Talk:Napierian logarithm

Unsourced definition

This article's definition of Napierian logarithms in the lead has been unsourced since the beginning. I added a source with a number recently, but it's also not clearly the right number for this context. It has a slightly different definition, and would lead to a very different lead. I'm not completely confident that I'd get this right, or make it acceptable to others if I just took this on, so I've asked for expert help to discuss and plan what to do. Also, the ref I added has a 2010 date, and an article by the same title with 2012 date was already there as external link; I don't know whether there are significant differences between the two. Dicklyon (talk) 03:38, 9 May 2018 (UTC)

The EL:

• Denis Roegel (2012) Napier’s Ideal Construction of the Logarithms, from the Loria Collection of Mathematical Tables.

In particular, the first ratio expression is just complicated, and using no specific log base is that much harder to interpret, though I agree it's equivalent to this next one, which may be correct but I haven't seen in any sources:

$\mathrm {NapLog} (x)=\log _{\frac {10^{7}}{10^{7}-1}}10^{7}-\log _{\frac {10^{7}}{10^{7}-1}}x$

The typically described procedure for table 1 of Constructio would suggest writing in term of a base slightly less than 1, $1-10^{-7}$, maybe like this:

$\mathrm {NapLog} (x)=\log _{1-10^{-7}}(x/10^{7})$

This is simpler, but doesn't necessarily makes things clear, and may still not be correct, if I understand the source. This is very close to what the source seems to say:

$\mathrm {NapLog} (x)=-10^{7}\log _{e}(x/10^{7})$

But it's not quite the same, because $(1-10^{-7})^{10^{7}}$ differs from e by about 1 in the 7th decimal place. It's not clear to me which is what. Notice the footnote at bottom of page 9 of the source: "Some authors write incorrectly that Napier took the logarithm of 9999999 to be 1. This is for instance the case of Delambre [42, vol. 1, p. xxxv]. The logarithm of 9999999 may have been 1 in Napier’s first experiments, but not in the actual Descriptio." The formula with base ${1-10^{-7}}=9999999/10000000$ would give 1, but Roegel says Napier found 1.00000005 and that the exact [value] with his more "natural" formula would be $10^{7}(\log(10^{7})-\log(9999999))=1.000000050000003$. Do I believe him?

Dicklyon (talk) 04:19, 9 May 2018 (UTC)

References

@Dicklyon: I've been lookin' over the paper you reference. Yes, I think we can believe him. Apparently people have been sayin' that Napier gave a logarithm of 1 to 9999999 for a long time (Delambre was 1821), and whoever started our Wikipedia article went with that. Apparently Napier meant that his logarithm should be

$\mathrm {NapLog} (x)=-10^{7}\log _{e}(x/10^{7})$

but the tables he calculated were not exact. For example, he estimated the logarithm of 9900000 to be 100503.3210291 whereas it should be 100503.35853501... (This corresponds to ln(.99)=−0.010050335853501...) That means that most of his logarithms must be too low by about 0.4 ppm.

The erroneous definition by which the logarithm of 9999999 is exactly 1 gives values that are 0.05 ppm lower than what Napier intended. So actually Napier's error is greater than the difference between the two definitions!

So go ahead and edit the article, and remove the call for expert opinion.

Eric Kvaalen (talk) 16:30, 9 May 2018 (UTC)

OK, maybe, but I remain unclear on how this works. I don't understand how it's possible that Napier meant what you and that author said, since e and natural logs didn't exist. So I'll need to read it some more and see if I get it. Dicklyon (talk) 19:54, 9 May 2018 (UTC)

@Dicklyon: Well, he basic'ly invented the natural logarithm, before calculus was discovered. That's why the natural logarithm is sometimes called the Naperian logarithm, even though he scaled the numbers by 10 million. Eric Kvaalen (talk) 05:27, 11 May 2018 (UTC)

Sure, I get that. But it's not exactly clear to me how. The story of base 999999/1000000 makes sense to me; just make a table of where at each step you shift by 7 places and subtract. And that's how he make his table 1. But then there were the other two tables, and interpolation methods, etc., and it remains unclear to me what made it come out as a "natural" logarithm, with factors of 10^7. I'll work on understanding that. Dicklyon (talk) 06:12, 11 May 2018 (UTC)

@Dicklyon: Note: in his first table he didn't just calculate powers of 0.9999999 -- he assigned logarithms to them which were multiples of 1.00000005, not 1. So he wasn't findin' log base 0.9999999. If he hadn't made some miscalculations he would have gotten a very good approximation of natural log. Of course, he could have gotten something even better by using even more nines, and so on. But you always have to draw the line somewhere. Eric Kvaalen (talk) 16:25, 12 May 2018 (UTC)

OK, looking at the Desciptio, I see the key defining line: 26. The logarithm of a given sine is that number which has increased arithmetically with the same velocity throughout as that with which radius began to decrease geometrically, and in the same time as radius has decreased to the given sine. From this, the "natural" interpretation follows. The rest is just scale factors. Easy peasy. Dicklyon (talk) 19:52, 12 May 2018 (UTC)

The claim that Napier described something distinctly different from what is today called the natural logarithm seems to rely on reading according to present notational conventions a work that long predates those conventions. Napier introduces the use of scaled numbers in step 3 of the Descriptio, where he states that the "largest sine" may be represented in a table as 100000, but the "more learned" use 10000000 to better express the "differences of all the sines" i.e. precision. Should one conclude from this that he believed that the ratio of a triangle's height to its hypotenuse approaches 100,000 or 10,000,000, or conclude instead that he is offering two computationally convenient notations for unity? If the latter, shouldn't his notation for the logarithm be interpreted the same way, and this article redirect to natural logarithm with appropriate historical discussion there? 2601:642:4600:3F80:80CC:941C:FBBB:E7C7 (talk) 05:20, 19 September 2023 (UTC)