The Information - James Gleick [45]
In the information-poor world, where any table of numbers was a rarity, centuries went by before people began systematically to gather different printed tables in order to check one against another. When they did, they found unexpected flaws. For example, Taylor’s Logarithms, the standard quarto printed in London in 1792, contained (it eventually transpired) nineteen errors of either one or two digits. These were itemized in the Nautical Almanac, for, as the Admiralty knew well, every error was a potential shipwreck.
Unfortunately, one of the nineteen corrections proved erroneous, so the next year’s Nautical Almanac printed an “erratum of the errata.” This in turn introduced yet another error. “Confusion is worse confounded,”♦ declared The Edinburgh Review. The next almanac would have to put forth an “Erratum of the Erratum of the Errata in Taylor’s Logarithms.”
Particular mistakes had their own private histories. When Ireland established its Ordnance Survey, to map the entire country on a finer scale than any nation had ever accomplished, the first order of business was to ensure that the surveyors—teams of sappers and miners—had 250 sets of logarithmic tables, relatively portable and accurate to seven places.♦ The survey office compared thirteen tables published in London over the preceding two hundred years, as well as tables from Paris, Avignon, Berlin, Leipzig, Gouda, Florence, and China. Six errors were discovered in almost every volume—and they were the same six errors. The conclusion was inescapable: these tables had been copied, one from another, at least in part.
Errors arose from mistakes in carrying. Errors arose from the inversion of digits, sometimes by the computers themselves and sometimes by the printer. Printers were liable to transpose digits in successive lines of type. What a mysterious, fallible thing the human mind seemed to be! All these errors, one commentator mused, “would afford a curious subject of metaphysical speculation respecting the operation of the faculty of memory.”♦ Human computers had no future, he saw: “It is only by the mechanical fabrication of tables that such errors can be rendered impossible.”
Babbage proceeded by exposing mechanical principles within the numbers. He saw that some of the structure could be revealed by computing differences between one sequence and another. The “calculus of finite differences” had been explored by mathematicians (especially the French) for a hundred years. Its power was to reduce high-level calculations to simple addition, ready to be routinized. For Babbage the method was so crucial that he named his machine from its first conception the Difference Engine.
By way of example (for he felt the need to publicize and explain his conception many times as the years passed) Babbage offered the Table of Triangular Numbers. Like many of the sequences of concern, this was a ladder, starting on the ground and rising ever higher:
1, 3, 6, 10, 15, 21 …
He illustrated the idea by imagining a child placing groups of marbles on the sand:
Suppose the child wants to know “how many marbles the thirtieth or any other distant group might contain.” (It is a child after Babbage’s own heart.) “Perhaps he might go to papa to obtain this information; but I much fear papa would snub him, and would tell him that it was nonsense—that it was useless—that nobody knew the number, and so forth.” Understandably papa knows nothing of the Table of Triangular Numbers published at the Hague by É. de Joncourt, professor of philosophy. “If papa fail to inform him, let him go to mamma, who will not fail to find means to satisfy her darling’s curiosity.”♦ Meanwhile,