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I was digging about the discovery of logarithms by John Napier. It is clear that he was looking for a function he could compute (at least approximately) that would map products into sums. He came up with an ingenious idea[1]. Of course, we can't know how he came up with it, but when I was thinking about it, it struck me that he must not have a continuous exponential function to use. Was it the case?

  1. https://www.maa.org/press/periodicals/convergence/logarithms-the-early-history-of-a-familiar-function-john-napier-introduces-logarithms
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  • $\begingroup$ Thinking in terms of our present notions of "continuous" and "function" is probably not going to be very helpful, maybe somewhat like wondering to what extent Mary Shelley had gender issues and climate change issues in mind when writing "Frankenstein". (What relevance to feminism was the fact that the monster was chosen male? What relevance to deforestation or climate change was implied by the monster's exposure to fire?) $\endgroup$ Mar 31 at 17:36
  • $\begingroup$ There were very few modern mathematicians in Europe before Napier. So you should be able to find out if any of them used exponents like $2^x$ with non-rational $x$. $\endgroup$
    – markvs
    Mar 31 at 20:20
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    $\begingroup$ Possibly of interest is the 7-part series of papers History of the exponential and logarithmic concepts by Florian Cajori, which is freely available through the links given in this MSE answer. $\endgroup$ Mar 31 at 21:13
  • $\begingroup$ You wrote "It is clear that he was looking for a function" but this is not correct. The modern concept of "function" did not exist at that time. $\endgroup$
    – Somos
    Apr 1 at 1:52

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I want to thank Dave L Renfro for his comment. After carefully looking at the two first papers in the series History of the exponential and logarithmic concepts by Florian Cajori, I found my answer.

The publication of Napier's work on logarithms was in 1614. By then, we didn't even have an established notation for exponents. People used to talk about geometric progressions, in words.

The notation we nowadays use for powers of a number was introduced by René Descartes' La géométrie in 1637 (hence, about 23 years after Napier's work). But this was only for natural numbers in the exponent, and Descartes still preferred to use $aa$ instead of $a^2$ for squares.

In 1656, John Walis writes, on his Arithmetica infinitorum, that the series $1/\sqrt 1, 1/\sqrt 2, 1/\sqrt 3, \ldots$ has index $-1/2$, while the series $1, 4, 9, \ldots$ has index $2$. Fractional exponents were explicit in 1676, following current notation, on a letter Newton wrote to H. Oldenburg and then forwarded to Leibniz. This was related to his famous binomial theorem.

As a final remark, I want to say that what Napier did was, in effect, define an exponential function with a continuous domain. In his work, he describes a point moving with a velocity proportional to the distance remaining in front of it. This is a differential equation (though this is before the development of calculus) whose solution is basically an exponential with base $e^{-1}$. He did not think about bases, though. He computed this function approximately, of course, so, in practice, he used an exponential with base $(1 - 1/10^7)^{10^7} \cong e^{-1}$.

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  • $\begingroup$ For the reader who wanders, Napier's goal was to make a table of a function that mapped products to sums, to simplify the long multiplications needed in astronomy. More specifically, to simplify the computations involving spherical trigonometry. $\endgroup$
    – Seno
    Apr 1 at 13:06

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