The Philosophical Breakfast Club

by Laura J. Snyder

  Herschel and Babbage had already availed themselves of the increased opportunity to travel in January of 1819, visiting France. The following year, Whewell and Richard Sheepshanks met Jones in Paris, and the three friends spent time together there—although, as Whewell complained, there were “no handsome women to be seen!”82 Jones remained behind in Paris while Whewell and Sheepshanks traveled on through Switzerland and Germany.

  In the summer of 1821, Herschel, Babbage, and Jones stayed for a time in Paris, where they met with the most important French savants of the day: Arago, Laplace, Biot. Herschel and Biot shared notes on experimental results in optics. The German naturalist Alexander von Humboldt was living in Paris at the time, and the men became acquainted with him at Laplace’s house. Herschel and Babbage then traveled southward through Dijon to French Switzerland, and spent a week in Geneva. Jones again lingered in Paris, enjoying the excellent wines. From Geneva, Babbage and Herschel went south through Chamonix, Aiguebelle, and Modane to Turin, where they visited the astronomer Giovanni Plana. They proceeded to Milan, and returned via Lake Como, Lake Maggiore, and the Simplon Pass.

  Along the way they climbed the Breithorn, a 13,000-foot mountain next to Mont Rosa. After five hours of hard walking in knee-deep snow, the high crest of the mountain still remained to be climbed. It was a sharp edge of snow “along which a cat might have walked,” as John wrote to his mother. Three out of their four Swiss guides refused to go any farther, but Herschel and Babbage persisted, along with one old man who had climbed the mountain before, and reached the summit.83

  The two men traveled with a full contingent of scientific instruments in their carriage. Wherever they went, they took barometric readings and temperature measurements, determined angles by a small pocket sextant, and filled entire notebooks with geological and mineralogical observations. When they returned, Babbage published a paper based on the readings they had made from the Staubbach Falls, in the valley of Lauterbrunnen. Each morning they had carried a thermometer and barometer from their inn over a small wooden bridge traversing a torrent, where they took the initial reading. They then followed a path along the bank, rising with a steep descent, to the top of the waterfall, where they took their second reading. This they did over a series of days, to compare the readings at the top and the bottom of the falls, trying to determine whether height could be determined by barometric pressure.84 Their results were inconclusive, but they considered the trip a scientific success nevertheless. They returned home in October 1821, Herschel having recovered from his broken heart. He resumed notations in his experimental notebook in November.

  BY THIS TIME, Babbage, Herschel, Jones, and Whewell had each graduated, and had begun to lead their scientific lives. For the next five decades these lives would come together and pull apart, like light rays focused and dispersed in an ongoing series of optical experiments.

  4

  MECHANICAL TOYS

  ON THE AFTERNOON OF DECEMBER 20, 1821, BABBAGE EXCITEDLY summoned Herschel. “Can you come to me in the evening as early as you like I want to explain my Arithmetical engine and to open to you sundry vast schemes which promise to reach the third and fourth generation—… Do let me see you for I cannot rest until I have communicated to you a world of new thought.”1

  The idea for an “arithmetical engine” had come to Babbage a few weeks earlier, soon after he and Herschel had returned from Europe. One morning the two sat opposite each other in Babbage’s house in Devonshire Street. They were bent over astronomical charts calculated by “computers”—the name given to men and women, usually schoolteachers, clergymen, or surveyors, who picked up extra income on their off-hours by performing routine mathematical calculations by hand, using a fixed procedure over and over again.2 Babbage and Herschel each held a set of data calculated by a different computer using the same formula. If all the figures had been worked out correctly, the two sets of data would match perfectly. Babbage read off one number at a time, waiting for Herschel to check whether it accorded with what was on his sheet, putting a mark in the margin whenever Herschel told him the figures differed. Again and again, the men found discrepancies, indicating error on the part of one or the other (or both) of the computers.

  The next year, when he wrote about this moment, Babbage could not recall who had first come up with the idea; but he knew that one of them had sighed in exasperation, “If only a steam-engine could be invented to make these calculations.” Decades later, in his autobiographical Passages from the Life of a Philosopher, Babbage told the story differently, taking credit for the notion of a calculating engine.3 Whoever may have initially voiced the idea, it was Babbage who became obsessed with the project of inventing and building such a machine, an enterprise he saw as fulfilling Francis Bacon’s call for renovating science and improving people’s lives. What Babbage eventually devised would be like nothing that had ever been created before.

  AIDS FOR CALCULATION went back centuries, of course, to counting pebbles and tokens, tally sticks, the abacus, and the slide rule. But all those methods relied on the human agent to move pebbles or sticks, slide balls across a wire, or manipulate rods, and then read off the results. In the fifteenth century, Leonardo da Vinci dreamed of a completely mechanical calculator, one that would minimize the role of the human operator and thus reduce the possibility of error. By the time of Bacon, in the sixteenth and seventeenth centuries, leading intellects in Europe had seized on this idea. Not only would mechanical calculation increase accuracy, it would free up the mathematician for more lofty operations. Gottfried Leibniz, the philosopher, mathematician, and rival of Newton, remarked that “it is beneath the dignity of excellent men to waste their time in calculation when any peasant could do the work just as accurately with the aid of a machine.”4

  At around the same time, workmen were gaining useful experience constructing mechanical devices to amuse the rich and entertain royalty. Mechanical automata—like Babbage’s “admirable dancer” two centuries later—were all the rage. The Smithsonian Institution in Washington has in its collection an automaton friar that may have been constructed as early as 1560. About fifteen inches tall, the friar, driven by a key-wound spring, walks the path of a square, striking his chest with his right arm, while raising and lowering a small wooden cross and rosary in his left hand, nodding his head, rolling his eyes, and mouthing silent prayers. In 1649 an artisan in France created a magnificent automaton for the young Louis XIV: a miniature coach and horses, with a footman, a page, and a seated lady, all exhibiting perfect motion. (The world would have to wait until 1737 for the first digesting automaton—a mechanical duck that seemed to eat and defecate, created by the French engineer Jacques de Vaucanson.) Such mechanical expertise would soon be harnessed to build calculating machines.

  The first mechanical calculating device known to have been constructed was designed by Wilhelm Schickard (1592–1635) of Wurttemberg, later part of Germany. Schickard was, impressively, Professor of Hebrew, Oriental Languages, Mathematics, Astronomy, and Geography at the University of Tubingen. Schickard was well acquainted with the famous astronomer Johannes Kepler, discoverer of the elliptical shape of planetary orbits, who had come to Tubingen to help defend his mother when she was accused of being a witch. When Kepler left Tubingen after his mother was freed in 1621, the two men kept up a lively and frequent correspondence. In letters to Kepler, Schickard described a calculating machine he had designed and built in 1623 (though no actual machine has ever been found).

  Schickard’s “Calculating Clock,” as it is called, was about the size of an old manual typewriter: twenty-two inches wide, fourteen and a half inches deep, and almost twenty-three inches high.5 It could add and subtract automatically, by the movement of geared wheels at the bottom part of the machine meshed together and linked to a display—much like a car’s odometer today. There were six dials at the bottom of the machine, each connected to a toothed wheel inside the machine. By turning the dials clockwise, the operator could perform addition; subtraction
was done by turning the dials counterclockwise.

  The machine could automatically carry tens during addition, for example when 1 was added to 9 to make 10. Every time a wheel rotated through a complete turn (passing 9), a single tooth would catch in an intermediate wheel, which would cause the next highest wheel to turn, increasing it by one. However, the force used to execute the carry came from the initial power of the first gear meshing with the next ones, so there was a limit to how many digits could be calculated before the necessary force would damage the initial gear; Shickard’s machine was designed with only six digits.6

  Schickard’s machine could not, by itself, perform multiplication and division. The operator of the machine would perform long multiplication and long division using the bottom part of the machine (the adding and subtracting part) in tandem with the top part of the machine, which contained six dials above window openings through which multiplication tables were visible.

  Before the letters discussing Schickard’s machine and the drawings of it came to light in 1957, it had been believed for centuries that the first mechanical calculator ever constructed was that of Blaise Pascal (1623–1662). Pascal is known to many college freshmen today for devising “Pascal’s Wager.” Countering atheism, Pascal argued that even though the existence of God cannot be proved by philosophical argument, it is still the most rational course to act as if (or bet that) God does exist—because if you are right you have everything to gain, and if you are wrong you have little to lose. Pascal turned to philosophy at the end of his life. Before that, he was known as a mathematical prodigy and one of the inventors of probability theory, so it is not surprising that one of his main philosophical tenets was expressed in terms of a gamble.

  When he was nineteen, in 1642, Pascal designed and built a mechanical calculator, the “Pascaline.” His father, Étienne, had been appointed tax commissioner in Rouen, and spent hours tediously calculating and recalculating taxes owed. Eager to help his father, the young Pascal devised a machine that could do the calculations for him.

  The machine was contained in a box about the size of a shoebox. On the top surface of the box was a series of windows, each showing a small drum on which the results digits were engraved. Below these windows were the setting mechanisms, which looked like wheels with spokes radiating out from the center, leading to numbers inscribed around the edges. Pascal’s machine could calculate results of up to eight digits.

  Unlike in Schickard’s machine, the gear wheels inside were able to turn in only one direction, so strictly speaking addition alone was possible, not subtraction. Subtraction was carried out by the method of arithmetical complements, a technique by which the subtraction of one number from another can be performed by the addition of positive numbers.7 Pascal’s machine could multiply and divide, but only by repeated additions and subtractions. For instance, to multiply a number by five, the operator would add the number to itself four times.

  Pascal, like everyone else in his day, was unaware of Schickard’s machine. But he realized that intermeshed toothed gears could not work as the carry mechanism if more than a few digits were involved. Instead, he devised a new mechanism that used the force of falling weights to perform the carry rather than the power from a long chain of geared wheels. A small lever was placed between each gear wheel. The lever was actually a small weight that was lifted up by two pins attached to the wheel as it rotated. When a wheel rotated from 9 to 0, the pins slipped out of the weight, allowing it to fall and, in the process, causing it to interact with the pins sticking out of the next wheel, driving it around one place. When a ripple carry was executed, the mechanism would make a “clunk, clunk, clunk” sound, one “clunk” for each successive carry.8

  Some fifty models of Pascal’s machine were constructed, in wood, ivory, and copper. One was presented to the king. Optimistically, Pascal obtained a “privilege” protection, the equivalent of a patent, on his invention. But only about fifteen were sold, mainly as decorative novelties to wealthy patrons. The machines were too expensive and too delicate to be used widely; the Pascaline was never taken seriously as a practical device. (Perhaps the disappointment was what drove Pascal to philosophy.)9

  The next important development in mechanical calculation came with the machine designed by Leibniz. In the course of his travels through France, Leibniz had heard of Pascal’s invention and its flaws. He decided he would construct a superior calculating machine. In the 1670s, Leibniz invented a mechanical calculator that could add and subtract, as well as carry out multiplication and division automatically, unlike in Schickard’s machine, and not just by repeated addition and repeated subtraction, as in Pascal’s machine.

  To carry out these operations, Leibniz designed a special sort of stepped drum gear, a cylinder in which gearing teeth were set at varying lengths along the cylinder: there were nine rows in total, the row corresponding to the digit 1 running one-tenth of the length of the cylinder, the row of the digit 2 running two-tenths of the length of the cylinder, and so on to 9. Because of these new kinds of drums, the machine is known as the “Stepped Reckoner.”

  The machine was twenty-six and a half inches long, ten and a half inches wide, and seven inches high, housed in an oak case. Inside were two rows of the stepped drums, one in the eight-digit setting mechanism or input section, and the other in the sixteen-digit calculating mechanism or accumulator. (Leibniz’s calculator could handle results up to sixteen digits long.) For addition, the crank handle on the side of the machine would be turned in the clockwise position after the number was dialed in. For subtraction, the crank handle would be turned in the opposite direction. The Stepped Reckoner had a special multiplier-setting disk and handle crank in the center of the machine, used for performing multiplication and division.

  Leibniz arranged for a French clockmaker named Olivier to construct the calculator for him. This machine worked, but it could not ripple a carry across several digits: while it could move from “09” to “10,” it could not move from “999” to “1000” without the intervention of the operator.10 Whenever a carry was pending, a pentagonal disk corresponding to that unit would have one of its points protruding from the top of the machine; when no carry was needed, a flat surface would be flush with the top of the machine. When the operator saw a point projecting from the top of the machine, he would have to reach over and give the pentagonal disk a push to cause the carry to be registered on the next digit.

  Leibniz devoted many years and an incredibly large sum of his own money to this endeavor. He recognized that the machine did not have much commercial potential; he wrote to the Dutch-Swiss mathematician Daniel Bernoulli that his invention “has not been made for those who sell vegetables or little fish, but for observatories or halls of computers, or others who can bear the cost easily and need to undertake many calculations.”11 Leibniz tried to persuade the Russian tsar Peter the Great to send a copy of the machine to China, to impress the Chinese emperor with the value of east-west trade, but this suggestion was ignored.

  Mechanical calculating devices were beautiful but basically ineffectual toys in the first two centuries of their history. Work continued until the nineteenth century, when the first commercially successful calculating machine was developed. It would be closely followed by Babbage’s much more remarkable invention, one that had the potential to alter science and everyday life forever.

  ON NOVEMBER 18, 1820, Charles Xavier Thomas de Colmar, a French insurance executive, was awarded a patent for his new “arithmometer.” His first machine took up an entire tabletop. It was similar to Leibniz’s Stepped Reckoner, with the same drum mechanism, and could add, subtract, multiply, and divide. Subtraction was performed using the method of complements, as in Pascal’s machine. Colmar’s earlier machines had a ribbon drive: to perform the calculation after setting the initial figures, the operator would pull on a ribbon to turn the drums. In Colmar’s later model the ribbon drive was replaced with a sturdier hand crank, and a new mechanism was incorporated allowing
subtraction to be performed without using the method of complements, but simply by turning the crank in the opposite direction.12

  After Colmar first introduced his machine, he did not work on it or do much to promote its commercial use until the 1840s. By then he had developed his later model, and this was produced in large numbers and sold all around the world. Eventually it became the progenitor of a long line of calculating machines, culminating in the small pocket calculators in use today.

  During Babbage and Herschel’s trip to Paris in the summer of 1821, when they met with the most important French mathematicians and savants, they probably heard talk of this new invention, considered at the time a scientific and technical wonder. In December of 1821—the very month that Babbage claimed to have worked out the idea for an “arithmetical engine”—a notice of the arithmometer appeared in the Monthly Magazine, or British Register. The popular publication announced that “by [this device] a person unacquainted with figures may be made to perform, with wonderful promptitude, all the rules of arithmetic. The most complicated calculations are done as readily and exactly as the most simple.… It will be very useful in the higher departments of science, and has long been a desideratum.”13 This description might have inspired Babbage to think of an arithmetical engine when he and Herschel were confronted by so many errors in the astronomical charts. But as soon as the idea occurred to him, Babbage had something far more ambitious in mind.