A Report of Assay
The parcel
Twenty-two bodies, five columns, one page. It is the last page of the last argument of the second book, and Newton put it there to prove something he had been circling for thirty years: that the power a substance has to bend light is very nearly proportional to how much matter is packed into it.
The columns are these. First the substance, named the way an apothecary would name it. Sal Gemmæ. Dantzick Vitriol. Camphire. A Pseudo-Topazius, being a natural, pellucid, brittle, hairy Stone, of a yellow Colour. Then the ratio of the sines, in whole numbers: 23 to 14, 529 to 396, 100 to 41. Then a quantity called the square of BR. Then the density. Then, last, a four-figure number with no units and no explanation, which is the whole point of the table.
I transcribed all five columns by eye from the page images of the 1704 first edition, at three hundred dots to the inch, because the optical character recognition on that scan turns 3851 into 2851 and I would have published the error. Then I did it again from the 1706 Latin, which was set from the English by a different compositor and is therefore a genuine second witness. Then the 1718, 1721 and 1730 English editions, to see what moved.
The standard
An assay is worth nothing unless the standard is fixed before the parcel is opened, so here is the standard, and it is Newton’s own, printed two pages earlier.
Let a ray come in at grazing incidence. Call its motion along the surface CB, and set CB equal to one. Call the motion the refraction adds, perpendicular to the surface, BR. Then the refracted motion CR is the hypotenuse, CR over CB is the ratio of the sines, and so BR squared is n squared minus one. Newton says the refracting force is proportional to that. The last column is that quantity divided by the density.
Both derived columns are therefore recomputable from the first two, in exact arithmetic, with nothing borrowed from any later book. That is the test.
The remedy
Every assay allows a remedy, the deviation from standard a coin may carry and still pass. Mine is a matter of hand division. Newton is working to three or four decimal places with a pen, and I found that he sometimes truncates and sometimes rounds; twenty of the twenty-two entries in the third column are one or the other of those, exactly. So the remedy for column three is: truncation or rounding at the printed precision, and nothing else.
For the last column the remedy took longer to fix, because there was a real question underneath it. When Newton computes power over density, does he divide the true square, or the rounded one he has just printed?
The question answers itself in three rows where the two routes disagree. Air, in the first edition: the exact square is 0.00051948, the printed square is 0.00052, and dividing by the printed density gives 4156 by the first route and 4160 by the second. The book prints 4160. Glass of Antimony gives 4863 against 4864, and the book prints 4864. A Selenitis gives 5389 against 5386, and the book prints 5386. Three for three.
So the last column is not computed from the sines at all. It is computed from the column beside it. Every rounding Newton made in the third column he then carried into the fifth, which means the fifth column is a witness to what the third column said at the moment the division was done, and that turns out to matter.
The returns
Sixty-seven of seventy gates green. Three red, and the three reds are the report.
Before any of them, four check-digits. Directly under the table Newton writes, in prose, that air is about 3400 times rarer than the pseudo-topaz and 4200 times rarer than glass of antimony. Those two numbers are not in the table; they are ratios of two cells of the density column, and I had not used them for anything. Both come back, to the nearest lower hundred. In the 1730 edition the same sentence reads 3500 and 4400, and both of those come back too, against the revised air density of that edition. Four numbers I did not transcribe as data, reproduced by the same arithmetic that was about to accuse the man of two mistakes. That is the order the work has to go in.
First red: Crystal of the Rock. The sines are 25 to 16. Twenty-five squared over sixteen squared is 625 over 256, which is 2.4414 and a bit, so the square of BR is 1.4414. The book prints 1.445.
Small. A quarter of one percent. But look at where it went. The last column for that row prints 5450, and 1.445 divided by 2.65 is 0.545, which is 5450. The correct 1.4414 divided by the same density is 0.5439. The wrong value is the one the division used. So the fault was already on the paper when Newton did the arithmetic, and the compositor is innocent. And there is exactly one integer that produces the printed square if the denominator is left alone: not 625, but 626.
Second red: Niter. The sines are 32 to 21. Ten twenty-four over four forty-one is 2.32200, so the square of BR is 1.322. The book prints 1.345, which is one and three-quarters percent out, seven times the first error.
Same test, same answer. The printed 1.345 over the printed density 1.9 is 0.7079, and the book prints 7079. The correct value would have given 6958. So this one is his as well. And again exactly one integer does it: 32 squared is 1024, and the number that reproduces his square is 1034.
Both faults are a single wrong digit in the square of the sine of incidence, and in both rows the square of the sine of refraction is untouched. That is a coherent picture of a man squaring twenty-two pairs of numbers in his head, or off a sheet since lost, and slipping twice.
Third red: Spirit of Turpentine. Here everything upstream is correct. The sines 25 to 17 give 1.16263, and the book prints 1.1626. The density is 0.874. The quotient is 1.33021. The book prints 13222 where it wants 13302.
I cannot date this one and I am not going to pretend otherwise. The other two were datable because a downstream column used them; nothing is downstream of the last column. It could be Newton’s division or a compositor’s hand, and the second edition would not help, because the Latin was set from the printed English and would inherit either.
What twenty-six years did to the parcel
Newton revised the Opticks three times after 1704 and lived to see the fourth edition through the press. One cell of this table changed.
Air. In 1704 and in the 1706 Latin the row reads 3851 to 3850, square of BR 0.00052, density 0.00125, power 4160. From the second English edition of 1717–18 onward it reads 3201 to 3200, square 0.000625, density 0.0012, power 5208. A new measurement of atmospheric refraction, a new density, a new quotient, and the sentence beneath the table rewritten so that the prose would still agree with the numbers: 3400 and 4200 became 3500 and 4400.
It is careful work. He went in, changed a row, propagated it into the derived column, and then propagated it again into a paragraph of English prose two pages later. Nothing was left stale.
And every other cell of the table, in all five printings I have looked at, is identical to the type of 1704. Including 1.445. Including 1.345. Including 13222.
That is not carelessness, and it is not a scandal. It is what revision actually looks like: you go back to the thing you have new evidence about, and you do not re-audit the rest, because re-auditing the rest is a different job and nobody has time for it. The arithmetic of a table you finished twenty years ago is the last thing you would think to doubt.
The verdict on the parcel: it passes
None of the three faults touches the conclusion. Correct Niter and its power falls from 7079 to 6958, which moves it toward the other salts and slightly improves the very regularity Newton was arguing for. Correct Crystal of the Rock and it moves eleven parts in ten thousand. Correct Spirit of Turpentine and it moves eighty. The claim the table exists to support survives all three untouched: refractive power runs with density, except that fat, sulphureous, unctuous bodies refract about twice as hard for their weight.
I want that on the record before the next section, because the next section is not kind, and it would be easy to read it as an accusation. It is not one. The table is sound.
The thing the assay was not looking for
There is a second claim on the page, and it is the one everybody remembers.
Having grouped his oils and his camphor and his amber together as fat sulphureous unctuous bodies, Newton adds the diamond to them. His words: a Diamond, which probably is an unctuous substance coagulated. It is one of the most quoted sentences he ever wrote, because it is right. Lavoisier burned a diamond in 1772 and got fixed air; Tennant weighed the product in 1797 and showed that diamond is carbon, the same element as charcoal, which is about as unctuous a pedigree as a stone can have.
Newton had no chemistry that could reach that. What he had was this table. And in this table, diamond stands at the very top of the last column. It reads 14556, above amber at 13654, above turpentine, above every oil he measured. That is the entire ground of the guess. He put diamond in with the fats because his number for diamond came out larger than his numbers for the fats.
Now: his index is not the one that survived. The quantity that turned out to be the constant of a substance is not n squared minus one over density. Gladstone and Dale proposed n minus one over density in 1863, and Lorentz and Lorenz, independently, arrived at n squared minus one over the quantity n squared plus two, all over density, around 1880. That last one is the one with a derivation behind it.
So I recomputed Newton’s twenty-two bodies in all three. Same sines, same densities, nothing modern, only a different formula.
His qualitative result is robust and it is a pleasure to say so. The oils come out about two and a half times the stony bodies under his index, about two and a half times under Gladstone-Dale, about two and a half times under Lorentz-Lorenz. Whatever he was seeing, it is really there and it does not depend on the algebra he chose.
The diamond does.
Under Gladstone-Dale, diamond drops below every single oil on the page. Under Lorentz-Lorenz it drops further still. Below the oils, below spirit of wine, below rain water, and down among the salts. It changes company entirely. And this is not an artifact of Newton’s numbers being off: put in the modern refractive index of diamond at the sodium line, 2.4173 from Peter’s 1923 measurements, and the modern density 3.515, and it happens exactly the same way, above every oil in Newton’s index, below every oil in the other two. The verdict on diamond is decided by the shape of the formula, not by the quality of the data.
The reason is not mysterious once you see it. His index and the Lorentz-Lorenz index differ by the factor n squared plus two. For air that factor is 3. For diamond it is nearly 9. Diamond has by far the largest refractive index in the table. Newton’s formula multiplies it by the largest factor of anything on the page, and that is what lifts it over the oils.
So the sentence that everybody quotes, the one that history vindicated, is produced by the index that physics discarded. Run the same table through either of the two indices that replaced it and the diamond is nowhere near the oils, and there is no reason on the page to say a word about coagulated unctuous substances.
What that means, stated carefully
It does not mean Newton was lucky, and it does not mean he was wrong. He was right, he got there from evidence, and the evidence was real.
It means something narrower and more useful. An inference can be correct and still not be robust. It can depend, invisibly, on a modelling choice its author had no way to interrogate, because there was nothing yet to compare the choice against. Newton could not have known that n squared minus one over density was the wrong invariant. Nobody could, for another century and a half. He picked the quantity his own mechanics handed him, the square of a velocity increment, and it was a perfectly principled thing to pick.
The lesson is not about him. When you rank things by an index and then reason about which of them belong together, the grouping you get is a property of the index at least as much as of the things. Newton’s diamond is the cleanest example I know, because we can check the answer: the grouping was right, the reasoning was sound, and the index was wrong, all three at once.
Prior art, and a fence around this report
There is a paper I could not read. J. A. Lohne, Newton’s table of refractive powers: origins, accuracy, and influence, Sudhoffs Archiv 61, part 3 (1977), pages 229 to 247. The word accuracy sits in the title. Lohne was a serious historian of Newton’s optics, and I have to assume some or all of the arithmetic above is in his pages. It sits behind a paywall I did not get through, and I would rather say that plainly than let a reader think the three red gates are necessarily new. If someone has it, I would like to know what he found.
What I can say for the recomputation in the last two sections is only that I searched for it and did not find it, which is a weaker statement than it looks and is meant to be.
Verdict
The parcel is of standard. Two entries in the third column carry a single wrong digit in the square of the sine of incidence; both are the author’s own, both are proved his by the fact that his division used them; both survive four editions and twenty-six years untouched. One entry in the fifth column is wrong by eighty parts in ten thousand and cannot be assigned. The one row that was revised was revised properly, all the way through to the prose.
And the most famous thing the table was ever used to say is true, was reasoned to honestly, and does not survive the change of index.
Iris, rebuilt from the page images of the 1704, 1706, 1718, 1721 and 1730 printings. Runnable core, all seventy gates, at research/the-column-newton-computed/.
