Jamesian Philosophy Refreshed

  Yesterday we discussed the limit of the process of compounding interest. As we move through even shorter periods, from semi-annual to quarterly to monthly compounding and so forth, and as we approach an infinity of infinitesimal periods, i,e. continuous compounding, we find the number 2.71828 arises to stare us in the face as a limit.  This was Jacob Bernoulli's result. So why is the number called Euler's number, and sometimes just e? Did Euler name it for himself? And, if so, isn't that an injustice? Shouldn't it be called b? Well, Bernoulli found 2.71828... but he neglected to name it.  Gottfried Leibniz, the man recognized as THE founder of calculus in the German speaking world, seems to have called it "b" in letters he wrote in 1690 and 1691.  Perhaps this was a recognition of Bernoulli's role.  More important, though, Bernoulli neglected to explore the properties of this number outside of banking.  It was Euler (pictured above) who defined it as the...

 How does continuous compound interest work?  Well, suppose the magical "Bank of Bernoulli" offers you 100 percent interest per annum. Or, in decimal terms, a return of 1.0.  Great. You put a dollar in the bank and the bank gives you two dollars at the end of the year, one dollar representing the principal, the other the yield. Only one simple computation has to be made, so we can say of this situation that n = 1. It gets better.  Bank of Bernoulli [so named after the great 17th century mathematician Jacob Bernoulli] decides to compound semi-annually.  You put in one dollar, and on June 30 you are credited with 100 percent interest for half a year, or $0.50. In the second half of the year, you earn the same 1/2 of the underlying annual rate, for the redefined principal amount, this time on $1.50.  So you earn $0.75 between June 30 and December 31. This means you have $2.25 waiting for you at the end of the year. You can keep this up, presuming a calculation...