The Difference Engine: Charles Babbage and the Quest to Build the First Computer

Chapter 1

THE TABLES CRISIS

I wish to God these calculations had been executed by steam.

— Charles Babbage, 1821

A carriage clatters to a halt outside No. 5 Devonshire Street, London. A well-dressed man in his late twenties alights with a bundle of manuscripts tucked under his arm. At the sound of the carriage, a tall gentleman comes out to greet him. The visitor is John Herschel, an astronomer, up in London from his father's house in Slough. His host is Charles Babbage, mathematician. It is summer 1821.

The two friends are pleased to see each other again. Their friendship had flowered during their undergraduate years at Cambridge, and they have stayed in close touch since. They exchange pleasantries as they go inside and swap some scientific gossip. Within a few minutes they sit down to their task in the drawing room.

Herschel divides the bundle of manuscripts equally between them. The sheets are covered with closely written calculations and lists of numbers - mathematical tables they are preparing for the Astronomical Society. The results in front of them have been calculated by 'computers'. These are not machines, but people who perform routine calculations by hand according to a fixed arithmetical procedure. The two stacks of papers contain the results of the same set of calculations carried out by different computers. If the computers have done their work perfectly, the two sets of results should be identical.

They settle in their chairs and start comparing the figures. Herschel reads a number from his sheet; Babbage checks it against the entry in front of him. Line by line they proceed in thispainstaking way. They find an error - the result in Herschel's manuscript differs from Babbage's. The computers have made a mistake. The two friends mark the entry and proceed. Concentration is intense. At times they lose track, and have to go back. More errors. Babbage becomes increasingly agitated. Finally, he can contain himself no longer. 'I wish to God these calculations had been executed by steam.'

This exasperated appeal was made when Babbage was twenty-nine. Its ramifications were to dominate the rest of his life.

The burdens of calculation had plagued many before him, and Babbage was voicing the frustrations of centuries. Relief from the tyranny of numbers was sought in early mechanical aids to counting - tally sticks, pebbles and tokens - which date back several thousand years. There was a large gap until the shift from counting to mechanical calculation seized the imagination of some of the leading intellects of seventeenth-century Continental Europe. Wilhelm Schickard, Blaise Pascal and Gottfried Leibniz devised and constructed mechanical calculators to relieve the drudgery of doing sums. Leibniz's 'reckoner' built in the late 1670s used a revolutionary device, the stepped drum, which dominated calculator design for the next two centuries. The calculators of Leibniz and Pascal were sensations. They were paraded in the palaces of royalty and adorned the drawing rooms of the aristocracy, savants and the wealthy. But they were more in the nature of ornamental curiosities - objets de salon - exquisite, delicate, unreliable and unsuited to daily use.

By 1821, when Babbage and Herschel sat down to check their manuscripts, the situation was not much improved. Thomas de Colmar had just introduced a new calculating device, the arithmometer. This was a small desktop instrument with dials, sliders and a handle which, through a series of manual operations, was capable of basic arithmetic. The early arithmometers were erratic, and it was decades before they made their mark as practical devices for routine use. Slide rules were a great boon for quick and convenient calculation, but the scales and divisions were read by eye and there was an element of subjective judgement in the last decimal places. Accuracy was typically limited to three or four figures, which was fine for some purposes, but not all.

It was not until the last decades of the nineteenth century that mechanical desktop calculators became reliable, robust and cheap enough for general arithmetic - addition, subtraction, multiplication and division. Until then, scientists, engineers, surveyors and architects relied on printed tables for mathematical functions or calculations requiring more than a few figures of accuracy. Journeymen, builders, tradesmen, merchants and excise officers turned to printed books of look-up lists and 'ready reckoners' for multiplication, multiples of fractions, conversion of units and a host of arithmetical tasks essential to their daily work. Bankers, investors, actuaries, moneylenders and clerks relied on tables of interest for returns on investment, annuities and assurance premiums. The bookshelves in studies, offices, ship's cabins and workshops groaned under the weight of volumes of tables for desk use, and well-thumbed pocket editions populated the bags and pouches of those on the move.

The need for tables and the reliance placed on them became especially acute during the first half of the nineteenth century, which witnessed a ferment of scientific invention and unprecedented engineering ambition - bridges, railways, shipbuilding, construction and architecture. The heroes of the age laid much of the foundation for modern scientific and industrial life - Michael Faraday, Charles Wheatstone, Humphry Davy, John Dalton, Isambard Kingdom Brunel, Joseph Whitworth and Charles Darwin. The nineteenth century was not only an age of reason. It was also an age of quantification in which science and engineering set about reducing the world to number. With the rise of science and the burgeoning Industrial Revolution, the need for accurate and convenient numerical calculations mushroomed.

There was one need for tables that was paramount - navigation. Navigators found their position on the open seas from the moon and the stars, and astronomical tables of the kind being checked by the two friends in 1821 were crucial for this purpose. Britain was a leading maritime nation, and accurate navigation was critical to the safety of the fleet both for the protection of the realm and for trade. The stakes were high. Capital, private fortunes and lives were at stake.

The problem was that tables were riddled with errors. Finance, trade, science and navigation were at risk from hidden dangers, and the insecurity of flawed tables undermined the certainties promised and sought by the burgeoning new sciences. When errors were found in published tables a correction sheet of errata was issued and included in the next edition, and these correction sheets give some clues to just how serious the problem was. Dionysius Lardner, professor of natural philosophy and astronomy at London University, and a prolific populariser of science, sought to expose the sorry state of affairs. He inspected a private collection of 140 volumes of tables (probably Babbage's) which had a printed area covering more than sixteen thousand square feet - the size of about six tennis courts. In a random selection of 40 volumes he found no fewer than 3,000 errors acknowledged in the errata sheets. Some of the correction sheets themselves contained rrors. Lardner ridiculed the need for errata of errata, and in the case of the Nautical Almanac - a standard volume of astronomical tables - he trumpeted the absurdity of errata of errata of errata. Lardner's case was that tables were generically flawed. Yet others argued that they were accurate enough. There was no absolutely certain way of verifying tables, and experts disagreed about whether there was a crisis at all, notwithstanding Lardner's painful expose.

The problem was not the errors already found and flagged in the correction sheets, but the insecurity of not knowing how many errors remained undetected. Herschel, writing in 1842 to the Chancellor of the Exchequer, Henry Goulburn, captured this deep anxiety of the unknown: 'an undetected error in a logarithmic table is like a sunken rock at sea yet undiscovered, upon which it is impossible to say what wrecks may have taken place'. News of shipwrecks was a constant reminder of the dire consequences to navigation of undetected errors. Scientific magazines buzzed with novel life-saving apparatus about which outlandish claims were made. One device offered a rubber suit with a buoyancy belt and weights to keep the hapless victim vertical. A floating store provided drinking water, food, reading material ('so that he may read the news to pass the time'), cigars, and a pipe and tobacco, as well as torches and rockets to signal the victim's position to rescuers. The kit included metal frames clad with rubber which fitted round the survivor's hands so that, when tired of reading, smoking, eating and drinking while bobbing around, he or she could go places by paddling. Another device was a buoy with sailcloth trousers, gumboots and metal hoops to protect against rocks and voracious fish. The upper section of the buoy was open and large enough to allow some freedom of movement of arms and head, as well as storage for a month's supply of food and water. A concertina cowling like that found on a modern convertible car could be pulled over as a roof in high sea. Yet other devices used pedal power for propulsion and lights for attracting attention.

The preoccupation with shipwrecks was not confined to scientific magazines and reports of sensationally impractical inventions. The Illustrated London News carried stark accounts of distressed ships often depicted in dramatic engravings. The foundering of the iron steamer Brigand was reported with an artist's rendering and a vivid eyewitness report by a survivor of 'this lamentable catastrophe'. The Brigand was only two years old, and had been built at a cost of £32,000 to ply between Liverpool and Bristol. Such reports rarely identified navigational error as the cause of disaster, nor did they specifically incriminate the accuracy of tables. But they were constant public reminders of the dangers of sea voyages, and provided a strong base from which to argue the dread consequences of erratic navigation.

The source of errors in tables was clear: human fallibility. The preparation of mathematical tables was a laborious, tedious and exacting task. For a start, mathematicians devised the formulae and the range of values the tables were to cover. This was the clever part. They then calculated the 'pivotal values' - series of numbers at relatively large intervals. Filling the gaps involved calculating all the values in between (called sub-tabulation). This was a task of awesome drudgery. Each entry in the table had to be calculated by hand, and the production of a complete table demanded the seemingly endless repetition of the same series of arithmetical steps performed on slightly different numbers. To spare themselves, the mathematicians farmed out the routine work to the 'computers' - a common reference to people who performed calculations. (Computers are not the only devices to have human antecedents. Later in the nineteenth century, 'typewriter' referred to a person who typed rather than to the machine itself. It was not unheard of for the boss to elope with his typewriter.)

It was standard practice to give the same task to two computers who would perform the same set of calculations without collaboration. The results, independently prepared in this way, would then be checked against each other. This system of error detection was not foolproof, and it was not unknown for both computers to commit the same mistake. But if independently calculated results were found to be in agreement, this established a high degree of confidence in the correctness of the entry. Any discrepancy in the two results signalled an error and the offending pair of numbers were checked for correctness. This was the stage that Babbage and Herschel had reached when Babbage was driven to proclaim what is perhaps one of the most resonant utterances of the nineteenth century in appealing to 'steam' to execute calculations.

Once checked, results were copied by hand into lists and the manuscripts given to printers for typesetting. Inevitably the process of transcription was itself vulnerable to error. The typesetter then set the results using loose metal type, in preparation for printing. A compositor read each numerical result and selected a separate piece of type for each digit in the number. Pieces of type were laid alongside one another to make up the digits of the desired number, and sets of type were blocked together in a frame to make up a page for use in a printing press. Typesetting acres of numbers is a monotonous task and one that, as ever, is liable to error. A compositor setting text into type can at least recognise meaningful groups of letters making up words. But numbers have no immediate meaning, and there is no intuitive sense of whether or not any one digit has a sensible relationship to the one before or after.

Even when printed sheets of tables finally rolled off the press, the job was not yet done. The printed copy still needed to be proof-read. Anyone faced with a sea of numbers will be daunted by the burden of verifying them. It is not only monumentally tedious, but the process of checking is itself vulnerable to error however conscientiously the task is carried out.

At each of the four stages in the manual production of tables - calculation, transcription, typesetting and proofing - errors wait in ambush. Human frailty is the enemy of accuracy. What confidence could anyone have in the integrity of the outcome?

But what of Babbage's appeal to steam? Britain was in the throes of industrial revolution. Products poured from the new manufactories, and there seemed no limit to the variety and invention of the industrial arts as new materials and processes of manufacture were exploited. Britain led the way and, as Benjamin Disraeli put it in 1838, was fast becoming 'the workshop of the world'. The essayist and historian Thomas Carlyle called the decades of upheaval 'the Age of Machinery'. Science, engineering and the flourishing new technologies held limitless promise. Machines were the obsession of the times, and the extent to which motive power and mechanism permeated life sometimes touched on the absurd. One scientific magazine proclaimed the benefits of a 'new domestic motor'. The illustration shows a woman sitting in a rocking chair darning a sock. Through a system of pulleys, levers and weights, the oscillating chair rocks a cradle with a cherubic sleeping infant, and operates a milk churn at the same time. The reporter noted:

By this means, it will be observed, the hands of the fair operator are left free for darning stockings, sewing, or other light work while the entire individual is completely utilised. Fathers of large families of girls, Mormons, and others blessed with a superabundance of the gentler sex, are thus afforded an effective method of diverting the latent feminine energy, usually manifested in the pursuit of novels, beaux, embroidery, opera-boxes, and bonnets, into channels of useful and profitable labour.

On an industrial scale the engines of change were not women, but the steam engines which powered the factories. At the time of Babbage's invocation of steam as the salvation of tables from the curse of human fallibility, steam was both the actual and symbolic agent of change and held the promise of prosperity for all.

The encounter with Herschel in 1821 when the two friends met to verify the calculations of the computers was the genesis episode in the history of automatic computation. The account given here is based on the description Babbage wrote some eighteen years after the meeting.

In a separate record written closer to the events, Babbage is less sure whether it was he or Herschel who made the seminal appeal to steam, and comments that, whoever it was, he was in any case half-joking. Despite the differences between Babbage's accounts, the essential features of the encounter are clear. The problem was the daunting handicap of human fallibility to reliable calculation. The solution was machines.

—From The Difference Engine: Charles Babbage and the Quest to Build the First Computer by Doron Swade. (c) September 2001, Viking, a division of Penguin Putnam, used by permission.