Let there be a quantity growing in such a way that the increment of its growth, during a given time, shall always be proportional to its own magnitude. This resembles the process of reckoning interest on money at some fixed rate; for the bigger the capital, the bigger the amount of interest on it in a given time.
Now we must distinguish clearly between two cases, in our calculation, according as the calculation is made by what the arithmetic books call “simple interest,” or by what they call “compound interest.” For in the former case the capital remains fixed, while in the latter the interest is added to the capital, which therefore increases by successive additions.
(1) At simple interest. Consider a concrete case. Let the capital at start be £, and let the rate of interest be per cent. per annum. Then the increment to the owner of the capital will be £ every year. Let him go on drawing his interest every year, and hoard it by putting it by in a
