APPENDIX TO CHAPTER XII 359 y ; 14 > ,and this sum in another six months will “amount” by al\2 compound interest to 1 XX 7) , which must be equal to 1 + 3, the “amount” of $1 in one year at the equivalent rate of interest, 7, reckoned annually; i.e. 1d imi li) rim 5) or, expanding and reducing, — 3 i=1+ er which is the same result obtained before when the rate of interest was regarded as the price of capital. Similarly, for the interest rate i’, reckoned quarterly, we NIANE may prove 1 +i = 1 XY D , and for the interest rate ®, F(r)\n reckoned n times a year, 1 + ¢ =(1 + 5 In other words, im " 7m a y(n) he 1 3 2 -[(1 + =¥s] n n As n increases indefinitely, the last expression approaches a limit. The limit of i™ is the “rate of interest per annum computed continuously,” called 8. The limit of the square bracket is the base of the Napierian system of logarithms called e; for : “ns os 1 Yd by the definition of e usually given, it is the limit of 1 + i) when % is any number increasing indefinitely. Evidently a is such a number, for n, by hypothesis, is to be increased 1 n indefinitely, and i evidently decreases. Hence at the limit the last formula becomes, 1 +4 ft =e; Or, substituting for e its numerical value, 1 + i= (2.7182818)% : Another proof of this formula could be given for the case where the rate of interest is conceived as the price of capital,