An introduction to the summation of differences of a function; an elementary exposition of the nature of the algebraic processes replaced by the abbreviations of the infinitesimal calculus, by B. F. Groat.

LIMITS

21

Hence the sum of the first nine terms of i + i differs from
e by less than 0.000004, and consequently will give the first six
figures of e provided the sixth decimal figure of the sum is less
than 6. For clearness the calculation is arranged as follows:
2.0000000 = I + I -I = 2.000000.5000000 = I 12.500000.1666666 < I - 13 <.166667.04I6666 < I L4 <.041667.0083333 < i 5 <.008334.0013888 < — 16 <.001389.0001984 < I -17 <.000199.0000248 < I [8 <.000025.0000027 < I- 9; I - 88 <.000004.0000002 < I - _o; 2.718285
2.7182814
Hence e lies between 2.7182814 and 2.718285, and consequently 2.71828 must be the first six figures of e. As a matter
of fact 2.718281 are the first seven figures; but to prove this
would require further investigation. By taking in more terms of
e, and carrying each term sufficiently far out, we may, by reasoning analogous to the above, find the value of c as nearly as
desired..
It will transpire that e is the base of the natural system of
logarithms, an absolutely incommensurable number. It is called
the Napierian Base in honor of Napier, the inventor of logarithms, and its value to thirty decimal places is
2.71828 18284 59045 23536 02874 7I353