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LXI. .Notices respecting New Boohs. 
The Axioms of Projective Geometry and the Axioms of Descriptive 
Geometry. By A. N. Whitehead, Sc.D., F.R.S. Cambridge 
University Press, 1907. 
HPHESE tracts, which form Nos. 4 and 5 of the Cambridge Tracts 
-*- in Mathematics and Mathematical Physics, deal with a branch 
of mathematics which has lately received much attention* from 
mathematicians, and which, although in its present form of com- 
paratively recent origin, has already attained a high state of 
development. So important has this subject — the Foundations 
of Mathematics — become that it is imperative on every mathe- 
matician to have some acquaintance with its developments in 
order to be in sympathy with much of the mathematical work of 
the present day. Dr. Whitehead's two tracts deal with the geo- 
metric side and form an excellent introduction to a detailed study 
of the whole field. They contain a clear exposition of the founda- 
tions of geometry, comprising as much of the theory as is of 
interest to the general mathematical reader, to whom presumably 
they are especially intended to appeal. They are, however, equally 
valuable to the specialist, since they give a fresh and connected 
view of a subject which is particularly confusing from the variety 
of ways in which it may be approached. 
Theory of Sets of Points. ByW. H. Young, M.A., Sc.D., and 
Grace Chisholm Youkg-, Phil. Doc. (Gott.). Cambridge Uni- 
versity Press, 1906. 
This is in many respects a unique book. Not only is it the 
first of its kind which has ever appeared in English, but in no 
other language apparently has an attempt been made to give a 
systematic exposition of the subject. The subject itself does not 
demand on the part of the reader a large stock of mathematical 
knowledge. The book may therefore be taken up at almost any 
point of his mathematical training, probably the sooner the better. 
After a brief account of rational and irrational numbers and of 
the manner in which numbers are represented as to order by points 
on a straight line, the authors enter upon the theory of linear sets 
and sequences leading up to the conceptions of potency, content, 
and order. Plane sets are introduced in Chapter VIII. ; and in 
Chapter IX. a succession of theorems deals with the conceptions 
of region, domain, boundary, and rim, leading up to the definition 
of a curve. It is here and in the later discussion of plane content 
and area that the student familiar with the ordinary geometrical 
assumptions regarding curves and areas will probably for the first 
time be fully impressed with the value of the theory of sets of 
points. There can be but one opinion as to the great service 
rendered by Dr. and Mrs. Young in placing in our hands this 
systematic treatise on the foundations of mathematics. An 
appendix running to fifteen pages gives the bibliography of the 
subject. 
