Page:Calculus Made Easy.pdf/160

call a logarithmic rate of growing. Unit logarithmic rate of growth is that rate which in unit time will cause $1$ to grow to $2.718281$. It might also be called the organic rate of growing: because it is characteristic of organic growth (in certain circumstances) that the increment of the organism in a given time is proportional to the magnitude of the organism itself.

If we take $100$ per cent. as the unit of rate, and any fixed period as the unit of time, then the result of letting $1$ grow arithmetically at unit rate, for unit time, will be $2$, while the result of letting $1$ grow logarithmically at unit rate, for the same time, will be $2.71828\ldots .$

A little more about Epsilon. We have seen that we require to know what value is reached by the expression $\left(1+{\dfrac {1}{n}}\right)^{n}$, when $n$ becomes indefinitely great. Arithmetically, here are tabulated a lot of values (which anybody can calculate out by the help of an ordinary table of logarithms) got by assuming $n=2$; $n=5$; $n=10$; and so on, up to $n=10,000$.

${\begin{alignedat}{2}&(1+{\tfrac {1}{2}})^{2}&&=2.25.\\&(1+{\tfrac {1}{5}})^{5}&&=2.489.\\&(1+{\tfrac {1}{10}})^{10}&&=2.594.\\&(1+{\tfrac {1}{20}})^{20}&&=2.653.\\&(1+{\tfrac {1}{100}})^{100}&&=2.704.\\&(1+{\tfrac {1}{1000}})^{1000}&&=2.7171.\\&(1+{\tfrac {1}{10,000}})^{10,000}&&=2.7182.\end{alignedat}}$