280 
‘science as they are in religion, or in wars 
which involve the life and death of individuals 
and of nations. But, even if we have the 
material at hand, is it possible to write the 
mathematical history of the nineteenth cen- 
tury at this early date? Do we not now lack 
the proper perspective? In reply, we admit at 
once that we can not now write a history 
which will satisfy mathematicians seventy-five 
or one hundred years hence. All we can hope 
to do is to render a service to the present and 
the next generation of mathematical students 
and investigators. Nor can the history of the 
nineteenth century be written seventy-five or 
a hundred years from now in a manner that 
will be fully acceptable to all posterity. The 
general proposition holds true that no decade 
ean write history which does not have to be 
rewritten later; no decade can write history 
for all future decades. There is an inevitable 
relativity of historical narrative. The reason 
for this relativity is obvious. The point of 
view changes; the attention of mathematicians 
will be directed to new concepts. Part of the 
history of mathematics will have to be re- 
written in order to give proper emphasis to 
these new concepts. If it were possible, after 
the lapse of a few centuries, to impart finality 
to a history, then assuredly the history of 
Greek mathematics should have been rigidly 
determined and fixed long ago. But the facts 
disclose no such finality. In recent years the 
history of Greek mathematics has been partly 
recast. Zeno of Elea, whose arguments on 
motion formerly received little or no attention 
from mathematicians and were completely 
ignored by Montucla, the great eighteenth 
century historian of mathematics—this Zeno 
who was berated by philosophical writers as an 
insincere dialectician or as the progenitor of 
modern pettifogging lawyers—has been inter- 
preted by Paul Tannery and other recent 
historians as having dealt sincerely and ably 
with questions of infinity now playing a lead- 
ing réle in modern mathematics. Geometric 
ideas of the last fifty years have brought into 
prominence the postulate of Eudoxus and 
Archimedes which the older historians of 
mathematics passed over in silence. The ad- 
SCIENCE 
[N. S. Vou. XLVIII. No. 1238 
vent of the non-Euclidean geometry has 
thrown LEuclid’s parallel postulate into a 
wholly different light. Euclid’s once criticized 
definition of equal ratios as contained in the 
fifth book of his Elements acquires a fresh 
interest when seen in the light of Dedekind’s 
theory of the irrational. Many other illustra- 
tions might be cited to prove that historical 
narrative is relative, that history can not be 
written by a historian of one age, however 
keen, to satisfy all succeeding ages. Since 
this is so, should historians in despair drop 
their pens, remain idle and permit mathe- 
maticians to labor without the stimulus and 
light which a history of the modern develop- 
ments of their science can give? Assuredly, 
no. A history of nineteenth century mathe- 
matics can be written acceptably to workers 
of to-day and tomorrow, but will probably be 
in need of revision on the day after to-morrow. 
To wait for the moment when a history of 
mathematics can be written that will answer 
the demands of all future time is to postpone 
the crossing of a great river until all the 
water has flowed by. 
An important step is the ascertainment of 
the magnitude of the task, the determination 
of the volume of mathematical literature to 
be penetrated. For purposes of comparison 
we have found it convenient to consider the 
mathematical productiveness during seven 
periods. Moritz Cantor’s four volumes of 
lectures on the history of mathematics furnish 
the data for the first four periods. Statistics 
which I gathered from the first three volumes 
of Poggendorfi’s “ Biographisches Handworter- 
buch ” were used to characterize the next two 
periods, while Professor H. 8. White’s “ Forty 
Years’ Fluctuations in Mathematical Re- 
search ””? that took place since 1870 furnished 
the figures for the last period. Cantor’s first 
volume gives the history of about 32 centuries 
down to 1200 a.p.; the second volume covers 
468 years, to 1668; the third volume gives the 
history during 90 years, to 1758; while the 
fourth volume is limited to 41 years and 
earries the history down to 1799. We shall . 
assume that the sizes of the volumes are ap- 
2Sormnce, N. S., Vol. 42, 1915, pp. 105-113. 
