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 ebe numerically approximated as0.43429without knowing e or any its properties?
Presuming that the histories are correct, there must be a mathematically interesting reason for both of those events.