184 
as learned, His six chapters range from elemen- 
tary arithmetic to differential equations, and touch 
upon such things as friendly numbers, magic 
squares, the transcendency of 7, Tartaglia’s 
rhymed rule for solving a cubic, and so on. Alto- 
gether, the book is written in a light and elegant 
style, reminding us of Lucas; it is neither so 
technical, nor so critical, as its title might sug- 
gest. 
Sometimes we are inclined to think that the 
author is poking fun; for instance (p. 50), he sug- 
gests that o (zero) is the initial letter of 
It is scarcely necessary to say that the Greeks 
adopted the decimal notation, including zero and 
the nine other digits, after it had been invented 
by Eastern (probably Indian) mathematicians ; and 
that our zero is almost certainly an enlargement 
of the dot which is still used by Oriental printers 
(unless the dot is a contraction for an older circle). 
“Bernouilli,” instead of Bernoulli, occurs so 
often that it can scarcely be condoned as a mis- 
print; “Neper” we let pass, as a traditional mis- 
spelling ; otherwise the names of authors seem ta 
be correct. 
important fact is that the author, besides being 
ovoev ! 
This is a small matter; a much more | 
| (5) Die 
interested in the discoveries of the ancients, is | 
fully awake to the merits of the moderns. For 
instance, we have the modern definition of “func- | 
tion”; references to the modern theories of irra- | 
like. 
“débutants en mathématiques.”’ 
it is certainly not one of formule; so it must be 
judged as one of ideas, and since it begins with a 
quotation from Plato’s “ Republic,” we may sup- 
pose that this is what the author means. 
gratulated, because the ideas which he suggests 
are eternal, though the forms under which he 
presents them are merely those which seem for 
the present the most convenient and suitable. One 
great advantage of the historical treatment of 
the subject is that it shows how what we may call 
the machinery of the subject has been improved 
and simplified. 
Prof.. Boutroux promises. us. another volume 
dealing mainly with analytical geometry, mathe- 
matical logic, and infinitesimal calculus; it will 
also deal with complex quantities and_ series. 
Teachers will note that the range of the whole 
work approximately covers the course of general 
mathematics in the science faculties of the higher 
educational bodies in France. 
ele 
NO) 2321, VOE. «924 
tional numbers, of functional equations, and the | 
ee 
It is interesting to see from the preface that 
this book is intended to be a “repertorium ” for | 
As a repertorium | 
of industrial organic analysis. 
_ series of commercial organic products have been 
_ selected, and detailed instructions are given of 
the analytical processes employed in determining 
From this point of view, the author may be con- | 
NATURE [APRIL 22; gom4 
ANALYTICAL AND SYNTHETICAL 
CHEMISTRY. 
(1) Industrial Organic Analysis: for the Use of 
Technical and Analytical Chemists and Stu- 
dents. By Paul S. Arup. With a foreword by 
Prot. J. C. Irvine. Pp. xii-- 3405 (Mendon® 
J. and A. Churchill,1913.)' Pricé 7s: 6d. net: 
(2) A Text-book of Quantitative Chemical Ana- 
lysis. By Dr. A.C) Cumming’ and Dr Se 
Kay. Pp. xi+382. (London: Gurney and 
Jackson; Edinburgh: Oliver and Boyd, 1913.) 
Price 7s. 6d. net. 
(3) The Sugars and their Simple Derivatives. By 
Dr. J: E. Mackenzie. Pp. xvi++242. (Londow 
Gurney and Jackson; Edinburgh: Oliver and 
Boyd, tgr3.). Price’7s. 6d. net. 
(4) The Silicates in Chemistry and Commerce: 
including the Exposition of a Hexite and Pen- 
tite Theory and of a Stereo-chemical Theory of 
General Application, By Dr. W. Asch and Dr. 
D. Asch. Translated, with critical notes and 
some additions, by Alfred B. Searle. Pp. xx+ 
456. (London: Constable and Co., Ltd., 1913.) 
Price 21s) mete 
Elemente der  siebenten Gruppe des 
periodischen Systems: aus Abegg’s der an- 
organischen Chemie. Vierter Band. Zweite 
Abteilung. Herausgegeben von Dr. Fr. Auer-_ 
Pp. x+904. (Leipzig: S. Hirzel, 1913.) 
Price 26 marks. 
HIS volume is intended for the use of 
students who, having received a 
grounding in theoretical and practical chemistry, 
are desirous of gaining an insight into the methods 
Eight typical 
bach. 
the industrial value of the materials under con- 
sideration. The subjects chosen include coal and 
coke, coal tar and its distillation products, the 
_petroleums, the fatty oils and fats, soap, milk, 
| butter, starch and its degradation products, flour, 
| barley, malt, and the preservatives and colouring 
matters introduced into foods. One very valu- 
able feature of the book is the references to the 
larger manuals and special monographs given at 
the end of each chapter. 
In the foreword Prof. Irvine deals with the 
controversial topic of the college training of in- 
dustrial chemists. The subject was not new 
thirty years ago, and although the discussion is 
' unending there is really no general problem to be 
faced; still less is there any general solution to be 
provided. Experts differ, and the weighty 
opinions of Prof. Martin (sic) Bogert, President 
