882 
in close touch with the masters of the past,® 
and this unity of purpose is almost essen- 
tial to secure real effective work in the 
immense field of mathematical endeavor. 
As a class of problems which are much 
more suitable for direct objects of research 
on the part of those who are not in close 
contact with a master in his field, we may 
mention the numerous prize subjects which 
are announced from year to year by foreign 
academies. 
Among the learned societies which an- 
nounce such subjects the Paris Academy of 
Sciences is probably most widely known, 
but there are many others of note. The 
subjects announced annually by these so- 
cieties cover a wide range of mathematical 
interests, but they are frequently beyond 
the reach of the young investigator.° It 
is very easy to obtain these subjects, since 
they generally appear in the ‘‘notes’”’ of 
many mathematical journals. In our 
country the Bulletin of the American 
Mathematical Society is rendering very 
useful service along this and many other 
lines. While some of these subjects are 
very general, there are others which indi- 
cate clearly the particular difficulties which 
must be overcome before further progress 
in certain directions seems possible and 
hence these subjects deserve careful study, 
especially on the part of the younger in- 
vestigators. 
As long as one is completely guided, in 
selecting subjects for research, by the 
standing problems or by the subjects an- 
nounced by learned bodies and those pro- 
posed individually by prominent investiga- 
tors, one is on safe ground. Real progress 
along any of these lines is welcomed by our 
° Darboux, Bulletin des Sciences Mathématiques, 
Vol. 32 (1908), p. 107. 
For solutions of such problems in pure mathe- 
maties by Americans, see Bulletin of the American 
Mathematical Society, Vol. 7 (1901), p. 190; Vol. 
16 (1910), p. 267. 
SCIENCE 
[N.S. Vou. XXXV. No. 910 
best journals, as such progress can easily 
be measured, and it fits into a general trend 
of thought which is easily accessible in view 
of the many developed avenues of approach. 
Notwithstanding these advantages, the real 
investigator should reach the time when he 
can select his own problems without advice 
or authority; when he feels free to look at 
the whole situation from a higher point of 
view and to assume the responsibility of an 
independent choice, irrespective of the fact 
that an independent choice may entail dis- 
trust and misgivings on the part of many 
who would have supported him nobly if he 
had remained on their plane. 
In looking at the whole situation from 
this higher point of view many new and 
perplexing questions confront us. Why 
should the developments of the past have 
followed certain routes? What is the 
probability that the development of the 
territory lying between two such routes 
will exhibit new points of contact and 
greater unity in the whole development? 
What should be some guiding principles in 
selecting one rather than another subject of 
investigation? What explanation can we 
give for the fact that some regions bear 
evidences of great activity in the past but 
are now practically deserted, while others 
maintained or increased their relative pop- 
ularity through all times? 
One of the most important tests that can 
be appled to a particular mathematical 
theory is whether it serves as a unifying 
and clarifying principle of wide applica- 
tions. Whether these applications relate 
to pure mathematics only or to related 
fields seems less important. In fact, the 
subjects of application may have to be de- 
veloped. If this is the case, it is so much 
the better provided always that the realm 
of thought whose relations are exhibited by 
the theory is extensive and that the rela- 
tions are of such a striking character as to 
