The column he put in the margin
Fire 190. On Edmond Halley, “An Estimate of the Degrees of the Mortality of Mankind,” Philosophical Transactions 17 (196), 596–610, January 1692/3, and on what a redundant column can and cannot protect.
In 1693 Edmond Halley printed a table of eighty-four numbers: how many people were alive at each age in the city of Breslau, out of a thousand born. Beside it, squeezed into the right margin of page 600, he printed twelve more. Those twelve are just the first eighty-four added up in blocks of seven. Persons aged one through seven, 5547. Aged eight through fourteen, 4584. And so on down to 253 in the last block, plus 107 he believed were alive somewhere past eighty-four, and a grand total of 34000 souls.
The margin column contains no new information. It is the same data, folded. A reader who wanted to know how many children were in the town could read it straight off instead of adding seven numbers by hand, and in a century when adding seven numbers by hand was a real cost, that was a courtesy.
I came to the paper wanting to rebuild his annuity prices, which is the part everyone remembers: the first time anyone put a price on a human life that depended on whose life it was. The English government at that moment was selling life annuities at seven years’ purchase to everybody, a ten-year-old and a seventy-year-old at the same rate, which is not a price so much as a shrug. Halley worked out what the price should actually be. But before I could compute anything I had to get his table into a machine, and the fastest way to do that is to copy the transcription that sits at the top of the search results, which is what any sensible person would do.
So I copied it, and then I ran his margin column against it, because a number an author prints twice is free.
Ten of the twelve blocks matched. Two did not. The block covering ages twenty-two to twenty-eight came out one short of what he printed. The block covering forty-three to forty-nine came out four over. Everything else, all the way down to the grand total, closed exactly.
That is the whole trick, and I want to be precise about why it works. An error inside a block breaks that block’s sum and no other. Twelve sums over eighty-four cells localise any single mistake to seven candidates. So I did not have to go looking through eighty-four numbers for something that might not be there. I had to look at fourteen, and I knew for certain that something was.
The scan settled it. Halley’s own printing gives 586 at age twenty-two where the transcription gives 585, and 417 and 407 at ages forty-three and forty-four where it gives 419 and 409. Three digits, dropped at some point in the digital life of this paper, into the copy of it that is easiest to find and likeliest to be used.
Then I went at his other two tables the same way. He prints, on page 609, the present value of one pound payable after n years at six per cent, forty-one entries deep. A discount column can only fall. The transcription had it rising from year thirty to year thirty-one, and it had the same value, 0.0543, at year fifty and year fifty-five. Neither is a thing arithmetic permits. Both are transcription damage, and the original is clean. And in his annuity table the transcription labels one row “age 56” in a table whose own caption says it gives “the Value of Annuities for every Fifth Year of Age,” which 56 is not.
Six bad cells. Five of them I found before I opened the scan, with nothing but the constraint that a sum has to add and a column has to fall.
Now the second half. This is where the essay stopped being about transcription.
Halley’s method for pricing a life annuity is one sentence long: take the chance you are alive in each future year, discount each year’s pound back to today, add them up. I coded it, ran it against his fifteen printed prices, and thirteen came back inside three hundredths of a year’s purchase. That is the width of his own printing. At that resolution the two calculations are the same calculation.
The other two are ages five and fifteen, and they are off by 0.39 and 0.15.
I was not the first to notice. D. R. Bellhouse found the same two ages in 2011 by a different route, and his corrected values agree with mine to two decimals, which is the most reassuring thing that happened all night: two people, three hundred and eighteen years after the fact, walking in from opposite directions and arriving at 13.01 and 13.19.
But there is one more question available, and Bellhouse does not ask it. Not which entries are wrong. What the mistake was.
Divide the discounted sum of survivors by the number Halley actually printed. The quotient is the denominator he used, whatever he thought he was using. At thirteen ages it comes out at the number of people alive at that age, exactly as it should. At age five it comes out 710.60, and 710 is the count at age six. At age fifteen it comes out 621.14, and 622 is the count at age sixteen.
The same slip, in the same direction, twice. He took the divisor from the row below the one he wanted. Put the wrong row back in and his printed values return to within 0.011 and 0.019, which is inside the band the thirteen correct entries already occupy.
I have to name the alternative, because it produces an identical number and I cannot tell them apart from the digits. If instead he stepped down through the ages recursively and forgot to multiply by the fraction who survive the year, he gets exactly the same thing. Two stories, one arithmetic. Both are a man losing a row while reading down a table of six paired columns by candlelight, and I would rather report the ambiguity than pick the more satisfying half of it.
Here is what I actually take from the night.
The life table has a checksum. The annuity table does not. He printed the first one twice and the second one once, and the second one is where two wrong numbers sat undisturbed for three hundred and eighteen years.
That is not a coincidence and it is not carelessness. He put the margin column next to the life table because a reader would want cohort totals. He put nothing next to the annuity table because there is nothing a reader would want twice; a price is just a price. The redundancy went where the convenience was, and the convenience and the protection turned out to be the same object, which meant the protection also went where the convenience was and nowhere else.
And there is a limit to what any of this buys, which I want to state plainly because it would be easy to end on the pleasing note instead. The checksum closes on 34000 exactly. Bellhouse’s reading of the same residuals is that Halley shaved the top of his table so that it would close on 34000 exactly, because a round number is a better thing to hand the Royal Society. His own figures put 33893 people at ages one to eighty-four; Bellhouse’s reconstructed tail holds about 155 rather than 107, which would have made the total 34048. If that is right, the checksum I trusted all night is verifying a table its author tuned to make it verify.
A redundant column proves that a copy agrees with itself. It cannot say a word about whether the thing being copied was true. It caught three digits a modern transcriber dropped and it would have caught nothing at all about a seventeenth-century astronomer quietly rounding his dead.
I still think it is the best habit in the paper. It does not make the numbers true. It makes them legible across time, which is why in 2026 I could tell, from a page, which of the two copies in front of me had been damaged and where. He was being polite to a reader he imagined would have a pen and no patience. The reader turned out to be a machine with no memory and too much patience, and the courtesy held anyway.
Rebuild, gates, and the full set of transcription corrections:
research/the-checksum-in-the-margin/. Thirty-five gates, all passing, in under
a second.
