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e in Euler's Earlier Work
1736 The first time that e actually appeared in print for the public was in Euler’s 1736 two-volume work on mechanics, Mechanica sive motus scientia analytice exposita [6, op E15, E16]. For simplicity, we will refer to it as the Mechanica. In this work Euler clarifies Newton’s laws of motion through the machinations of calculus . The symbol e also appears in various articles by Euler in the years 1747 and 1751 .
However, it was not until after the Mechanica that Euler used e exclusively. In his earlier writings on this subject, he first used the letter c to denote the constant in question. In the Mechanica Euler defines e in terms almost identical to what he would later use in the Introductio . Euler’s use of c was picked up and used by D’Alembert and the astronomer Daniel Melandri, with Laplace using it as late as 1812. However Daniel Bernoulli’s use of e in a 1760 publication may have helped solidify e’s place in mathematics .
1731 (1843) In correspondence dated 25 November, 1731 to Christian Goldbach regarding the integral of (1 - x1/n)p [6, letter XV, op 00729], Euler used “e to denote that number whose hyperbolic logarithm is = 1.” [12, p. 97] However, these letters did not get published until 1843.
1727 (1862) Euler wrote so much that often later material was published before earlier, the story being that in later life his son or a colleague would pick up the top several papers on his “done” pile and send them to be published. Thus, older papers were sometimes left on the bottom of the pile. So when looking at Euler’s work, we must look at the date the work was written, since the publication date means little as far as chronology is concerned. A case in point is Euler’s Meditatio in experimenta explosione tormentorum nuper institute (Meditation upon experiments made recently on the firing of Cannon). This work was written in 1727, when Euler was 20 years old, but not published until 1862, 79 years after his death. In this work Euler uses e for the first time in his recorded work .
If the Meditatio had been published within a few years of being written, we might be more likely to attribute the use of e to this 1727 work instead of his 1748 Introductio. In either case, we still credit Euler with giving us Euler’s number, 2.71828… And he was indeed the one to give us the notation e. But is it fair to refer to it as “Euler’s number”? Not to be confused with: the Euler Number (of a finite complex K), χ(K); the Euler or Euler-Mascheroni Constant, γ; the Euler Numbers or zig numbers ; the Eulerian Numbers; or finally, Euler’s Characteristic, also known as Euler’s Number .
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.