Authors: James Essinger
Jacquard’s Web
prominent French scientists, mathematicians, and astronomers, Jean Arago among them.
It was very likely during this first visit to Paris that Babbage first heard of an ambitious French project undertaken at the turn of the century to make a set of reliable mathematical tables for the French Ordnance Survey. The production of the new tables, known as
Les Tables du Cadastre
(a cadastre is a register of property showing the extent, value, and ownership of land for taxation), was, at the time, the most ambitious effort of mass-calculation ever undertaken.
The project had been overseen by the eminent civil engineer Baron Gaspard Clair François Marie Riche de Prony. As his impressive name suggests, he was an aristocrat by birth. Despite often being in great jeopardy, he had managed to survive the Revolution’s Reign of Terror—the period when law and justice ran amok in France, and guillotines in France worked round the clock. De Prony’s survival was mainly due to certain influential Revolutionaries—chief among them Lazare Carnot himself—
admiring his scientific talents.
The purpose of de Prony’s undertaking was to calculate the logarithms of the numbers from
1
to
200 000
and make them available to the Ordnance Survey so that it could more easily carry out mathematical calculation. The logarithm (log) of any given quantity is the number to the power of which
10
must be raised to arrive at the quantity in question. The great advantage of logarithms is that if you know the logs of two numbers you want to multiply together, all you need to do is add the two logs together, note this total, and go back to the logarithm tables to see which number has the logarithm you end up with. That number is your answer. Eventually, tables of ‘antilogarithms’
were produced which simplified the business of finding the actual numeric result from the logarithmic result, but these were not available in Babbage’s day.
De Prony was flattered to be asked to undertake this great project. Not surprisingly, he was terrified of failing. He urgently 60
From weaving to computing
cast his mind around for an inspired idea that would give him a better-than-evens chance of success.
Finally, while in a second-hand bookshop, he came across a copy of Adam Smith’s
The Wealth of Nations
, published in
1776
.
An enormously influential book, it had taught a whole generation of British industrialists about the importance of the free market and the huge increases in productivity that could result from the division of labour, with different workers specializing in different elements of the overall task.
In a famous passage, Adam Smith relates how the productivity of a pin factory he had visited had been maximized by groups of workers specializing in different stages of the production of the pins. One group of workers had, for example, straightened the wire, another cut the wire, another sharpened the tips of the pins, and so on. In this way, Adam Smith explained, the total output of the pin factory would be many times greater than that which could have been produced if each individual worker had handled every stage of the pin-making process.
De Prony decided to use the same principle to give himself the best chance of making his vast set of tables with the greatest accuracy, and within a reasonable time-frame. After planning his approach carefully, he decided to divide his human calculators into three teams.
The first team would oversee the entire undertaking. This would involve investigating and furnishing the different formulae for each function to be calculated and setting down the simple steps of the calculation process. The team would be made up of half a dozen of the best mathematicians in France, including Carnot himself and Adrien-Marie Legendre, who was famous for important work on elliptic integrals, which provided basic analytic tools for mathematical physics.
The second team of seven or eight human calculators converted the formulae into key intermediate numbers which would be the basis for the actual calculations of the values to be set down in the tables.
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Jacquard’s Web
The third team consisted of sixty to eighty clerks whose mathematical ability was largely limited to being able to add and subtract. By virtue of the way the huge project was organized, this was all they needed to do in order to perform their necessary calculations. Curiously enough, many of the clerks were former hairdressers to the aristocracy. These hairdressers found themselves unemployed after the Revolution, for most of the aristocratic heads on which they had practised their art were no longer connected to their owners’ necks. Besides, hairdressing was not immediately a top priority in post-Revolutionary France.
The tables produced by de Prony’s pioneering technique occupied seventeen large folio volumes and had a reputation for being reliable. The tables impressed Babbage enormously. Their reliability was such that they were used by the French Army as late as
1940
to assist with calculations relating to surveys of terrain. But even though de Prony’s tables were regarded by French mathematicians as an enormously useful asset for more than a century, they were never actually
printed
. Instead, they remained in manuscript form. De Prony arranged for the French Government to make an arrangement with a Parisian publishing company to prepare
1200
printing plates on which a substantial portion of the tables would be printed. But the printing never went ahead, apparently for cost reasons. After the huge operation to prepare the figures, the fact that the tables were never printed meant that they were not particularly useful, because only people with access to the original manuscripts could actually use them.
De Prony’s mass-production approach to his enormous calculation assignment struck a chord deep within Babbage’s mind. When Babbage developed his first cogwheel calculator, he decided to base his machine on a method that would reduce the extremely complex business of tables calculation to its simplest essentials, much as de Prony had. Furthermore, Babbage knew about the problems de Prony had experienced with getting his tables printed. Babbage was determined to incorporate a printing 62
From weaving to computing
mechanism
within
his machine. This would allow the machine to produce a printed output onto paper automatically, eliminating the possibility of human error.
Babbage’s plan, in fact, was that the machine itself should make printing plates that could be used as many times as required. The calculating and printing machine was to be called the Difference Engine, for reasons we explore shortly. The first great professional challenge Babbage set himself was to build it.
Meeting this challenge, and grappling with the practical and conceptual difficulties it involved, took Babbage into a realm of almost inconceivably complex and original inventiveness. He started by planning his Difference Engine; he ended by designing what was nothing less than a computer controlled by the very same type of cards that programmed the Jacquard loom.
But why was Babbage so keen to build a calculating machine in the first place?
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1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 1 ❚ 1 1 1 1 1 1
2 ❚ 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2 2
3 3 3 3 ❚ 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3 3
4 4 4 ❚ 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4 4
5 5 5 5 5 5 5 5 5 5 ❚ 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5 5
6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6 ❚ ❚ 6 6 6 6 6 6 6 6 6 6 6 6 6 6 6
7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 ❚ 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7 7
8 8 8 8 8 8 8 8 8 8 8 ❚ 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8 8
9 9 9 9 9 ❚ 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9 9
0 0 0 0 0 0 0 0 0 0 0 0 0 0 ❚ 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0 0
It was not long before I made my first acquaintance with what is perhaps the most celebrated icon in the prehistory of computing, all that Babbage built of the Difference Engine No.
1
—the finished portion of the unfinished engine … True to its inventor’s promise, it calculates without error. It works impeccably to this day and in defiance of Babbage’s detractors, provides compelling evidence for the feasibility of his early designs.
Doron Swade,
The Cogwheel Brain
,
2000
To understand why Charles Babbage chose to devote much of his life to trying to build a cogwheel calculation machine, we need to go back to an evening in the summer of
1821
, when he was working in his home at Devonshire Street with his friend Herschel.
The two gentleman mathematicians had been asked by the Royal Astronomical Society to help improve the accuracy of some important astronomical tables it had produced.
The Royal Astronomical Society, which still exists, admirably preserves its records of its illustrious past. In Babbage’s day the 65
Jacquard’s Web
Society made extensive use of astronomical tables for the purposes of calculating the movements of heavenly bodies. Compiling these tables required many elaborate and tedious arithmetic calculations.
Babbage and Herschel were not, in fact, carrying out the calculations themselves. Instead, they were merely checking over work that had already been carried out by clerks. Curiously enough, clerks who undertook arithmetical calculations of the complex kind necessary to compile astronomical and mathematical tables were known at the time as ‘computers’.
For Babbage and Herschel, even the mere checking process was extremely tedious and difficult. They were checking over the results of two independent human computers. The problem was that if they found a discrepancy between the two results they obviously had no way of knowing which was correct and so had to perform a calculation of their own. But how could they be sure that their result or one of the computers’ result was right? The job really was nothing less than a mathematical nightmare.
Quite understandably, at some point during the painful labour he was sharing with his friend, Babbage suddenly exclaimed:
‘My God, Herschel! How I wish these calculations could be executed by steam!’
The cry may have been a simple expression of frustration, but Babbage was not a man of empty words. Babbage was above all a practical man, and when a purpose and decisive course of action had caught fire within his mind, he possessed the energy and financial means to take steps to turn it into reality, or at least into as much of a reality as could be achieved. Later that very evening, Babbage started thinking seriously about how his objective might be attained. It was necessity that had driven Joseph-Marie Jacquard to seek to build a revolutionary type of loom. It was 66
The Difference Engine
now necessity that was powering Charles Babbage’s inventive imagination forward.
In the early nineteenth century, the lack of a reliable means of carrying out mathematical calculations was a serious problem, and the more extensive the role technology was playing in society, the more serious the problem became. As Britain’s Industrial Revolution gathered momentum, the difficulty of performing complex calculations accurately became a grave limiting factor to the changes this revolution was bringing to Britain’s industry and economy.
The compilation of mathematical tables such as astronomical tables, or the logarithmic tables Gaspard de Prony had produced for post-Revolutionary France, was essential for facilitating accurate calculations. But the whole problem was that
the unreliability
of calculations also extended to the compilation of mathematical tables.
And of course, if you were using inaccurate logarithmic tables to carry out an important calculation, your calculation was doomed from the start.
Even worse, you had no way of knowing exactly where an inaccuracy in the mathematical tables might lie. This fact created a disturbing climate of psychological insecurity among scientists, astronomers, and mathematicians. Under the circumstances it was hardly surprising that John Herschel, writing in
1842
to the Chancellor of the Exchequer Henry Goulburn, bitterly observed: 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.
From our perspective today, with cheap electronic calculators readily at our disposal and even available on our mobile phones and watches, and with every desktop computer featuring a powerful calculator function that provides completely reliable results in less time than it takes to click a mouse, it is difficult for us to imagine
not
having access to all this calculating power. We 67
Jacquard’s Web
need to make an even greater effort of imagination to empathize with the fact that early nineteenth-century scientists could not trust the mathematical tables they were using. Errors could literally appear anywhere, and there might be a terrible mistake in the midst of an otherwise perfect column or row.
A particularly painful illustration of this problem is seen in the work of the English mathematician William Shanks. In
1853
Shanks announced that he had successfully calculated to an astounding total of
530
decimal places the mathematical quantity
. This, the ratio between the circumference of a circle and its diameter, is an irrational number that begins
3
.
14159
and proceeds with a never-ending series of decimal places. Shanks devoted the next twenty years of his life to extending the approximation further in order to take the evaluation into new and undiscovered realms of mathematical achievement. But unfortunately for Shanks, an error in the
528
th place meant that
all
his subsequent work was entirely wasted.
Babbage was not the first inventor to try to build a machine for facilitating calculation. In
1642
the French scientist and philosopher Blaise Pascal had constructed an adding machine to aid him in computations for his father’s business accounts. The machine consisted of a train of number wheels whose positions could be observed through windows in the cover of a box that enclosed the mechanism. Numbers were entered by means of dial wheels. But Pascal’s machine turned out to be unreliable. It never made any impact in mechanizing calculation.