The Room Inside a Number

A figure on a page is not a number. It is a claim about an interval, and the width of that interval is whatever the printer set in type. That sounds like pedantry until you ask what a rounded table can prove, at which point it becomes the only question there is: the answer is exact, it needs no tolerance and no curve fitting, and for the most famous table in physics it comes out to four decimal places in one edition and to nothing at all in the one before.

Here is the whole idea. 0.124 does not mean 124 / 1000. It means the quantity lies somewhere in [0.1235, 0.1245), because that is the set of numbers that print as 0.124. Every published figure is a room of that kind, and the room has a size you can compute from the digits alone.

The last two chips are the ones that matter for anyone writing a check. A helper that takes a number rather than the printed string has already lost: JavaScript renders 41.50 as 41.5, so the helper sizes the interval ten times too wide, and it reads the exponent in 1.5e-7 as decimal places, so it sizes that one three hundred times too wide. Both mistakes make a check pass when it should not. This site had exactly that helper, in exactly one stratum, until tonight.

What a table proves

Now the part that is a theorem rather than a caution. Suppose a table prints periods T and distances a for several bodies and asserts that T is proportional to a^p. Which values of p are consistent with the page? A pair (k, p) works when every row admits some true period and some true distance inside its own printed interval obeying T = k a^p. Write that out and the constant drops away, leaving two inequalities for each pair of rows and nothing else:

So point the instrument at the tables. Newton states the three-halves power three times, in 1687, 1713 and 1726, and revises the numbers under it every time. Below, each row he printed, solved.

Three things fall out, and none of them needed a judgement call.

The first edition's own arithmetic does not close. The last row of Newton's satellite table is his own computation, labelled Ex temporibus periodicis. In 1687 that row is inconsistent with the periods printed six lines above it, under every exponent: the floor its pairs impose sits above the ceiling. Anchor it the way he actually worked, on Flamsteed's eclipse distance for the first satellite, and two of the four cells miss. He printed 14,168 where his own periods give 14.1701…14.1710, and 24,968 where they give 24.9600…24.9615.

The second edition's does. By 1713 the periods are printed to the second rather than to a fifth of a minute, the computed row is redone, and all four cells land inside the interval the printed periods allow. That row now certifies the three-halves power to four decimal places: with the anchor the eighteenth-century commentators document, the admitted exponents are [1.4999832, 1.5000277] and nothing else.

What the observations prove is a hundred times weaker than what the digits suggest. Cassini's eclipse row, the best in any edition, admits [1.49306, 1.50664]: the three-halves power to about seven parts in a thousand. The tightest observational pinning anywhere in these tables is not in a table at all. It is five figures in a sentence in the body of Phaenomenon II, and it gives ±0.00064.

Newton did not fit anything, which is worth saying because it is what makes these rows testable at all. He assumed one observed distance and carried the rest across by the law, and Le Seur and Jacquier's commentary of 1739–42 records which distance he assumed for which system: the first satellite at 5 2/3 for Jupiter, the fourth at 8 ring-semidiameters for Saturn. So the sharpest test available does not need their testimony either. Ask instead whether any single anchor reconciles a computed row with the periods printed on its own page. Five such rows exist across the three editions.

So there is no tidy arc of improvement. Jupiter's row is repaired between the first and second editions and then reprinted unchanged. Saturn's is sound in the second edition and breaks in the third, where the fifth satellite is printed 23,35 and the anchor at 8 gives 23.3135. That is three units in the last printed place, in a row whose other four cells land exactly, which is the signature of one slip rather than a method that does not work.

A published claim, and what the printed rows say to it

Asking what exponent Newton's data support is not a new question. Quayshawn Spencer asked it in 2004, in a paper about whether Newton's rules of reasoning guarantee truth, took the slope of log T against log r across three successive pairs of Jupiter's satellites, got 1.48, 1.44 and 1.38, and concluded:

“An appeal to rule 1 does not work since seven fifths, ten sevenths, and three halves are all simple fractions and are all equally supported by the data (since the imprecision in data exceeds these fractions).”

Run the same question on the rows Newton actually printed, with each figure read as the interval its own last digit denotes, and it comes out the other way. Of the twelve distance rows printed for Jupiter's satellites across the three editions, two admit ten sevenths, and they are the same row: Cassini's naked-telescope estimates, four whole numbers, 5 8 13 23, printed in 1687 and reprinted in 1713. Every other row, every row given to two decimals or better, excludes ten sevenths and seven fifths both. Borelli's thirds exclude them. Cassini's eclipse row excludes them. Newton's computed row excludes them by a factor of two hundred in the width of the interval. Use the selector above to watch each verdict appear.

Two honest qualifications, because they matter. Spencer's distances (5.46, 8.61, 13.71, 24.42) match no row Newton printed in any edition, so this is not a disagreement about arithmetic; it is a different table. And read at their own printed precision his four figures are not consistent with any single power law either, which is the same thing as saying his three slopes disagree with one another by more than two decimals can absorb. His philosophical point about Rule 1 may well stand on other ground. What does not stand is the parenthesis: for Newton's own printed rows, the imprecision in the data does not exceed the gap between three halves and ten sevenths.

Phaenomenon IV, cell by cell

The third edition is the first to print the planets' periodic times, which is the only reason this next check is possible at all: periods and distances finally appear on the same page, page 393. Newton gives three rows of mean distances, Kepler's, Boulliau's, and his own computed from the periods, with the Earth at 100000. That last figure is the unit and not a measurement, so it has no interval: nothing was rounded to produce it. Everything else is a room.

None of the three rows is consistent with any power law at the precision printed, and for the two observation rows that is the finding rather than a problem: distances known to three or four figures are printed to six, so the digits claim more than the observations knew, and the contradiction is the measure of the overclaim. Kepler's Saturn misses by 2798 units, Boulliau's Jupiter by 2410.

Newton's own row is a different animal, because he computed it. Three of its five non-unit cells follow from his printed periods to the last digit: Mars, Mercury, and the Earth. Three do not. Measured edge to edge, so these are minimum discrepancies: Saturn is 207 units high, Venus 17 high, Jupiter 13 low.

Which of the two is wrong

An interval test can say that two cells disagree. It cannot say which is at fault, and it should not pretend to. But the disagreement can be pushed back the other way: invert each printed distance into the period it implies, and then a value measured since gets a vote. Used only for that, never to define the disagreement.

The two failures are of opposite kinds. For Saturn the printed period is right, near enough, and the distance cell is the error: 954006 implies a period between 10762.775 and 10762.793 days, where his own page says 10759.275 and the modern value is 10759.22. For Venus it is the other way round: the distance cell 72333 implies between 224.697 and 224.703 days, a window that contains the modern 224.701, while the period printed on the same page, 224.6176, is wrong by 0.083 of a day. One cell of that table is a bad distance and another is a bad period, and the difference is invisible to any comparison that treats the figures as numbers.

On the Saturn cell there is one more wrinkle, and it is typographic. In both scans consulted, of two different physical copies, the fourth digit of that figure prints as an open c rather than a closed zero: a defective sort in the type, not damage to one book. Six independent later witnesses read it as 954006, so 954006 it is. It does not matter for the result, which is the useful thing about interval arithmetic: every reading of 954?06 misses the implied window, so the conclusion is the same whatever that glyph was meant to be.

The same question, asked of this site

An instrument that only measures other people is a weapon, not an instrument. So: this repository contains no shared assertion library, which means nearly every verifier declares its own comparator and picks its own tolerance by hand. Measured in units of the last printed place of the figure being compared against, here is what those tolerances are worth.

What the number means, precisely, is the resolution of the check: how far a computed value could stray from the literal and still pass. It is not a verdict on the author. A tolerance of one per cent against the Feigenbaum constant printed to fourteen places carries two trillion units of slack and is perfectly sensible, because it is testing whether a numerical method converges, not whether the constant's digits are right. The slack figure is only a verdict when the literal itself is the thing under test, which is why a hand-checked subset of sites quoting named publications is reported separately below.

Two of those twelve derive the tolerance from the source's own precision, and one of the two does it in code rather than by hand. The rest range from two units to two hundred. In several cases the file says plainly that it is comparing two different methods rather than testing the source's digits, and there a loose tolerance is the right choice; the slack figure then simply states, correctly, that the check could not notice a wrong digit. That is worth knowing either way.