54 



A new Method for computing Interest. 



By means of formula (12) 1 have computed the values in the 

 third column of the following table. The first column gives the 

 time in days; the second is the simple interest of $1, at 7 per 

 cent, per annum ; the fourth column gives the difference between 

 the simple interest and the instantaneous compound interest. 



Days. 



Amount of $1 at sim- 

 ple interest, the rate 

 percent, per annum 

 being 7. 



2 

 3 



4 

 5 

 6 



7 



8 



9 



10 



20 



30 



40 



50 



60 



70 



80 



90 



100 



200 



300 



365 



— 



1000192 

 1-000384 

 1-000575 

 1-000767 



1000959 

 1001151 

 1001342 

 1001534 

 1-001726 

 1001918 

 1-003836 

 1005753 



1-007671 

 1009589 

 1011507 

 1013425 

 1015342 

 1-017260 

 1-019178 

 1038356 

 1057534 

 1-070000 



Amount of $1 atcom- 



pound interest, the 



interest being com- Difference 



pounded at the end, 



of every instant, — ' 

 . the rate per cent. 



per annum being 



676537. 



1000183 

 1-000371 

 1-000556 

 1000742 



1 -000927 

 1001113 



1-001298 

 1-001484 

 1-001670 

 1001855 

 1-003714 



1005576 

 1007442 

 1009311 

 1011884 

 1013060 

 1014940 

 1016823 

 1-018709 

 1037769 

 1057185 

 1070000 



between 

 the simple interest 

 and the compound 

 interest. 



0-000009 

 0-000013 



000019 

 000025 

 000032 

 000038 

 000044 

 0000050 

 0-000056 

 0-000063 

 000122 

 0000177 

 000229 

 0-000278 

 000323 

 0000365 

 000402 

 000437 

 000469 

 0000587 

 000349 

 000000 



By the foregoing table we see that the greatest difference, as 

 given in the fourth column, is $0.000587, which corresponds to 

 200 days. If we wish to know the exact time when this differ- 

 ence is a maximum, we proceed as follows : The amount at sim- 



$ 



is 



+ 



The amount at instantaneous compound interest at r 7 per cent 



per annum for $1 



e 



T't 



The difference is therefore expressed by 



1-Wr 



e 



T't 



In this expression t is the only variable. 



(13). 



