The story of e

Greek numerals went no further than the myriad M, equal to ten thousand, which was enough to count the army of Darius the Great. But this system was powerless to count grains of sand, as the poet Pindar put it: "Sand [arena in Greek] escapes number."

To remedy this, Archimedes devised in his work The Sand Reckoner a system equivalent to our decimal system, using the myriad of myriads as its base, denoted here by N = M². The "first numbers" run from 1 to N = 10⁸, the "second numbers" from N to N² = 10¹⁶, and so on up to $A=N^N={10}^{8\times {10}^8},A\; =\; N^N\; =\; 10^\{8\; \backslash times\; 10^8\},$ the last of the "N^th numbers." Archimedes then continued the process with A as the new base to demonstrate the power of his notation.

He considered a "first period" from 1 to A, and so on up to $B=A^N={10}^{8\times {10}^{16}}\mathrm{.}B\; =\; A^N\; =\; 10^\{8\; \backslash times\; 10^\{16\}\}.$ In this way, he defined a number greater than the number of grains of sand in a "Universe completely filled with sand." Archimedes, strikingly modern in this respect, was thus the first to use an exponential principle.

The role of trigonometry -------------------------

Since antiquity, it has been known that, in a geometric sequence *u_n = qⁿ with common ratio q, the product of the n*^th term and the m^th term equals the (n+m)^th term. Exponential notation makes this replacement of multiplication by addition particularly clear: