464 WILLIAM HALLOWES MILLER. 



volume of only eighty-six pages, which the author modestly called 

 " A Tract on Crystallography." 



Miller began his study of crystallography with the same materials 

 as Naumann ; but, in addition, he adopted the beautiful method of 

 Franz Ernst Neumann of referring the faces of a crystal to the sur- 

 face of a circumscribed sphere by means of radii drawn perpendic- 

 ular to the faces. The points where the radii meet the spherical 

 surface are the poles of the faces, and the arcs of great circles con- 

 necting these poles may obviously be used as a measure of the angles 

 between the crystal faces. This invention of Neumann's was the 

 germ of Miller's system of crystallography ; for it enabled the Eng- 

 lish mathematician to apply the elegant and compendious methods of 

 spherical trigonometry to the solution of crystallographic problems ; 

 and Professor Miller always expressed his great indebtedness to Neu- 

 mann not only for this simple mode of defining the position of the 

 faces of a crystal, but also for his method of representing the relative 

 position of the poles of the faces on a plane surface by a beautiful 

 application of the methods of stereographic and gnomonic projection. 

 This method of representing a crystal shows very clearly the relations 

 of the parts, and was undoubtedly of great aid to Miller in assisting 

 him to generalize his deductions. 



From the outset. Professor Miller apprehended more clearly than 

 any previous writer the all-embracing scope of the great law of crys- 

 tallography. He opens his Treatise with its enunciation, and from 

 this law as the fundamental principle of the subject the whole of his 

 system of crystallography is logically developed. Beyond this, all that 

 is peculiar to Miller's system is involved in two or three general 

 theorems. The rest of his Treatise consists of deductions from these 

 principles and their application to particular cases. 



One of the most important of these principles, and one which in the 

 Treatise is involved in the enunciation of the fundamental law of 

 crystallography, is in its essence nothing but an analytical device. As 

 we have already stated, Weiss had shown that, if a : b : c represent 

 the ratio of the interce])ts of any plane of a crystal on the three axes 

 X, y, and z respectively, the intercepts of any other possible plane must 

 satisfy the proportion — 



A : B : C= ma: nb : pc 



in which tn, ?i, and p are simjile whole numbers. The irrational 

 values a, b, and c are fundamental magnitudes for every crystalline 



