These silver figures were the chef d’oeuvres of the artist. They had cost him years of unwearied labour, and were not even then finished, like the engines that Babbage was to design.
Four years after Byron first went to Cambridge in 1805, Babbage attended the same college as Ada’s father, Trinity. His Cambridge years couldn’t have been further removed from a cauldron of moral turpitude or a pile of mounting debts, however. He keenly played chess and recalls, for example, how he and some friends founded a Ghost Club, designed to collect evidence about the existence of ghosts. They also founded a club which they called The Extractors, designed to help its members should any of them be the subject of a petition to get them sent to a lunatic asylum. It is difficult not to feel that merely belonging to such a club ought really to help qualify a member for being the subject of that very kind of petition.
His wildest time was a sailing trip of several days during term time. As Babbage relates in his autobiography:
I was very fond of boating, not of the more manual labour of rowing, but the more intellectual art of sailing. I kept a beautiful light, London built boat, and occasionally took long voyages down the river, beyond Ely into the Fens. To accomplish these trips, it was necessary to have two or three strong fellows to row when the wind failed or was contrary. These were useful friends upon my aquatic expeditions, but not being of exactly the same calibre as my friends of the Ghost Club, were very cruelly and disrespectfully called by them ‘my Tom fools’ …
I also directed my servant to order the cook to send me a large well-seasoned meat pie, a couple of fowls, etc. These were packed in a hamper with three or four bottles of wine and one of noyeau [this is in fact Noyaux, an almond liqueur]. We sailed when the wind was fair, and rowed when there was none. Whittlesea Mere was a very favourite resort for sailing, fishing and shooting. Sometimes we reached [King’s] Lynn.
The truth is that Babbage, having been encouraged in all his enthusiasms, was something of a dilettante. He dabbled, and what was worse, he rather seems to have enjoyed dabbling. Unlike his friend Charles Dickens, his father’s money had always cushioned him from the need to ever finish anything. And much like Byron, Babbage had a disdain for making money off his inventions, let alone turning them into a commercial enterprise.
Any goals would be set by Babbage himself. By the time he went up to Trinity College, Cambridge, Britain was in the midst of its unprecedented technology revolution. Transport, communications and above all the application of steam power to industry were giving mankind the opportunity to use levels of power thousands of times greater than that which the horse, or the human hand, could produce. Babbage felt he would like to take part in that revolution in some capacity or other, so he withdrew from the curriculum for the Senate House Exam and pursued his own mathematical and scientific agenda. At the time, gentlemen scholars were permitted to do this.
With two friends he had met at Cambridge – John Herschel (son of the famous astronomer Sir William Herschel who had, in 1781, sensationally discovered Uranus, the first new planet) and George Peacock – Babbage also helped form what he called the ‘Analytical Society.’ Its main objective was to overhaul the study of calculus at Cambridge and replace the notation of Isaac Newton with what Babbage and his friends regarded as the much more efficient notation invented by Leibniz. The campaign was, in the end, successful, although it would not be won until after Babbage graduated from Cambridge in 1814. (He would later become the eleventh Lucasian Professor in Mathematics at the university; Newton had been the second holder of this prestigious chair.) But the vigour of the arguments put forward to support the change forced the outside mathematical world to start to take notice of the founders of the Analytical Society, particularly Babbage and Herschel. It was an important contribution to science, and one of the few Babbage ever saw through to its end.
The friendship between Babbage and Herschel was the first serious intellectual friendship either had. They were touchingly good companions, and Babbage was to name his oldest son Benjamin, after his father and Herschel. They addressed each other as ‘Dear Herschel’ or ‘Dear Babbage’ in letters: an extremely intimate salutation by the formal standards of the time. The informality of their letters (which usually contained abundant mathematical formulae as well as personal material) is neatly explained by a comment Herschel made at the start of a letter he wrote to Babbage on February 25, 1813:
When men with common pursuits in which they are deeply interested, correspond on the subject of those pursuits, the trifling ceremonials of an ordinary correspondence may in great measure be waived.
Babbage’s friendship with Herschel frequently sustained and supported Babbage during a life with many setbacks.
While his father was still alive, Babbage pretended to have an interest in earning a living. But he never found any jobs that he either wanted to do, or was appointed to do. Despite Benjamin’s moans and complaints, he gave Charles and Georgiana enough money to live on in reasonable comfort. They moved into their marital home at 5 Devonshire Street, Portland Place, London in the middle of September 1815.
At the same time, Babbage wasted no time in making his mark on the scientific scene. During 1815 – the year of Ada’s birth – he gave a series of lectures on astronomy to the Royal Institution. In the spring of 1816, he was elected a member of the Royal Society, a learned assembly of great scientists. The full official name of this august body, which still exists today, is the Royal Society of London for Improving Natural Knowledge, but the Royal Society was, from the outset, open to scientists from around Britain. For the next few years, Babbage’s work was mainly mathematical. He published more than a dozen mathematical papers, all of which were regarded as highly competent, though not of enormous importance.
Babbage, like Byron, doted on his children but after his wife Georgiana died in 1827 they were farmed out.
Herschel, his eldest son, stayed during the school holidays with Babbage’s mother Elizabeth who lived (until 1844) in London. Babbage’s beloved only daughter, Georgiana (who was two years younger than Ada), also stayed with his mother at her home at 5 Devonshire Street, a ten-minute walk from Babbage’s house on Dorset Street.
Although he could easily afford to travel in style, Babbage was not the kind of man to idle time on a frivolous Grand Tour. Instead, Babbage and Herschel made their first trip to France in 1819, the first of many such excursions they made to exchange information and ideas with French men of science. With Herschel’s father Sir William’s reputation opening doors for the two young men, they were able to meet several prominent French scientists, mathematicians and astronomers.
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 Ordinance Survey.
The project had been overseen by the eminent civil engineer Baron Gaspard de Prony. Despite often being in great jeopardy, he had managed to survive the French Revolution’s Reign of Terror mainly due to certain influential revolutionaries – chief among them Lazare Carnot himself, ‘L’Organisateur de la Victoire’ (‘the Organiser of the Victory’) – who admired his scientific talents.
The French Ordinance Survey required prodigious amounts of multi-digit multiplication and it needed accurate tables to simplify and speed up its work. Logarithms (a seventeenth-century invention by John Napier) were the shortcut to such otherwise time-consuming multiplications. A logarithm converts each number into a calculation of the number 10. The great advantage of this is that if you look up the logarithms (logs) of the two numbers which you want to multiply, all you need to do is add their 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 the product of the multiplication of the two original numbers.
While the principle appears simple, it is easier said than done. The purpose of De Prony’s undertaking was to calculate the logarithm
s of the numbers from 1 to 200,000, a massive undertaking even though 200,000 is not a particularly high number. For any numbers higher than 200,000, French surveyors would be obliged to do the laborious sums manually.
De Prony, not surprisingly, was terrified of failing as the Reign of Terror continued to claim the heads of its victims. It was only when De Prony came across a copy of Adam Smith’s The Wealth of Nations, published some two decades before in 1776, that he found his answer.
In a famous passage, Adam Smith relates how the productivity of a pin factory he had visited had been maximised by groups of workers specialising 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 make 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 (famous for important work on elliptic integrals, which provided basic analytic tools for mathematical physics).
The second team of seven or eight human calculators would convert the formulae into key numbers that would be the basis for the actual calculations of the values to be set down in the tables.
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 organised, 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’s thirst for decapitation.
The tables produced by De Prony’s pioneering technique occupied seventeen large folio volumes and had a reputation for being reliable. 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. The tables impressed Babbage enormously.
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, apparently for cost reasons. Only surveyors at the French Ordinance Survey with access to the original volumes could actually use them.
De Prony’s mass-production approach to his enormous calculation assignment struck a chord deep within Babbage’s analytical mind. When Babbage developed his first cogwheel calculator, he decided to base his machine on the Method of Differences that would reduce the extremely complex business of tables calculation to its simplest essentials, much as De Prony had. Instead of 6 digits (200,000), he planned his tables to run to up to 30 digits, if ever completed.
Furthermore, Babbage knew about the problems De Prony had experienced with getting his tables printed. Babbage was determined to incorporate a printing 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.
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, even if it never got further than the seventh part of the machine that Ada first saw in 1833. He started by planning his Difference Engine, he ended by designing what was nothing less than a calculator controlled by punched cards.
12
The Analytical Engine
On Monday, December 19, 1834, just five days after Ada turned nineteen, she spent an evening with Babbage, Lady Byron and Mary Somerville that led to her feeling even more excited about Babbage and his inventions than she already did. Babbage, too, was thrilled about an extraordinary new horizon that had opened up in his own mind. In his mad-scientist way, he communicated his excitement most successfully and dramatically to his guests.
We can get a good idea of just how excited he was from a journal entry Lady Byron made late that evening. According to her, Babbage spoke about his discovery in metaphorical terms rather than seeking to explain it in precise detail. The first glimpse of his discovery had aroused in his mind a sensation that was something like ‘throwing a bridge from the known to the unknown world.’ According to Lady Byron’s journal, Babbage also said that the breakthrough made him feel that he was standing on a mountain peak and watching mist in a valley below start to disperse, revealing a glimpse of a river whose course he could not follow, but which he knew would be bound to leave the valley somewhere. Writing in her journal, Lady Byron later noted ‘I understand it to include means of solving equations that hitherto had been considered unsolvable.’
In fact, what was happening was that Babbage was telling his guests about his invention of a completely new machine: the Analytical Engine, which was, in Babbage’s thinking, soon to supersede the Difference Engine, though he never entirely stopped tinkering with his plans for the Difference Engine.
Babbage’s only daughter, Georgiana, had died suddenly at the age of seventeen, just over three months earlier, on September 26, and in his grief he dedicated himself to solving an important mechanical issue of the Difference Engine.* But he had gone well beyond the solving of that problem – and his brilliant conception of the Analytical Engine was the result.
Babbage was utterly devastated by Georgiana’s death. As always, he tried to escape personal misery by plunging himself into his work. The result was the Analytical Engine, which has a claim to be the most brilliant theoretical invention (as even today no Analytical Engine has ever been built) of the nineteenth century.
The genesis of the device Babbage called the Analytical Engine – echoing the society he had founded with Herschel at Cambridge – can be traced back to a second paper on the Difference Engine that Babbage had read to the Royal Astronomical Society on December 13, 1822, when Ada had just turned seven.
In this paper, he explained to his audience that useful as the Difference Engine was, it was always going to be handicapped by the need to reset the machine for each new set of calculations. The point was that the initial numbers that were entered on the cogwheels had to be entered into the Difference Engine by hand. Once the engine was set up, the handle could be turned to ensure that the calculation process went on automatically. In principle, the calculations would follow regularly without further intervention by whoever was operating it. But unfortunately, in some calculations, the results would start to become inaccurate as the table production progressed. The machine wasn’t to blame for this. It stemmed from the fact that the calculations were based on a mathematical formula which would not in every case be precisely accurate for every single desired numerical result due to the fact that certain numbers would need to be rounded off as they consisted of an infinite number of decimals (one third is a common division, for example, but as a number the Difference Engine could only approximate it as 0.3333, etc.; the number of cogwheels limited the number of threes and could never, by definition, be infinite).
What was really needed was a machine that would not feature this continual slight reduction in accuracy; a machine, moreover, that could do far more than simply calculate mathematical tables. He named this new endeavour the Analytical Engine. It would soon supersede the Difference Engine into whose design the government had poured the equivalent of two frigates and Babbage himself
more than a decade of his life.
Babbage pursued the notion of the Analytical Engine relentlessly during the months that followed his evening with Ada, Lady Byron and Mary Somerville.
The new machine Babbage envisaged would be enormous, about the size of a small steam locomotive in his day or a large van today. It would have contained perhaps as many as 20,000 cogwheels, some mounted in vertical columns like the Difference Engine but others used in a variety of other configurations. Thousands of gear-shafts, camshafts and power transmission rods would have enabled calculations carried out in one part of the machine to be mechanically relayed to other parts. In sketch-books containing his ideas, he made an enormous number of drawings and diagrams for the Analytical Engine, and completed the actual manufacture of many small working cogwheel components designed to be used in its mechanism.
Above all, the entire operation of the Analytical Engine would be controlled by a punched-card system. The punched-card system was not Babbage’s idea, but – as Babbage freely acknowledged (he was always generous with his credits) – based on the Jacquard loom. It was this part that was to provide an important catalyst for Ada’s understanding of what the Analytical Engine could really achieve – an understanding that even eluded Charles Babbage himself.
The struggle Babbage had to come to terms with the magnitude of his vision of the Analytical Engine can be surmised from the fact that it took him two years from his initial vision of the Analytical Engine to decide precisely what control system he should use. In Volume II of notebooks that Babbage himself described as his ‘Scribbling Books’ – his extensive hand-written journals preserved at the London Science Museum – there is an entry for June 30, 1836, which contains the brief but momentous comment:
Ada's Algorithm Page 10