66 
Notices respecting New Boohs. 
crystals and the angles they make with each other. The expressions 
which in this treatise have thus been obtained are remarkable for 
their symmetry and simplicity, and are all adapted to logarithmic 
computation. They are, it is believed, for the most part new.” 
It is not possible for us to give any detailed account of Professor 
Miller’s methods ; hut we may observe that each face of a crystal 
is determined by the portions cut off from the three axes of the cry- 
stal, and is expressed by a symbol (h k V) in which the indices de- 
pend upon these portions. When several contiguous faces have 
their intersections parallel, they may be considered as belonging to a 
zone ; and this zone is indicated by its symbol [u v w] . Some of the 
simplest methods* of determining the law of derivation of a proposed 
face consist in referring it to such zones. Thus if we have, given, 
the symbols of two zones [p q r], [u v w], the symbol of the face 
common to the two zones (h k 1) is known from the equations 
7i = vr — wq, ^ = wp — ur, 7 = uq — vp. 
The mathematical student of crystallography cannot fail to be 
delighted with the completeness and symmetry with which, in Pro- 
fessor Miller’s Treatise, formulae of this kind are obtained for each 
system of crystallization ; and with the great and instructive variety 
of examples to which they are applied. It will be found, by atten- 
tion to these examples, that the methods employed in this work are 
not only, analytically speaking, the most general and symmetrical, 
but also practically the most compendious and convenient for the 
determination of the laws of derivation of any proposed form. 
We cannot help thinking, however, which we do with regret, 
that this book, mathematically so admirable, will be a sealed book 
to a large body of crystallographical students. It is written with a 
rigorous brevity, w'orthy of the ancient mathematicians ; a quality, 
in itself, doubtless, a beauty, but one of those stern beauties which 
repel, rather than attract, common beholders. There is not a single 
phrase in which the author shows any sympathy for those of his 
readers who have not been disciplined in mathematics to the extent 
which his investigations require. And this requisite discipline is, 
in truth, not slight ; for though the knowledge which he presup- 
poses in his reader does not go beyond the doctrines of spherical 
trigonometry, no one can follow Prof. Miller’s reasonings with any 
facility, except his habits of mathematical generalization and abs- 
traction have been well matured. And even the method of indi- 
cating the positions of the faces of crystals by their poles upon a 
sp)here of projection, although it much simplifies the calculation, ob- 
scures our conception of the relations of the crystalline form ; at 
least it does this when we are first called upon to employ the me- 
thod, and before it is become familiar to us. This, however, is an 
inconvenience attendant ujion most simplifications of physical pro- 
blems, and we speak to regret rather than to blame it in the present 
instance. But jicrhaps we might venture to express a wish that the 
])ractical rules for the calculation of crystals had been separated 
from the mathematical investigations which contain the demonstra- 
