TRANSACTIONS OV SECTION C. 633 



risle, Ilaliy, Molis, aod Weiss that the position of any single face of any crystal 

 can be exactly defined by means of two sets of quantities : firstly, three lines or 

 axes, of which the lengths and mutual inclinations are characteristic of the substance 

 itself; secondly, three whole numbers or indices, rarely rising higher in magnitude 

 than the nutnber 6 ; further, an empirical arrangement of crystals into systems 

 had been based by Mohs and Weiss on the relative lengths and inclinations of the 

 axes. And a long series of observations of the optical characters of crystals had 

 revealed to Brewster the fact that the boundaries oi the classes of optically isotropic, 

 uniaxal and biaxal crystals form part of the boundaries of the empirical systems. 

 But whereas only three optical classes of crystals had been recognised, it was certain 

 that there were at least four geometrical systems, and it was a matter of contro- 

 versy as to whether the independence of two others should not be regarded as 

 geometrically established. 



The first important discovery following the issue of Whewell's lleport was one 

 which proved that the two doubted systems are natural ones. It was found by 

 Herschel and Neumann that the biaxal crystals are not optically similar, as had 

 hitherto been supposed, but are of three kinds. In crystals of one kind — for 

 example, barytes — the two lines bisecting the angle of the optic axes internally and 

 externally, and a third line perpendicular to both, are constant in direction in the 

 crystal whatever the colour of the light ; in a second kind — for instance, selenite — 

 only one of these lines is constant when the colour varies ; in a third kind — for 

 instance, axinile — none of the three lines has any constancy of direction. And these 

 three kiuds of biaxal crystals correspond exactly in their facial development to the 

 three systems of crystallisation of which the independence had already been asserted 

 by some crystallographers on geometrical grounds. From this time the arrange- 

 ment of crystals into the six systems has been regarded as a natural one ; and the 

 optical method based on the figures seen in plates when examined in convergent 

 polarised light has been in constant use, and is an invaluable aid in the determination 

 of the system of crystallisation. 



Crystalloijraphic Notation. — For a simple method of expressing the relative 

 positions of crystal faces by a symbol, crystallographers are infinitely indebted to 

 the late Professor Miller of Cambridge. The symbols introduced by Mohs, Weiss, 

 L^vy, Naumann, and the modification of the latter suggested by Dana, though 

 interesting, are not to be compared for legibility, pronounceability, or utility in 

 calculation, with the simple symbol which is associated with the name of Professor 

 Miller. Though the symbol was not invented by him, he was the one who, so to 

 saj', gave it life. He discovered and made known its many 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 calcula- 

 tion which render the advantages of the symbol particularly manifest. It may be 

 here remarked that in that treatise the rationality of the anharmonic ratios of anj' 

 four tautozonal planes of a crystal was first made known, and the property was 

 largely used in the simplification of the methods of calculation : the fact that the 

 fraction was of the kind which had been 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 a system of weights 

 and measures, involves such serious practical difficulties that many years passed 

 away before the Millerian symbol received abroad the consideration which it 

 deserved. Now, at last, no Continental text-book of Mineralogy fails to introduce 

 the Millerian indices, even if the symbols of L6vy or of Naumann 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 crystalline 

 symbol into another form to make it intelligible 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 crystallo- 

 graphic notation. For this result credit is largely due to Professor Groth of 

 Munich, whose adoption of the Millerian symbol in the ' Zeitschrift fiir Krystallo- 

 graphie ' has done much to bring home its advantages to the foreign worker. It is 

 to be hoped that Professor Groth will earn the further gratitude of students by 



