Mathematical philosophy, a study of fate and freedom; lectures for educated laymen, by Cassius J. Keyser.

444 MATHEMATICAL PHILOSOPHY
bodied as cofactor of toil in enduring achievements thus
survives the dead and works as living capital for augmentation and transmission to posterity, is the secret and
process of progressive civilization-building. The question is: What is the Law thereof-the natural law?
What its general type is you apprehend at once; it is like
that of a rapidly increasing geometric progression-if P
be the progress made in a given generation, conveniently
called the "first," and if R denote the ratio of improvement, then the progress made in the second generation is
PR, that in the third is PR2, and that made in the single
Tth generation will be PR"' 1. Observe that R is a large
number,-how large we do not know,-and that the
time T enters as an exponent-and so the expression
PRT-1 iS called an exponential function of Time, and it
makes evident, even to the physical eye, the involution of
time in the life of man. This is an amazing function, as
every student of the Calculus knows; as T increases,
which it is always doing, the function not only increases
but it does so at a rate which itself increases according
to a similar law, and the rate of increase of the rate of
increase again increases in like manner, and so on endlessly, thus sweeping on towards infinity in a way that
baffles all imagination and all descriptive speech. Yet
such is approximately the law,-the natural law,-for
the advancement of Civilization, immortal offspring of
the spiritual marriage of Time and human Toil. I have
said "approximately," for it does not represent adequately
the natural law for the progress of civilization; it does
not, however, err by excess, it errs by defect; for, upon
a little observation and reflection, it is evident that R,
the ratio of improvement, is not a constant, as above contemplated, but it is a variable that grows larger and larger
as time increases, so that the function PRT-" increases