226 
present one is welcome as a record, of progress, 
even in what may be called the elements of the 
subject, and as an original work by three distin- 
guished experts. It is divided into three parts, 
for each of which one author is mainly responsible. 
Part i, by Prof. G. A. Miller, deals with the 
general properties of groups, beginning with sub- 
stitution-groups, and going on to the abstract 
definition by generators; there are special chapters 
on Abelian groups, on groups of order p™ with 
p prime, on the polyhedral groups, on isO- 
morphisms, and on solvable groups. Part ii., by 
Prof. H. F. Blichfeldt, is on linear groups, and a 
valuable summary of the present stage of that 
theory; in particular, there is a chapter on char- 
acteristics. Part iii, by Prof. L. E. Dickson, is 
on applications, and is naturally of a more 
elementary character; there are three chapters on 
the Galoisian theory of equations, one on rule and 
compass constructions, one on the inflexions of a 
plane cubic, one on the 27 lines of a cubic sur- 
face, and one which is a scrap on solutions of 
equations by a standard form F(z, k) =o, involving 
one parameter. 
The outstanding novelty is the early proof of 
Sylow’s theorem, which actually begins on p. 27. 
The proof is led up to by the definition and dis- 
cussion of “double co-sets,” which, it appears, 
were first used by Cauchy, and long afterwards 
taken up by Frobenius. In group-theory Sylow’s 
theorem occupies a place something like that of 
the law of quadratic reciprocity in arithmetic; it 
is of a fundamental character, and each distinct 
proof of it marks an advance in the general theory. 
It may be noted, however, that the proof given by 
Prof. Miller assumes the group considered to be 
given in the form of a substitution-group ; this is 
not a real limitation, because every group is iso- 
morphic with a set of substitutions—a theorem 
proved on p. 63. We may, perhaps, be justified 
in thinking that the “genuine” proof of Sylow’s 
theorem has yet to be discovered; that is to say, 
a proof based on the abstract definitions of a 
group, without using any special image of it, and 
also a proof which comes at the proper place in 
the sequence of the theory. 
The authors give a considerable number of exer- 
cises, including some which are really easy. This is 
important, because every mathematical student 
ought to know the elements of group-theory ; it 
is the only thing which gives unity to a host of 
scattered results in elementary algebra, trigo- 
nometry, and analytical geometry, and it is often 
a guide to us when we wish to estimate before- 
hand the complexity of a particular problem. In 
the case of some of the harder exercises hints are 
given to help the reader. 
How far this treatise will suit a beginner, it is 
difficult to say. The subject is, for most students, 
a hard one, except in its very early stages, and 
Prof. Miller’s contribution is concise as well as 
abstract. In any case, those who have made some 
progress in the theory, and wish to know its pre- 
sent condition, will find the work of great interest 
and value; a judicious skipper who begins at 
NO. 2456, voL. 98] 
NATURE 
eS SS 
eee 
[NovEMBER 23, 1916 
2° eS eee 
enjoy himself 
than a con- 
p- 321 or thereabout will perhaps 
more, and make more progress, 
scientious plodder with a bookmark. 
The treatise is appropriately dedicated to M. 
Camille Jordan, to whom the authors justly assign. 
the credit of having mainly helped to establish. 
group-theory as a leading branch of mathematics. 
G. B. M. 
AN INTERNATIONAL GEOGRAPHICAL. 
EXCURSION. 
Memorial Volume of the Transcontinental Excur- 
sion of 1912 of the American Geographical 
Society of New York. Pp. xi+407. (New 
York: American Geographical Society, 1915;); 
Price 3 dollars. 
W Ga excursion of which this volume is @ 
memorial was organised by Prof. W. M. 
of Harvard University, to celebrate two 
of the foundation 
of New 
Davis, 
things—the sixtieth anniversary 
of the American Geographical Society 
York, and the entry of that society into its new 
building in Broadway at 156th Street. The 
members of the excursion were mainly geo- 
graphers invited from nearly every European 
country, and these were taken in a special train. 
and by other means of communication over routes 
amounting in all to nearly 13,000 miles, first west-- 
wards through the northern tier of States to 
Seattle, thence south to San F rancisco, and back 
through the middle States, but going so far south 
as Birmingham, Ala., in rounding the south of the 
Appalachians. Besides the European members of 
the party, there were about a dozen American 
geographers who went the whole round; and 
numerous other American geographers, geolo- 
gists, and others capable of furnishing informa- 
tion about different parts of the United States. 
joined the party for shorter or longer stages. 
In the course of the excursion discussions were 
constantly being held with regard to the geo- 
graphical features of the districts visited; and, 
seeing that Prof. Davis was the leader as well as. 
the organiser of the party, it is only natural that 
those discussions should have frequently turned 
on the interpretation of the features according to” 
the terminology which he has introduced so widely 
into geography. The contents of this volume are 
mainly made up of articles, written in German, 
English, French, and Italian, by members of the 
party, and it would have been interesting to find 
in one of these an example of the applica- 
tion of that terminology to geographical descrip- 
tion, the purpose to which its author contends it 
is pre-eminently suited. But there is none. 
the other hand, there are two or three in which 
the morphogenetic nomenclature of Davis is dis- 
cussed, and more or less criticised, as by Prof. 
Ricchieri (pp. 63-5), Prof. Jaeger in his (some- 
what maccaronic—intermingled German and Eng- 
lish) article entitled ‘“‘Bemerkungen zur system- 
atischen Beschreibung der Landformen,” and inci- 
dentally by Waldbaur in his “Bemerkungen tiber 
Stufeniandschaften ” (bottom of p. 86, etc.). Even 
On 
