The name is absent

ON THE ANALYSIS OF LIBRARY GROWTH

exponential growth proceeding at the same rate as the 1210 to 1500
growth.² Thus shortly after the invention of printing a major
change in the growth rate of the number of universities occurred.
One interpretation of this phenomenon is that printing made the
textbook cheaper and more available and thus permitted an in-
creased student-to-teacher ratio, decreasing the necessity for
founding new institutions. After 300 years, the natural growth
in the student population increased this ratio beyond levels that
could be efficiently maintained even with the availability of in-
expensive texts, thus encouraging the foundation of more univer-
sities, at the previously observed rate. This explanation is offered
solely as a possibility; research would have to be done to see if it
is consistent with the growth of student populations. An indication
that this explanation may not be farfetched is the contemporary
effort to introduce television and teaching machines to permit
greater student-to-teacher ratios; both of these devices are im-
proved means for transmitting information, which of course the
printed text is too.

3. Stable Growth

The growth with time of a population (of people, or books, or
any other quantity) is said to be exponential if its rate of change
is proportional to the population ; the constant of proportionality is
the growth exponent. In the standard notations of the differential
calculus, this is expressed by writing

_n                    dP(t) = a P(t) ,

^,                             dt

where P(t) denotes the population at time t, and α₀ is the growth
exponent. The solution to this differential equation is

2)                   P(t) = P(t_o)e^a° ^ft'^t°^, .

where t₀ is any conveniently chosen time. If a population does
grow according to this equation, then its logarithm varies linearly
with the time, that is,

IogPflJ = IogPfr_oJ + a_n(t-t_o) ;

here log denotes the natural logarithm function. This means that
exponential growth is represented by straight lines on semi-
logarithmic graph paper, such as is used for Figures 1 and 2, for
instance.

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