Abacus
An abacus, also called a counting frame, is a hand-operated calculating tool which was used from ancient times, in the ancient Near East, Europe, China, and Russia, until largely replaced by handheld electronic calculators, during the 1980s, with some ongoing attempts to revive their use. An abacus consists of a two-dimensional array of slidable beads. In their earliest designs, the beads could be loose on a flat surface or sliding in grooves. Later the beads were made to slide on rods and built into a frame, allowing faster manipulation.
Each rod typically represents one digit of a multi-digit number laid out using a positional numeral system such as base ten. Roman and East Asian abacuses use a system resembling bi-quinary coded decimal, with a top deck representing fives and a bottom deck representing ones. Natural numbers are normally used, but some allow simple fractional components, and a decimal point can be imagined for fixed-point arithmetic.
Any particular abacus design supports multiple methods to perform calculations, including addition, subtraction, multiplication, division, and square and cube roots. The beads are first arranged to represent a number, then are manipulated to perform a mathematical operation with another number, and their final position can be read as the result.
In the ancient world, abacuses were a practical calculating tool. It was widely used in Europe as late as the 17th century, but fell out of use with the rise of decimal notation and algorismic methods. Although calculators and computers are commonly used today instead of abacuses, abacuses remain in everyday use in some countries. The abacus has an advantage of not requiring a writing implement and paper or an electric power source. Merchants, traders, and clerks in some parts of Eastern Europe, Russia, China, and Africa use abacuses. The abacus remains in common use as a scoring system in non-electronic table games. Others may use an abacus due to visual impairment that prevents the use of a calculator. The abacus is still used to teach the fundamentals of mathematics to children in many countries such as Japan and China.
Etymology
The word abacus dates to at least 1387 AD when a Middle English work borrowed the word from Latin that described a sandboard abacus. The Latin word is derived from ancient Greek ἄβαξ which means something without a base, and colloquially, any piece of rectangular material. Alternatively, without reference to ancient texts on etymology, it has been suggested that it means "a square tablet strewn with dust", or "drawing-board covered with dust ". While the table strewn with dust definition is popular, some argue evidence is insufficient for that conclusion. Greek ἄβαξ probably borrowed from a Northwest Semitic language like Phoenician, evidenced by a cognate with the Hebrew word ʾābāq, or "dust".Both abacuses and abaci are used as plurals. The user of an abacus is called an abacist.
History
Ancient Near East
Mesopotamia
The Sumerian abacus appeared between 2700 and 2300 BC. It held a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system. Primitive forms of the abacus existed in Sumeria, such as the non-fixed bead abacus, a line of strings fixed to a board, counting strings consisting of beads, ropes and knots, and counting boards. The exact form and function of these devices are scattered around different sources, and their exact relation to the modern abacus is unknown but it is almost certain there is a relationship with masihatu, which functioned similar to an abacus utilizing multiple strings and sets of beads.Some scholars point to a character in Babylonian cuneiform that may have been derived from a representation of the abacus. It is the belief of Old Babylonian scholars, such as Ettore Carruccio, that Old Babylonians "seem to have used the abacus for the operations of addition and subtraction; however, this primitive device proved difficult to use for more complex calculations".
Egypt
Greek historian Herodotus mentioned the abacus in Ancient Egypt. He wrote that the Egyptians manipulated the pebbles from right to left, opposite in direction to the Greek left-to-right method. Archaeologists have found ancient disks of various sizes that are thought to have been used as counters. However, there are no known illustrations of this device.Persia
At around 600 BC, Persians first began to use the abacus, during the Achaemenid Empire. Under the Parthian, Sassanian, and Iranian empires, scholars concentrated on exchanging knowledge and inventions with the countries around them – India, China, and the Roman Empire – which is how the abacus may have been exported to other countries.Europe
Greece
The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC. Demosthenes complained that the need to use pebbles for calculations was too difficult. A play by Alexis from the 4th century BC mentions an abacus and pebbles for accounting, and both Diogenes and Polybius use the abacus as a metaphor for human behavior, stating "that men that sometimes stood for more and sometimes for less" like the pebbles on an abacus. The Greek abacus was a table of wood or marble, pre-set with small counters in wood or metal for mathematical calculations. This Greek abacus was used in Achaemenid Persia, the Etruscan civilization, Ancient Rome, and the Western Christian world until the French Revolution.The Salamis Tablet, found on the Greek island Salamis in 1846 AD, dates to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble in length, wide, and thick, on which are 5 groups of markings. In the tablet's center is a set of 5 parallel lines equally divided by a vertical line, capped with a semicircle at the intersection of the bottom-most horizontal line and the single vertical line. Below these lines is a wide space with a horizontal crack dividing it. Below this crack is another group of eleven parallel lines, again divided into two sections by a line perpendicular to them, but with the semicircle at the top of the intersection; the third, sixth and ninth of these lines are marked with a cross where they intersect with the vertical line. Also from this time frame, the Darius Vase was unearthed in 1851. It was covered with pictures, including a "treasurer" holding a wax tablet in one hand while manipulating counters on a table with the other.
Rome
The normal method of calculation in ancient Rome, as in Greece, was by moving counters on a smooth table. Originally pebbles were used. Marked lines indicated units, fives, tens, etc. as in the Roman numeral system.Writing in the 1st century BC, Horace refers to the wax abacus, a board covered with a thin layer of black wax on which columns and figures were inscribed using a stylus.
One example of archaeological evidence of the Roman abacus, shown nearby in reconstruction, dates to the 1st century AD. It has eight long grooves containing up to five beads in each and eight shorter grooves having either one or no beads in each. The groove marked I indicates units, X tens, and so on up to millions. The beads in the shorter grooves denote fives resembling a bi-quinary coded decimal system related to the Roman numerals. The short grooves on the right may have been used for marking Roman "ounces".
Medieval Europe
The Roman system of 'counter casting' was used widely in medieval Europe, and persisted in limited use into the nineteenth century. Wealthy abacists used decorative minted counters, called jetons.Due to Pope Sylvester II's reintroduction of the abacus with modifications, it became widely used in Europe again during the 11th century It used beads on wires, unlike the traditional Roman counting boards, which meant the abacus could be used much faster and was more easily moved.
Russia
The Russian abacus, the schoty, usually has a single slanted deck, with ten beads on each wire. 4-bead wire was introduced for quarter-kopeks, which were minted until 1916. The Russian abacus is used vertically, with each wire running horizontally. The wires are usually bowed upward in the center, to keep the beads pinned to either side. It is cleared when all the beads are moved to the right. During manipulation, beads are moved to the left. For easy viewing, the middle 2 beads on each wire usually are of a different color from the other eight. Likewise, the left bead of the thousands wire may have a different color.The Russian abacus was in use in shops and markets throughout the former Soviet Union, and its usage was taught in most schools until the 1990s. Even the 1874 invention of mechanical calculator, Odhner arithmometer, had not replaced them in Russia. According to Yakov Perelman, some businessmen attempting to import calculators into the Russian Empire were known to leave in despair after watching a skilled abacus operator. Likewise, the mass production of Felix arithmometers since 1924 did not significantly reduce abacus use in the Soviet Union. The Russian abacus began to lose popularity only after the mass production of domestic microcalculators in 1974.
The Russian abacus was brought to France around 1820 by mathematician Jean-Victor Poncelet, who had served in Napoleon's army and had been a prisoner of war in Russia. To Poncelet's French contemporaries, it was something new. Poncelet used it, not for any applied purpose, but as a teaching and demonstration aid. The Turks and the Armenian people used abacuses similar to the Russian schoty. It was named a coulba by the Turks and a choreb by the Armenians.
East Asia
China
The earliest known written documentation of the Chinese abacus dates to the 2nd century BC.The Chinese abacus, also known as the suanpan, comes in various lengths and widths, depending on the operator. It usually has more than seven rods. There are two beads on each rod in the upper deck and five beads each in the bottom one, to represent numbers in a bi-quinary coded decimal-like system. The beads are usually rounded and made of hardwood. The beads are counted by moving them up or down towards the beam; beads moved toward the beam are counted, while those moved away from it are not. One of the top beads is 5, while one of the bottom beads is 1. Each rod has a number under it, showing the place value. The suanpan can be reset to the starting position instantly by a quick movement along the horizontal axis to spin all the beads away from the horizontal beam at the center.
The prototype of the Chinese abacus appeared during the Han dynasty, and the beads are oval. The Song dynasty and earlier used the 1:4 type or four-beads abacus similar to the modern abacus including the shape of the beads commonly known as Japanese-style abacus.
In the early Ming dynasty, the abacus began to appear in a 1:5 ratio. The upper deck had one bead and the bottom had five beads. In the late Ming dynasty, the abacus styles appeared in a 2:5 ratio. The upper deck had two beads, and the bottom had five.
Various calculation techniques were devised for Suanpan enabling efficient calculations. Some schools teach students how to use it.
In the long scroll Along the River During the Qingming Festival painted by Zhang Zeduan during the Song dynasty, a suanpan is clearly visible beside an account book and doctor's prescriptions on the counter of an apothecary's.
The similarity of the Roman abacus to the Chinese one suggests that one could have inspired the other, given evidence of a trade relationship between the Roman Empire and China. However, no direct connection has been demonstrated, and the similarity of the abacuses may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2. Incidentally, this ancient Chinese calculation system 市用制 allows use with a hexadecimal numeral system which is used for traditional Chinese measures of weight ..
Another possible source of the suanpan is Chinese counting rods, which operated with a decimal system but lacked the concept of zero as a placeholder. The zero was probably introduced to the Chinese in the Tang dynasty when travel in the Indian Ocean and the Middle East would have provided direct contact with India, allowing them to acquire the concept of zero and the decimal point from Indian merchants and mathematicians.