_ January 6, 1923] 
NATURE II 

set forth in the account of two insects of the orchard, 
by Mr. R. E. Snodgrass, or to learn a lesson from 
Prof. Lameere’s lecture on the origin of insect societies. 
Or one might get practical hints on the suppression of 
insect pests by a better utilisation of birds from Mr. 
W. L. McAtee, or excite oneself over the adventures in 
the life of a fiddler crab, so delightfully told by Mr. 
O. W. Hyman. But we must reluctantly pass all 
these, and pass too Mr. Bassler on the little Polyzoa, 
Mr. Gilmore on the mighty horned dinosaurs, Mr. 
Maxon on the Botanical Gardens of Jamaica, Mr. 
Safford’s strange study ‘of the narcotic Daturas, and 
the richly illustrated articles on Hopi Indians, modern 
Mexicans, and racial groups and figures, by Fewkes, 
Genin, and Hough. We must end, but we permit 
ourselves the perhaps too obvious comment, that this 
publication is indeed an admirable example of “ the 
diffusion of knowledge among men.” 

Our Bookshelf. 
Introduction a la géométrie non-Euclidienne. Par 
Dr. A. MacLeod. Pp. 433. (Paris: J. Hermann, 
1922.) 20 francs. 
In the theory of relativity, on which so much has been 
written during the last few years, one of the main 
difficulties encountered by most readers is the un- 
familiar conception of space and time involved. Apart 
from the difficulty in the conception of a space-time 
continuum, the notions that space as we know it may 
possibly be only of limited extent, and that the sum of 
the angles of a triangle is not. necessarily equal to two 
right angles, are apt to prove only too bewildering to 
readers whose knowledge of geometrical matters is con- 
fined to the Euclidean system. 
The question whether the axioms imposed by Euclid 
are necessary for building up a logical system of 
geometry has long engaged the attention of mathe- 
maticians. In the non-Euclidean system, largely 
developed by Gauss, the absolute, 7.e. the “ circular 
points at infinity ” of Euclidean geometry, is replaced 
by a non-degenerate conic. All this entails revised 
definitions of such terms as “distance” and “ right 
angle.” 
_ Dr. MacLeod in the work before us presents the 
subject with strict logical precision, the reasoning 
which leads to the various results being given fully and 
accurately. The actual amount of ground covered 
is not so great as in Mr. Coolidge’s treatise, a book 
which occasionally suffers from over-condensation. Un- 
initiated readers will be interested in noticing that the 
proof of a familiar proposition, that the greater angle 
of a triangle is opposite the greater side, requires six 
pages of reasoning. The book would have been im- 
proved by more diagrams, but these can be supplied 
without difficulty. It can be recommended as an 
excellent introduction to the subject. 
Wi E.sHB: 
NO. 2775, VOL. 111] 
History of the Theory of Numbers. By Prof. L. E, 
Dickson. Volume II. (Publication No. 256, Vol. 
Il.). Pp. xxvi+ 803. (Washington: Carnegie 
Institution of Washington, 1920.) 7.50 dollars. 
Tue arithmetical questions treated by Diophantus of 
Alexandria, who flourished about the year 250 A.D., 
included such problems as the solution of the equations 
x+y+s=6, xyts=u?, xy—-s=v? 
in rational numbers. Little attention was given to this 
type of problem from Diophantus’s time till that of 
Fermat (1650), the founder of modern Diophantine 
analysis. The most general arithmetical question to 
which the peculiar methods of Diophantine analysis 
apply is the determination of all the solutions in rational 
numbers of a system of algebraic equations. 
Rot, ‘Bigg <8 = Pon = Ope E— 15°25, 5... TR; 
there being more unknowns than equations. Particular 
problems of this type have attracted the attention of a 
very large number of workers. 
Prof. Dickson, in the second volume of his History, 
gives an account of what has been accomplished in this 
field of thought. Original memoirs have been carefully 
scrutinised and abstracted. Naturally, in such a com- 
pilation, there is much matter which would not now be 
regarded as of any great scientific importance, and, in 
fact, the main value of many of the reports is on the 
side of historical development. 
Scientifically, the most important chapters in the 
present volume are those on (i.) partitions of numbers, 
(ii.) representation of numbers as sums of squares, (iii.) 
Pellian equations, (iv.) indeterminate equations of the 
third degree, and (v.) Fermat’s last theorem. It is 
to be trusted that the mathematical world will duly 
appreciate the immense amount of labour expended by 
Prof. Dickson in the preparation of such a book. 
W.-H: B: 
Penrose’s Annual: The Process Year Book. Review 
of the Graphic Arts. Edited by William Gamble. 
Vol. 25. Pp.xvit+110+plates+64. (London: Percy 
Lund, Humphries and Co., Ltd., 1923.) 8s. net. 
Mr. GAMBLE, in his editorial review of process work, 
looks back twenty-seven years to the first volume of 
this annual and remarks upon the improvement of the 
process block since then. He considers that it is now 
so perfect that there is little if any possibility of advance 
in this direction. ‘‘ The signs of the times are that the 
process block has passed its prime and that there will 
be a slow and steady diminution of its employment.” 
Rotary photogravure and off-set lithography are 
improving, and collotype is reviving, its most important 
application being in the highest grade of colour work. 
The superseding of type composition by a photo- 
graphic method occupies a prominent position in the 
volume. The “ photoline process” of Mr. Arthur 
Dutton, though the machinery for it is not yet on the 
market, is so far perfected that we have here good 
examples of solid text, tabular matter, title pages, 
ornamental work, and a demonstration that any size 
of letter can be obtained from one master alphabet. 
The body of the book contains several articles of 
exceptional value following the editor’s general 
summary. Mr. Stanley Morison contributes a long 
and well illustrated historical article on “ Printing in 
France,” and “ Printing in China ” is dealt with in a 
