nd its immediate
_raison d'etre_ in something other than the meaning that is gone or the
meaning that is not yet here. It is true that the barest facts do not
lack meaning, though a meaning which has been theirs in the past is
lost. The meaning, however, that is still theirs is confessedly
inadequate, otherwise there would be no scientific problem to be solved.
Thus, when older theories of the spread of infectious diseases lost
their validity because of instances where these explanations could not
be applied, the diagnoses and accounts which could still be given of the
cases of the sickness themselves were no explanation of the spread of
the infection. The facts of the spread of the infection could be brought
neither under a doctrine of contagion which was shattered by actual
events nor under a doctrine of the germ theory of disease, which was as
yet unborn. The logical import of the dependence of these facts upon
observation, and hence upon the individual experience of the scientist,
I shall have occasion to discuss later; what I am referring to here is
that the conscious growth of science is accompanied by the appearance of
this sort of material.

There were two fields of ancient science, those of mathematics and of
astronomy, within which very considerable advance was achieved, a fact
which would seem therefore to offer exception to the statement just
made. The theory of the growth of mathematics is a disputed territory,
but whether mathematical discovery and invention take place by steps
which can be identified with those which mark the advance in the
experimental sciences or not, the individual processes in which the
discoveries and inventions have arisen are almost uniformly lost to view
in the demonstration which presents the results. It would be improper to
state that no new data have arisen in the development of mathematics, in
the face of such innovations as the minus quantity, the irrational, the
imaginary, the infinitesimal, or the transfinite number, and yet the
innovations appear as the recasting of the mathematical theories rather
than as new facts. It is of course true that these advances have
depended upon problems such as those which in the researches of Kepler
and Galileo led to the early concepts of the infinitesimal procedure,
and upon such undertakings as bringing the combined theories of geometry
and algebra to bear upon the experiences of continuous change. For a
century after the formulation of