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

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.

• This is not the only calculational error in book III, Garisto found another one in Proposition VIII. There Newton calculated the relative mass of the Earth for 11" parallax, but then changed the parallax to 10.5" in the third edition, while forgetting to change the relative mass accordingly (which he did calculate in the manuscript). Is this error present in all editions? Commented Nov 12, 2020 at 20:53
• yes, I knew that story (from Garisto himself on twitter). Until now, I only checked, the english (Motte) & french (du Châtelet) translated version. But now that you're asking, I looked into the latin version (I found the annotated first edition here cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00001/1 and the one typed in the wikisource (not sure of the edition) la.wikisource.org/wiki/…) : the calculation is not there, the whole first part of Prop.XIX, Prob.III is missing (there Prop.XIX is only Prob.II)... interesting... Commented Nov 13, 2020 at 10:10
• Someone knows where to find the 2nd & 3rd editions online ? Commented Nov 13, 2020 at 10:17
• OK! I have the (annotated) 2nd ed. (cudl.lib.cam.ac.uk/view/PR-ADV-B-00039-00002/851) & it's enlightening! The printed version has an earth radius of 19,695,539 and a (correct) computed velocity of 1436.223. But he changed the radius by hand to 19,622,659 & the velocity (still correct) to 1430.085. That hints to the answer: at the time of the 2nd ed. he's not decided which value to choose for the radius! Note that none of these are the value used in the translation of the 3rd ed. Note also that the circumference given page 848 IS the one used in the translated editions ... Commented Nov 13, 2020 at 12:55
• Is there an online version of volume II, 3rd edition? Commented Nov 13, 2020 at 13:42

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

• It may be some little help to transcribe with original spelling, word-endings and punctuation, as follows: == "Corpus in circulo, ad distantiam pedum 19695539 a centro, singulis diebus sidereis horarum 23, 56’. 4” uniformiter revolvens, tempore minuti unius secundi describit arcum pedum 1436,223, cujus sinus versus est pedum 0.05236558, seu linearum 7.54064." == This translates as: ../.. Commented Jun 9 at 21:31
• == This translates as: == "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.} Commented Jun 9 at 21:31