62 A new Method for computing Interest. 



months ? or than any fractional part of a year's interest which 

 may have accrued at any other period ? The true way, in my 

 opinion, is to consider the interest due at any and all periods of 

 time ; or in other words, that a sum of money at interest should 

 be constantly augmenting; that is, I would have all sums of 

 money, whether for great or small portions of time, at compound 

 interest, the interest being added in at the end of every instant 

 This would give a unity to all cases of casting interest ; no mis- 

 understanding could then possibly arise as to the amount of in- 

 terest due. I shall show that it is not difficult to compute inter- 

 est tables on this principle. The per cent, per annum may be so 

 taken as to make the interest due at the end of one year, the same 

 as in the case of simple interest. 



I do not suppose that such a method of casting interest will 

 ever come into general use ; all I wish is, to show that such a 

 method is practicable, and if adopted would at once settle all dis- 

 putes in respect to usury. Interest would then be a uniformly 

 increasing quantity, not limited to any particular epoch for re- 

 ceiving its increments. 



Let us now endeavor to find a formula for the amount of a 

 given sum at compound interest, when compounded at the end 

 of every instant. 



Let P=the principal. 



r'=the rate per cent, per anuum. 

 t=X\\e time in years. 

 a=the amount. 



It is obvious that in the next instant dt, the increment of a, 

 would be ar'dt; but by the principles of the calculus this incre- 

 ment of interest is represented by da. Therefore we have this 

 condition, 



da= ar'dt (1), 



where a and t are considered as variable, and r' as constant. In- 

 tegrating (1) and adding the arbitrary constant c, we have 

 log. a^r't-^c (2). 



Now, when i=0, a=P, these values substituted in (2) we find 

 c--log. P. Hence condition (2) becomes 



log.|=r'i (3), 



or a=Pe'-'« (4), 



where e=2.7I8281828459, &c. 



