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Encyclopedia > Abacus
A Chinese abacus
A Chinese abacus
Calculating-Table by Gregor Reisch: Margarita Philosophica, 1508
Calculating-Table by Gregor Reisch: Margarita Philosophica, 1508

An abacus also called a counting frame, is a calculating tool for performing arithmetic processes. Nowadays, abaci are often constructed as a wooden frame with beads sliding on wires, but originally they were beads or stones moved in grooves in sand or on tablets of wood, stone, or metal. The abacus was in use centuries before the adoption of the written modern numeral system and is still widely used by merchants and clerks in China, Japan, Africa, India and elsewhere. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... I like cream cheese, it tastes good on toast. ... A world map showing the continent of Africa Africa is the worlds second-largest and second most-populous continent, after Asia. ...


The user of an abacus is called an abacist; he or she slides the beads of the abacus by hand.[1]

Contents

Etymology

The use of the word abacus dates from before 1387, when a Middle English work borrowed the word from Latin to describe a sandboard abacus. The Latin word came from abakos, the Greek genitive form of abax ("calculating-table"). Because abax also had the sense of "table sprinkled with sand or dust, used for drawing geometric figures", some linguists speculate that the Greek word may be derived from a Semitic root (cf. Phoenician abak, "sand", Hebrew ābāq (pronounced "a-vak"), "dust").[citation needed] The preferred plural of abacus is a subject of disagreement, but both abacuses[2] and abaci[3] are in use. Middle English is the name given by historical linguistics to the diverse forms of the English language spoken between the Norman invasion of 1066 and the mid-to-late 15th century, when the Chancery Standard, a form of London-based English, began to become widespread, a process aided by the... For other uses, see Latin (disambiguation). ... This article does not cite any references or sources. ... 14th century BC diplomatic letter in Akkadian, found in Tell Amarna. ... The root is the primary lexical unit of a word, which carries the most significant aspects of semantic content and cannot be reduced into smaller constituents. ... Phoenician was a language originally spoken in the coastal region of what is now Lebanon. ... Hebrew redirects here. ...


Mesopotamian abacus

The period 2700–2300 BCE saw the first appearance of the Sumerian abacus, a table of successive columns which delimited the successive orders of magnitude of their sexagesimal number system.[4]


Babylonians may have used the abacus for the operations of addition and subtraction. However, this primitive device proved difficult to use for more complex calculations.[5] Some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus.[6]


Egyptian abacus

The use of the abacus in ancient Egypt is mentioned by the Greek historian Crabertotous, who writes that the manner of this disk's usage by the Egyptians was opposite in direction when compared with the Greek method. Archaeologists have found ancient disks of various sizes that are thought to have been used as counters. However, wall depictions of this instrument have not been discovered, casting some doubt over the extent to which this instrument was used.[7] The pyramids are the most recognizable symbols of the civilization of ancient Egypt. ...


Greek abacus

The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BCE.[8] 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 see use in Achaemenid Persia, the Etruscan civilization, Ancient Rome and, until the French Revolution, the Western Christian world.


A tablet found on the Greek island Salamis in 1846 dates back to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble 149 cm long, 75 cm wide, and 4.5 cm thick, on which are 5 groups of markings. In the center of the tablet is a set of 5 parallel lines equally divided by a vertical line, capped with a semi-circle 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 semi-circle 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. Salamis (Greek, Modern: Σαλαμίνα Salamína, Ancient/Katharevousa: Σαλαμίς Salamís) is the largest Greek island in the Saronic Gulf, about 1 nautical mile (2 km) off-coast from Piraeus. ...


Roman abacus

Reconstructed Roman Abacus
Reconstructed Roman Abacus
Main article: Roman abacus

The normal method of calculation in ancient Rome, as in Greece, was by moving counters on a smooth table. Originally pebbles, calculi, were used. Later, and in medieval Europe, jetons were manufactured. Marked lines indicated units, fives, tens etc. as in the Roman numeral system. This system of 'counter casting' continued into the late Roman empire and in medieval Europe, and persisted in limited use into the nineteenth century.[9] Reconstruction of a Roman Abacus, made by the RGZ Museum in Mainz, 1977. ... Reconstruction of a Roman Abacus, made by the RGZ Museum in Mainz, 1977. ... The Romans developed the so-called fucken Roman abacus, or rather a portable counting board, based on previous Greek counting boards. ... Integral and differential calculus is a central branch of mathematics, developed from algebra and geometry. ... Jetons were a token or coin-like medal produced across Europe during the 16th and 17th centuries. ... The system of Roman numerals is a numeral system originating in ancient Rome, and was adapted from Etruscan numerals. ...


Writing in the 1st century BCE, 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.[10]


The earliest archaeological evidence of the Roman abacus, shown here in reconstruction, dates to the 1st century CE.[10] 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—five units, five tens etc., essentially in a bi-quinary coded decimal system, obviously related to the Roman numerals. The short grooves on the right may have been used for marking Roman ounces. The Romans developed the so-called fucken Roman abacus, or rather a portable counting board, based on previous Greek counting boards. ... Bi-quinary coded decimal is a numeral encoding scheme used in many abacuses and in some early computers, including the Colossus. ... Roman numerals are a numeral system originating in ancient Rome, adapted from Etruscan numerals. ...


Chinese abacus

Suanpan (the number represented in the picture is 6,302,715,408)
Suanpan (the number represented in the picture is 6,302,715,408)
Main article: Suanpan

The earliest known documentation of the Chinese abacus dates to the 2nd century BCE.[11] Scanned and uploaded by Malcolm Farmer Source: Article for abacus, 9th edition Encyclopedia Britannica, volume 1 (1875) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... Scanned and uploaded by Malcolm Farmer Source: Article for abacus, 9th edition Encyclopedia Britannica, volume 1 (1875) This image has been released into the public domain by the copyright holder, its copyright has expired, or it is ineligible for copyright. ... Suanpan (the number represented in the picture is 6,302,715,408) An extended version of a suanpan The suanpan (Simplified Chinese: ; Traditional Chinese: ; Pinyin: ) is an abacus of Chinese origin first described in a 190 CE book of the Eastern Han Dynasty, namely Supplementary Notes on the Art of...


The Chinese abacus known as the suànpán is typically 20 cm tall and it comes in various 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 for both decimal and hexadecimal computation. Modern abacuses have one bead on the top deck and four beads on the bottom deck. The beads are usually rounded and made of a hardwood. The beads are counted by moving them up or down towards the beam. If you move them high, you count their value. If you move them down, you don't count their value. The suanpan can be reset to the starting position instantly by a quick jerk along the horizontal axis to spin all the beads away from the horizontal beam at the center. For other uses, see Decimal (disambiguation). ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ... Beech is a typical temperate zone hardwood For the record label, see Hardwood Records. ...


Suanpans can be used for functions other than counting. Unlike the simple counting board used in elementary schools, very efficient suanpan techniques have been developed to do multiplication, division, addition, subtraction, square root and cube root operations at high speed. In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... In mathematics, a square root (√) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... Plot of y = In mathematics, the cube root of a number, denoted or x1/3, is the number a such that a3 = x. ...


In the famous long scroll Riverside Scenes at Qingming Festival painted by Zhang Zeduan (1085-1145) during the Song Dynasty (960-1297), a suanpan is clearly seen lying beside an account book and doctor's prescriptions on the counter of an apothecary's (Feibao). Zhang Zeduan (Traditional: 張擇端; Simplified: 张择端; pinyin: ; Wade-Giles: Chang Tse-tuan) was a Chinese painter. ... Zhang Zeduan (Traditional: 張擇端; Simplified: 张择端; pinyin: ; Wade-Giles: Chang Tse-tuan) was a Chinese painter. ... April 2 - Emperor Zhezong became emperor of Song Dynasty. ... Events Pope Lucius II is succeeded by Pope Eugene III Nur ad-Din ascends to power in Syria Construction begins on Notre-Dame dChartres in Chartres, France Korean historian Kim Pusik compiled the historical text Samguk Sagi. ... Northern Song in 1111 AD Capital Bianjing (汴京) (960–1127) Linan (臨安) (1127–1276) Language(s) Chinese Religion Buddhism, Taoism, Confucianism Government Monarchy Emperor  - 960–976 Emperor Taizu  - 1126–1127 Emperor Qinzong  - 1127–1162 Emperor Gaozong  - 1278–1279 Emperor Bing History  - Zhao Kuangyin taking over the throne of the Later Zhou... Events Edgar the Peaceable crowned King of England. ... Events 8 January - Monaco gains independence. ... Interior of an apothecarys shop. ...


The similarity of the Roman abacus to the Chinese one suggests that one could have inspired the other, as there is some evidence of a trade relationship between the Roman Empire and China. However, no direct connection can be demonstrated, and the similarity of the abaci may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model (like most modern Japanese) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2, allowing use with a hexadecimal numeral system. Instead of running on wires as in the Chinese and Japanese models, the beads of Roman model run in grooves, presumably making arithmetic calculations much slower. The Romans developed the so-called fucken Roman abacus, or rather a portable counting board, based on previous Greek counting boards. ... For other uses, see Roman Empire (disambiguation). ... In mathematics and computer science, hexadecimal, base-16, or simply hex, is a numeral system with a radix, or base, of 16, usually written using the symbols 0–9 and A–F, or a–f. ...


Another possible source of the suanpan is Chinese counting rods, which operated with a decimal system but lacked the concept of zero as a place holder. The zero was probably introduced to the Chinese in the Tang Dynasty (618-907) when travel in the Indian Ocean and the Middle East would have provided direct contact with India and Islam allowing them to acquire the concept of zero and the decimal point from Indian and Islamic merchants and mathematicians. The counting rods (Traditional Chinese: , Simplified Chinese: , pinyin: chou2) were used by ancient Chinese before the invention of the abacus. ... Decimal, or denary, notation is the most common way of writing the base 10 numeral system, which uses various symbols for ten distinct quantities (0, 1, 2, 3, 4, 5, 6, 7, 8 and 9, called digits) together with the decimal point and the sign symbols + (plus) and − (minus) to... For other senses of this word, see zero or 0. ... For the band, see Tang Dynasty (band). ... A map showing countries commonly considered to be part of the Middle East The Middle East is a region comprising the lands around the southern and eastern parts of the Mediterranean Sea, a territory that extends from the eastern Mediterranean Sea to the Persian Gulf. ... For people named Islam, see Islam (name). ... The decimal separator is used to mark the boundary between the integer and the fractional parts of a decimal numeral. ...


The Chinese abacus migrated from China to Korea around the year 1400. Koreans call it jupan (주판), supan (수판) or jusan (주산). [12] This article is about the Korean civilization. ...


Indian abacus

1st century sources, such as the Abhidharmakosa describe the knowledge and use of abacus in India.[13] Around the 5th century, Indian clerks were already finding new ways of recording the contents of the Abacus.[14] Hindu texts used the term shunya(means Zero) to indicate the empty column on the abacus.[15] The 1st century was that century that lasted from 1 to 100 according the Gregorian calendar. ... Europe in 450 The 5th century is the period from 401 to 500 in accordance with the Julian calendar in the Christian Era. ...


Korean and Japanese abaci

Japanese soroban
Japanese soroban

The abacus migrated from China to Korea around the year 1400 and later Japan, around the year 1600.[16] Korea's version of the abacus is called jupan (주판) or supan (수판) or jusan (주산). [17] reduce size of image File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... reduce size of image File history Legend: (cur) = this is the current file, (del) = delete this old version, (rev) = revert to this old version. ... This article is about the Korean civilization. ... This article is about the Korean civilization. ...


The Chinese suanpan was called soroban (算盤, そろばん, lit. "Counting tray") in Japan, imported via Korea around 1600.[18] Like the suanpan, the soroban is still used in Japan today, even with the proliferation, practicality, and affordability of pocket electronic calculators. This article is about the calculator. ...


Native American abaci

Representation of an Inca quipu
Representation of an Inca quipu

Some sources mention the use of an abacus called a nepohualtzintzin in ancient Aztec culture. This Mesoamerican abacus used a 5-digit base-20 system. Image File history File links From Meyers Konversationslexikon of 1888 - Public Domain This image is from the 4th edition of the Meyers Konversationslexikon. ... Image File history File links From Meyers Konversationslexikon of 1888 - Public Domain This image is from the 4th edition of the Meyers Konversationslexikon. ... For other meanings of Inca, see Inca (disambiguation). ... Inca Quipu. ... Image File history File links Question_book-3. ...


The quipu of the Incas was a system of knotted cords used to record numerical data, like advanced tally sticks—but not used to perform calculations. Calculations were carried out using a yupana (quechua for "counting tool"; see figure) which was still in use after the conquest of Peru. The working principle of a yupana is unknown, but in 2001 an explanation of the mathematical basis of these instruments was proposed. By comparing the form of several yupanas, researchers found that calculations were based using the Fibonacci sequence 1,1,2,3,5 and powers of 10, 20 and 40 as place values for the different fields in the instrument. Using the Fibonacci sequence would keep the number of grains within any one field at minimum.
Inca Quipu. ... For other meanings of Inca, see Inca (disambiguation). ... Tally sticks are an ancient mnemonic device (memory aid) to record and document numbers or quantities even messages. ... It has been suggested that this article or section be merged with Quechuan languages. ... In mathematics, the Fibonacci numbers form a sequence defined recursively by: In words: you start with 0 and 1, and then produce the next Fibonacci number by adding the two previous Fibonacci numbers. ...


Russian abacus

Russian abacus
Russian abacus

The Russian abacus, the schoty (счёты), usually has a single slanted deck, with ten beads on each wire (except one wire which has four beads, for quarter-ruble fractions). This wire is usually near the user. (Older models have another 4-bead wire for quarter-kopeks, which were minted until 1916.) The Russian abacus is often used vertically, with wires from left to right in the manner of a book. The wires are usually bowed to bulge upward in the center, in order to keep the beads pinned to either of the two sides. 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 (the 5th and 6th bead) usually have a colour different from the other 8 beads. Likewise, the left bead of the thousands wire (and the million wire, if present) may have a different color. Image File history File links Download high resolution version (919x1064, 993 KB) A russian abacus. ... Image File history File links Download high resolution version (919x1064, 993 KB) A russian abacus. ...


The Russian abacus was in use in all shops and markets throughout the former Soviet Union, and the usage of it was taught in most schools till 1990s. Today it is regarded as an archaism and replaced by microcalculator. In school the usage of calculator is taught since 1990s.


School abacus

School abacus used in Danish elementary school. Early 20th century.
School abacus used in Danish elementary school. Early 20th century.

Around the world, abaci have been used in pre-schools and elementary schools as an aid in teaching the numeral system and arithmetic. In Western countries, a bead frame similar to the Russian abacus but with straight wires and a vertical frame has been common (see image). It is still often seen as a plastic or wooden toy. Download high resolution version (1675x1793, 317 KB) Wikipedia does not have an article with this exact name. ... Download high resolution version (1675x1793, 317 KB) Wikipedia does not have an article with this exact name. ... This article is about different methods of expressing numbers with symbols. ... Arithmetic tables for children, Lausanne, 1835 Arithmetic or arithmetics (from the Greek word αριθμός = number) is the oldest and most elementary branch of mathematics, used by almost everyone, for tasks ranging from simple day-to-day counting to advanced science and business calculations. ...


The type of abacus shown here is often used to represent numbers without the use of place value. Each bead and each wire has the same value and used in this way it can represent numbers up to 100.


The most significant educational advantage of using an abacus, rather than loose beads or counters, when practicing counting and simple addition is that it gives the student an awareness of the groupings of 10 which are the foundation of our number system. Although adults take this base 10 structure for granted, it is actually difficult to learn. Many 6-year-olds can count to 100 by rote with only a slight awareness of the patterns involved.


Abacus in Medieval Pictures

Uses by the blind

An adapted abacus, invented by Helen Keller, called a Cranmer abacus is still commonly used by individuals who are blind. A piece of soft fabric or rubber is placed behind the beads so that they do not move inadvertently. This keeps the beads in place while the users feel or manipulate them. They use an abacus to perform the mathematical functions multiplication, division, addition, subtraction, square root and cubic root. This article is about the visual condition. ... In mathematics, multiplication is an elementary arithmetic operation. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ... 3 + 2 = 5 with apples, a popular choice in textbooks[1] This article is about addition in mathematics. ... 5 - 2 = 3 (verbally, five minus two equals three) An example problem Subtraction is one of the four basic arithmetic operations; it is the inverse of addition. ... In mathematics, a square root (√) of a number x is a number r such that , or in words, a number r whose square (the result of multiplying the number by itself) is x. ... In mathematics, the cube root (∛) of a number is a number which, when cubed (multiplied by itself and then multiplied by itself again), gives back the original number. ...


Although blind students have benefited from talking calculators, the abacus is still very often taught to these students in early grades, both in public schools and state schools for the blind. The abacus teaches math skills that can never be replaced with talking calculators and is an important learning tool for blind students. Blind students also complete math assignments using a braille-writer and Nemeth code (a type of braille code for math) but large multiplication and long division problems can be long and difficult. The abacus gives blind and visually impaired students a tool to compute math problems that equals the speed and mathematical knowledge required by their sighted peers using pencil and paper. Many blind people find this number machine a very useful tool throughout life. The Nemeth Braille Code for Mathematics is a method of encoding mathematical and scientific notation linearly using standard six-dot Braille cells for tactile reading by the visually impaired. ...


Notes

  1. ^ "abacist", "abacus", in Merriam-Webster's Third New International Dictionary Unabridged, 2000, Version 2.5.
  2. ^ Oxford English Dictionary, 1989
  3. ^ Merriam-Webster's 2003
  4. ^ Ifrah 2001:11
  5. ^ Carruccio, page 14
  6. ^ Crump, page 188
  7. ^ Smith, page 160
  8. ^ Ifrah 2001:15
  9. ^ Pullan, page 18
  10. ^ a b Ifrah 2001:18
  11. ^ Ifrah 2001:17
  12. ^ http://enc.daum.net/dic100/contents.do?query1=b19j3727a
  13. ^ Stearns, page 44
  14. ^ Körner, page 232
  15. ^ Mollin, page 3
  16. ^ http://www.thocp.net/hardware/abacus.html
  17. ^ http://enc.daum.net/dic100/contents.do?query1=b19j3727a
  18. ^ http://www.ee.ryerson.ca/~elf/abacus/history.html

A Chinese abacus Calculating-Table by Gregor Reisch: Margarita Philosophica, 1508 For other uses, see Abacus (disambiguation). ... A Chinese abacus Calculating-Table by Gregor Reisch: Margarita Philosophica, 1508 For other uses, see Abacus (disambiguation). ... A Chinese abacus Calculating-Table by Gregor Reisch: Margarita Philosophica, 1508 For other uses, see Abacus (disambiguation). ... A Chinese abacus Calculating-Table by Gregor Reisch: Margarita Philosophica, 1508 For other uses, see Abacus (disambiguation). ...

References

  • Ifrah, Georges (2001), The Universal History of Computing: From the Abacus to the Quantum Computer, New York: John Wiley & Sons.
  • Reilly, Edwin D.; William Leonard Langer (2004). Concise Encyclopedia of Computer Science. John Wiley and Sons. ISBN ISBN 0470090952. 
  • Körner, Thomas William; William Leonard Langer (1996). The Pleasures of Counting. Houghton Mifflin Books. ISBN ISBN 0521568234. 
  • Mollin, Richard Anthony (1998). Fundamental Number Theory with Applications. CRC Press. ISBN ISBN 0849339871. 
  • Smith, David Eugene. History of Mathematics (Volume 2). Courier Dover Publications. ISBN ISBN 0486204308. 
  • Crump, Thomas (1992). The Japanese Numbers Game: The Use and Understanding of Numbers in Modern Japan. Routledge. ISBN ISBN 0415056098. 
  • Carruccio, Ettore (2006). Mathematics And Logic in History And in Contemporary Thought. Aldine Transaction. ISBN ISBN 0202308502. 
  • Stearns, Peter N.; William Leonard Langer (2001). The Encyclopedia of World History: Ancient, Medieval, and Modern, Chronologically Arranged. Houghton Mifflin Books. ISBN ISBN 0395652375. 
  • Peng Yoke Ho (2000). Li, Qi and Shu: An Introduction to Science and Civilization in China. Courier Dover Publications. ISBN ISBN 0486414450. 
  • (2003) Merriam-Webster's Collegiate Dictionary, 11th edition, Merriam-Webster, Inc. ISBN 0877798095. 
  • "abacus". Oxford English Dictionary. Oxford University Press. 2nd ed. 1989.
  • Pullan, J. M. (1968). The History of the Abacus. London: Books That Matter. ISBN 0-09-089410-3. 

The CRC Press, LLC is a publishing group which specializes in producing technical books in a wide range of subjects. ...

See also

In logic, an abacus is an instrument, often called the logical machine, analogous to the mathematical abacus. ... The abacus system of mental calculation is a system where users mentally visualize an abacus to do calculations. ... Chisanbop or chisenbop (from Korean chi finger + sanpŏp calculation [1] ) is an abacus-like finger counting method used to perform basic mathematical operations. ... Napiers bones are an abacus invented by John Napier for calculation of products and quotients of numbers. ...

Further reading

  • Menninger, Karl W. (1969). Number Words and Number Symbols: A Cultural History of Numbers. MIT Press. ISBN 0-262-13040-8. 
  • Kojima, Takashi (1954). The Japanese Abacus: its Use and Theory. Tokyo: Charles E. Tuttle. ISBN 0-8048-0278-5. 

External links

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Wikipedia does not have an article with this exact name. ... Image File history File links Wikibooks-logo. ... Image File history File links Wikiquote-logo. ... Image File history File links Wikisource-logo. ... Image File history File links Commons-logo. ... Image File history File links WikiNews-Logo. ... Image File history File links Wikiversity-logo-Snorky. ...

Tutorials

Abacus curiosities

cut-the-knot is an educational website maintained by Alexander Bogomolny and devoted to popular exposition of a great variety of topics in mathematics. ...

  Results from FactBites:
 
ABACUS: ABACUS VERSUS TALKING CALCULATOR (3485 words)
This device is an abacus and is an adaptation of the Japanese abacus.
For the child with a visual impairment the abacus is comparable to the sighted child's pencil and paper, and should be considered a fundamental component of his math instruction.
The abacus has also been great to "back up" a teenager when she has had great difficulty with concepts that were taught in grade school-----but perhaps she missed or passed over at the time.
Abacus - LoveToKnow 1911 (588 words)
In Romanesque architecture the abacus is square with the lower edge splayed off and moulded or carved, and the same was retained in France during the medieval period; but in England, in Early English work, a circular deeply moulded abacus was introduced, which in the 14th and 15th centuries was transformed into an octagonal one.
The diminutive of Abacus, Abaciscus, is applied in architecture to the chequers or squares of a tessellated pavement.
In the abacus the combinations are inscribed each on a single slip of wood or similar substance, which is moved by a key; incompatible combinations can thus be mechanically removed at will, in accordance with any given series of premises.
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