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 $$n=10,000$$.