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Mathematical Devices, Mechanical

Mathematical Devices, Mechanical

The earliest mechanical mathematical devices were crude, slow, and subject to errors. Users had to adjust the device manually for some functions, such as carrying in addition, just as did the counting table and abacus. The ability to build faster and more accurate devices was constrained by the limited technology of the time.

Schickards's Mechanical Calculator

For many years the French mathematician and philosopher Blaise Pascal (16231662) was believed to have built the first mechanical calculator. However, research in the 1950s and 1960s revealed that an earlier mechanical calculator was built by Wilhelm Schickard (15921635), a German professor and minister.

Schickard also had an interest in mathematics, and was a friend of Johann Kepler (15711630), the prominent astronomer. By 1623 Schickard had produced the first workable mechanical adding machine capable of carrying and borrowing from one column to the next. The machine consisted of a set of Napier's bones drawn on cylinders so that any bone could be selected by turning a dial.

Horizontal slides exposed different sections of the bones to show single-digit multiples of the selected number. The result would be stored in a set of wheels called the accumulator. Whenever one of the wheels made a complete rotation by passing from 9 to 0, a tooth would increase the next highest digit in the accumulator by one.

In earlier computing devices, such as the abacus, digits were added separately and the user manually carried from one digit to the next higher digit. Carrying from one place to the next without human intervention was the greatest problem facing designers of mechanical digital calculators, but it could be accomplished by having two gears with a gear ratio of 10:1. When the first counter added 9 + 1, the second counter went from 0 to 1, and the first counter went back to 0. The number 10 was the result.

The accumulator in Schickard's machine had only six digits because of difficulties in propagating "carries" to several places, as in adding 99,999 + 1 to produce 100,000. Each subsequent carry put force on each previous gear so that when several carries were propagated, the force on the units gear could damage it.

Schickard wrote letters to Kepler telling him that he had built two copies of the machine, one for himself and one for Kepler. He included detailed drawings of his machine in these letters. The machine Schickard made for Kepler was destroyed by fire. Schickard and his entire family died in a plague and no trace of his machine has ever been found. However, Schickard's machine was reconstructed in 1960 based on the drawings Schickard had sent to Kepler.

Pascal's Calculating Machine

Pascal (16231662) produced the next important calculating machine. The son of a tax collector, Pascal invented the machine to help his father with the large number of calculations his work required. Pascal's machine, called the Pascaline, had a carry mechanism completely different from Schickard's.

In the Pascaline, weights were placed between each two accumulator wheels. When one accumulator wheel moved from 9 to 0, a weight would drop and move the accumulator wheel for the next higher digit by one place. This worked better than Schickard's carry mechanism, but it was very sensitive to external movements and would sometimes generate extra carries when the device was bumped.

The wheels in the Pascaline turned in only one direction and therefore could not subtract directly. In order to subtract, the nines complement of a number was added. For example, to subtract 461 from 1,090, first subtract 461 from 9,999. The result is 9,538. Then add 9,538 and 1,090 giving 10,628. Take the leading 1 and add it to the units place, giving 629. This is the result of the original subtraction.

This formula works because first 9,999 is added, then 10,000 is subtracted, then 1 is added, giving a net change of zero. The disadvantage of this process is that the nines complement had to be computed by the person operating the machine, which was slow and subject to human error. Few people of that time were well educated and hence most machine operators were unable to perform such calculations.

Although the Pascaline was not commercially successful, it was influential in the designs of susequent calculating machines. Pascal produced about fifty different models of his mechanical calculating machines during his life, all based on the same design. Many of them have survived to the twenty-first century. However, they are delicate and produce unreliable results.

Leibniz's Mechanical Multiplier

The Pascaline could multiply only by repeated addition and divide by repeated subtraction. This could be time-consuming if the multiplier and divisor were large numbers. Multiplying numbers directly became possible in about 1671 when the German mathematician (and co-developer of calculus ) Gottfried Leibniz (16461716) invented a mechanical multiplier.

To multiply a number by 3, for example, in Leibniz's multiplier, the number to be multiplied was entered into the machine by setting pointers that controlled the gears. Then a crank was turned three times. The machine had two layers so a multiplier could have two digits. To multiply a number by 73, the crank would be turned three times, then the top layer of the machine was shifted one decimal place to the left and the crank was turned seven more times.

Leibniz's machine included one of his inventions, the stepped drum. This drum had nine cogs on its exterior, each of a different length. The longest cog signified the number 9, and the shortest signified 1. A gear above the drum could change positions along the drum's length and engage a different number of cogs depending on where the drum was located. The gear could represent any number from 1 to 9 on the drum simply by moving.

Leibniz's machine, though ingenious, could not carry from one digit to the next higher digit without the help of the human operator. A pointer would indicate when a carry needed to be made, and the operator of the machine would push the pointer down, which would cause a carry to the next higher digit. If this carry caused another carry, another pointer would go up and the process would be repeated.

Leibniz had high hopes for his machine. He sent a copy of it to Peter the Great of Russia, asking the czar to send it to the emperor of China to encourage the emperor to increase trade with European nations.

Despite its limitations, the stepped-drum design endured for many years. Leibniz's basic idea had been sound, but the engineering and manufacturing technology of his time did not allow him to build accurate, reliable machines. In the years after Leibniz's death, engineering and design practice improved.

Later Calculating Devices

The first commercially successful calculating machine was built around 1820. This machine, called the "arithmometer," was developed by Charles Xavier Thomas of France. Thomas was able to build on Leibniz's innovations and develop an efficient, accurate carry mechanism for his arithmometer. Machines having the same general design as the arithmometer were sold for about 90 years.

The varying lengths of the cogs allowed calculating devices based on the stepped drum to represent any number with a fixed number of cogs (nine). In 1885 an American, Frank Baldwin, and W. T. Odhner, a Swede, simultaneously but independently developed a gear with a variable number of teeth. The gears in this machine consisted of flat disks with movable pins that could be retracted into the disk or made to protrude.

Machines made of variable-toothed gears were reliable, but their main benefit was that the disks were very thin and so they were much smaller than the stepped-drum machines. A machine using stepped drums might be as large as an entire desk surface, whereas the variable-toothed gear machines would fit on the corner of a desk. These smaller machines were much more practical and more widely used.

The machines with variable-toothed gears, however, retained one drawback of the earlier machines. Performing calculations was a two-step process, which made them time-consuming. The operator first had to set levers or gears to enter a number and then pull a lever or crank to make the machine perform the calculation.

Because of the time required to enter and operate on numbers, the calculating machines produced up to this point were better suited for the work of scientists than that of office workers. Scientific calculations often required many operations on a few numbers, whereas business applications often required adding long lists of numbers.

Between 1850 and 1885 a number of patents were granted for devices that greatly simplified the work required to operate them; all were either never constructed or of too limited a capacity to be useful. Then, the year after Baldwin and Odhner built their variable-toothed machines, an American machinist named Dorr E. Felt produced a working calculating machine that had a range of uses.

Felt's machine combined the operations of entering a number and performing the calculation using that number into a stroke of a key. This machine was called a comptometer and required that the operator only push a key corresponding to numbers in order to add them. For example, to add 367, the operator would push the 3 in the hundreds column, the 6 in the tens column, and the 7 in the units column.

Desktop key-driven mechanical calculators remained in common use until electronic hand-held calculators replaced them in the 1970s. Though many improvements were made that would make them cheaper, smaller, and more reliable, the basic design of Felt's machine remained the same.

Babbage's Machine

One of the most influential mechanical mathematical machines was never built. In the late-eighteenth century, mathematical tables (such as trigono-metric and logarithmic) were major tools of mathematicians and scientists. These tables contained many errors in calculation, copying, and printing. In 1827 an eccentric British genius named Charles Babbage (17911871) published the most accurate set of mathematical tables produced up to that time.

Years earlier, while a student of mathematics at Cambridge, Babbage had dreamed of building a machine that would quickly and accurately compute and print mathematical tables. Babbage was able to obtain funding from the British government for his machine, which he called the Difference Engine. This machine would operate on the principle of differences. For example, consider the simple polynomial x 2. The first difference is the difference between the square of a number and the square of the previous number. The second difference is the difference between the first difference of a number and the first difference of the previous number.

x x 2 First difference Second difference
1 1
2 4 3
3 9 5 2
4 16 7 2
5 25 9 2

Note that the second differences of x 2 are all the same. Any second-degree polynomial has constant second differences, third-degree polynomials have constant third differences, and so on. This makes it possible to compute polynomial functions of large numbers simply by knowing the value for smaller numbers and adding to them.

This method of differences was useful for constructing mathematical tables because many functions, such as logarithms and trigonometric functions, do not have constant differences but can be approximated by polynomials that do. Babbage then proposed to build a machine that would use this method to construct accurate mathematical tables.

But financial and design problems plagued Babbage's Difference Engine and it was never builtif it had been constructed, it would have weighed about 2 tons.*

*In 1991 researchers in London's Science Museum completed a working model of Babbage's Difference Engine.

While working on designs for the Difference Engine, Babbage devised the idea of an improved machine, the Analytical Engine, which could compute any mathematical function, even those not based on common differences. This engine was designed with a memory and would have been controlled by a program that was punched into cards. The Analytical Engine was never built, but its concept was the forerunner of twentieth-century computers.

see also Babbage, Charles; Computers, Evolution of Electronic; Mathematical Devices, Early; Pascal, Blaise.

Loretta Anne Kelley


Aspray, William, ed. Computing before Computers. Ames, IA: Iowa State University Press, 1990.

Harmon, Margaret. Stretching Man's Mind: A History of Data Processing. New York: Mason/Charter Publishers, Inc., 1975.

Shurkin, Joel. Engines of the Mind. New York: W. W. Norton & Company, 1996.

Turck, J. A. V. Origin of Modern Calculating Machines. New York: Arno Press, 1972.

Williams, Michael R. A History of Computing Technology, 2nd ed. Los Alamitos, CA: IEEE Computer Society Press, 1997.


In 1853, a Swedish father-andson team, Pehr and Edvard Scheutz, developed a tabulating machine based on Charles Babbage's designs. The Scheutz machine used the method of differences to produce several hundred mathematical tables.

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