Enchantress of Numbers

by Jennifer Chiaverini

  How delightful it was to be present at a party where such fascinating subjects were discussed! Just as I was about to work my way into the circle of attentive listeners, a burst of laughter caught my attention. Turning toward the sound, I spied a gentleman who looked to be not much older than my mother speaking earnestly to a group of four gentlemen and three ladies who seemed both amused and intrigued. The gentleman stood while his companions sat in a half circle around him, as if he were a teacher addressing a class. He was slightly taller than average height, and animated, with broad shoulders and a barrel chest. His square face bore an expression that suggested rationality and a jaw that implied stubbornness. My mother and her phrenologist friends would have frowned at his brow, which suggested intelligence without reverence, and at his mouth, which warned of an impatient nature. Fortunately, I was no phrenologist.

  Glancing back the way I had come, I searched the crowd for my mother, only to find that so many acquaintances had stopped her for a chat that she had not proceeded very far beyond the entrance. She seemed perfectly content without me, so rather than wait for her to make introductions, I approached the lively gentleman and his enthralled audience alone.

  “And so after years of dangerous, difficult surveying, the meridian was measured from the North Pole to the equator, passing through Paris, naturally, and from thence we derived the meter,” the gentleman was saying as I drew closer. “For almost thirty-five years we have benefited from having a standard distance of measurement, as Delambre himself said, ‘for all people, for all time.’ But what of all of the old tables, which until then had been used to make complex scientific calculations?”

  “They would be no good anymore, I suppose,” said one of the ladies. “They don’t conform to the meter.”

  “Exactly right, Mrs. Crawford,” the gentleman replied. “So our intrepid Frenchmen were faced with a daunting conundrum: How on earth could they draw up new tables when the number of calculations required to create them vastly exceeded the total number of calculations that could be completed by all the mathematicians in France?”

  “The simple answer is that they could not,” remarked the eldest gentleman in the circle.

  “Yes, Lord Mitchell, quite so. It was a physical impossibility. Or so it was until the French mathematician Gaspard Riche de Prony was inspired by the economist Adam Smith’s study of the division of labor.”

  “The Wealth of Nations,” I heard myself say. His companions suddenly shifted their gazes to me, and I felt myself blush beneath their scrutiny.

  “Yes, indeed, miss,” said the gentleman, smiling approvingly. Emboldened, I seated myself in an empty chair on the edge of the circle, and when one of the gentlemen moved to make room and Mrs. Crawford smiled and beckoned, I drew my chair closer. “De Prony was fascinated by Smith’s description of the manufacture of pins. One man drew out the wire, a second straightened it, the third cut it, the fourth sharpened it to a point, and so on and so forth so efficiently that twenty pounds of pins could be produced in a day.”

  “And de Prony thought this process could be applied to creating the new mathematical tables?” queried another woman.

  “Yes, Lady Addicott, he did. Why should he not manufacture logarithms as one manufactures pins?” Smiling, he spread his hands and shrugged as if the idea were so obvious, someone should have thought of it years before. “With Delambre’s help, he contrived a sort of mathematics factory. The first room was assigned to the smallest group of workers, the most highly skilled, and therefore, the most expensive: the professional mathematicians.”

  “You would include yourself in this most exclusive group, I presume?” teased Mrs. Crawford.

  “I think not,” said Lord Mitchell. “Mr. Babbage is meant to be Monsieur de Prony in this scenario, the master in charge of all.”

  As laughter went up from the circle, Mr. Babbage grinned and waggled a finger at the gentleman. “You know me too well, all of you.”

  “What was the function of these mathematicians?” I inquired.

  “The most important one, Miss . . .”

  “Byron,” I supplied, pretending not to notice the raised eyebrows and significant glances that greeted the revelation.

  “Ah, yes. Miss Byron. Their role was to determine the formulae that were required to solve a particular calculation, and then to reduce each formula to its smallest components, to forms of mathematical operations simple enough to be used by non-mathematicians.”

  “Who were seated in the next room?” ventured Lady Addicott.

  Mr. Babbage nodded. “Exactly so. The second room, larger than the first, accommodated the calculators. They were not as skilled as the mathematicians, but still were quite accomplished, and there were more of them. Their role was to determine the range of values for the calculation in question, as well as the layout of the table.”

  As he spoke, I imagined a bustling workshop, much like the engraving of a mill I had once seen, but instead of workers laboring over their looms, clerks were seated at rows of tables, diligently toiling with pencil and paper. I wondered if de Prony allowed women to work as calculators. Women worked in the mills, enduring arduous labor in dusty, loud, and potentially dangerous conditions; why couldn’t a sufficiently bright young lady work in a mathematics factory too?

  I wasn’t thinking of myself, of course. Peeresses did not work in factories, and I wanted to be the next Mrs. Mary Somerville, not a mere calculator.

  “The third room was necessarily the largest,” Mr. Babbage continued. “Therein worked roughly sixty to eighty computers, the least qualified—and least expensive—employees. Their task was to compute the results of the calculation in question using the formulae and values provided to them.”

  “You say they were the least qualified,” said Lord Mitchell, brow furrowed, “but I must assume they had some knowledge of arithmetic.”

  “Certainly. The same basic skills any shopkeeper, weaver, or laborer must possess in order to conduct business—simple addition, subtraction, multiplication, and division.” Mr. Babbage’s smile turned wry. “De Prony had an ample supply of suitably qualified and recently unemployed workers eager for these new jobs, and you’ll never guess what their previous occupation had been.”

  After a moment, one of the gentlemen said, “Clerks of some sort, I gather?”

  “Hairdressers,” said Mr. Babbage, a dark undercurrent in his voice. “After the Revolution, so many aristocratic heads had been parted from their unfortunate owners that the demand for workers proficient in wig powdering and pompadour constructing no longer existed. Thus many former hairdressers became computers instead.”

  For a moment a chilling shadow seemed to fall over the circle, and I shared the sudden apprehension. What had happened in France would never happen in England, I reminded myself firmly. My mother, Dr. King, and many other intelligent people whose opinions I trusted said it was so. And wasn’t my mother, through her tireless efforts to educate the children of the working poor, doing all she could to help the lower classes improve their lot without the violent chaos of revolution?

  “But as I have said—” Mr. Babbage sighed comically, and the shadow lifted. “The computers and the calculators were not skilled mathematicians, and errors could be introduced at any stage of the process, and those errors would be compounded as the process moved along. When I visited Paris—oh, what was it, about ten years ago—and I observed de Prony’s mathematical factory, I was intrigued, but I thought to myself: How much more wonderful it would be if those computations had been accomplished by machine!”

  Mrs. Crawford and Lady Addicott smiled indulgently, and one of the gentlemen chuckled, but I perched on the edge of my chair, enthralled.

  Mr. Babbage’s eyes were bright with excitement. “I knew that a machine could do everything the computers did—”

  “Except arrange hair,” one of the gentlemen broke in.

  Ev
eryone laughed, including Mr. Babbage. “Fair enough, but that skill is not required in mathematics. What is needed is the ability to calculate finite differences. Using this method, one can create a multitude of different numerical tables using simple addition and subtraction.” He rubbed his hands together and beamed, brimming over with enthusiasm. “A machine performing the series of steps that comprise the technique of finite differences could carry out mathematical operations indefinitely and flawlessly.”

  “De Prony’s calculators and computers could not do this sufficiently well?” queried Lord Mitchell.

  My thoughts flew to the obvious answer, but before I could offer it, Mr. Babbage shook his head. “Even the best men require rest and make occasional mistakes. One of the great advantages that we may derive from machinery is the check that it affords against the inattention, the idleness, or the dishonesty of human agents.”

  Lord Mitchell frowned. “Perhaps, but if you were to replace the calculators and computers with machines, then you’ve reintroduced the problem of mass unemployment.”

  Several of Mr. Babbage’s listeners nodded, but before he could address that concern, Mrs. Crawford spoke. “Forgive me, Mr. Babbage, but if de Prony’s mathematics factory has already produced these tables, why would a calculating machine be necessary?”

  “Ah, yes.” He held up a finger, smiling, clearing enjoying the debate. “But de Prony’s factory hasn’t created all the tables the world shall ever need, nor could it. His system is limited to six digits, whereas my machine would go up to thirty.” His smile broadened as his listeners exchanged glances, clearly impressed. “Also, de Prony has never been able to publish his tables. The cost would be prohibitive, and the problem of the likelihood of introducing error is insurmountable. His tables remain in manuscript form, and only those few individuals with access to the volumes held at the French Ordnance Survey can make use of them. My machine would have a printing mechanism built into it, so that any result could be immediately produced on paper, quickly and efficiently, with no risk of transcription or typesetting errors.”

  “Mr. Babbage,” I said, “you speak with such certainty. Do you mean to say that you already know how to build such a machine?”

  Everyone else smiled. “My dear Miss Byron,” said Lady Addicott, amused, “you must be the only person in London who has not heard of Mr. Babbage’s famous Difference Engine.”

  Difference Engine. The phrase had an aura of magic about it.

  “Not everyone has heard of it,” Mr. Babbage demurred, eyes twinkling. “Not yet, anyway. Give me time.”

  “Then you have built it,” I exclaimed. “How marvelous!”

  “I’ve built a demonstration model only,” he replied. “The full-scale version of my Difference Engine is, shall we say, a work in progress.”

  “It shall be completed, Babbage,” the youngest gentleman said stoutly, and all except Mr. Babbage, Lord Mitchell, and I myself nodded. I did not because I had only just heard of the Difference Engine and its aspiring creator, and I could not possibly judge. I don’t know why Lord Mitchell declined to offer Mr. Babbage reassurances, unless he sympathized with the hairdressers who would be sacked if Mr. Babbage succeeded.

  “It shall be completed,” Mr. Babbage agreed, sighing. “One day the stars shall align, and the tightfisted government will release the funds they have promised me, and my recalcitrant engineer will resume the manufacture of the parts I need, and then we shall see what we shall see.”

  I longed to be there to see it.

  The circle broke up and its members dispersed, as will happen at a party, but I lingered, eager to hear more about Mr. Babbage’s wondrous Difference Engine. My mind raced with the variety and number of practical benefits that would result from such a machine. The tables it could produce for sailors navigating on the open sea, the tedious mental labors that could be delegated to a device that suffered neither boredom nor fatigue—the possibilities were themselves incalculable.

  I approached Mr. Babbage, and we had scarcely exchanged greetings when my mother joined us and I promptly introduced them. “I observed from across the room that you had my daughter quite enthralled, Mr. Babbage,” my mother said, smiling. “I must assume that you were discussing mathematics, music, or horses to have captivated her so.”

  “Mr. Babbage has invented the most wonderful machine,” I told my mother, my words tumbling over one another in my excitement. “He calls it the Difference Engine, and it uses the technique of finite differences to tabulate polynomial functions.”

  “How delightfully well you describe it, Miss Byron,” said Mr. Babbage, a trifle surprised. “I had no idea I was addressing a fellow mathematician.”

  His compliment pleased me immensely, but I knew my mother would not want me to seem proud. “I am a student only, with a great deal yet to learn,” I said modestly, “but I find your Difference Engine absolutely fascinating.”

  “You must come and see it, then,” he said, nodding to us both. “I give weekly soirées—which I confess I exploit to introduce people to my work—and I would be honored if you would attend.”

  I wanted to blurt out that we would not miss it for the world, but I wisely held my tongue to give my mother a moment to consider. “Thank you, Mr. Babbage. We would be delighted,” she said, and if I had been only a few years younger, I might have jumped up and down for joy.

  He assured us that we should expect his invitation without delay, and we parted with great satisfaction on all sides. Thus even though I had yet to meet Mrs. Somerville, I could no longer consider the party in any way a disappointment.

  The following evening, in order to provide a contrast to the intelligent, edified conversation we had enjoyed at Lord and Lady Copley’s the day before, my mother allowed me to attend the opera with Lady Elizabeth Byron. My mother did not join us; she disliked opera but said that since I loved music so dearly, I might go and judge for myself.

  I cannot recall precisely what I had expected, but I was surprised by what I saw. The celebrated Italian soprano Giuditta Angiola Maria Costanza Pasta starred in Anna Bolena, in a role Donizetti had written specifically for her voice. She was quite wonderful, both as a singer and as an actress, and was especially gifted in expressing intense passions, but I confess the unfamiliar conventions of opera, which the other members of the audience took for granted, were quite lost on me.

  But that was not the fault of the singer or the music. I did not understand what I beheld, and my thoughts were elsewhere—on Mr. Babbage, his Difference Engine, and the invitation I fervently hoped he would not forget to send.

  Chapter Fourteen

  What Wondrous New Machines Have Late Been Spinning!

  June 1833

  How I rejoiced when Mr. Babbage’s invitation duly came and my mother agreed that we should accept. In the days following Lord and Lady Copley’s party, we had discovered that Mr. Babbage’s soirées were all the fashion in London among scientists, philosophers, and authors, as well as the aristocrats interested in knowing them. In Mr. Babbage’s drawing room and salons, duchesses mingled with doctors, engineers with earls. We had heard, too, that Mr. Babbage and Mrs. Somerville were good friends, so I held out hope that he would introduce me to her after she returned from abroad.

  My mother had, of course, made discreet inquires about Mr. Babbage before accepting his invitation. She learned that he had studied mathematics at Trinity College and Peterhouse at Cambridge, where he had been regarded as the top mathematician. After graduation, he had sought several academic posts and had been chosen for none, and so he had worked as an independent scholar in mathematics, astronomy, and electrodynamics and had dabbled in actuarial and insurance businesses. He had been elected a fellow of the Royal Society in 1816, and upon the death of his father, a prosperous banker, he had inherited a substantial estate worth one hundred thousand pounds. Such a fortune impressed even my mother.

  On Monday evenin
g, 17 June, my mother and I rode to Mr. Babbage’s residence at 1 Dorset Street in Marylebone, a relatively quiet, remote neighborhood near Regent’s Park. He had moved there in 1828 after the deaths the previous year of his father; his beloved wife, Georgiana; and two of their eight children, of whom two others had perished earlier in childhood. His eldest son was away studying engineering and architecture, and his daughter and two younger sons lived with his mother elsewhere in London.

  Mr. Babbage’s Georgian home, aesthetically pleasing in its symmetry and proportions, was white stone and redbrick outside, all polished wood, bright windows, and gleaming brass within. Although it missed the feminine touches his late wife might have brought to each room, it was gracious, comfortable, and tastefully decorated, indicative of a master who preferred clean lines and admired beauty from function.

  The soirée was already merrily in progress when my mother and I arrived. We found Mr. Babbage mere moments after we entered, bustling about in the role of genial host, encouraging his guests to partake of food, drink, and fascinating conversation. His face lit up when he spotted us, and he quickly worked his way to our side. “Welcome, Lady Byron, Miss Byron,” he said. “What a great honor and a pleasure it is to have you join us.”

  “I don’t suppose Mrs. Somerville has come?” I said, ignoring the exasperated look this evoked from my mother. “I would so like to meet her.”

  “I’m sorry, Miss Byron,” said Mr. Babbage as he led us into a spacious sitting room with tall windows overlooking the front garden. “As far as I know, Mrs. Somerville is in Paris.”

  “I had hoped she had returned.”

  “Not yet, I’m afraid. I’m happy to count Mrs. Somerville as a good friend and a frequent guest, however, so perhaps another time you visit, I shall be able to introduce you.”