The Philosophical Breakfast Club (14 page)

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.
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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.
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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.)
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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.
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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.”
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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.

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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.
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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.”
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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.

Babbage knew that a machine that could merely add, subtract, multiply, and divide was not of the utmost importance for scientific or commercial endeavors. What was really needed was a machine that could compute—and do so accurately—the kinds of tables used not only by men of science, but also by workers in nearly every field, from captains of ships to “captains of industry,” as Thomas Carlyle would soon scornfully call powerful businessmen.

Numerical tables were hugely important in the days before electronic computers. Such tables were used to look up figures that otherwise would require onerous calculations each and every time. There were tables of actuarial statistics for insurance agents; tables of astronomical data for astronomers and navigators; tables of taxation rates for excise officers; tables of compound interest rates and investment returns for bankers, investors, moneylenders, and clerks; tables with figures relating to strength of materials and distances for engineers, architects, surveyors, and builders; tables with figures relating to mapping a spherical earth on a flat surface for cartographers. There were tables of logarithms, tables of multiplication, tables of multiples of fractions, tables of conversions of units, and others. Such tables saved inordinate amounts of time, and therefore money. And, if calculated correctly, such tables could save lives.

For example, the computers of the
Nautical Almanac
provided sailors with tables containing precalculated predictions of lunar distances for every three hours of every day of the year. Lunar distances were used in determining a ship’s longitude, which allowed navigators to plot the fastest, safest course to the ship’s destination. The problem of determining longitude at sea had plagued explorers and navigators for centuries. Latitude, or north-south position, was fairly easy to determine from the positions of stars in the sky. But figuring out longitude—east-west position—was a more vexing quandary. It was known that the earth rotates at a rate of 15 degrees per hour. So if the local time could be compared to the time at another fixed point, then the difference between the two could be used to calculate longitude (with each hour of difference corresponding to 15 degrees of longitude difference). But finding the time at another fixed point presented a challenge.

If a clock set to Greenwich Time could be brought on board at the start of the voyage, and could continue to run accurately, the ship’s captain would be able to keep track of Greenwich Time and use that for his comparison with local time. However, clocks were run by pendulums, which were thrown off by the roiling of the sea, and which rusted easily in the salty air. In the mid-1700s, John Harrison designed a clock that could keep time on a long sea voyage, an invention that seemed to solve the longitude problem. But the Harrison chronometer was incredibly expensive to produce, and it was not until 1840 or later that most British ships carried one. Until then, the older method of lunar distances was routinely used to calculate the difference between Greenwich Time and local time.
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A navigator would use a sextant to measure the angle between the moon and a star (this was taking a lunar distance). He would consult the tables in the
Nautical Almanac
, which would give him the distances between the moon and nine easily observed stars, and the times at Greenwich at which those distances would occur. By comparing his observation with this table, the navigator would be able to determine Greenwich Time. He would then ascertain local time by using the sextant to observe the altitude of a star. Longitude could then be easily established, by comparing Greenwich Time with local time.

Having the Greenwich Time lunar distances precalculated in the tables saved even the most experienced sailors from having to spend over four hours making the calculations at sea in order to determine
longitude, and decreased the chances of error. But the figures printed in the tables required extremely difficult calculations, which were themselves subject to error. Each month of the year required 1,365 calculations using logarithms applied to sexagesimal numbers, that is, numbers in base-60 (celestial distances are calculated in degrees, minutes, and seconds, where there are sixty seconds of arc in each minute, sixty minutes of arc in each degree, and 360 degrees in the celestial sphere). Although the results were checked, the printed tables in the
Almanac
still inevitably contained many mistakes.
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Such errors could—and sometimes did—lead ships to lose their way, even to be wrecked at sea. Babbage knew this well: his friend Whewell had survived a shipwreck in 1819, while trying to go from Brighton to Calais (luckily, another ship was nearby that safely delivered all the passengers back to Brighton). Later, Herschel would prey on the public’s fear of shipwrecks to prod the government to fund Babbage’s invention: “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,” he warned.
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But it wasn’t only the specter of error on the high seas that inspired Babbage to create his table-calculating machine. It was also hearing about the great French table-making project, the eighteen-volume
Tables du Cadastre
(the tables for the French Ordnance Survey), which had been supervised in the 1790s by the mathematician and civil engineer Gaspard-Clair-François-Marie Riche, Baron de Prony (1755–1839). De Prony had been commissioned to produce a definitive set of logarithmic and trigonometric tables for the newly introduced metric system in France, to facilitate the accurate measurement of property as a basis for taxation.

De Prony had recently read Adam Smith’s
Wealth of Nations
, originally published in 1776. In his book, Smith discussed the importance of a division of labor in the manufacture of pins. It made no sense, Smith cautioned, to have a man who was talented enough to temper iron also turn the grinding wheel, which could be done by an unskilled boy. It was a waste not only of talent, but of money, as the skilled man needed to be paid more per day than the unskilled boy. Smith’s analysis was taken up in Britain, leading to the establishment of the factory system of manufacturing there; instead of having finished products made, one at a time, by workers who made each one from start to finish, manufacturers
began to divide up the labor into parts in something like a modern-day assembly line.