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The magic words are
See the article whose title is the above quotation by Barry Cipra, SIAM News, July 1994, 1, 12-13.
The works of Napier and Briggs contain many tables of logarithms to various bases to facilitate the calculations. Napier’s original 1614 work was, like most scientific works of the time, written in Latin. Englishman Edward Wright translated Napier’s text into English. This version of the Descriptio was printed in two editions, 1616 and 1618. The second edition contains an appendix of tables of logarithms, one entry of which assigns 2.302585 to 10. In modern notation this gives us approximately log2.718 10 = 2.302585 . In general, this appendix contains logs of the form (in modern notation) 106 logex. Thus this is a table of what we would now call natural logarithms. Thus, buried in the work of Napier is the first implicit use of e as a constant. (Keep in mind that when I refer to e, it was not given that designation until Euler over a century later.) However, it is commonly believed that this appendix was not written by Napier, but was compiled by William Oughtred .
William Oughtred (1574-1660) is known for giving us such notation as X for multiplication and :: for proportion . In relation to his work on logarithms, Oughtred capitalized on Napier’s definition of the log in 1622 to create a simple device for performing arithmetical operations. By placing two rulers in logarithmic scale next to each other, he found he could perform multiplications and other operations by sliding these two rulers along one another, creating the first slide rule [6, 9]. Mitchell and Strain  conjecture that this contribution to the development of natural logarithms and e is largely forgotten due to the fact that Briggs’s revision of Napier’s work was much more efficient, thus leaving Napier’s earlier works and, along with them, the appendix of Oughtred, largely unused.
Shell-Gellasch, Amy, "Napier's e," Loci (December 2008), DOI: 10.4169/loci003209
rational approximation of an irrational number
students find it really curious, and more intriguing when i tell them that (1+(1/n))^n in the limit of n tending to infinity is an irrational number 'e'. Here n is a positive integer. They find it really difficult to believe that a rational expression on the left hand side tends to an irrational number. How do I convince them ?
Geometrical introduction of Napier's e
Usually Ludolph's number is introduced geometrically as the "perimeter to diameter number" of any curve of constant distance to a center-point. So why shouldn't we introduce Napier's number in a more or less similar way as the "tangent-point-height to base-point-height number" of any curve of constant relative growth ? Here we refer to the following property: Chose an arbitrary point T on an arbitrary growth curve (exponential curve) g with base line (asymptote) b. Let B be the point of intersection of b and the tangent line at T. Let B' be the point of intersection of g and the perpendicular to b containing B; and let T' be the point of intersection of b and the perpendicular to b containing T. It can be proved that line segment T'T measured with BB' gives Napier's number. I would like to learn about publications in which this geometrical approach is mentioned.