188 



NATURE 



{7a7i. 7, 1875 



fundamental to crystallography. This may be expressed 

 by saying that the angles of a crystal are symmetrically 

 repeated. 



The study of crystallography in its aspect as the science 

 of chemical morphology thus resolves itself into the dis- 

 covery of the laws which regulate the repetition of planes, 

 the directions of which in space, and not their relative 

 magnitudes, result from that geometrical instinct which 

 guides the molecules of every individual substance as 

 they become colligated into the symmetrical structure of 

 a crystal. 



The lecturer then went on to point out that the features 

 of a crystal the symmetrical recurrence of which had to be 

 studied were the faces, the cds^cs, and the ijuoins (or solid 

 angles) ; and he entered on a general geometrical review 

 of the conditions under which faces in meeting produce 

 an edge, or a quoin, or a series of edges or of quoins ; 

 and after showing the mode by which the angular inclina- 

 tion of two faces was measured, he dilated on one in par- 

 ticular among the various modes in whicli faces might 

 meet, namely, that in which three or more faces intersect 

 with each other in the same line or edge, or in edges 

 parallel to the same line. For the crystallographer such 

 groups of planes possess the highest significance; a group 

 thus presenting parallel edges he denominates a zone, and 

 it is clear that the direction of the line to which all the 

 edges that can possibly be formed by the intersections of 

 any and every pair of the planes belonging to the zone is 

 indicated when we know the direction of any one of tliese 

 edges. A considerable part of the earlier among the 

 ensuing lectures will have to be devoted to the considera- 

 tion of this subject of zones : and the development of the 

 relations between the planes of a zone, under the restric- 

 tions imposed by a simple and beautiful law, will be found 

 to involve fundamental principles regarding the symmetry 

 which controls at once the morphological and the physical 

 properties of the crystal in such a manner that all the 

 systems, the symmetrical forms, and the general character 

 of the optical, thermal, magnetic, electric and mechanical 

 properties of the crystaUised substance hang, as it were, 

 suspended from that simple law by a chain, each link of 

 which is a simple deduction from the link in the argu- 

 ment immediately above it. 



Then taking a crystal of the mineral barytes. Prof. 

 Maskelyne pointed out that certain planes upon it were 

 repeated, some in parallel pairs, and others four times, 

 but also in pairs that were parallel, while all of these 

 planes presented the property already stated to be cha- 

 racteristic of a zone : their edges were parallel. Then, 

 supposing a lapidary's wheel to have been passed through 

 the middle of the crystal perpendicularly to all these 

 edges, and therefore perpendicularly to the faces them- 

 selves, he proceeded to deal with the profile of the 

 planes of the zone as they would be seen in such a 

 section. He first defined such a section as the plane 

 of the zone, or the zone-plane ; and characterised it 

 as a plane perpendicular to the edges of the zone. 

 Then drawing a figure to represent this profile or zone- 

 plane, he pointed out that two of the planes of the zone 

 being perpendicular to each other, he might draw two 

 lines through a point within the crystal and in the zone- 

 plane parallel to the traces of those two planes, and 

 therefore perpendicular to each other, and that now he 

 could use these lines as axes, or as an artificial scaffold- 

 ing, to which he could refer the traces of the other faces 

 of the zone, and by the aid of which he might determine 

 the relative directions of those faces. 



The circumstance already established by the scrutiny 

 of many crystals, namely, that the faces of the crystal 

 might be drawn nearer or further from a point within 

 the crystal indifferently, justified the lecturer in drawing 

 the traces of two of the faces in the zone so as to inter- 

 sect in the same point on one of the two axes thus chosen. 

 They would thus intercept on the other axis two different 



portions of that axis. Calling the former of these axes 

 Z and the latter X, we may say that the ratio of the 

 ntercept by either of the two planes on the Z axis to the 

 intercept on the X axis by the same plane is the tangent 

 of the angle formed by the trace of the plane in question 

 with that of the plane parallel to the axis of X, or the co- 

 tangent of the angle it forms with the trace of the plane 

 parallel to the axis Z. This tangent for the plane in 

 question, which gave an angle of 51° S' by measurement 

 for the angle on the axis A", had a value \'2\oi. The 

 other face of the zone, being represented by the line which 

 met the axis of .V at an angle of 68° 4', would thus yield a 

 corresponding tangent of 2'4834. It will be seen, there- 

 fore, that the ratios of the intercepts for the two planes 

 would be, for the first plane, 



the A' intercept : the .2" intercept :: i : r2407 

 for the second plane, 



the A' intercept : the Z intercept : : i : 2'4834 



If the first of these ratios be called that of a : c, the 



second will be that of a : 2c, i.e. of - ■ -. The co-tan- 



2 ■ I 

 gents of the angles would of course yield similar ratios for 

 the distances on the axes A' and Z a.t which the two planes 

 intersect with them : but the common intercept on the Z 

 axis would in this case be unity. The ratios would be 



.r^ntercept ^^^ ^j^^ ^^.^^ j 

 Z mtercept 



ane = 



o'8o594 _ a 



_ o'40267 



Ditto for the second plane = 



A third plane in the zone treated in the same way 

 would give an angle the tangent of which would lead to 



a ratio for the intercepts corresponding to - : - , and 



S 2 

 if the same process were extended to all the planes 

 in the zone, it would be found that all of them would 

 yield, by the simple process of measuring their in- 

 clinations and taking the tangents of their angles on 

 the plane represented by the axis A', values that may 



be represented by the proportion j '■ j, where a and c 



are in the ratio above determined, and where // and / 

 always are capable of representation by rational and 

 generally, nay, almost always, by very small whole num- 

 bers. This law thus simply enunciated for the faces of a 

 single zone, as referred to two axes parallel to two faces 

 of the zone here taken as perpendicular to each other, 

 will be found, when the faces of the crystal are referred to 

 three axes instead of two, not in the same plane, and also 

 when they are inclined to one another at other angles 

 than right angles, still to control the inclinations of the 

 (aces of the crystal, provided only that the axes X y Z 

 thus taken be lines of crystallographic significance, such 

 as lines p.irallel to edges formed by faces of the crystal ; 

 while the ratios a : l> : c represent the intercepts on those 

 axes taken in the order X i' Z o( a. fourth face of the 

 crystal and arc the numerators, while letters such as /i k I 

 stand for the numerical denominators in the fractions 

 that represent the ratios of the intercepts of any other 

 fifth plane of the crystal. Any three numbers in the 

 ratios a : b : c represent the intercepts on the axes of the 

 fourth or standard plane, and are called the parameters 

 of the crystal ; one parameter in particular being 

 generally taken as unity. The numbers by which 

 the parameters have to be divided in order to assign 

 the ratios of the intercepts to any fifth plane of the 

 system, namely, the simplest numbers expressive of the 

 ratios h : k : I, are called the indices of that plane ; and 

 when these indices are united into what is termed the 

 symbol of the plane, by being written in brackets as 

 {h k I), (321), &c., one understands by this that 



