What Just Happened: A Chronicle From the Information Frontier

by James Gleick

  0 1 2 3 4 5 … (base 2 logarithms)

  1 2 4 8 16 32 … (natural numbers)

  and the result is a crude table of logarithms—crude, because the whole-number exponents are the easy ones. A useful table of logarithms had to fill in the gaps, with many decimal places of accuracy.

  In Napier’s mind was an analogy: differences are to ratios as addition is to multiplication. His thinking crossed over from one plane to another, from spatial relationships to pure numbers. Aligning these scales side by side, he gave a calculator a practical means of converting multiplication into addition—downshifting, in effect, from the difficult task to the easier one. In a way, the method is a kind of translation, or encoding. The natural numbers are encoded as logarithms. The calculator looks them up in a table, the code book. In this new language, calculation is easy: addition instead of multiplication, or multiplication instead of exponentiation. When the work is done, the result is translated back into the language of natural numbers. Napier, of course, could not think in terms of encoding.

  Briggs revised and extended the necessary number sequences and published a book of his own, Logarithmicall Arithmetike, full of pragmatic applications. Besides the logarithms he presented tables of latitude of the sun’s declination year by year; showed how to find the distance between any two places, given their latitudes and longitudes; and laid out a star guide with declinations, distance to the pole, and right ascension. Some of this represented knowledge never compiled and some was oral knowledge making the transition to print, as could be seen in the not-quite-formal names of the stars: the Pole Starre, girdle of Andromeda, Whales Bellie, the brightest in the harpe, and the first in the great Beares taile next her rump.♦ Briggs also considered matters of finance, offering rules for computing with interest, backward and forward in time. The new technology was a watershed: “It may be here also noted that the use of a 100 pound for a day at the rate of 8, 9, 10, or the like for a yeare hath beene scarcely known, till by Logarithms it was found out: for otherwise it requires so many laborious extractions of roots, as will cost more paines than the knowledge of the thing is accompted to be worth.”♦ Knowledge has a value and a discovery cost, each to be counted and weighed.

  Even this exciting discovery took several years to travel as far as Johannes Kepler, who employed it in perfecting his celestial tables in 1627, based on the laboriously acquired data of Tycho Brahe. “A Scottish baron has appeared on the scene (his name I have forgotten) who has done an excellent thing,” Kepler wrote a friend, “transforming all multiplication and division into addition and subtraction.”♦ Kepler’s tables were far more accurate—perhaps thirty times more—than any of his medieval predecessors, and the accuracy made possible an entirely new thing, his harmonious heliocentric system, with planets orbiting the sun in ellipses. From that time until the arrival of electronic machines, the majority of human computation was performed by means of logarithms.♦ A teacher of Kepler’s sniffed, “It is not fitting for a professor of mathematics to manifest childish joy just because reckoning is made easier.”♦ But why not? Across the centuries they all felt that joy in reckoning: Napier and Briggs, Kepler and Babbage, making their lists, building their towers of ratio and proportion, perfecting their mechanisms for transforming numbers into numbers. And then the world’s commerce validated their pleasure.

  Natural Numbers Logarithms base 2

  1 0

  2 1

  3 1.5850

  4 2

  5 2.3219

  6 2.5850

  7 2.8074

  8 3

  9 3.1699

  10 3.3219

  11 3.4594

  12 3.5850

  13 3.7004

  14 3.8074

  15 3.9069

  16 4

  17 4.0875

  18 4.1699

  19 4.2479

  20 4.3219

  21 4.3923

  22 4.4594

  23 4.5236

  24 4.5850

  25 4.6439

  26 4.7004

  27 4.7549

  28 4.8074

  29 4.8580

  30 4.9069

  31 4.9542

  32 5

  33 5.0444

  34 5.0875

  35 5.1293

  36 5.1699

  37 5.2095

  38 5.2479

  39 5.2854

  40 5.3219

  41 5.3576

  42 5.3923

  43 5.4263

  44 5.4594

  45 5.4919

  46 5.5236

  47 5.5546

  48 5.5850

  49 5.6147

  50 5.6439

  Charles Babbage was born on Boxing Day 1791, near the end of the century that began with Newton. His home was on the south side of the River Thames in Walworth, Surrey, still a rural hamlet, though the London Bridge was scarcely a half hour’s walk even for a small boy. He was the son of a banker, who was himself the son and grandson of goldsmiths. In the London of Babbage’s childhood, the Machine Age made itself felt everywhere. A new breed of impresario was showing off machinery in exhibitions. The shows that drew the biggest crowds featured automata—mechanical dolls, ingenious and delicate, with wheels and pinions mimicking life itself. Charles Babbage went with his mother to John Merlin’s Mechanical Museum in Hanover Square, full of clockwork and music boxes and, most interesting, simulacra of living things. A metal swan bent its neck to catch a metal fish, moved by hidden motors and cams. In the artist’s attic workshop Charles saw a pair of naked dancing women, gliding and bowing, crafted in silver at one-fifth life size. Merlin himself, their elderly creator, said he had devoted years to these machines, his favorites, still unfinished. One of the figurines especially impressed Charles with its (or her) grace and seeming liveliness. “This lady attitudinized in a most fascinating manner,”♦ he recalled. “Her eyes were full of imagination, and irresistible.” Indeed, when he was a man in his forties he found Merlin’s silver dancer at an auction, bought it for £35, installed it on a pedestal in his home, and dressed its nude form in custom finery.♦

  The boy also loved mathematics—an interest far removed from the mechanical arts, as it seemed. He taught himself in bits and pieces from such books as he could find. In 1810 he entered Trinity College, Cambridge—Isaac Newton’s domain and still the moral center of mathematics in England. Babbage was immediately disappointed: he discovered that he already knew more of the modern subject than his tutors, and the further knowledge he sought was not to be found there, maybe not anywhere in England. He began to acquire foreign books—especially books from Napoleon’s France, with which England was at war. From a specialty bookseller in London he got Lagrange’s Théorie des fonctions analytiques and “the great work of Lacroix, on the Differential and Integral Calculus.”♦

  He was right: at Cambridge mathematics was stagnating. A century earlier Newton had been only the second professor of mathematics the university ever had; all the subject’s power and prestige came from his legacy. Now his great shadow lay across English mathematics as a curse. The most advanced students learned his brilliant and esoteric “fluxions” and the geometrical proofs of his Principia. In the hands of anyone but Newton, the old methods of geometry brought little but frustration. His peculiar formulations of the calculus did his heirs little good. They were increasingly isolated. The English professoriate “regarded any attempt at innovation as a sin against the memory of Newton,”♦ one nineteenth-century mathematician said. For the running river of modern mathematics a student had to look elsewhere, to the Continent, to “analysis” and the language of differentiation as invented by Newton’s rival and nemesis, Gottfried Wilhelm Leibniz. Fundamentally, there was only one calculus. Newton and Leibniz knew how similar their work was—enough that each accused the other of plagiarism. But they had devised incompatible systems of notation—different languages—and in practice these surface differences mattered more than the underlying sameness. Symbols and operators were what a mathematician h
ad to work with, after all. Babbage, unlike most students, made himself fluent in both—“the dots of Newton, the d’s of Leibnitz”♦—and felt he had seen the light. “It is always difficult to think and reason in a new language.”♦

  Indeed, language itself struck him as a fit subject for philosophical study—a subject into which he found himself sidetracked from time to time. Thinking about language, while thinking in language, leads to puzzles and paradoxes. Babbage tried for a while to invent, or construct, a universal language, a symbol system that would be free of local idiosyncrasies and imperfections. He was not the first to try. Leibniz himself had claimed to be on the verge of a characteristica universalis that would give humanity “a new kind of an instrument increasing the powers of reason far more than any optical instrument has ever aided the power of vision.”♦ As philosophers came face to face with the multiplicity of the world’s dialects, they so often saw language not as a perfect vessel for truth but as a leaky sieve. Confusion about the meanings of words led to contradictions. Ambiguities and false metaphors were surely not inherent in the nature of things, but arose from a poor choice of signs. If only one could find a proper mental technology, a true philosophical language! Its symbols, properly chosen, must be universal, transparent, and immutable, Babbage argued. Working systematically, he managed to create a grammar and began to write down a lexicon but ran aground on a problem of storage and retrieval—stopped “by the apparent impossibility of arranging signs in any consecutive order, so as to find, as in a dictionary, the meaning of each when wanted.”♦ Nevertheless he felt that language was a thing a person could invent. Ideally, language should be rationalized, made predictable and mechanical. The gears should mesh.

  Still an undergraduate, he aimed at a new revival of English mathematics—a suitable cause for founding an advocacy group and launching a crusade. He joined with two other promising students, John Herschel and George Peacock, to form what they named the Analytical Society, “for the propagation of d’s” and against “the heresy of dots,” or as Babbage said, “the Dot-age of the University.”♦ (He was pleased with his own “wicked pun.”) In their campaign to free the calculus from English dotage, Babbage lamented “the cloud of dispute and national acrimony, which has been thrown over its origin.” Never mind if it seemed French. He declared, “We have now to re-import the exotic, with nearly a century of foreign improvement, and to render it once more indigenous among us.”♦ They were rebels against Newton in the heart of Newton-land. They met over breakfast every Sunday after chapel.

  “Of course we were much ridiculed by the Dons,” Babbage recalled. “It was darkly hinted that we were young infidels, and that no good would come of us.” Yet their evangelism worked: the new methods spread from the bottom up, students learning faster than their teachers. “The brows of many a Cambridge moderator were elevated, half in ire, half in admiration, at the unusual answers which began to appear in examination papers,”♦ wrote Herschel. The dots of Newton faded from the scene, his fluxions replaced by the notation and language of Leibniz.

  Meanwhile Babbage never lacked companions with whom he could quaff wine or play whist for six-penny points. With one set of friends he formed a Ghost Club, dedicated to collecting evidence for and against occult spirits. With another set he founded a club called the Extractors, meant to sort out issues of sanity and insanity according to a set of procedures:

  Every member shall communicate his address to the Secretary once in six months.

  If this communication is delayed beyond twelve months, it shall be taken for granted that his relatives had shut him up as insane.

  Every effort legal and illegal shall be made to get him out of the madhouse [hence the name “Extractors”].

  Every candidate for admission as a member shall produce six certificates. Three that he is sane and three others that he is insane.♦

  But the Analytical Society was serious. It was with no irony, all earnestness, that these mathematical friends, Babbage and Herschel and Peacock, resolved to “do their best to leave the world a wiser place than they found it.” They rented rooms and read papers to one another and published their “Transactions.” And in those rooms, as Babbage nodded over a book of logarithms, one of them interrupted: “Well, Babbage, what are you dreaming about?”

  “I am thinking that all these Tables might be calculated by machinery,”♦ he replied.

  Anyway that was how Babbage reported the conversation fifty years later. Every good invention needs a eureka story, and he had another in reserve. He and Herschel were laboring together to produce a manuscript of logarithm tables for the Cambridge Astronomical Society. These very logarithms had been computed before; logarithms must always be computed and recomputed and compared and mistrusted. No wonder Babbage and Herschel, laboring over their own manuscript at Cambridge, found the work tedious. “I wish to God these calculations had been executed by steam,” cried Babbage, and Herschel replied simply, “It is quite possible.”

  Steam was the driver of all engines, the enabler of industry. If only for these few decades, the word stood for power and force and all that was vigorous and modern. Formerly, water or wind drove the mills, and most of the world’s work still depended on the brawn of people and horses and livestock. But hot steam, generated by burning coal and brought under control by ingenious inventors, had portability and versatility. It replaced muscles everywhere. It became a watchword: people on the go would now “steam up” or “get more steam on” or “blow off steam.” Benjamin Disraeli hailed “your moral steam which can work the world.” Steam became the most powerful transmitter of energy known to humanity.

  It was odd even so that Babbage thought to exert this potent force in a weightless realm—applying steam to thought and arithmetic. Numbers were the grist for his mill. Racks would slide, pinions would turn, and the mind’s work would be done.

  It should be done automatically, Babbage declared. What did it mean to call a machine “automatic”? For him it was not just a matter of semantics but a principle for judging a machine’s usefulness. Calculating devices, such as they were, could be divided into two classes: the first requiring human intervention, the second truly self-acting. To decide whether a machine qualified as automatic, he needed to ask a question that would have been simpler if the words input and output had been invented: “Whether, when the numbers on which it is to operate are placed in the instrument, it is capable of arriving at its result by the mere motion of a spring, a descending weight, or any other constant force.”♦ This was a farsighted standard. It eliminated virtually all the devices ever used or conceived as tools for arithmetic—and there had been many, from the beginning of recorded history. Pebbles in bags, knotted strings, and tally sticks of wood or bone served as short-term memory aids. Abacuses and slide rules applied more complex hardware to abstract reckoning. Then, in the seventeenth century, a few mathematicians conceived the first calculating devices worthy of the name machine, for adding and—through repetition of the adding—multiplying. Blaise Pascal made an adding machine in 1642 with a row of revolving disks, one for each decimal digit. Three decades later Leibniz improved on Pascal by using a cylindrical drum with protruding teeth to manage “carrying” from one digit to the next.♦♦ Fundamentally, however, the prototypes of Pascal and Leibniz remained closer to the abacus—a passive register of memory states—than to a kinetic machine. As Babbage saw, they were not automatic.

  It would not occur to him to use a device for a one-time calculation, no matter how difficult. Machinery excelled at repetition—“intolerable labour and fatiguing monotony.”♦ The demand for computation, he foresaw, would grow as the uses of commerce, industry, and science came together. “I will yet venture to predict, that a time will arrive, when the accumulating labour which arises from the arithmetical application of mathematical formulae, acting as a constantly retarding force, shall ultimately impede the useful progress of the science, unless this or some equivalent method is devised for relieving it fr
om the overwhelming incumbrance of numerical detail.”♦

  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.”