The **abacus** is an ancient calculating instrument, used as an aid to perform mathematical operations; it is the first tool used for calculations since the 21st century BC in China and the Fertile Crescent, and later used also among the Greeks and Romans. The term “abacus” is derived from the Latin abacus, via the genitive form ἄβακας of the Greek ἄβαξ, which in turn comes from the Hebrew חשבונייה, “dust.” In fact, the original term referred to the earliest abacuses consisting of a tablet on which to scatter sand dust.

The Roman abacus consisted, in the most perfected form, of a series of rods in which balls or discs were inserted. The rods were arranged on a rectangular frame divided into two sections of different sizes. While the balls sliding on the sticks of the major section always indicated the units of the order corresponding to each stick, those of the minor section indicated a multiple of this unit, usually five. The use of the abacus spread in the West around the year 1000 by Gerbert of Aurillac (later Pope Sylvester II) and was until the seventeenth century the most widely used calculating instrument in commercial activities. The abacus fell into disuse because of the ease of calculation allowed by Arabic numerals. Still used in China and Japan, it survives in the West with a didactic function in the form of an abacus.

## Operation

An abacus, in its most common form, is a rectangular-shaped tablet consisting of a series of parallel guides (wires, grooves, etc.), which conventionally indicate units, tens, hundreds, and so on. Along each guide can be moved stones (called calculations, hence the modern term of mathematical meaning) or other movable objects to perform arithmetic operations. The materials used for the construction of the abacuses and their construction style vary greatly depending on the place and the historical period, but the operation is always based on the fundamental principle that the value of a configuration of calculi depends on the place it occupies, i.e. the guide on which it is placed. According to this principle, the stones on different lines indicate quantities of different order, even fractional. This principle will be the basis of every positional numbering system.

The operations facilitated by the use of the abacus are not only additions and subtractions, but also multiplications and divisions, seen respectively as repeated additions and subtractions. Furthermore, with appropriate physical configurations of the instrument and appropriate techniques, the speed of execution of calculations can be remarkable. However, the abacus cannot be considered a mechanical calculator since it has no mechanisms. The operator must manually perform all operations, nothing is done automatically.

## Types

The typology of the abacus has been enriched over the centuries with various forms. We can distinguish two main classes:

- powder abacus;
- column abacus.

### Powder abacus

The powder abacus is a system for representing mathematical operations in written form and geometrical figures in drawn form. It consists primarily of a rectangular wooden or clay tablet. On one of the two faces of the tablet, sand powder is spread. Then, using the fingers or a chopstick, the powder is moved in such a way as to draw signs representing mathematical operations and geometrical figures. No longer in use today, the powder abacus was used in antiquity by the Phoenicians, Hebrews, Greeks, Etruscans and Romans. Various authors of the past speak of the powder abacus.

### Columnar abacus

The columnar abacus is a system for representing, in visual form, numbers used in very simple mathematical operations such as addition and subtraction. In the columnar abacus, elements of various kinds (pebbles, tokens, rings, etc.) are aligned on a series of parallel columns. Following precise conventions in positioning these elements, numbers are represented. The column abacus is still used today. The following main types of column abacus can be distinguished:

**abacus with lapilli**: it consists primarily of a rectangular tablet which has, on one of the two faces, a series of parallel grooves. Inside the grooves, following precise conventions, are placed lapilli, or pebbles of other minerals, in order to represent numbers. Today no longer used, the lapilli abacus was used in antiquity by the Greeks, Etruscans and Romans. The pebbles were called “psephoi” by the Greeks and “calculi” by the Romans. The lapilli abacus is also a type of counting table;**button abacus**: it is a rectangular bronze tablet that has a series of parallel slots. In the slots are inserted sliders that have a button-shaped end. Acting with the fingers on these ends, the cursors slide and, following precise conventions relating to their position, represent the numbers. No longer used today, the button abacus was used in antiquity by the Greeks and Romans. The button abacus is also a type of counting table;**token abacus**: it is mainly made of a rectangular tablet which has, on one of the two faces, a series of parallel grooves. Inside the grooves, following precise conventions, are placed tokens in order to represent numbers. The tokens are small discoidal objects which have numbers on their faces. Became famous in the eleventh century the abacus of Gerbert of Aurillac (later known as Pope Sylvester II) on 27 columns and a thousand tokens with the natural numbers from 1 to 9 (missing the symbol of 0 replaced by a blank space) that allowed to perform more quickly mathematical operations. The coin operated abacus is no longer used today. The token abacus is also a type of account table;**ring abacus**: consists of a series of parallel rods, or parallel wires, attached to a support. A series of rings are threaded through each rod, or wire. The rings are free to slide along the rods or along the wires. Following precise conventions in placing the rings, numbers are represented. The ring abacus is still used today. Types of ring abacuses include the following:**Suanpan**: is the ring abacus used in China. In its modern form, it consists of a series of parallel rods attached to a rectangular support. Seven rings are threaded through each rod. By means of a horizontal rod, the five rings at the bottom are separated from the two rings at the top. The suanpan is still in use today. The oldest suanpan, currently preserved and known, dates back to the 16th century and was found inside a tomb.**Soroban**: it is the ring abacus used in Japan. It consists of a series of parallel rods attached to a rectangular support. Five rings are threaded through each rod. By means of a horizontal rod, the four lowest rings are separated from the highest ring. The soroban is derived from suanpan and is still used today.**Tschoty**: is the ring abacus used in Russia. It consists of a series of parallel rods attached to a rectangular support. Ten rings are threaded through each rod. The tschoty is still used today.**Choreb**: is the ring abacus used in Armenia. It is similar to the tschoty.**Culba**: it is the ring abacus used in Turkey. It is similar to the tschoty.

In Korea, the use of abacuses continued until the nineteenth century; in China and Japan, the use of this instrument lasted a long time (longer than in the West), so much so that even in the second half of the twentieth century many Japanese shopkeepers used an abacus to do math. In the Japanese abacus the groove is double: the lower part contains four objects and the upper one only one, making the operations reminiscent in a sense of those with Roman numerals. Nowadays the use of the abacus for practical purposes is increasingly restricted, if not already completely disappeared. We can recall the use of a variant (the schoty) in the countries of the former Soviet Union. Vice versa, the abacus, especially in its variant abacus, is often used as a didactic game for children. Some elementary schools adopt it to teach children to count and to perform some simple additions and subtractions.