216 



SCIENCE 



[N. S. Vol. XXXII. No. 815 



• aries, nor in the directions of the axes which 

 lay indifferently to each other as well as to the 

 milky nucleus which appeared in the section 

 as a lot of air bubbles of different size. 



I venture to hope that my attempt will 

 -call forth similar researches and would be 

 ^lad if any other might have a chance to 

 conserve or to study some bigger or more 

 peculiar hailstones than I, and in this way 

 improve our deficient notions about the origin 

 ■of hail and the details of its formation. 



Boris 'Wed^beeg 

 Phtsical Labokatoet, 

 Technological Institute of Tomsk 



j a speculation in crystallography 



The conception of six systems of crystalli- 

 zation, which has been prevalent for nearly a 

 century, has doubtless impressed many stu- 

 dents of the subject as somewhat arbitrary. 

 Especially does it appear so when it assumes 

 four axes for the hexagonal system and only 

 three for each of the others. To be sure, these 

 axes are simply lines of reference, and the 

 systems are ingeniously formulated so as to 

 include all possible forms, which can be pro- 

 duced by the regular arrangement of particles 

 having uniform size and similar shape in each 

 particular case. While the scheme of classifi- 

 cation is comprehensive and practical, it has 

 really no more foundation in nature than the 

 Linnean orders and families of plants. 



The real misuse of these systems, however, 

 has been when the crystallographic axes have 

 been assumed to correspond to molecular 

 bonds, or to similarly related axes in each 

 molecule of a given substance, so that the 

 molecules of a crystal are conceived to be ar- 

 ranged in straight lines corresponding to 

 these axes. While this idea, which is often 

 used in text-books, may be helpful in the ex- 

 planation of crystals and of crystalline forms, 

 yet it can readily be shown to be arbitrary 

 and unnatural and therefore liable to mislead. 



The purpose of this paper ia to outline a 

 more rational explanation of crystal structure, 

 constancy of angles, cleavage and other phys- 

 ical properties. 



If we should use globules of uniform size to 



represent molecules, or ultimate particles of 

 an isometric substance, and allow them to 

 take their most compact form, as they would 

 by their mutual attraction, or under the in- 

 fluence of uniform external pressure, they 

 would not arrange themselves in lines corre- 

 sponding to rectangular axes, as is so often 

 indicated in crystallographic diagrams and 

 models. Instead of each one touching its 

 neighbor at six points, as it would in that case, 

 it touches at twelve points. Nature often 

 shows a similar fact in the globular cells of 

 organic tissues. When such are crowded to- 

 gether until the intervening spaces are ob- 

 literated, each cell takes the form of a rhom- 

 hic dodecahedron. 



We, therefore, take this form as a promis- 

 ing suggestion and look at it more carefully. 

 We see that if we draw lines through the 

 centers of opposite planes, or, if with globules 

 compactly arranged, we draw lines through 

 the centers of each and through opposite 

 points of contact, each will be traversed by 

 six lines, and if space be filled with such dode- 

 cahedrons, or with equal-sized globules, such 

 space will be traversed with straight lines 

 running in six directions. Every such line, 

 or axis, will form an angle of 60° with four of 

 the others and 90° with the sixth. 



If space permitted, we might show that by 

 assuming such an arrangement of isometric 

 molecules all the different planes of that 

 system may be as logically derived as they can 

 be from the commonly postulated arrange- 

 ment parallel with rectangular axes. Start- 

 ing with a single molecule, if we should add 

 successive layers of similar molecules equally 

 in all directions, the result would be a rhom- 

 bic dodecahedron. If the obtuse interfacial 

 angles, eight in number, were modified by one 

 plane, because of some variation of molecular 

 attraction, or because of different density of 

 the generating solution, octohedral planes 

 would appear. If, on the other hand, the 

 acute angles, of which there are six, should 

 be similarly modified, we should have cubic 

 planes, and so on through all the forms of the 

 isometric system. 



But, as we should naturally expect from the 



