JUNE 7, 1912] 
thought roads have been proved to be safe 
and they always lead to some prominent 
objective points. Hence they primarily 
serve to economize thought. The number 
of objects of mathematical thought is in- 
finite, and these roads enable a finite mind 
to secure an intellectual penetration into 
some parts of this infinitude of objects. It 
should also be observed that mathematics 
consists of a connected network of thought 
roads, and mathematical progress means 
that other such connected or connecting 
roads are being established, which either 
lead to new objective points of interest or 
exhibit new connections between known 
roads. 
The network of thought roads called 
mathematics furnishes a very interesting 
chapter in the intellectual history of the 
world, and in recent years an increasing 
number of investigators have entered the 
field of mathematical history. The results 
are very encouraging. In fact, there are 
very few other parts of mathematics where 
the progress during the last twenty years 
has been as great as in this history. This 
progress is partly reflected by special 
courses in this subject in the leading uni- 
versities of the world. While the earliest 
such course seems to have been given only 
about forty years ago, a considerable 
number of universities are now offering 
regular courses in this subject, and these 
courses have the great advantage that they 
establish another point of helpful contact 
between mathematics and other fields. 
Mathematical thought roads may be dis- 
tinguished by the facts that by means of 
certain assumptions they have been proved 
to lead safely to certain objective points 
of interest, and each of them connects, at 
least in one point, with a network of other 
such roads which were called mathematics, 
pabhnuara by the ancient Greeks. The math- 
ematical investigator of the present day is 
SCIENCE 
885 
pushing these thought roads into domains 
which were totally unknown to the older 
mathematicians. Whether it will ever be 
possible to penetrate all scientific knowl- 
edge in this way and thus to unify all the 
advanced scientific subjects of study under 
the general term of mathematics, as was 
the case with the ancient Greeks," is a 
question of deep interest. 
The scientific world has devoted much 
attention to the collection and the classifica- 
tion of facts relative to material things, and 
has secured already an immensely valuable 
store of such knowledge. As the number 
of these facts increases, stronger and 
stronger means of intellectual penetration 
are needed. In many cases mathematics 
has already provided such means in a large 
measure; and, judging from the past, one 
may reasonably expect that the demand for 
such means will continue to increase as long 
as scientific knowledge continues to grow. 
On the other hand, the domain of logic has 
been widely extended through the work of 
Russell, Poincaré and others; and Russell’s 
conclusion that any false proposition im- 
plies all other propositions whether true 
or false is of great general interest. 
During the last two or three centuries 
there has been a most remarkable increase 
in facilities for publication. Not only have 
academies and societies started journals for 
the use of their members, but numerous 
journals inviting suitable contributions 
from the public have arisen. The oldest of 
the latter type is the Journal des Scavans, 
which was started at Paris in 1665, while 
the Transactions of the Royal Society of 
London, started in the same year, should 
probably be regarded as the oldest of the 
former type. These journals have done an 
%The term mathematics was first used with its 
present restricted meaning by the Peripatetic 
School. Cantor, ‘‘Vorlesungen iiber Geschichte 
der Mathematik,’’ Vol. I. (1907), p. 216. 
