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Chapter Four

To Throw the Powers of Thought into Wheel-Work

(Lo, the Raptured Arithmetician)

Light almost solar has been extracted from the refuse of fish; fire has been sifted by the lamp of Davy; and machinery has been taught arithmetic instead of poetry.

—Charles Babbage (1832)

NO ONE DOUBTED THAT Charles Babbage was brilliant. Nor did anyone quite understand the nature of his genius, which remained out of focus for a long time. What did he hope to achieve? For that matter, what, exactly, was his vocation? On his death in London in 1871 the Times obituarist declared him “one of the most active and original of original thinkers” but seemed to feel he was best known for his long, cranky crusade against street musicians and organ-grinders. He might not have minded. He was multifarious and took pride in it. “He showed great desire to inquire into the causes of things that astonish childish minds,” said an American eulogist. “He eviscerated toys to ascertain their manner of working.” Babbage did not quite belong in his time, which called itself the Steam Age or the Machine Age. He did revel in the uses of steam and machinery and considered himself a thoroughly modern man, but he also pursued an assortment of hobbies and obsessions—cipher cracking, lock picking, lighthouses, tree rings, the post—whose logic became clearer a century later. Examining the economics of the mail, he pursued a counterintuitive insight, that the significant cost comes not from the physical transport of paper packets but from their “verification”—the calculation of distances and the collection of correct fees—and thus he invented the modern idea of standardized postal rates. He loved boating, by which he meant not “the manual labor of rowing but the more intellectual art of sailing.” He was a train buff. He devised a railroad recording device that used inking pens to trace curves on sheets of paper a thousand feet long: a combination seismograph and speedometer, inscribing the history of a train’s velocity and all the bumps and shakes along the way.

As a young man, stopping at an inn in the north of England, he was amused to hear that his fellow travelers had been debating his trade:

“The tall gentleman in the corner,” said my informant, “maintained you were in the hardware line; whilst the fat gentleman who sat next to you at supper was quite sure that you were in the spirit trade. Another of the party declared that they were both mistaken: he said you were travelling for a great iron-master.”

“Well,” said I, “you, I presume, knew my vocation better than our friends.”

“Yes,” said my informant, “I knew perfectly well that you were in the Nottingham lace trade.”

He might have been described as a professional mathematician, yet here he was touring the country’s workshops and manufactories, trying to discover the state of the art in machine tools. He noted, “Those who enjoy leisure can scarcely find a more interesting and instructive pursuit than the examination of the workshops of their own country, which contain within them a rich mine of knowledge, too generally neglected by the wealthier classes.” He himself neglected no vein of knowledge. He did become expert on the manufacture of Nottingham lace; also the use of gunpowder in quarrying limestone; precision glass cutting with diamonds; and all known uses of machinery to produce power, save time, and communicate signals. He analyzed hydraulic presses, air pumps, gas meters, and screw cutters. By the end of his tour he knew as much as anyone in England about the making of pins. His knowledge was practical and methodical. He estimated that a pound of pins required the work of ten men and women for at least seven and a half hours, drawing wire, straightening wire, pointing the wire, twisting and cutting heads from the spiral coils, tinning or whitening, and finally papering. He computed the cost of each phase in millionths of a penny. And he noted that this process, when finally perfected, had reached its last days: an American had invented an automatic machine to accomplish the same task, faster.

Babbage invented his own machine, a great, gleaming engine of brass and pewter, comprising thousands of cranks and rotors, cogs and gearwheels, all tooled with the utmost precision. He spent his long life improving it, first in one and then in another incarnation, but all, mainly, in his mind. It never came to fruition anywhere else. It thus occupies an extreme and peculiar place in the annals of invention: a failure, and also one of humanity’s grandest intellectual achievements. It failed on a colossal scale, as a scientific-industrial project “at the expense of the nation, to be held as national property,” financed by the Treasury for almost twenty years, beginning in 1823 with a Parliamentary appropriation of £1,500 and ending in 1842, when the prime minister shut it down. Later, Babbage’s engine was forgotten. It vanished from the lineage of invention. Later still, however, it was rediscovered, and it became influential in retrospect, to shine as a beacon from the past.

Like the looms, forges, naileries, and glassworks he studied in his travels across northern England, Babbage’s machine was designed to manufacture vast quantities of a certain commodity. The commodity was numbers. The engine opened a channel from the corporeal world of matter to a world of pure abstraction. The engine consumed no raw materials—input and output being weightless—but needed a considerable force to turn the gears. All that wheel-work would fill a room and weigh several tons. Producing numbers, as Babbage conceived it, required a degree of mechanical complexity at the very limit of available technology. Pins were easy, compared with numbers.

It was not natural to think of numbers as a manufactured commodity. They existed in the mind, or in ideal abstraction, in their perfect infinitude. No machine could add to the world’s supply. The numbers produced by Babbage’s engine were meant to be those with significance: numbers with a meaning. For example, 2.096910013 has a meaning, as the logarithm of 125. (Whether every number has a meaning would be a conundrum for the next century.) The meaning of a number could be expressed as a relationship to other numbers, or as the answer to a certain question of arithmetic. Babbage himself did not speak in terms of meaning; he tried to explain his engine pragmatically, in terms of putting numbers into the machine and seeing other numbers come out, or, a bit more fancifully, in terms of posing questions to the machine and expecting an answer. Either way, he had trouble getting the point across. He grumbled:

On two occasions I have been asked,—“Pray, Mr. Babbage, if you put into the machine wrong figures, will the right answers come out?” In one case a member of the Upper, and in the other a member of the Lower, House put this question. I am not able rightly to apprehend the kind of confusion of ideas that could provoke such a question.

Anyway, the machine was not meant to be a sort of oracle, to be consulted by individuals who would travel from far and wide for mathematical answers. The engine’s chief mission was to print out numbers en masse. For portability, the facts of arithmetic could be expressed in tables and bound in books.

To Babbage the world seemed made of such facts. They were the “constants of Nature and Art.” He collected them everywhere. He compiled a Table of Constants of the Class Mammalia: wherever he went he timed the breaths and heartbeats of pigs and cows. He invented a statistical methodology with tables of life expectancy for the somewhat shady business of life insurance. He drew up a table of the weight in Troy grains per square yard of various fabrics: cambric, calico, nankeen, muslins, silk gauze, and “caterpillar veils.” Another table revealed the relative frequencies of all the double-letter combinations in English, French, Italian, German, and Latin. He researched, computed, and published a Table of the Relative Frequency of the Causes of Breaking of Plate Glass Windows, distinguishing 464 different causes, no less than fourteen of which involved “drunken men, women, or boys.” But the tables closest to his heart were the purest: tables of numbers and only numbers, marching neatly across and down the pages in stately rows and columns, patterns for abstract appreciation.

A book of numbers: amid all the species of information technology, how peculiar and powerful an object this is. “Lo! the raptured arithmetician!” wrote Élie de Joncourt in 1762. “Easily satisfied, he asks no Brussels lace, nor a coach and six.” Joncourt’s own contribution was a small quarto volume registering the first 19,999 triangular numbers. It was a treasure box of exactitude, perfection, and close reckoning. These numbers were so simple, just the sums of the first n whole numbers: 1, 3 (1+2), 6 (1+2+3), 10 (1+2+3+4), 15, 21, 28, and so on. They had interested number theorists since Pythagoras. They offered little in the way of utility, but Joncourt rhapsodized about his pleasure in compiling them and Babbage quoted him with heartfelt sympathy: “Numbers have many charms, unseen by vulgar eyes, and only discovered to the unwearied and respectful sons of Art. Sweet joy may arise from such contemplations.”

Tables of numbers had been part of the book business even before the beginning of the print era. Working in Baghdad in the ninth century, Abu Abdullah Mohammad Ibn Musa al-Khwarizmi, whose name survives in the word algorithm, devised tables of trigonometric functions that spread west across Europe and east to China, made by hand and copied by hand, for hundreds of years. Printing brought number tables into their own: they were a natural first application for the mass production of data in the raw. For people in need of arithmetic, multiplication tables covered more and more territory: 10 × 1,000, then 10 × 10,000, and later as far as 1,000 × 1,000. There were tables of squares and cubes, roots and reciprocals. An early form of table was the ephemeris or almanac, listing positions of the sun, moon, and planets for sky-gazers. Tradespeople found uses for number books. In 1582 Simon Stevin produced Tafelen van Interest, a compendium of interest tables for bankers and moneylenders. He promoted the new decimal arithmetic “to astrologers, land-measurers, measurers of tapestry and wine casks and stereometricians, in general, mint masters and merchants all.” He might have added sailors. When Christopher Columbus set off for the Indies, he carried as an aid to navigation a book of tables by Regiomontanus printed in Nuremberg two decades after the invention of moveable type in Europe.

Joncourt’s book of triangular numbers was purer than any of these— which is also to say useless. Any arbitrary triangular number can be found (or made) by an algorithm: multiply n by n + 1 and divide by 2. So Joncourt’s whole compendium, as a bundle of information to be stored and transmitted, collapses in a puff to a one-line formula. The formula contains all the information. With it, anyone capable of simple multiplication (not many were) could generate any triangular number on demand. Joncourt knew this. Still he and his publisher, M. Husson, at the Hague, found it worthwhile to set the tables in metal type, three pairs of columns to a page, each pair listing thirty natural numbers alongside their corresponding triangular numbers, from 1(1) to 19,999(199,990,000), every numeral chosen individually by the compositor from his cases of metal type and lined up in a galley frame and wedged into an iron chase to be placed upon the press.

Why? Besides the obsession and the ebullience, the creators of number tables had a sense of their economic worth. Consciously or not, they reckoned the price of these special data by weighing the difficulty of computing them versus looking them up in a book. Precomputation plus data storage plus data transmission usually came out cheaper than ad hoc computation. “Computers” and “calculators” existed: they were people with special skills, and all in all, computing was costly.

Beginning in 1767, England’s Board of Longitude ordered published a yearly Nautical Almanac, with position tables for the sun, moon, stars, planets, and moons of Jupiter. Over the next half century a network of computers did the work—thirty-four men and one woman, Mary Edwards of Ludlow, Shropshire, all working from their homes. Their painstaking labor paid £70 a year. Computing was a cottage industry. Some mathematical sense was required but no particular genius; rules were laid out in steps for each type of calculation. In any case the computers, being human, made errors, so the same work was often farmed out twice for the sake of redundancy. (Unfortunately, being human, computers were sometimes caught saving themselves labor by copying from one other.) To manage the information flow the project employed a Comparer of the Ephemeris and Corrector of the Proofs. Communication between the computers and comparer went by post, men on foot or on horseback, a few days per message.

A seventeenth-century invention had catalyzed the whole enterprise. This invention was itself a species of number, given the name logarithm. It was number as tool. Henry Briggs explained:

Logarithmes are Numbers invented for the more easie working of questions in Arithmetike and Geometrie. The name is derived of Logos, which signifies Reason, and Arithmos, signifying Numbers. By them all troublesome Multiplications and Divisions in Arithmetike are avoided, and performed onely by Addition in stead of Multiplication, and by Subtraction in stead of Division.

In 1614 Briggs was a professor of geometry—the first professor of geometry—at Gresham College, London, later to be the birthplace of the Royal Society. Without logarithms he had already created two books of tables, A Table to find the Height of the Pole, the Magnetic Declination being given and Tables for the Improvement of Navigation, when a book came from Edinburgh promising to “take away all the difficultie that heretofore hath beene in mathematical calculations.”

There is nothing (right well beloved Students in the Mathematickes) that is so troublesome to Mathematicall practice, not that doth more molest and hinder Calculators, then the Multiplications, Divisions, square and cubical Extractions of great numbers, which besides the tedious expence of time, are for the most part subject to many slippery errors.

This new book proposed a method that would do away with most of the expense and the errors. It was like an electric flashlight sent to a lightless world. The author was a wealthy Scotsman, John Napier (or Napper, Nepair, Naper, or Neper), the eighth laird of Merchiston Castle, a theologian and well-known astrologer who also made a hobby of mathematics. Briggs was agog. “Naper, lord of Markinston, hath set my head and hands a work,” he wrote. “I hope to see him this summer, if it please God, for I never saw book, which pleased me better, and made me more wonder.” He made his pilgrimage to Scotland and their first meeting, as he reported later, began with a quarter hour of silence: “spent, each beholding other almost with admiration before one word was spoke.”

Briggs broke the trance: “My Lord, I have undertaken this long journey purposely to see your person, and to know by what engine of wit or ingenuity you came first to think of this most excellent help unto astronomy, viz. the Logarithms; but, my Lord, being by you found out, I wonder nobody else found it out before, when now known it is so easy.” He stayed with the laird for several weeks, studying.

In modern terms a logarithm is an exponent. A student learns that the logarithm of 100, using 10 as the base, is 2, because 100 = 10². The logarithm of 1,000,000 is 6, because 6 is the exponent in the expression 1,000,000 = 10⁶. To multiply two numbers, a calculator could just look up their logarithms and add those. For example:

100 × 1,000,000 = 10² × 10⁶ = 10^{(2 + 6)}

Looking up and adding are easier than multiplying.

But Napier did not express his idea this way, in terms of exponents. He grasped the thing viscerally: he was thinking in terms of a relationship between differences and ratios. A series of numbers with a fixed difference is an arithmetic progression: 0, 1, 2, 3, 4, 5 . . . When the numbers are separated by a fixed ratio, the progression is geometric: 1, 2, 4, 8, 16, 32 . . . Set these progressions side by side,

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.

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 had 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:

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

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

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

4 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.¹ 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 from 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.”

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, Babbage answers the question by means of a table of differences. The first column contains the number sequence in question. The next columns are derived by repeated subtractions, until a constant is reached—a column made up entirely of a single number.

Any polynomial function can be reduced by the method of differences, and all well-behaved functions, including logarithms, can be effectively approximated. Equations of higher degree require higher-order differences. Babbage offered another concrete geometrical example that requires a table of third differences: piles of cannonballs in the form of a triangular pyramid—the triangular numbers translated to three dimensions.

The Difference Engine would run this process in reverse: instead of repeated subtraction to find the differences, it would generate sequences of numbers by a cascade of additions. To accomplish this, Babbage conceived a system of figure wheels, marked with the numerals 0 to 9, placed along an axis to represent the decimal digits of a number: the units, the tens, the hundreds, and so on. The wheels would have gears. The gears along each axis would mesh with the gears of the next, to add the successive digits. As the machinery transmitted motion, wheel to wheel, it would be transmitting information, in tiny increments, the numbers summing across the axes. A mechanical complication arose, of course, when any sum passed 9. Then a unit had to be carried to the next decimal place. To manage this, Babbage placed a projecting tooth on each wheel, between the 9 and 0. The tooth would push a lever, which would in turn transmit its motion to the next wheel above.

At this point in the history of computing machinery, a new theme appears: the obsession with time. It occurred to Babbage that his machine had to compute faster than the human mind and as fast as possible. He had an idea for parallel processing: number wheels arrayed along an axis could add a row of the digits all at once. “If this could be accomplished,” he noted, “it would render additions and subtractions with numbers having ten, twenty, fifty, or any number of figures, as rapid as those operations are with single figures.” He could see a problem, however. The digits of a single addition could not be managed with complete independence because of the carrying. The carries could overflow and cascade through a whole set of wheels. If the carries were known in advance, then the additions could proceed in parallel. But that knowledge did not become available in timely fashion. “Unfortunately,” he wrote, “there are multitudes of cases in which the carriages that become due are only known in successive periods of time.” He counted up the time, assuming one second per operation: to add two fifty-digit numbers might take only nine seconds in itself, but the carrying, in the worst case, could require fifty seconds more. Bad news indeed. “Multitudes of contrivances were designed, and almost endless drawings made, for the purpose of economizing the time,” Babbage wrote ruefully. By 1820 he had settled on a design. He acquired his own lathe, used it himself and hired metalworkers, and in 1822 managed to present the Royal Society with a small working model, gleaming and futuristic.

BABBAGE’S WHEEL-WORK

He was living in London near the Regent’s Park as a sort of gentleman philosopher, publishing mathematical papers and occasionally lecturing to the public on astronomy. He married a wealthy young woman from Shropshire, Georgiana Whitmore, the youngest of eight sisters. Beyond what money she had, he was supported mainly by a £300 allowance from his father—whom he resented as a tyrannical, ungenerous, and above all close-minded old man. “It is scarcely too much to assert that he believes nothing he hears, and only half of what he sees,” Babbage wrote his friend Herschel. When his father died, in 1827, Babbage inherited a fortune of £100,000. He briefly became an actuary for a new Protector Life Assurance Company and computed statistical tables rationalizing life expectancies. He tried to get a university professorship, so far unsuccessfully, but he had an increasingly lively social life, and in scholarly circles people were beginning to know his name. With Herschel’s help he was elected a fellow of the Royal Society.

Even his misfires kindled his reputation. On behalf of The Edinburgh Journal of Science Sir David Brewster sent him a classic in the annals of rejection letters: “It is with no inconsiderable degree of reluctance that I decline the offer of any Paper from you. I think, however, you will upon reconsideration of the subject be of opinion that I have no other alternative. The subjects you propose for a series of Mathematical and Metaphysical Essays are so very profound, that there is perhaps not a single subscriber to our Journal who could follow them.” On behalf of his nascent invention, Babbage began a campaign of demonstrations and letters. By 1823 the Treasury and the Exchequer had grown interested. He promised them “logarithmic tables as cheap as potatoes”—how could they resist? Logarithms saved ships. The Lords of the Treasury authorized a first appropriation of £1,500.

As an abstract conception the Difference Engine generated excitement that did not need to wait for anything so mundane as the machine’s actual construction. The idea was landing in fertile soil. Dionysius Lardner, a popular lecturer on technical subjects, devoted a series of public talks to Babbage, hailing his “proposition to reduce arithmetic to the dominion of mechanism,—to substitute an automaton for a compositor,—to throw the powers of thought into wheel-work.” The engine “must, when completed,” he said, “produce important effects, not only on the progress of science, but on that of civilization.” It would be the rational machine. It would be a junction point for two roads—mechanism and thought. Its admirers sometimes struggled with their explanations of this intersection: “The question is set to the instrument,” Henry Colebrooke told the Astronomical Society, “or the instrument is set to the question.” Either way, he said, “by simply giving motion the solution is wrought.”

But the engine made slower progress in the realm of brass and wrought iron. Babbage tore out the stables in back of his London house and replaced them with a forge, foundry, and fireproofed workshop. He engaged Joseph Clement, a draftsman and inventor, self-educated, the son of a village weaver who had made himself into England’s preeminent mechanical engineer. Babbage and Clement realized that they would have to make new tools. Inside a colossal iron frame the design called for the most intricate and precise parts—axles, gears, springs, and pins, and above all figure wheels by the hundreds and then thousands. Hand tools could never produce the components with the needed precision. Before Babbage could have a manufactory of number tables, he would have to build new manufactories of parts. The rest of the Industrial Revolution, too, needed standardization in its parts: interchangeable screws of uniform thread count and pitch; screws as fundamental units. The lathes of Clement and his journeymen began to produce them.

A WOODCUT IMPRESSION (1853) OF ASMALL PORTION OF THE DIFFERENCE ENGINE

As the difficulties grew, so did Babbage’s ambitions. After ten years, the engine stood twenty-four inches high, with six vertical axles and dozens of wheels, capable of computing six-figure results. Ten years after that, the scale—on paper—had reached 160 cubic feet, 15 tons, and 25,000 parts, and the paper had spread, too, the drawings covering more than 400 square feet. The level of complexity was confounding. Babbage solved the problem of adding many digits at once by separating the “adding motions” from the “carrying motions” and then staggering the timing of the carries. The addition would begin with a rush of grinding gears, first the odd-numbered columns of dials, then the even columns. Then the carries would recoil across the rows. To keep the motion synchronized, parts of the machine would need to “know” at critical times that a carry was pending. The information was conveyed by the state of a latch. For the first time, but not the last, a device was invested with memory. “It is in effect a memorandum taken by the machine,” wrote his publicizer, Dionysius Lardner. Babbage himself was self-conscious about anthropomorphizing but could not resist. “The mechanical means I employed to make these carriages,” he suggested, “bears some slight analogy to the operation of the faculty of memory.”

In ordinary language, to describe even this basic process of addition required a great effulgence of words, naming the metal parts, accounting for their interactions, and sorting out interdependencies that multiplied to form a long chain of causality. Lardner’s own explanation of “carrying,” for example, was epic. A single isolated instant of the action involved a dial, an index, a thumb, an axis, a trigger, a notch, a hook, a claw, a spring, a tooth, and a ratchet wheel:

Now, at the moment that the division between 9 and 0 on the dial B² passes under the index, a thumb placed on the axis of this dial touches a trigger which raises out of the notch of the hook which sustains the claw just mentioned, and allows it to fall back by the recoil of the spring, and drop into the next tooth of the ratchet wheel.

Hundreds of words later, summing up, Lardner resorted to a metaphor suggesting fluid dynamics:

There are two systems of waves of mechanical action continually flowing from the bottom to the top; and two streams of similar action constantly passing from the right to the left. The crests of the first system of adding waves fall upon the last difference, and upon every alternate one proceeding upwards. . . . The first stream of carrying action passes from right to left along the highest row and every alternate row.

This was one way of abstracting from the particular—the particulars being so intricate. And then he surrendered. “Its wonders, however, are still greater in its details,” he wrote. “We despair of doing it justice.”

Nor were ordinary draftsman’s plans sufficient for describing this machine that was more than a machine. It was a dynamical system, its many parts each capable of several modes or states, sometimes at rest and sometimes in motion, propagating their influence along convoluted channels. Could it ever be specified completely, on paper? Babbage, for his own purposes, devised a new formal tool, a system of “mechanical notation” (his term). This was a language of signs meant to represent not just the physical form of a machine but its more elusive properties: its timing and its logic. It was an extraordinary ambition, as Babbage himself appreciated. In 1826 he proudly reported to the Royal Society “On a Method of Expressing by Signs the Action of Machinery.” In part it was an exercise in classification. He analyzed the different ways in which something—motion, or power—could be “communicated” through a system. There were many ways. A part could receive its influence simply by being attached to another part, “as a pin on a wheel, or a wheel and pinion on the same axis.” Or transmission could occur “by stiff friction.” A part might be driven constantly by another part “as happens when a wheel is driven by a pinion”—or not constantly, “as is the case when a stud lifts a bolt once in the course of a revolution.” Here a vision of logical branching entered the scheme: the path of communication would vary depending on the alternative states of some part of the machine. Babbage’s mechanical notation followed naturally from his work on symbolic notation in mathematical analysis. Machinery, like mathematics, needed rigor and definition for progress. “The forms of ordinary language were far too diffuse,” he wrote. “The signs, if they have been properly chosen, and if they should be generally adopted, will form as it were an universal language.” Language was never a side issue for Babbage.

He finally won a university post, at Cambridge: the prestigious Lucasian Professorship of Mathematics, formerly occupied by Newton. As in Newton’s time, the work was not onerous. Babbage did not have to teach students, deliver lectures, or even live in Cambridge, and this was just as well, because he was also becoming a popular fixture of London social life. At home at One Dorset Street he hosted a regular Saturday soirée that drew a glittering crowd—politicians, artists, dukes and duchesses, and the greatest English scientists of the age: Charles Darwin, Michael Faraday, and Charles Lyell, among others.² They marveled at his calculating machine and, on display nearby, the dancing automaton of his youth. (In invitations he would write, “I hope you intend to patronise the ‘Silver Lady.’ She is to appear in new dresses and decorations.”) He was a mathematical raconteur—that was no contradiction, in this time and place. Lyell reported approvingly that he “jokes and reasons in high mathematics.” He published a much-quoted treatise applying probability theory to the theological question of miracles. With tongue in cheek he wrote Alfred, Lord Tennyson, to suggest a correction for the poet’s couplet: “Every minute dies a man, / Every minute one is born.”

I need hardly point out to you that this calculation would tend to keep the sum total of the world’s population in a state of perpetual equipoise, whereas it is a well-known fact that the said sum total is constantly on the increase. I would therefore take the liberty of suggesting that in the next edition of your excellent poem the erroneous calculation to which I refer should be corrected as follows: “Every moment dies a man / And one and a sixteenth is born.” I may add that the exact figures are 1.167, but something must, of course, be conceded to the laws of metre.

Fascinated with his own celebrity, he kept a scrapbook—“the pros and cons in parallel columns, from which he obtained a sort of balance,” as one visitor described it. “I was told repeatedly that he spent all his days in gloating and grumbling over what people said of him.”

But progress on the engine, the main source of his fame, was faltering. In 1832 he and his engineer Clement produced a working demonstration piece. Babbage displayed it at his parties to guests who found it miraculous or merely puzzling. The Difference Engine stands—for a replica works today, in the Science Museum in London—as a milestone of what could be achieved in precision engineering. In the composition of its alloys, the exactness of its dimensions, the interchangeability of its parts, nothing surpassed this segment of an unfinished machine. Still, it was a curio. And it was as far as Babbage could go.

He and his engineer fell into disputes. Clement demanded more and more money from Babbage and from the Treasury, which began to suspect profiteering. He withheld parts and drawings and fought over control of the specialized machine tools in their workshops. The government, after more than a decade and £17,000, was losing faith in Babbage, and he in the government. In his dealing with lords and ministers Babbage could be imperious. He was developing a sour view of the Englishman’s attitude toward technological innovation: “If you speak to him of a machine for peeling a potato, he will pronounce it impossible: if you peel a potato with it before his eyes, he will declare it useless, because it will not slice a pineapple.” They no longer saw the point.

“What shall we do to get rid of Mr. Babbage and his calculating machine?” Prime Minister Robert Peel wrote one of his advisers in August 1842. “Surely if completed it would be worthless as far as science is concerned. . . . It will be in my opinion a very costly toy.” He had no trouble finding voices inimical to Babbage in the civil service. Perhaps the most damning was George Biddell Airy, the Astronomer Royal, a starched and methodical figure, who with no equivocation told Peel precisely what he wanted to hear: that the engine was useless. He added this personal note: “I think it likely he lives in a sort of dream as to its utility.” Peel’s government terminated the project. As for Babbage’s dream, it continued. It had already taken another turn. The engine in his mind had advanced into a new dimension. And he had met Ada Byron.

CHARLES BABBAGE (1860)

In the Strand, at the north end of the Lowther shopping arcade, visitors thronged to the National Gallery of Practical Science, “Blending Instruction with Amusement,” a combination toy store and technology show set up by an American entrepreneur. For the admission price of a shilling, a visitor could touch the “electrical eel,” listen to lectures on the newest science, and watch a model steamboat cruising a seventy-foot trough and the Perkins steam gun emitting a spray of bullets. For a guinea, she could sit for a “daguerreotype” or “photographic” portrait, by which a faithful and pleasing likeness could be obtained in “less than One Second.” Or she could watch, as young Augusta Ada Byron did, a weaver demonstrating the automated Jacquard loom, in which the patterns to be woven in cloth were encoded as holes punched into pasteboard cards.

Ada was “the child of love,” her father had written, “—though born in bitterness, and nurtured in convulsion.” Her father was a poet. When she was barely a month old, in 1816, the already notorious Lord Byron, twenty-seven, and the bright, wealthy, and mathematically knowledgeable Anne Isabella Milbanke (Annabella), twenty-three, separated after a year of marriage. Byron left England and never saw his daughter again. Her mother refused to tell her who her father was until she was eight and he died in Greece, an international celebrity. The poet had begged for any news of his daughter: “Is the Girl imaginative?—at her present age I have an idea that I had many feelings & notions which people would not believe if I stated them now.” Yes, she was imaginative.

She was a prodigy, clever at mathematics, encouraged by tutors, talented in drawing and music, fantastically inventive and profoundly lonely. When she was twelve, she set about inventing a means of flying. “I am going to begin my paper wings tomorrow,” she wrote to her mother. She hoped “to bring the art of flying to very great perfection. I think of writing a book of Flyology illustrated with plates.” For a while she signed her letters “your very affectionate Carrier Pigeon.” She asked her mother to find a book illustrating bird anatomy, because she was reluctant “to dissect even a bird.” She analyzed her daily situation with a care for logic.

Miss Stamp desires me to say that at present she is not particularly pleased with me on account of some very foolish conduct yesterday about a simple thing, and which she said was not only foolish but showed a spirit of inattention, and though today she has not had reason to be dissatisfied with me on the whole yet she says that she can not directly efface the recollection of the past.

She was growing up in a well-kept cloister of her mother’s arranging. She had years of sickliness, a severe bout of measles, and episodes of what was called neurasthenia or hysteria. (“When I am weak,” she wrote, “I am always so exceedingly terrified, at nobody knows what, that I can hardly help having an agitated look & manner.”) Green drapery enclosed the portrait of her father that hung in one room. In her teens she developed a romantic interest in her tutor, which led to a certain amount of sneaking about the house and gardens and to lovemaking as intimate as possible without, she said, actual “connection.” The tutor was dismissed. Then, in the spring, wearing white satin and tulle, the seventeen-year-old made her ritual debut at court, where she met the king and queen, the most important dukes, and the French diplomat Talleyrand, whom she described as an “old monkey.”

A month later she met Charles Babbage. With her mother, she went to see what Lady Byron called his “thinking machine,” the portion of the Difference Engine in his salon. Babbage saw a sparkling, self-possessed young woman with porcelain features and a notorious name, who managed to reveal that she knew more mathematics than most men graduating from university. She saw an imposing forty-one-year-old, authoritative eyebrows anchoring his strong-boned face, who possessed wit and charm and did not wear these qualities lightly. He seemed a kind of visionary— just what she was seeking. She admired the machine, too. An onlooker reported: “While other visitors gazed at the working of this beautiful instrument with the sort of expression, and I dare say the sort of feeling, that some savages are said to have shown on first seeing a looking-glass or hearing a gun, Miss Byron, young as she was, understood its working, and saw the great beauty of the invention.” Her feeling for the beauty and abstractions of mathematics, fed only in morsels from her succession of tutors, was overflowing. It had no outlet. A woman could not attend university in England, nor join a scientific society (with two exceptions: the botanical and horticultural).

Ada became a tutor for the young daughters of one of her mother’s friends. When writing to them, she signed herself, “your affectionate & untenable Instructress.” On her own she studied Euclid. Forms burgeoned in her mind. “I do not consider that I know a proposition,” she wrote another tutor, “until I can imagine to myself a figure in the air, and go through the construction & demonstration without any book or assistance whatever.” She could not forget Babbage, either, or his “gem of all mechanism.” To another friend she reported her “great anxiety about the machine.” Her gaze turned inward, often. She liked to think about herself thinking.

AUGUSTA ADA BYRON KING, COUNTESS OF LOVELACE, AS PAINTED IN 1836 BY MARGARET CARPENTER.“I CONCLUDE SHE IS BENT ON DISPLAYING THEWHOLE EXPANSE OF MY CAPACIOUS JAW BONE, UPON WHICH I THINK THE WORD MATHEMATICSSHOULD BE WRITTEN.”

Babbage himself had moved far beyond the machine on display in his drawing room; he was planning a new machine, still an engine of computation but transmuted into another species. He called this the Analytical Engine. Motivating him was a quiet awareness of the Difference Engine’s limitations: it could not, merely by adding differences, compute every sort of number or solve any mathematical problem. Inspiring him, as well, was the loom on display in the Strand, invented by Joseph-Marie Jacquard, controlled by instructions encoded and stored as holes punched in cards.

What caught Babbage’s fancy was not the weaving, but rather the encoding, from one medium to another, of patterns. The patterns would appear in damask, eventually, but first were “sent to a peculiar artist.” This specialist, as he said,

punches holes in a set of pasteboard cards in such a manner that when those cards are placed in a Jacquard loom, it will then weave upon its produce the exact pattern designed by the artist.

The notion of abstracting information away from its physical substrate required careful emphasis. Babbage explained, for example, that the weaver might choose different threads and different colors—“but in all these cases the form of the pattern will be precisely the same.” As Babbage conceived his machine now, it raised this very process of abstraction to higher and higher degrees. He meant the cogs and wheels to handle not just numbers but variables standing in for numbers. Variables were to be filled or determined by the outcomes of prior calculations, and, further, the very operations—such as addition or multiplication—were to be changeable, depending on prior outcomes. He imagined these abstract information quantities being stored in cards: variable cards and operation cards. He thought of the machine as embodying laws and of the cards as communicating these laws. Lacking a ready-made vocabulary, he found it awkward to express his fundamental working concepts; for example,

how the machine could perform the act of judgment sometimes required during an analytical inquiry, when two or more different courses presented themselves, especially as the proper course to be adopted could not be known in many cases until all the previous portion had been gone through.

He made clear, though, that information—representations of number and process—would course through the machinery. It would pass to and from certain special physical locations, which Babbage named a store, for storage, and a mill, for action.

In all this he had an intellectual companion now in Ada, first his acolyte and then his muse. She married a sensible and promising aristocrat, William King, her senior by a decade and a favorite of her mother. In the space of a few years he was elevated to the peerage as earl of Lovelace— making Ada, therefore, a countess—and, still in her early twenties, she bore three children. She managed their homes, in Surrey and London, practiced the harp for hours daily (“I am at present a condemned slave to my harp, no easy Task master”), danced at balls, met the new queen, Victoria, and sat for her portrait, self-consciously (“I conclude [the artist] is bent on displaying the whole expanse of my capacious jaw bone, upon which I think the word Mathematics should be written”). She suffered terrible dark moods and bouts of illness, including cholera. Her interests and behavior still set her apart. One morning she went alone in her carriage, dressed plainly, to see a model of Edward Davy’s “electrical telegraph” at Exeter Hall

& the only other person was a middle-aged gentleman who chose to behave as if I were the show [she wrote to her mother] which of course I thought was the most impudent and unpardonable.—I am sure he took me for a very young (& I suppose he thought rather handsome) governess. . . . He stopped as long as I did, & then followed me out.— I took care to look as aristocratic & as like a Countess as possible. . . . I must try & add a little age to my appearance. . . . I would go & see something everyday & I am sure London would never be exhausted.

Lady Lovelace adored her husband but reserved much of her mental life for Babbage. She had dreams, waking dreams, of something she could not be and something she could not achieve, except by proxy, through his genius. “I have a peculiar way of learning,” she wrote to him, “& I think it must be a peculiar man to teach me successfully.” Her growing desperation went side by side with a powerful confidence in her untried abilities. “I hope you are bearing me in mind,” she wrote some months later, “I mean my mathematical interests. You know this is the greatest favour any one can do me.—Perhaps, none of us can estimate how great. . . .”

You know I am by nature a bit of a philosopher, & a very great speculator, —so that I look on through a very immeasurable vista, and though I see nothing but vague & cloudy uncertainty in the foreground of our being, yet I fancy I discern a very bright light a good way further on, and this makes me care much less about the cloudiness & indistinctness which is near.—Am I too imaginative for you? I think not.

The mathematician and logician Augustus De Morgan, a friend of Babbage and of Lady Byron, became Ada’s teacher by post. He sent her exercises. She sent him questions and musings and doubts (“I could wish I went on quicker”; “I am sorry to say I am sadly obstinate about the Term at which Convergence begins”; “I have enclosed my Demonstration of my view of the case”; “functional Equations are complete Will-o-the-wisps to me”; “However I try to keep my metaphysical head in order”). Despite her naïveté, or because of it, he recognized a “power of thinking . . . so utterly out of the common way for any beginner, man or woman.” She had rapidly mastered trigonometry and integral and differential calculus, and he told her mother privately that if he had encountered “such power” in a Cambridge student he would have anticipated “an original mathematical investigator, perhaps of first rate eminence.” She was fearless about drilling down to first principles. Where she felt difficulties, real difficulties lay.

One winter she grew obsessed with a fashionable puzzle known as Solitaire, the Rubik’s Cube of its day. Thirty-two pegs were arranged on a board with thirty-three holes, and the rules were simple: Any peg may jump over another immediately adjacent, and the peg jumped over is removed, until no more jumps are possible. The object is to finish with only one peg remaining. “People may try thousands of times, and not succeed in this,” she wrote Babbage excitedly.

I have done it by trying & observation & can now do it at any time, but I want to know if the problem admits of being put into a mathematical Formula, & solved in this manner. . . . There must be a definite principle, a compound I imagine of numerical & geometrical properties, on which the solution depends, & which can be put into symbolic language.

A formal solution to a game—the very idea of such a thing was original. The desire to create a language of symbols, in which the solution could be encoded—this way of thinking was Babbage’s, as she well knew.

She pondered her growing powers of mind. They were not strictly mathematical, as she saw it. She saw mathematics as merely a part of a greater imaginative world. Mathematical transformations reminded her “of certain sprites & fairies one reads of, who are at one’s elbows in one shape now, & the next minute in a form most dissimilar; and uncommonly deceptive, troublesome & tantalizing are the mathematical sprites & fairies sometimes; like the types I have found for them in the world of Fiction.” Imagination—the cherished quality. She mused on it; it was her heritage from her never-present father.

We talk much of Imagination. We talk of the Imagination of Poets, the Imagination of Artists &c; I am inclined to think that in general we don’t know very exactly what we are talking about. . . .

It is that which penetrates into the unseen worlds around us, the worlds of Science. It is that which feels & discovers what is, the real which we see not, which exists not for our senses. Those who have learned to walk on the threshold of the unknown worlds . . . may then with the fair white wings of Imagination hope to soar further into the unexplored amidst which we live.

She began to believe she had a divine mission to fulfill. She used that word, mission. “I have on my mind most strongly the impression that Heaven has allotted me some peculiar intellectual-moral mission to perform.” She had powers. She confided in her mother:

I believe myself to possess a most singular combination of qualities exactly fitted to make me pre-eminently a discoverer of the hidden realities of nature. . . . The belief has been forced upon me, & most slow have I been to admit it even.

She listed her qualities:

Firstly: Owing to some peculiarity in my nervous system, I have perceptions of some things, which no one else has; or at least very few, if any. . . . Some might say an intuitive perception of hidden things;—that is of things hidden from eyes, ears & the ordinary senses. . . .

Secondly;—my immense reasoning faculties;

Thirdly; . . . the power not only of throwing my whole energy & existence into whatever I choose, but also bring to bear on any one subject or idea, a vast apparatus from all sorts of apparently irrelevant & extraneous sources. I can throw rays from every quarter of the universe into one vast focus.

She admitted that this sounded mad but insisted she was being logical and cool. She knew her life’s course now, she told her mother. “What a mountain I have to climb! It is enough to frighten anyone who had not all that most insatiable & restless energy, which from my babyhood has been the plague of your life & my own. However it has found food I believe at last.” She had found it in the Analytical Engine.

——

Babbage meanwhile, restless and omnivorous, was diverting his energies to another burgeoning technology, steam’s most powerful expression, the railroad. The newly formed Great Western Railway was laying down track and preparing trial runs of locomotive engines from Bristol to London under the supervision of Isambard Kingdom Brunel, the brilliant engineer, then just twenty-seven years old. Brunel asked Babbage for help, and Babbage decided to begin with an information-gathering program— characteristically ingenious and grandiose. He outfitted an entire railway carriage. On a specially built, independently suspended table, rollers unwound sheets of paper a thousand feet long, while pens drew lines to “express” (as Babbage put it) measurements of the vibrations and forces felt by the carriage in every direction. A chronometer marked the passage of time in half seconds. He covered two miles of paper this way.

As he traversed the rails, he realized that a peculiar danger of steam locomotion lay in its outracing every previous means of communication. Trains lost track of one another. Until the most regular and disciplined scheduling was imposed, hazard ran with every movement. One Sunday Babbage and Brunel, operating in different engines, barely avoided smashing into each other. Other people, too, worried about this new gap between the speeds of travel and messaging. An important London banker told Babbage he disapproved: “It will enable our clerks to plunder us, and then be off to Liverpool on their way to America at the rate of twenty miles an hour.” Babbage could only express the hope that science might yet find a remedy for the problem it had created. (“Possibly we might send lightning to outstrip the culprit.”)

As for his own engine—the one that would travel nowhere—he had found a fine new metaphor. It would be, he said, “a locomotive that lays down its own railway.”

Bitter as he was about England’s waning interest in his visionary plans, Babbage found admirers on the continent, particular in Italy—“the country of Archimedes and Galileo,” as he put it to his new friends. In the summer of 1840 he gathered up his sheaves of drawings and journeyed by way of Paris and Lyon, where he watched the great Jacquard loom at Manufacture d’Étoffes pour Ameublements et Ornements d’Église, to Turin, the capital of Sardinia, for an assembly of mathematicians and engineers. There he made his first (and last) public presentation of the Analytical Engine. “The discovery of the Analytical Engine is so much in advance of my own country, and I fear even of the age,” he said. He met the Sardinian king, Charles Albert, and, more significantly, an ambitious young mathematician named Luigi Menabrea. Later Menabrea was to become a general, a diplomat, and the prime minister of Italy; now he prepared a scientific report, “Notions sur la machine analytique,” to introduce Babbage’s plan to a broader community of European philosophers.

As soon as this reached Ada Lovelace, she began translating it into English, correcting errors on the basis of her own knowledge. She did that on her own, without telling either Menabrea or Babbage.

When she finally did show Babbage her draft, in 1843, he responded enthusiastically, urging her to write on her own behalf, and their extraordinary collaboration began in earnest. They sent letters by messenger back and forth across London at a ferocious pace—“My Dear Babbage” and “My Dear Lady Lovelace”—and met whenever they could at her home in St. James’s Square. The pace was almost frantic. Though he was the eminence, fifty-one years old to her twenty-seven, she took charge, mixing stern command with banter. “I want you to answer me the following question by return of post”; “Be kind enough to write this out properly for me”; “You were a little harum-scarum and inaccurate”; “I wish you were as accurate and as much to be relied on as myself.” She proposed to sign her work with her initials—nothing so forward as her name—not to “proclaim who has written it,” merely to “individualize and identify it with other productions of the said A.A.L.”

Her exposition took the form of notes lettered A through G, extending to nearly three times the length of Menabrea’s essay. They offered a vision of the future more general and more prescient than any expressed by Babbage himself. How general? The engine did not just calculate; it performed operations, she said, defining an operation as “any process which alters the mutual relation of two or more things,” and declaring: “This is the most general definition, and would include all subjects in the universe.” The science of operations, as she conceived it,

is a science of itself, and has its own abstract truth and value; just as logic has its own peculiar truth and value, independently of the subjects to which we may apply its reasonings and processes. . . . One main reason why the separate nature of the science of operations has been little felt, and in general little dwelt on, is the shifting meaning of many of the symbols used.

Symbols and meaning: she was emphatically not speaking of mathematics alone. The engine “might act upon other things besides number.” Babbage had inscribed numerals on those thousands of dials, but their working could represent symbols more abstractly. The engine might process any meaningful relationships. It might manipulate language. It might create music. “Supposing, for instance, that the fundamental relations of pitched sounds in the science of harmony and of musical composition were susceptible of such expression and adaptations, the engine might compose elaborate and scientific pieces of music of any degree of complexity or extent.”

It had been an engine of numbers; now it became an engine of information. A.A.L. perceived that more distinctly and more imaginatively than Babbage himself. She explained his prospective, notional, virtual creation as though it already existed:

The Analytical Engine does not occupy common ground with mere “calculating machines.” It holds a position wholly its own. . . . A new, a vast, and a powerful language is developed . . . in which to wield its truths so that these may become of more speedy and accurate practical application for the purposes of mankind than the means hitherto in our possession have rendered possible. Thus not only the mental and the material, but the theoretical and the practical in the mathematical world, are brought into more intimate and effective connexion with each other.

. . . We may say most aptly, that the Analytical Engine weaves algebraical patterns just as the Jacquard-loom weaves flowers and leaves.

For this flight of fancy she took full responsibility. “Whether the inventor of this engine had any such views in his mind while working out the invention, or whether he may subsequently ever have regarded it under this phase, we do not know; but it is one that forcibly occurred to ourselves.”

She proceeded from the poetic to the practical. She set forth on a virtuoso excursion through a hypothetical program by which this hypothetical machine might compute a famously deep-seated infinite series, the Bernoulli numbers. These numbers arise in the summing of numbers from 1 to n raised to integral powers, and they occur in various guises all through number theory. No direct formula generates them, but they can be worked out methodically, by expanding certain formulas further and further and looking at the coefficients each time. She began with examples; the simplest, she wrote, would be the expansion of

and another approach would be via

but she would take a more challenging path, because “our object is not simplicity . . . but the illustration of the powers of the engine.”

She devised a process, a set of rules, a sequence of operations. In another century this would be called an algorithm, later a computer program, but for now the concept demanded painstaking explanation. The trickiest point was that her algorithm was recursive. It ran in a loop. The result of one iteration became food for the next. Babbage had alluded to this approach as “the Engine eating its own tail.” A.A.L. explained: “We easily perceive that since every successive function is arranged in a series following the same law, there would be a cycle of a cycle of a cycle, &c. . . . The question is so exceedingly complicated, that perhaps few persons can be expected to follow. . . . Still it is a very important case as regards the engine, and suggests ideas peculiar to itself, which we should regret to pass wholly without allusion.”

A core idea was the entity she and Babbage called the variable. Variables were, in hardware terms, the machine’s columns of number dials. But there were “Variable cards,” too. In software terms they were a sort of receptacle or envelope, capable of representing, or storing, a number of many decimal digits. (“What is there in a name?” Babbage wrote. “It is merely an empty basket until you put something in it.”) Variables were the machine’s units of information. This was quite distinct from the algebraic variable. As A.A.L. explained, “The origin of this appellation is, that the values on the columns are destined to change, that is to vary, in every conceivable manner.” Numbers traveled, in effect, from variable cards to variables, from variables to the mill (for operations), from the mill to the store. To solve the problem of generating Bernoulli numbers, she choreographed an intricate dance. She worked days and sometimes through the night, messaging Babbage across London, struggling with sickness and ominous pains, her mind soaring:

That brain of mine is something more than merely mortal; as time will show; (if only my breathing & some other et-ceteras do not make too rapid a progress towards instead of from mortality).

Before ten years are over, the Devil’s in it if I have not sucked out some of the life-blood from the mysteries of this universe, in a way that no purely mortal lips or brains could do.

No one knows what almost awful energy & power lie yet undevelopped in that wiry little system of mine. I say awful, because you may imagine what it might be under certain circumstances. . . .

I am doggedly attacking & sifting to the very bottom, all the ways of deducing the Bernoulli Numbers. . . . I am grappling with this subject, & connecting it with others.

She was programming the machine. She programmed it in her mind, because the machine did not exist. The complexities she encountered for the first time became familiar to programmers of the next century:

How multifarious and how mutually complicated are the considerations which the working of such an engine involve. There are frequently several distinct sets of effects going on simultaneously; all in a manner independent of each other, and yet to a greater or less degree exercising a mutual influence. To adjust each to every other, and indeed even to perceive and trace them out with perfect correctness and success, entails difficulties whose nature partakes to a certain extent of those involved in every question where conditions are very numerous and inter-complicated.

She reported her feelings to Babbage: “I am in much dismay at having got into so amazing a quagmire & botheration.” And nine days later: “I find that my plans & ideas keep gaining in clearness, & assuming more of the crystalline & less & less of the nebulous form.” She knew she had achieved something utterly new. Ten days later still, struggling over the final proofs with “Mr Taylors Printing Office” in Fleet Street, she declared: “I do not think you possess half my forethought, & power of foreseeing all possible contingencies (probable & improbable, just alike).— . . . I do not believe that my father was (or ever could have been) such a Poet as I shall be an Analyst; (& Metaphysician); for with me the two go together indissolubly.”

Who would have used this machine? Not clerks or shopkeepers, said Babbage’s son, many years later. Common arithmetic was never the purpose—“It would be like using the steam hammer to crush the nut.” He paraphrased Leibniz: “It is not made for those who sell vegetables or little fishes, but for observatories, or the private rooms of calculators, or for others who can easily bear the expense, and need a good deal of calculation.” Babbage’s engine had not been well understood, not by his government and not by the many friends who passed through his salon, but in its time its influence traveled far.

In America, a country bursting with invention and scientific optimism, Edgar Allan Poe wrote, “What shall we think of the calculating machine of Mr. Babbage? What shall we think of an engine of wood and metal which can . . . render the exactitude of its operations mathematically certain through its power of correcting its possible errors?” Ralph Waldo Emerson had met Babbage in London and declared in 1870, “Steam is an apt scholar and a strong-shouldered fellow, but it has not yet done all its work.”

It already walks about the field like a man, and will do anything required of it. It irrigates crops, and drags away a mountain. It must sew our shirts, it must drive our gigs; taught by Mr. Babbage, it must calculate interest and logarithms. . . . It is yet coming to render many higher services of a mechanico-intellectual kind.

Its wonders met disapproval, too. Some critics feared a rivalry between mechanism and mind. “What a satire is that machine on the mere mathematician!” said Oliver Wendell Holmes Sr. “A Frankenstein-monster, a thing without brains and without heart, too stupid to make a blunder; which turns out results like a corn-sheller, and never grows any wiser or better, though it grind a thousand bushels of them!” They all spoke as though the engine were real, but it never was. It remained poised before its own future.

Midway between his time and ours, the Dictionary of National Biography granted Charles Babbage a brief entry—almost entirely devoid of relevance or consequence:

mathematician and scientific mechanician; . . . obtained government grant for making a calculating machine . . . but the work of construction ceased, owning to disagreements with the engineer; offered the government an improved design, which was refused on grounds of expense; . . . Lucasian professor of mathematics, Cambridge, but delivered no lectures.

Babbage’s interests, straying so far from mathematics, seeming so miscellaneous, did possess a common thread that neither he nor his contemporaries could perceive. His obsessions belonged to no category—that is, no category yet existing. His true subject was information: messaging, encoding, processing.

He took up two quirky and apparently unphilosophical challenges, which he himself noted had a deep connection one to the other: picking locks and deciphering codes. Deciphering, he said, was “one of the most fascinating of arts, and I fear I have wasted upon it more time than it deserves.” To rationalize the process, he set out to perform a “complete analysis” of the English language. He created sets of special dictionaries: lists of the words of one letter, two letters, three letters, and so on; and lists of words alphabetized by their initial letter, second letter, third letter, and so on. With these at hand he designed methodologies for solving anagram puzzles and word squares.

In tree rings he saw nature encoding messages about the past. A profound lesson: that a tree records a whole complex of information in its solid substance. “Every shower that falls, every change of temperature that occurs, and every wind that blows, leaves on the vegetable world the traces of its passage; slight, indeed, and imperceptible, perhaps, to us, but not the less permanently recorded in the depths of those woody fabrics.”

In London workshops he had observed speaking tubes, made of tin, “by which the directions of the superintendent are instantly conveyed to the remotest parts.” He classified this technology as a contribution to the “economy of time” and suggested that no one had yet discovered a limit on the distance over which spoken messages might travel. He made a quick calculation: “Admitting it to be possible between London and Liverpool, about seventeen minutes would elapse before the words spoken at one end would reach the other extremity of the pipe.” In the 1820s he had an idea for transmitting written messages, “enclosed in small cylinders along wires suspended from posts, and from towers, or from church steeples,” and he built a working model in his London house. He grew obsessed with other variations on the theme of sending messages over the greatest possible distances. The post bag dispatched nightly from Bristol, he noted, weighed less than one hundred pounds. To send these messages 120 miles, “a coach and apparatus, weighing above thirty hundred weight, are put in motion, and also conveyed over the same space.” What a waste! Suppose, instead, he suggested, post towns were linked by a series of high pillars erected every hundred feet or so. Steel wires would stretch from pillar to pillar. Within cities, church steeples might serve as the pillars. Tin cases with wheels would roll along the wires and carry batches of letters. The expense would be “comparatively trifling,” he said, “nor is it impossible that the stretched wire might itself be available for a species of telegraphic communication yet more rapid.”

During the Great Exhibition of 1851, when England showcased its industrial achievement in a Crystal Palace, Babbage placed an oil lamp with a moveable shutter in an upstairs window at Dorset Street to create an “occulting light” apparatus that blinked coded signals to passersby. He drew up a standardized system for lighthouses to use in sending numerical signals and posted twelve copies to, as he said, “the proper authorities of the great maritime countries.” In the United States, the Congress approved $5,000 for a trial program of Babbage’s system. He studied sun signals and “zenith-light signals” flashed by mirrors, and Greenwich time signals for transmission to mariners. For communicating between stranded ships and rescuers on shore, he proposed that all nations adopt a standard list of a hundred questions and answers, assigned numbers, “to be printed on cards, and nailed up on several parts of every vessel.” Similar signals, he suggested, could help the military, the police, the railways, or even, “for various social purposes,” neighbors in the country.

These purposes were far from obvious. “For what purposes will the electric telegraph become useful?” the king of Sardinia, Charles Albert, asked Babbage in 1840. Babbage searched his mind for an illustration, “and at last I pointed out the probability that, by means of the electric telegraphs, his Majesty’s fleet might receive warning of coming storms. . . .”

This led to a new theory of storms, about which the king was very curious. By degrees I endeavoured to make it clear. I cited, as an illustration, a storm which had occurred but a short time before I left England. The damage done by it at Liverpool was very great, and at Glasgow immense. . . . I added that if there had been electric communication between Genoa and a few other places the people of Glasgow might have had information of one of those storms twenty-four hours previously to its arrival.

As for the engine, it had to be forgotten before it was remembered. It had no obvious progeny. It rematerialized like buried treasure and inspired a sense of puzzled wonder. With the computer era in full swing, the historian Jenny Uglow felt in Babbage’s engines “a different sense of anachronism.” Such failed inventions, she wrote, contain “ideas that lie like yellowing blueprints in dark cupboards, to be stumbled on afresh by later generations.”

Meant first to generate number tables, the engine in its modern form instead rendered number tables obsolete. Did Babbage anticipate that? He did wonder how the future would make use of his vision. He guessed that a half century would pass before anyone would try again to create a general-purpose computing machine. In fact, it took most of a century for the necessary substrate of technology to be laid down. “If, unwarned by my example,” he wrote in 1864, “any man shall undertake and shall succeed in really constructing an engine embodying in itself the whole of the executive department of mathematical analysis upon different principles or by simpler mechanical means, I have no fear of leaving my reputation in his charge, for he alone will be fully able to appreciate the nature of my efforts and the value of their results.”

As he looked to the future, he saw a special role for one truth above all: “the maxim, that knowledge is power.” He understood that literally. Knowledge “is itself the generator of physical force,” he declared. Science gave the world steam, and soon, he suspected, would turn to the less tangible power of electricity: “Already it has nearly chained the ethereal fluid.” And he looked further:

It is the science of calculation—which becomes continually more necessary at each step of our progress, and which must ultimately govern the whole of the applications of science to the arts of life.

Some years before his death, he told a friend that he would gladly give up whatever time he had left, if only he could be allowed to live for three days, five centuries in the future.

As for his young friend Ada, countess of Lovelace, she died many years before him—a protracted, torturous death from cancer of the womb, her agony barely lessened by laudanum and cannabis. For a long time her family kept from her the truth of her illness. In the end she knew she was dying. “They say that ‘coming events cast their shadows before,’ ” she wrote to her mother. “May they not sometimes cast their lights before?” They buried her next to her father.

She, too, had a last dream of the future: “my being in time an Autocrat, in my own way.” She would have regiments, marshaled before her. The iron rulers of the earth would have to give way. And of what would her regiments consist? “I do not at present divulge. I have however the hope that they will be most harmoniously disciplined troops;—consisting of vast numbers, & marching in irresistible power to the sound of Music. Is not this very mysterious? Certainly my troops must consist of numbers, or they can have no existence at all. . . . But then, what are these Numbers? There is a riddle—”

¹Leibniz dreamed grandly of mechanizing algebra and even reason itself. “We may give final praise to the machine,” he wrote. “It will be desirable to all who are engaged in computations . . . the managers of financial affairs, the administrators of others’ estates, merchants, surveyors, geographers, navigators, astronomers. . . . For it is unworthy of excellent men to lose hours like slaves in the labor of calculation.”

²Another guest, Charles Dickens, put something of Babbage into the character of Daniel Doyce in Little Dorrit. Doyce is an inventor mistreated by the government he tries to serve: “He is well known as a very ingenious man. . . . He perfects an invention (involving a very curious secret process) of great importance to his country and his fellow-creatures. I won’t say how much money it cost him, or how many years of his life he had been about it, but he brought it to perfection.” Dickens added: “A composed and unobtrusive self-sustainment was noticeable in Daniel Doyce—a calm knowledge that what was true must remain true.”