are twice as 

 far from each other as the particles of the same pair. 



Sohncke accordingly suggested a more general definition 

 than that of Bravais for the regularity of the arrangement, a 

 definition which had been proposed some years before by 

 Wiener — -namely, that the grouping relative to any one particle 

 is identical with that relative to any other. This definition 

 admits of the possibility of the hexagonal arrangement just 

 mentioned ; further, it allows of the orientation of the particles 

 themselves being different in adjacent lines. Following a 

 mathematical process which had been already employed by 

 Jordan, Sohncke deduced all the possible modes of grouping 

 consistent with the new definition, and for a lime was under the 

 impression that the types of symmetry found by him to be 

 mathematically possible are exactly identical with those already 

 referred to ; and this without introducing the auxiliary hypo- 

 thesis relative to partial symmetry of the elementary panicles of 

 merosymmetrical crystals, except in cases of hemimorphism. 

 It was, however, pointed out by Wulff, who has himself made 

 valuable contributions to the iubject, that though no unknown 



