The Abacus

Mathematical concepts and their offspring, arithmetical operations, were considered for thousands of years a pure intellectual exercise which could not be duplicated or performed by a man-made artifact. Even the Abacus, which appeared in Asia Minor 2500 years ago and is still in use today, is only a memory-helping device rather than a real calculating machine.

The Abacus is an ingenious counting device based on the relative positions of two sets of beads moving on parallel strings. The first set contains five beads on each string and allows counting from 1 to 5, while the second set has only two beads per string representing the numbers 5 and 10. The Abacus system seems to be based on a radix of five. Using a radix of five makes sense since humans started counting objects on their fingers.

Sometime, between 100 BC and 65 BC, a Greek ship carrying a load of bronze and marble statues and other artifacts from Rhodes to Rome sunk close to the coast of Antikythera, a small island of Greece in the Aegean Sea located between Crete and Kythera.  It remained in the bottom of the sea under 140 ft of water for two millenniums, until discovered in 1901 by native sponge divers.

The remains, maintained at the National Museum in Athens, include an ancient geared mechanism, now known as the Antikythera Calculator.

This interesting device, made up of 32 gear-wheels and measuring about 6.5 inches, resembles the mechanism of an 18th Century clock, and was used to calculate the movements of the Sun and the Moon.

Another interesting invention is Napier’s bones, a clever multiplication tool invented in 1617 by mathematician John Napier (1550-1617), of Scotland.

The bones are a set of vertical rectangular rods, each one divided in 10 squares. The top square contains a digit and the remaining squares contain the first 9 multiples of the digit. Each multiple has its digits separated by a diagonal line. When a number is constructed by arranging side by side the rods with the corresponding digits on the top, then its multiple can be easily obtained by reading the corresponding row of multiples from left to right while adding the digits found in the parallelograms formed by the diagonal lines. No wonder John Napier is also the inventor of the logarithms, a concept used to change multiplication into addition.

Napier's bones were very successful and were widely used in Europe until mid 1960's.

Logarithms were also the basis for the invention of the slide rule by William Oughtred (1574-1660), of England, in 1633.

Nature has countless examples of mechanical solutions to practical problems, so it comes as no surprise that the first attempt to design a calculating machine was probably made by the master of machine artifacts, Leonardo da Vinci (1452-1519).

Leonardo da Vinci got many of his ingenious ideas from a careful observation of the mechanics involved in the movements of living organisms. Interestingly enough, nature did not evolve the wheel as a solution to this problem; that solution was left to human ingenuity. It is interesting to note that the wheel has been the base for most of the mechanical devices used to replicate the thought process involved in arithmetic operations. As George Chase [1] said, "The history of mechanical computing machinery in its essence is the story of the numeral wheel and the devices that rotate it to register digital and tens-carry values."

The first calculating machines were built by gifted mathematicians moved by an intense desire to simplify the repetitive nature of arithmetical operations.

The first known adding machine was made by Wilhelm Schickard (1592-1635). In 1623, Schickard, a polymath and then professor at the University of Tübingen in Wuerttemberg, now part of Germany, designed and constructed a mechanical device which he called the Calculating Clock. Able to add and subtract up to six-digit numbers, the artifact was based on the movement of six dented wheels geared through a "mutilated" wheel which with every full turn allowed the wheel located at the right to rotate 1/10th of a full turn. An overflow mechanism rang a bell. The adding feature was devised to help performing multiplication with a set of Napier's cylinders included in the upper half of the machine. According to his notes, a prototype of this machine was destroyed by a fire. It seems that another prototype existed at the time but it has never been found.

A friend of the great astronomer Johannes Kepler (1571-1630), Schickard sent him several letters in 1623 and 1624 briefly describing his invention. Schickard and his family did not survive the bubonic plague and his detailed notes remained unknown until discovered in 1935 and 1956 by historian Franz Hammer. Mathematician Bruno Von Freytag from the University of Tübingen used them to reconstruct the machine in 1960. One unit is in the Deutsches Museum in München.

Blaise Pascal (1623-1662) was only 18 years old when he conceived the Pascaline in 1642. A precocious French mathematician and philosopher, Pascal discovered at the age of 12 that the sum of the angles in a triangle is always 180 degrees. Later on, he set the basis for the probability theory and made significant contributions to the science of hydraulics. The Pascaline, built in 1643, was possibly the first mechanical adding device actually used for a practical purpose. It was built by Pascal to help his father, Etienne Pascal, a tax collector, with the tedious activity of adding and subtracting large sequences of numbers. However the machine was difficult to use and probably not very useful because of the French currency system which was not base 10. A livre had 20 sols and a sol had 12 deniers.

Pascal was not aware of Schickard’s machine, and his solution was not as elegant and efficient. As Paul E. Dune said "had Schickard’s ideas found a wide audience then Pascal’s machine would not have been invented."

It was built on a brass rectangular box, where a set of notched dials moved internal wheels in a way that a full rotation of a wheel caused the wheel at the left to advance one 10^th. Although the first prototype contained only 5 wheels, later units were built with 6 and 8 wheels. A pin was used to rotate the dials. As opposed to Schickard’s machine, the wheels moved only clockwise and were designed only to add numbers. Subtraction was done by applying a cumbersome technique based on the addition of the nine’s complement.

Although the machine attracted a lot of attention in those days, it did not get wide acceptance because it was expensive, unreliable as well as difficult to use and manufacture. By 1652 about 50 units had been made but less than 15 had been sold. Initially, Pascal got a lot of interest in his invention and he even obtained a "privilege" protection (medieval equivalent of a patent) for his idea in 1649, but his interest in science and "material" pursuits ended when he retreated to a Jansensist convent in 1655 concentrating all his attention on philosophy. He died in 1662.

During a period of 30 years after Pascal's invention, several persons built calculating machines based on this design. The most notorious was the adding machine of Sir Samuel Morland (1625-1695), from England. This machine invented in 1666 had a duodecimal scale based on the English currency, and required human intervention to enter the carry displayed in an auxiliary dial.

It is interesting to note that even at the beginning of the 20th Century, several companies introduced models based directly on Pascal's design. One example is the Lightning Portable Adder introduced in 1908 by the Lightning Adding Machine Co. of Los Angeles. Another example is the Addometer introduced in 1920 by the Reliable Typewriter and Adding Machine Co. of Chicago. None of them achieved commercial success.

It was 1672 when the famous German polymath, mathematician and philosopher, Gottfried Wilhelm Von Leibniz (1646-1716), co-inventor of the differential calculus, decided to build a machine able to perform the four basic arithmetical operations. He was inspired by a steps-counting device (pedometer) he saw while on a diplomatic mission in Paris.

Like Pascal, Leibniz was a child prodigy. He learned Latin by the age of 8 and got his second doctorate when he was 19. As soon as he knew about Pascal’s design, he absorbed all its details and improved the design so as to allow for multiplication and division. By 1674 his design was complete and he commissioned the building of a prototype to a craftsman from Paris named Olivier.

The Stepped Reckoner, as Leibniz called his machine, used a special type of gear named Stepped Drum or Leibniz Wheel which was a cylinder with nine bar-shaped teeth of incrementing length parallel to the cylinder’s axis. When the drum is rotated by using a crank, a regular ten-tooth wheel, fixed over a sliding axis, is rotated zero to nine positions depending on its relative position to the drum. As in the Pascal device, there is one set of wheels for each digit. This allows the user to slide the mobile axis so that when the drum is rotated it generates in the regular wheels a movement proportional to their relative position. This movement is then translated by the device into multiplication or division depending on which direction the stepped drum is rotated.

There is no evidence that more than two prototypes of this machine were ever made. Even though Leibniz was one of the most outstanding polymaths of his time, he died in poverty and unrewarded. His machine remained in the attic of the University of Göttingen until a worker found it in 1879 while fixing a leak in the roof. Now it is in the State Museum of Hanover; another one is in the Deutsches Museum in München.

Pascal's and Leibniz’s designs were the basis for most of the mechanical calculators built during the 18^th Century. Giovanni Poleni made one in 1709, Lépine in 1725, Antonius Braun in 1725, Jacob Leupold in 1727, Hillerin de Boistissandau in 1730, C.L. Gersten in 1735, Jacob Isaac Pereire in 1750, Phillip Mathieus Hahn in Germany in 1773. Charles, the third Earl Stanhope, of England, in 1775; Johan Helfreich Müller in 1783, Jacob Auch in 1790, and Reichhold in 1792. [4].

Special consideration deserves the Parson Phillip Mathieus Hahn (1730-1790) who developed in 1773 the first functional calculator based on Leibniz's Stepped Drum. Hahn's calculator had a set of 12 drums in a circular arrangement actuated by a crank located in the axis of the arrangement. Hahn made these machines until his death in 1790, however, his two sons and his brother-in-law, Johann Christopher Schuster, continued with the manufacture probably as late as 1820.

By the end of the 18th Century, calculating machines were still curiosities used for display purposes, rather than for actual use. The limitations imposed by the technology made it impossible to meet Pascal's dream of making them a practical calculation device.