238 WATURE 
various special types of determinants with their 
corresponding properties. 
Too much stress can scarcely be laid on the 
matter of notation; again and again Dr. Muir, in 
his notes on a paper, brings out the essential 
meaning of its results by translating it, so to 
speak, into a more perspicuous language. On the 
whole, the merit of choosing a suitable notation 
appears to be mainly due to Cayley, although 
Cauchy and Grassmann must not be forgotten; 
their symbolical expressions are often extremely 
convenient, and are in every case nearly as econo- 
mical as the current ones, although not quite so 
convenient typographically. The crowning advan- 
tage of the current notation is that |ap,| can be 
used as a symbol for a general determinant of n 
rows and columns. 
As an expounder of symmetrical algebra, Cayley 
is unsurpassed, both in power and elegance; it is 
not surprising, therefore to find that nearly fifty 
of his papers are referred to. Mainly, these give 
examples of the application of determinants to 
particular problems; but as a contribution to the 
general theory we have the initial discussion of 
skew determinants, and various important notes 
on orthogonants. 
On the other hand, Sylvester occupies a very 
curious position. The properties of compound 
determinants are a very important part of the 
general theory; some of them were discovered by 
Cauchy, and Sylvester more than once asserts that 
he had discovered a general theorem, including 
Cauchy’s as particular cases, and virtually com- 
prehending all properties of determinants. Now, 
when we turn to Sylvester’s published papers on 
compound determinants, we are more than usually 
baffled by his inveterate habits of inaccuracy, 
stating theorems without proof, and hastily jump- 
ing at conclusions. Even Dr. Muir confesses 
(p. 197) that he cannot find or conjecture this all- 
embracing theorem. He does, however, point out 
what Sylvester has stated which is really true, and 
has put it into a more intelligible form; in fact, 
all his notes on Sylvester will be very useful to 
those who study the original papers. 
Turning to other writers, we have Hesse and 
Jacobi, who have left their names permanently 
connected with the subject, developing it from the 
sides of geometry and differential equations; 
Hermite also, in his memoirs on ternary quadratic 
forms, contributes to the theory of orthogonants. 
Oddly enough, there is only one paper by Clebsch, 
although (at a later period) he applied determinants 
to geometrical theory with a mastery equal to 
Cayley’s. Among the Italians we may specially 
note Brioschi and Faa di Bruno. 
As to the arrangement of Dr. Muir’s book, we 
NO. 2219, VOL. 89] 
[May 9, 1912 
may note that chapter i. contains references to two 
unimportant papers omitted in vol. i.; chap. ii. is 
on determinants in general, and the remaining 
fourteen deal with special types, such as alter- 
nants, &c. 
There is a list of authors, showing at a glance 
to which parts of the subject each person has con- 
tributed; to find out any particular theorem, the 
reader has to rely on the table of contents, and 
this means looking through one or more sections. 
But from the nature of the case this is about all 
that could be expected; it is practically impossible 
to index a set of analytical theorems. 
As a mathematical history, so far as can be 
judged by one who has not gone over the same 
ground in detail, Dr. Muir’s work seems irre- 
proachable in the cardinal points of proportion, 
completeness, and lucidity. He displays no bias, 
and his critical remarks are always to the point; 
while his analyses of the various papers are re- 
markably concise as well as clear. G. B. M. 
CHEMISTRY OF CELLULOSE. 
Die Chemie der Cellulose unter besonderer 
Beriicksichtigung der Textil- und Zellstoffindus- 
trien. By Prof. C. G. Schwalbe. Zweite 
Halfte (Schluss des Werkes). Pp. 273-666+ 
xii. (Berlin: Gebriider Borntraeger, 1911.) 
Price 14.80 marks. 
HIS second volume is closely consistent with 
ae: the former, reviewed in NaTuRE, vol. 
Ixxxv., p. 67, and as a completed work we must 
assign it a high place in technical literature, prin- 
cipally, however, from the point of view of the 
German technologist, who now finds himself for 
the first time in possession of a full bibliography in 
propria lingua of this rapidly growing subject. 
Of an exhaustive and most carefully edited biblio- 
graphy, in which equal prominence is given to all 
recorded investigations, and the authors allowed to 
tell their own story, little more can be said to 
commend it to the specialist as an indispensable 
addition to his library equipment. 
A more important function, however, of author- 
ship in this field is to influence the rising genera- 
tion of students and workers, and for this a bold 
and critically constructive handling of the material 
was called for. 
We may illustrate by selecting a section of the 
work dealing with the quantitative estimation of 
cellulose (pp. 613-624). 
It should have been shown that the analytical pro- 
cesses available are sharply defined by the general 
principles of classifications now recognised; and 
that they are ‘““normal” so far as they are based 
upon reactions, specific to the non-cellulose, that 
is, to its characteristic groups, and quantitatively 
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