Can anyone find Newton's calculation error in Principia, Book III, Proposition XIX?

Question

Musing about the historical evolution of the notation for the gravitational constant ($f$, $G$, $\kappa$, $\kappa^2$), I found myself digging for the first time in my life into Newton's Principia, looking for the data he could have used (had he thought algebrically) to estimate his constant (see this question). In the process, I noticed a funny calculation error in a simple division in Book III, Proposition XIX, Problem III (e.g. p.406 of Motte's translation, but I could find it in every version of the book I checked). Here it is:

the circumference of the earth is 123,249,600 feet and its semidiameter 19615800 Paris feet, upon the supposition that the earth is of a spherical figure. [...] A body in every sidereal day of 23h 56 4s uniformly revolving in a circle at the distance of 19615800 feet from the centre, in one second of time describes an arc of 1433.46 feet.

Dividing 123,249,600 feet by the 86,164 seconds of the sideral day, he should have found 1430.407! The error was acknowledged e.g. by MacDougal in his undergrads lecture in 2012 p.172 where he noted in a footnote that:

Using Newton’s numbers for circumference and time, our calculations show the velocity value to be 1,430.4 Paris feet per second, about 3 feet per second less than Newton’s value. It is unknown why there is this small discrepancy against Newton’s value.

You can also find it in Harper's book (2011) p.254 footnote 74. Harper made no comments about Newton's value, and had the wrong number of seconds in a day (86,160) so his result doesn't count, but he made me wonder if this calculation is mathematically cursed. At least in that case the mistake is obvious. But, and that is my question here, what happened with Isaac's division?

I find doubtful the possibility that 1433.46 instead of 1430.40(7) is a typo (2 non-adjacent digits are involved). I tried to play with the inputs, but again you need to have more that one digit wrong to get Newton's answer. (I'm making the implicit assumption that Isaac did at most one silly mistake.) So there may be an obvious calculation error somewhere, but I can't find anything plausible, so any hint is welcome.

I know it is not really an interesting physical question (neither mathematical, nor historical) but I am kind of haunted by it now, and I needed to do something to get rid of it.

Answer 1

It's certainly in the 1726 (3^rd) edition.

However, the first edition appears to not have this passage at all. The only mention of the semidiameter of the earth as 19615800 Parisienne feet (pedum Parifienfium, where the "f" is the long s sound ) is on page 424 see also here at Gunnerus Library and deals with calculation of the rotational periods of the planets. In this edition it goes from Prop XIX Prob. II to Prop. XX, Prob III. I was unable to find any passage similar to the one you described either earlier or later in the same section in this edition.

The second edition (at Bibliotek Zurich) appears to have a version of the third edition, possibly also with an incorrect calculation on page 379 under Proposition XIX Problema III:

Corpus in circulo, ad diftantium pedum 19695539 a centro, fingulus deibus fidereus horarum 23. 56'. 4" uniformiter revolvens, tempore minuti unius fecundi defcribit arcum pedum 1436,223,...

My rough translation (with my very little Latin): "A circular body with a distance of 19695539 feet from a central point, with a sidereal day of hours 23, 56 min, 4 s in uniform revolution will describe an arc of 1436.223 feet in one second,..."

Edited to add @Terry-s translation, which is much better than mine:

"A body revolving uniformly in a circle at a distance of 19695539 feet from the center, [making each of its revolutions] in single sidereal days of 23 h, 56m, 4s, describes in one second an arc of 1436.223 feet, of which the versed sine is 0.05236558 feet, or 7.54064 lines." {The words in square brackets have to be understood for the sense: the 1999 Cohen/Whitman English translation of the 3rd edition does essentially the same.}{'Lines' were 12 per inch, 144 per foot.}