360 NATURE OF CAPITAL AND INCOME by following to the limit the method of this Appendix, § 1 above. The number e, or 2.7182818, plays almost as important a role in mathematics as the number 3.141592, called x, which expresses the ratio of the circumference of a circle to its diameter, but the meaning of e is less familiar to most students. Various definitions and interpretations may be given. To the economist, the most interesting is the following: e is the “amount” of $1 put at compound interest during the “purchase period,” the latter being derived on the assumption that the rate of interest is payable continuously. This proposition is implicitly contained in the demonstration of the equation 1 + ¢ = €%, as given above. The following is a more explicit statement, with actual illustrative figures: — If we consider the rate of interest 4%, or oo payable annually, the purchase period is 25 years, or w and the amount of $1 put at 49 interest for these 25 years will be 1 + .04). If next we take 49, payable semi-annually, the amount of $1 during the purchase period of 25 ( 1+ 8" years will be 2 Similarly, the amount of $1 put at interest at (4 L04\ 49, payable quarterly is ( 4 ) : And if the interest is reckoned as payable n 1 + 2%. times a year, the amount is n At the limit, we have the amount of $1 at interest at 4% for 25 years when the rate of interest is payable continuously. The limit of the above expression, when n is made indefinitely great, is the definition of e. The distinction between the different rates of interest may be shown by a diagram. In Figure 33, let the curve B'AB represent a “ discount curve,” any two ordinates of which represent exchangeable goods situated at two corresponding points of time, as a and b, that is, the sum a4 of “present ” goods will buy the sum bB of goods which lie in the future a time interval ab, beyond the “present.” If we take these two points, a and b, a year apart, the rate of interest as reckoned annually is the