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Paraphrasing MacTutor's "The Story of E", Wikipedia gives this history of Euler's constant e:

The first references to the constant were published in 1618 in the table of an appendix of a work on logarithms by John Napier. However, this did not contain the constant itself, but simply a list of logarithms to the base e.

How is this possible? Presumably, any table of natural logarithms would contain e as the entry that gives 1.0000. And how could a table of logarithms be computed without knowing its base?

The article says that logarithms were not then recognized as the inverse of an exponential, so how could e would arise implicitly. It was not yet known as the limit of a compound interest formula (which occurred in the late 1600s), not yet recognized as an integral of a reciprocal (calculus didn't arise until the late 1600s), not recognized as function equal to its own derivative (also in the late 1600s).

The MacTutor history goes on to say:

A few years later, in 1624, again e almost made it into the mathematical literature, but not quite. In that year Briggs gave a numerical approximation to the base 10 logarithm of e but did not mention e itself in his work.

How could a person compute a numerical approximation of the common log of a number without knowing the number itself? How could you even present the result? "The base 10 logarithm of some important number yet to be is discovered is 0.43429, here is my work to compute it but the number itself is no where to be found and won't be seen for another 50 years. And also, I already have a table of logs at hand but haven't looked up the antilogarithm of the approximation."

The questions:

  • How (and why) could a table of natural logs arise without knowing e or any its properties?

  • How (and why) can log₁₀ e be numerically approximated as 0.43429 without knowing e or any its properties?

Presuming that the histories are correct, there must be a mathematically interesting reason for both of those events.

Raymond Hettinger
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  • The natural log of y can be defined as the integral of 1/x from 1 to y. e can then be defined as the number that makes the natural log equal to 1 – Barry Carter Aug 06 '22 at 01:16
  • @BarryCarter That explanation doesn't follow the timeline. Integrals weren't invented yet and that definition of e didn't arise until much later. AFAICT the early tables of natural logarithms and the approximation of log10(e) predates any thinking about "area under the curve." – Raymond Hettinger Aug 06 '22 at 01:21
  • You can approximate area without calculus, but, if it isn't that, I'm stumped – Barry Carter Aug 06 '22 at 01:26
  • "Unlike the logarithms used today, Napier's logarithms are not really to any base although in our present terminology it is not unreasonable (but perhaps a little misleading) to say that they are to base 1/e" says https://mathshistory.st-andrews.ac.uk/Biographies/Napier/ which goes on to explain how he thought about logs – Barry Carter Aug 06 '22 at 01:29
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    Roughly speaking: if a particle moves along a line segment AB with speed proportional to the distance from B, the time it takes is essentially the integral of 1/x, even though it's not seen as an area – Barry Carter Aug 06 '22 at 01:35
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    I recommend you read (at least) Chapter 6 of C.H. Edwards's The Historical Development of the Calculus. The title of that chapter is "Napier's Wonderful Logarithms." – Ted Shifrin Aug 06 '22 at 17:21
  • "Presumably, any table of natural logarithms would contain e as the entry that gives 1.0000." Here is an example of a table of natural logarithms. As you can verify for yourself, there is no entry that gives 1.0000. – David K Aug 06 '22 at 18:03
  • @DavidK It's unsurprising three place inputs only gives three places of accuracy, where 1.00063 corresponds to 2.72. Also, these table were normally used with interpolation. In this case (1 - 0.996949) / (1.00063 - 0.996949) * 0.01 + 2.71 gives 2.7182885085574573. You could also use a better table that with more detail. – Raymond Hettinger Aug 06 '22 at 23:10
  • Ah, it's the "any true table of natural logarithms" argument, a variation of "any true Scotsman." But in any case, the MacTutor and Wikipedia articles greatly oversimplify the story, which could be misleading. For one thing, Napier didn't use decimal points. There's a better account under Motivation for Napier's Logarithms. – David K Aug 07 '22 at 03:06

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