Mathematical Devices, Early

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Mathematical Devices, Early

Early humans counted and performed simple calculations using tools such as their fingers, notches in sticks, knotted strings, and pebbles. Most early cultures evolved some form of a counting board or abacus to perform calculations. Pencil and paper eventually replaced these early counting boards, but a modern form of the abacus may still be seen in use in parts of Russia and Asia in the twenty-first century.

Counting Boards

Ancient cultures such as the Greeks, Babylonians, and Romans marked parallel lines on a table and placed pebbles on the lines for counting. In the Western hemisphere, the Mayans, Aztecs, and Incas used kernels of grain as counters. The parallel lines represented numbers, and pebbles or other counters placed on the lines denoted multiples of that number. Since the value assigned to a counter depended on the line on which it was placed, these early counting devices used a place value system. Some of the cultures that used these place value devices for computations then recorded the results of these calculations using a number system that did not use place value, such as Roman numerals.

The pebbles the ancient Romans used for their counting boards were called calculii. Our modern words "calculate" and "calculus" come from this root word.

Very few counting tables still exist. However, they must have been common because they are often mentioned in wills and inventories. In the fifth century b.c.e., the Greek historian Herodotus (c. 485 b.c.e.–425 b.c.e.) described counting tables that used pebbles and wrote examples of the calculations for which they could be used. One such calculation was computing the interest due on a loan.

One of the few counting tables still in existence was found on the Greek island of Salamis. It is now in two pieces, but it was once a very large marble slab, approximately 5 feet by 2 1/2 feet. The table is marked with 11 vertical lines, a blank space between them, and horizontal lines crossing the vertical ones. Greek symbols appear along the top and bottom of the tablet. No one is certain what it was used for, but it could have been used for addition, subtraction, multiplication, and division.

To add numbers using a counting board or table, counters would be placed on the appropriate lines to denote the first number to be added. Additional counters were placed on appropriate lines to make up subsequent numbers to be added. If there were numbers to be carried, counters were removed from one line and an additional counter was placed on the line to represent the next higher number. At the end of the operations, the total value of the counters on the table indicated the sum. For subtraction, counters would be taken away, with any borrowing done manually. Since negative numbers were not used at the time, smaller numbers would be subtracted from larger ones.

By the thirteenth century a standard form of the counting table was prevalent in Europe. It was a table upon which lines were drawn to represent the place value of the counters to be put on the lines. The bottom line was the units place and each subsequent line represented ten times the value of the line below it. These lines formed a base-10 system . Each space between two lines represented numbers having five times the value of the line below the space. As soon as five counters appeared on a line or space, they were removed and replaced by one counter on the next higher space or line.

Early counters were usually pebbles, but by the thirteenth century in Europe, counters resembled coins. These later counters came to be called jetons from the French verb jeter, meaning "to throw." They were quite common and at one time manufacturing jetons was a major industry in Europe.

The Abacus

The more modern wire and bead abacus began in the Middle East during the early Middle Ages, c. 500 c.e.–1000 c.e. The abacus is believed to have spread from Europe along trade routes to the east. It was first adopted by the merchants in each society because they had to perform many calculations in their daily business activities.

The Greeks used the word "abax" to denote the surface on which they placed their counting lines. This may have come from the Semitic word abaq, meaning dust. This term spread to Rome where counting boards were called abaci. The abacus was called a choreb by the Turks and a stchoty by the Russians. As the abacus was used in more societies, its form changed, but the principles of computation remained the same.

By 1300 a device resembling the modern abacus was in common use in China. It consisted of a rectangular wooden frame with a bar running down its length dividing the abacus into two parts. The upper part, smaller than the lower, was sometimes referred to as "heaven," and the lower part as "Earth." Dowels were placed through the dividing bar, perpendicular to it.

The "heaven" part of each dowel was strung with 2 beads, each representing 5 times the place value of the number corresponding to the dowel. The "Earth" part of each dowel contained 5 beads, each representing the place value of the corresponding number. The initial position of each bead was touching either the outer frame or a bead touching the outer frame. The beads were used for counting or computing by touching them either to the bar or to a bead touching the bar. Any number from 0 to 15 could be represented by the beads on one dowel, although numbers greater than 9 would be carried to the next higher dowel. The Chinese called this device suan pan, or counting table.

Around 1500, the wire and bead abacus spread from China to Japan, where it was called the soroban. The modern soroban has only one bead in heaven and four in Earth, so 9 is the highest number that can be represented on a dowel.

The abacus does not multiply and divide as efficiently as it adds and subtracts. Multiplying when one of the factors is a small number can be done by repeated addition. It is a little trickier to multiply by larger numbers. For example, to multiply 141 by 36, first multiply 141 by 3 by adding 141 three times. Then multiply that result by 10. In a base-10 system, this would involve shifting each digit to the left to add a zero to the end. Then multiply 141 by 6, by adding 141 six times, and then add the two results together to get the product of 141 and 36. The world would have to wait for a further mathematical development in order to have tools that could multiply, divide, raise numbers to powers, and extract square roots.

Napier's Bones

The Scottish mathematician John Napier (1550 c.e.–1617 c.e.) wanted to simplify the work involved in calculations. He accomplished this by inventing a calculating device, "Napier's bones," so-called because the better quality instruments were made of bone or ivory. Napier's bones consisted of flat rods with a number 0 through 9 at the top of each. Underneath the top number on each bone are nine squares, each divided in half by a diagonal from upper right to lower left. The first square contains the product of the number and 1, the second the product of the number and 2, and so on. The tens place is in the upper half, the units place in the lower half.

To multiply a multi-digit number by a single digit, the rods corresponding to the larger number are placed side by side. The solution is found in the row corresponding to the multiplier. The rightmost digit of the product is in the lower half square of the rightmost rod. The next digit is the sum of the number in the upper half of the rightmost rod and the lower half of the rod to its left, and so on. If a sum is more than 9, it is carried to the next higher digit; hence, the person using the rods must keep track of the numbers to be carried.

Napier published a description of his invention in 1617, the year of his death. The bones became used widely in Europe and spread to China. Several improvements were made to Napier's bones over the years. One was the Genaille-Lucas ruler, which was similar to Napier's bones but designed to eliminate the need to carry from one digit to another. Napier's bones, and other related devices, could also be used for division and extracting square and cube roots. Napier's bones were used to build the first workable mechanical adding machine in 1623.

Logarithms and the Slide Rule

Another of Napier's inventions—logarithms—had a more lasting effect on simplifying calculations than his mechanical multiplier.* Logarithms are exponents, the power to which a number, such as 10 (called the base), is raised to yield a given number. Since exponents are added when two powers are multiplied together, the logarithm of a product is the sum of the logarithms of the factors. Likewise, when one power is divided by another, the exponent of the divisor is subtracted from the exponent of the dividend.

*John Napier was the first to use and then popularize the decimal point to separate the whole number part from the fractional part of a number.

Calculations using logarithms involve adding and subtracting instead of multiplying and dividing. Logarithms can also be used to raise numbers to powers or extract roots by multiplying and dividing. Using logarithms replaces more complicated computations with simpler ones.

After Napier devised his system of logarithms, English mathematician Henry Briggs (1561–1631) developed extensive tables of logarithms. Within a few decades scientists and mathematicians throughout the world were using logarithms for their calculations. Anyone using logarithms for computations had to use the tables to look up the logarithm of each number in the calculation. This could be a tedious task (though not as tedious as calculations without logarithms), and the tables contained errors.

But logarithms had yet another contribution to make. An English astronomy and mathematics teacher named Edmund Gunter (1581–1626) plotted logarithms of numbers on a line (called Gunter's Line of Numbers) and multiplied and divided numbers by adding and subtracting lengths on the line.

An English clergyman named William Oughtred (1574–1660) refined Gunter's line by using two pieces of wood that slid against each other. Each piece of wood contained a scale in which the distance of a number from the end of its line is proportional to its logarithm. To multiply two numbers, one of the numbers is lined up with 1 and the product appears opposite the other number. Division reverses the process. Unfortunately, the slide rule was not accurate to many decimal places. It also required the user to keep track of where the decimal point belonged.

Despite its drawbacks, the slide rule was enormously successful. It eliminated the need for using tables of logarithms. The slide rule was used by scientists and mathematicians, as well as students, for over 300 years until it was replaced by the electronic hand-held calculator. Human computing ability has come a long way from sticks and pebbles.

see also Abacus; Bases; Logarithms; Mathematical Devices, Mechanical; Slide Rule.

Loretta Anne Kelley