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As Napier calculated the logarithms of numbers in the base $\left(1-\frac{1}{10^7}\right)^{10^7}$, I expected to find the number $e$ in the tables of his Mirifici Logarithmorum Canonis Descriptio. That is, the number 3,678,794, which is equal to the integer part of $10^7\times\left(1-\frac{1}{10^7}\right)^{10^7}$, as $\left(1-\frac{1}{10^7}\right)^{10^7}$ gets close to $ \frac{1}{e}$ and Napier multiplies the result by $10^7$.

However, the number that I find on the last row of the last table is 3,465,735 rather than 3,678,794. So, this number is certainly not $1/e \times 10^7$. Is there a way of identifying the number $e$ in the tables?

Note: This question is related to this answer, which however refers to the book Constructio of Napier. I refer here to the book Descriptio, which is the first book of Napier on logarithms.

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    $\begingroup$ Why would you expect to see $e$ in a $\ln\sin$ table? The $3,465,735$ is there because it is $-10\ln(1/\sqrt{2})$, not because it is an inaccurate approximation of $e$. The closest you can get is an approximation of $\arcsin(1/e)$ (in degrees) in the entry where $-\ln\sin$ is close to $1$. $\endgroup$
    – Conifold
    Commented Oct 1, 2023 at 4:58
  • $\begingroup$ Thank you, @Conifold. Could you expand your comment in an answer? Maybe this is the way to go to identifying the number e from the tables $\endgroup$
    – Andre
    Commented Oct 1, 2023 at 9:55

2 Answers 2

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Q: "Can I find the number e in the tables of Napier?" (and) "The number that I find on the last row of the last table is 3,465,735 rather than 3,678,794. Is there a number closer to e in the tables?"

To begin by cutting quickly to the chase, I see no mistake on Napier's part with the number 3465735 in the tables (which the question suggests is a mistake by Napier). When converted to modern form, with sign and decimal point, this number is at most 1 unit (in the last place) different from the natural-logarithmic sine (or cosine) of 45 degrees -- which is what it ought to be, for its place in the layout of Napier's tables.

John Napier of Merchistoun (various spellings survive) gave his first tables in his book with title 'Mirifici Logarithmorum Canonis Descriptio' (1614) -- translated four years later by Edward Wright as 'Description of the admirable table of logarithms'. Napier tabulated what can readily be recognized now as natural (or Napierian, or base-e) logarithms of trigonometric functions, but printed them according to conventions of his day, so that (for example) one does not look for decimal points or for signed numbers.

In the main tables, for example, each page-header gives an angle in degrees, and the successive rows are for functions of the header degree supplemented by 0, 1, 2, (etc) minutes. The first column of tabular values in any row then gives the (natural) sine of the angle for that row, and the second column gives the (natural, Napierian or base-e) logarithm of that sine. The remaining columns give natural-logarithmic tangent and cosine, then the natural cosine, and finally, on the right, the minutes in the complement to 90 degrees of the angle identified by the supplement of minutes in the left column (when the right-hand column angle is read as a supplement of minutes with the number of degrees shown this time at the foot of the table-page).

The result has an economy of layout, so that the tables as a whole appear to cover primarily a semi-quadrant (i.e. a range of 45 degrees). But when read in the appropriate direction for any chosen angle, i.e. downwards from the top and from left to right for angles in the range 0 to 45 degrees, but upwards from the foot and from right to left for angles in the range 45 to 90 degrees, they give the whole gamut of logarithmic sines, tangents, and cosines for an entire quadrant (thus also natural-logarithmic cotangents), although modern names (apart from 'sinus' and 'logarithmi') do not appear.

Although 8 digits are printed, the accuracy in those early tables does not always reach 6 digits e.g. relative to a precise natural-logarithmic sine value.

So, to the extent that one can find tabulated angles and the natural, base-e, logarithms of their sines and other trigonometric functions to the accuracy indicated, then yes, one can say that the base of the natural logarithms that we call 'e' is implicitly found there throughout.

One does not find the value of 'e' given explicitly in Napier's tables. But then it also doesn't appear explicitly in a modern table of natural logarithms or natural-logarithmic trig functions, either -- unless one counts its presence in a table of $e^x$, which is not strictly a logarithm table at all.

There can conceivably be other ways of looking at this question -- perhaps for example by asking whether there is a number close to but not equal to 'e', to which Napier's numerical values relate more closely than to 'e' itself. One could perhaps also say that in modern terms Napier has tabulated $-10^7 * ln (sin(angle))$ rather than the $ln(sin(angle))$ itself, and so on. But there does not seem to be evidence that Napier thought in ways that match either of those ideas. The non-modern way in which he did think can probably best be gathered through reading his description or at least its 17th-c. English translation.

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  • $\begingroup$ Thank you for your answer. There is nothing in the question that suggests that Napier made a mistake. I pointed out a number that cannot be interpreted as number close to e. Of course, the number means something else, as you stated in your answer (I will revise the question to make it more explicit). I am looking for ways of finding e in the tables $\endgroup$
    – Andre
    Commented Oct 1, 2023 at 9:52
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    $\begingroup$ @Andre , you wrote "the number that I find on the last row of the last table is 3,465,735 rather than 3,678,794" -- directly implying that you expected to find 3,678,794, and implying it should have been there. My answer explains the makeup of the tables, making it clear that the number e itself is not expected to be tabulated and is not explicitly there. But as also explained, e is implicitly there (to the accuracy indicated) in the 'ln-trig' relations between the given angles and the tabulated quantities. If you are expecting to find e expressly in the table, could you explain why? $\endgroup$
    – terry-s
    Commented Oct 1, 2023 at 10:12
  • $\begingroup$ I expected to find 3,678,794, but finding another number meant, of course, that my expectation was wrong :). Your answer is good to understand what is in the columns of the tables. The number e is implicitly in the base, but it could be retrieved from a number stated in the tables (the number e, 1/e, or another number that could be related in the easiest possible way for a moder reader to e). I am looking for this. The reason is to make the findings of Napier closer to a modern reader and to clarify the evolution of the ideas that brought the definition of the number e $\endgroup$
    – Andre
    Commented Oct 1, 2023 at 10:18
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I received an answer to this question in the Math StackExchange (link). For completeness, I include the answer below and expand it further.

The tables constructed by Napier have rows with the first three columns given by the angle $x$, the sine of $x$ and a number called logarithm by him. This number is calculated from the sine of $x$, given in the second column. All numbers are multiplied by $10^7$ to avoid decimals. Here is the table starting with 30 degrees and continuing for each additional minutue in the rows:

Fig 1

The logarithm of Napier is the number $L$ that solves $n^L = y$, given $y$ in the second column, and $n = (1-1/z)^z$, $z = 10^7$. $z$ is so large that $n$ is close to $1/e$. When $L = 1$, then $y = 1/e$. The number $L = 1$ appears on the tables for 21 degrees, between 35 and 36 minutes:

enter image description here

The tables have $L = 10000685$ for $y = 3678541$ and $L = 9993335$ for $y = 3681246$. A linear interpolation yields $y = 3678793$. So, the number $e$ from the tables is given by $10^7/y$, which is equal to $2.71828$. Equal to $e$ from a calculator up to the 5th decimal place. Beautiful.

My goal with this exercise is to make the findings of Napier closer to a modern reader and to clarify the evolution of the ideas that brought the definition of the number $e$. I thank your participation.

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    $\begingroup$ Thank you for posting the extra material here. The question you asked on the math page was different than the question asked here, which was: "Can I find the number e in the tables of Napier?" The shortest answer to that question here is that the number e itself is not in Napier's tables. But the earlier answer here did point out that e "is implicitly found there". On the math page you asked "Is there a way of retrieving the number e from the tables of Napier?". and the answer you got shows how e can be retrieved from where it is only implicit. Two questions, two answers. $\endgroup$
    – terry-s
    Commented Apr 26 at 11:11

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