Astronomy in the medieval Islamic world

From Wikipedia, the free encyclopedia

An 18th-century Persian astrolabe, kept at the Whipple Museum of the History of Science in Cambridge, England.

Islamic astronomy comprises the astronomical developments made in the Islamic world, particularly during the Islamic Golden Age (9th–13th centuries), and mostly written in the Arabic language. These developments mostly took place in the Middle East, Central Asia, Al-Andalus, and North Africa, and later in the Far East and India. It closely parallels the genesis of other Islamic sciences in its assimilation of foreign material and the amalgamation of the disparate elements of that material to create a science with Islamic characteristics. These included Greek, Sassanid, and Indian works in particular, which were translated and built upon.

Islamic astronomy played a significant role in the revival of Byzantine and European astronomy following the loss of knowledge during the early medieval period, notably with the production of Latin translations of Arabic works during the 12th century. Islamic astronomy also had an influence on Chinese astronomy and Malian astronomy.

A significant number of stars in the sky, such as Aldebaran, Altair and Deneb, and astronomical terms such as alidade, azimuth, and nadir, are still referred to by their Arabic names. A large corpus of literature from Islamic astronomy remains today, numbering approximately 10,000 manuscripts scattered throughout the world, many of which have not been read or catalogued. Even so, a reasonably accurate picture of Islamic activity in the field of astronomy can be reconstructed.

History

A Large Persian Brass Celestial Globe with an ascription to Hadi Isfahani and a date of 1197 AH/ 1782-3 AD of typical spherical form, the globe engraved with markings, figures and astrological symbols, inscriptive details throughout

Ahmad Dallal notes that, unlike the Babylonians, Greeks, and Indians, who had developed elaborate systems of mathematical astronomical study, the pre-Islamic Arabs relied entirely on empirical observations. These observations were based on the rising and setting of particular stars, and this area of astronomical study was known as anwa. Anwa continued to be developed after Islamization by the Arabs, where Islamic astronomers added mathematical methods to their empirical observations. According to David King, after the rise of Islam, the religious obligation to determine the qibla and prayer times inspired more progress in astronomy for centuries.

Donald Hill (1993) divided Islamic Astronomy into the four following distinct time periods in its history:

Early Islam

Following the Islamic conquests, under the early caliphate, Muslim scholars began to absorb Hellenistic and Indian astronomical knowledge via translations into Arabic (in some cases via Persian).

The first astronomical texts that were translated into Arabic were of Indian and Persian origin. The most notable of the texts was Zij al-Sindhind, an 8th-century Indian astronomical work that was translated by Muhammad ibn Ibrahim al-Fazari and Yaqub ibn Tariq after 770 CE with the assistance of Indian astronomers who visited the court of caliph Al-Mansur in 770. Another text translated was the Zij al-Shah, a collection of astronomical tables (based on Indian parameters) compiled in Sasanid Persia over two centuries. Fragments of texts during this period indicate that Arabs adopted the sine function (inherited from India) in place of the chords of arc used in Greek trigonometry.

Golden Age

The Tusi-couple is a mathematical device invented by Nasir al-Din al-Tusi in which a small circle rotates inside a larger circle twice the diameter of the smaller circle. Rotations of the circles cause a point on the circumference of the smaller circle to oscillate back and forth in linear motion along a diameter of the larger circle.

The House of Wisdom was an academy established in Baghdad under Abbasid caliph Al-Ma'mun in the early 9th century. From this time, independent investigation into the Ptolemaic system became possible. According to Dallal (2010), the use of parameters, sources and calculation methods from different scientific traditions made the Ptolemaic tradition "receptive right from the beginning to the possibility of observational refinement and mathematical restructuring". Astronomical research was greatly supported by the Abbasid caliph al-Mamun through The House of Wisdom. Baghdad and Damascus became the centers of such activity. The caliphs not only supported this work financially, but endowed the work with formal prestige.

The first major Muslim work of astronomy was Zij al-Sindh by al-Khwarizmi in 830. The work contains tables for the movements of the sun, the moon and the five planets known at the time. The work is significant as it introduced Ptolemaic concepts into Islamic sciences. This work also marks the turning point in Islamic astronomy. Hitherto, Muslim astronomers had adopted a primarily research approach to the field, translating works of others and learning already discovered knowledge. Al-Khwarizmi's work marked the beginning of nontraditional methods of study and calculations.

In 850, al-Farghani wrote Kitab fi Jawani (meaning "A compendium of the science of stars"). The book primarily gave a summary of Ptolemic cosmography. However, it also corrected Ptolemy based on findings of earlier Arab astronomers. Al-Farghani gave revised values for the obliquity of the ecliptic, the precessional movement of the apogees of the sun and the moon, and the circumference of the earth. The book was widely circulated through the Muslim world, and even translated into Latin.

In addition to Alfraganus's findings, Egyptian Astronomer Ibn Yunus was actually the first Astronomer to really find valid fault in Ptolemy's calculations about the planet's movements and their peculiarity in the late 10th century. Ptolemy calculated that Earth's wobble, otherwise known as precession, varied 1 degree every 100 years. Ibn Yunus contradicted this finding by calculating that it was instead 1 degree every 70 years. This was impossible to believe, however, since it was still thought that the Earth was the center of the universe. Ibn Yunus and Ibn al-Shatir's findings were part of Copernicus's calculations to figure out that the sun was the center of the universe.

An illustration from al-Biruni's astronomical works, explains the different phases of the moon.

The period when a distinctive Islamic system of astronomy flourished. The period began as the Muslim astronomers began questioning the framework of the Ptolemaic system of astronomy. These criticisms, however, remained within the geocentric framework and followed Ptolemy's astronomical paradigm; one historian described their work as "a reformist project intended to consolidate Ptolemaic astronomy by bringing it into line with its own principles."

Between 1025 and 1028, Ibn al-Haytham wrote his Al-Shukuk ala Batlamyus (meaning "Doubts on Ptolemy"). While maintaining the physical reality of the geocentric model, he criticized elements of the Ptolemic models. Many astronomers took up the challenge posed in this work, namely to develop alternate models that resolved these difficulties. In 1070, Abu Ubayd al-Juzjani published the Tarik al-Aflak. In his work, he indicated the so-called "equant" problem of the Ptolemic model. Al-Juzjani even proposed a solution for the problem. In Al-Andalus, the anonymous work al-Istidrak ala Batlamyus (meaning "Recapitulation regarding Ptolemy"), included a list of objections to the Ptolemic astronomy.

Later period

Notable astronomers from the later medieval period include Mu'ayyad al-Din al-'Urdi (c. 1266), Nasir al-Din al-Tusi (1201–74), Qutb al-Din al Shirazi (c. 1311), Sadr al-Sharia al-Bukhari (c. 1347), Ibn al-Shatir (c. 1375), and Ali al-Qushji (c. 1474).

In the late 13th century, Nasir al-Din al-Tusi created the Tusi Couple, as pictured above. The model would later be viable to Copernicus's understanding of these movements in his work during the Renaissance period.

In the fifteenth century, the Timurid ruler Ulugh Beg of Samarkand established his court as a center of patronage for astronomy. He himself studied it in his youth, and in 1420 ordered the construction of an observatory, which produced a new set of astronomical tables, as well as contributing to other scientific and mathematical advances.

Influences in East Asia

China

Layout of the Beijing Ancient Observatory.

Islamic influence on Chinese astronomy was first recorded during the Song dynasty when a Hui Muslim astronomer named Ma Yize introduced the concept of 7 days in a week and made other contributions.

Islamic astronomers were brought to China in order to work on calendar making and astronomy during the Mongol Empire and the succeeding Yuan Dynasty. The Chinese scholar Yeh-lu Chu'tsai accompanied Genghis Khan to Persia in 1210 and studied their calendar for use in the Mongol Empire. Kublai Khan brought Iranians to Beijing to construct an observatory and an institution for astronomical studies.

Several Chinese astronomers worked at the Maragheh observatory, founded by Nasir al-Din al-Tusi in 1259 under the patronage of Hulagu Khan in Persia. One of these Chinese astronomers was Fu Mengchi, or Fu Mezhai. In 1267, the Persian astronomer Jamal ad-Din, who previously worked at Maragha observatory, presented Kublai Khan with seven Persian astronomical instruments, including a terrestrial globe and an armillary sphere, as well as an astronomical almanac, which was later known in China as the Wannian Li ("Ten Thousand Year Calendar" or "Eternal Calendar"). He was known as "Zhamaluding" in China, where, in 1271, he was appointed by Khan as the first director of the Islamic observatory in Beijing, known as the Islamic Astronomical Bureau, which operated alongside the Chinese Astronomical Bureau for four centuries. Islamic astronomy gained a good reputation in China for its theory of planetary latitudes, which did not exist in Chinese astronomy at the time, and for its accurate prediction of eclipses.

Some of the astronomical instruments constructed by the famous Chinese astronomer Guo Shoujing shortly afterwards resemble the style of instrumentation built at Maragheh. In particular, the "simplified instrument" (jianyi) and the large gnomon at the Gaocheng Astronomical Observatory show traces of Islamic influence. While formulating the Shoushili calendar in 1281, Shoujing's work in spherical trigonometry may have also been partially influenced by Islamic mathematics, which was largely accepted at Kublai's court. These possible influences include a pseudo-geometrical method for converting between equatorial and ecliptic coordinates, the systematic use of decimals in the underlying parameters, and the application of cubic interpolation in the calculation of the irregularity in the planetary motions.

Hongwu Emperor (r. 1368-1398) of the Ming Dynasty (1328–1398), in the first year of his reign (1368), conscripted Han and non-Han astrology specialists from the astronomical institutions in Beijing of the former Mongolian Yuan to Nanjing to become officials of the newly established national observatory.

That year, the Ming government summoned for the first time the astronomical officials to come south from the upper capital of Yuan. There were fourteen of them. In order to enhance accuracy in methods of observation and computation, Hongwu Emperor reinforced the adoption of parallel calendar systems, the Han and the Hui. In the following years, the Ming Court appointed several Hui astrologers to hold high positions in the Imperial Observatory. They wrote many books on Islamic astronomy and also manufactured astronomical equipment based on the Islamic system.
The translation of two important works into Chinese was completed in 1383: Zij (1366) and al-Madkhal fi Sina'at Ahkam al-Nujum, Introduction to Astrology (1004).

In 1384, a Chinese astrolabe was made for observing stars based on the instructions for making multi-purposed Islamic equipment. In 1385, the apparatus was installed on a hill in northern Nanjing.

Around 1384, during the Ming Dynasty, Hongwu Emperor ordered the Chinese translation and compilation of Islamic astronomical tables, a task that was carried out by the scholars Mashayihei, a Muslim astronomer, and Wu Bozong, a Chinese scholar-official. These tables came to be known as the Huihui Lifa (Muslim System of Calendrical Astronomy), which was published in China a number of times until the early 18th century, though the Qing Dynasty had officially abandoned the tradition of Chinese-Islamic astronomy in 1659. The Muslim astronomer Yang Guangxian was known for his attacks on the Jesuit's astronomical sciences.

Korea

Korean celestial globe based on the Huihui Lifa.

In the early Joseon period, the Islamic calendar served as a basis for calendar reform owing to its superior accuracy over the existing Chinese-based calendars. A Korean translation of the Huihui Lifa, a text combining Chinese astronomy with Islamic astronomy works of Jamal ad-Din, was studied in Korea under the Joseon Dynasty during the time of Sejong in the 15th century. The tradition of Chinese-Islamic astronomy survived in Korea up until the early 19th century.

Observatories

Medieval manuscript by Qutb al-Din al-Shirazi depicting an epicyclic planetary model.

The first systematic observations in Islam are reported to have taken place under the patronage of al-Mamun. Here, and in many other private observatories from Damascus to Baghdad, meridian degrees were measured, solar parameters were established, and detailed observations of the Sun, Moon, and planets were undertaken.

In the 10th century, the Buwayhid dynasty encouraged the undertaking of extensive works in astronomy, such as the construction of a large-scale instrument with which observations were made in the year 950. We know of this by recordings made in the zij of astronomers such as Ibn al-Alam. The great astronomer Abd Al-Rahman Al Sufi was patronised by prince Adud o-dowleh, who systematically revised Ptolemy's catalogue of stars. Sharaf al-Daula also established a similar observatory in Baghdad. Reports by Ibn Yunus and al-Zarqall in Toledo and Cordoba indicate the use of sophisticated instruments for their time.

It was Malik Shah I who established the first large observatory, probably in Isfahan. It was here where Omar Khayyám with many other collaborators constructed a zij and formulated the Persian Solar Calendar a.k.a. the jalali calendar. A modern version of this calendar is still in official use in Iran today.

The most influential observatory was however founded by Hulegu Khan during the 13th century. Here, Nasir al-Din al-Tusi supervised its technical construction at Maragha. The facility contained resting quarters for Hulagu Khan, as well as a library and mosque. Some of the top astronomers of the day gathered there, and from their collaboration resulted important modifications to the Ptolemaic system over a period of 50 years.

The Ulugh Beg Observatory in Samarqand.

In 1420, prince Ulugh Beg, himself an astronomer and mathematician, founded another large observatory in Samarkand, the remains of which were excavated in 1908 by Russian teams.
And finally, Taqi al-Din Muhammad ibn Ma'ruf founded a large observatory in Ottoman Constantinople in 1577, which was on the same scale as those in Maragha and Samarkand. The observatory was short-lived however, as opponents of the observatory and prognostication from the heavens prevailed and the observatory was destroyed in 1580. While the Ottoman clergy did not object to the science of astronomy, the observatory was primarily being used for astrology, which they did oppose, and successfully sought its destruction.

Instruments

Work in the observatorium of Taqi al-Din.

Our knowledge of the instruments used by Muslim astronomers primarily comes from two sources: first the remaining instruments in private and museum collections today, and second the treatises and manuscripts preserved from the Middle Ages. Muslim astronomers of the "Golden Period" made many improvements to instruments already in use before their time, such as adding new scales or details.

Celestial globes and armillary spheres

Celestial globes were used primarily for solving problems in celestial astronomy. Today, 126 such instruments remain worldwide, the oldest from the 11th century. The altitude of the sun, or the Right Ascension and Declination of stars could be calculated with these by inputting the location of the observer on the meridian ring of the globe.

An armillary sphere had similar applications. No early Islamic armillary spheres survive, but several treatises on "the instrument with the rings" were written. In this context there is also an Islamic development, the spherical astrolabe, of which only one complete instrument, from the 14th century, has survived.

Astrolabes

Brass astrolabes were a Hellenistic invention. The first Islamic astronomer reported as having built an astrolabe is Muhammad al-Fazari (late 8th century). Astrolabes were popular in the Islamic world during the "Golden Age", chiefly as an aid to finding the qibla. The earliest known example is dated to 927/8 (AH 315).

The instruments were used to read the time of rise of the Sun and fixed stars. al-Zarqali of Andalusia constructed one such instrument in which, unlike its predecessors, did not depend on the latitude of the observer, and could be used anywhere. This instrument became known in Europe as the Saphea.

Mechanical calendar

Abu Rayhan Biruni made an instrument he called "Box of the Moon", which was a mechanical lunisolar calendar, employing a gear train and eight gear-wheels. This was an early example of a fixed-wired knowledge processing machine.

Sundials

The Timbuktu Manuscripts showing both mathematics and astronomy.

 

Muslims made several important improvements to the theory and construction of sundials, which they inherited from their Indian and Greek predecessors. Khwarizmi made tables for these instruments which considerably shortened the time needed to make specific calculations.

Sundials were frequently placed on mosques to determine the time of prayer. One of the most striking examples was built in the 14th century by the muwaqqit (timekeeper) of the Umayyad Mosque in Damascus, ibn al-Shatir.

Quadrants

Ibn al-Shatir's model for the appearances of Mercury, showing the multiplication of epicycles using the Tusi-couple, thus eliminating the Ptolemaic eccentrics and equant.

Several forms of quadrants were invented by Muslims. Among them was the sine quadrant used for astronomical calculations, and various forms of the horary quadrant, used to determine time (especially the times of prayer) by observations of the sun or stars. A center of the development of quadrants was ninth-century Baghdad.

Equatoriums

The Equatorium is an Islamic invention from Al-Andalus. The earliest known was made in the 11th century.  It is a mechanical device for finding the positions of the moon, sun, stars and planets, without calculation using a geometrical model to represent the celestial body's mean and anomalistic position.

Indian astronomy

From Wikipedia, the free encyclopedia

Indian astronomy has a long history stretching from pre-historic to modern times. Some of the earliest roots of Indian astronomy can be dated to the period of Indus Valley Civilization or earlier. Astronomy later developed as a discipline of Vedanga or one of the "auxiliary disciplines" associated with the study of the Vedas, dating 1500 BCE or older. The oldest known text is the Vedanga Jyotisha, dated to 1400–1200 BCE (with the extant form possibly from 700–600 BCE).

 

Indian astronomy was influenced by Greek astronomy beginning in the 4th century BCE and through the early centuries of the Common Era, for example by the Yavanajataka and the Romaka Siddhanta, a Sanskrit translation of a Greek text disseminated from the 2nd century.

Indian astronomy flowered in the 5th-6th century, with Aryabhata, whose Aryabhatiya represented the pinnacle of astronomical knowledge at the time. Later the Indian astronomy significantly influenced Muslim astronomy, Chinese astronomy, European astronomy, and others. Other astronomers of the classical era who further elaborated on Aryabhata's work include Brahmagupta, Varahamihira and Lalla.

An identifiable native Indian astronomical tradition remained active throughout the medieval period and into the 16th or 17th century, especially within the Kerala school of astronomy and mathematics.

History

Some of the earliest forms of astronomy can be dated to the period of Indus Valley Civilization, or earlier. Some cosmological concepts are present in the Vedas, as are notions of the movement of heavenly bodies and the course of the year. As in other traditions, there is a close association of astronomy and religion during the early history of the science, astronomical observation being necessitated by spatial and temporal requirements of correct performance of religious ritual. Thus, the Shulba Sutras, texts dedicated to altar construction, discusses advanced mathematics and basic astronomy. Vedanga Jyotisha is another of the earliest known Indian texts on astronomy, it includes the details about the sun, moon, nakshatras, lunisolar calendar.

Greek astronomical ideas began to enter India in the 4th century BCE following the conquests of Alexander the Great. By the early centuries of the Common Era, Indo-Greek influence on the astronomical tradition is visible, with texts such as the Yavanajataka and Romaka Siddhanta. Later astronomers mention the existence of various siddhantas during this period, among them a text known as the Surya Siddhanta. These were not fixed texts but rather an oral tradition of knowledge, and their content is not extant. The text today known as Surya Siddhanta dates to the Gupta period and was received by Aryabhata.

The classical era of Indian astronomy begins in the late Gupta era, in the 5th to 6th centuries. The Pañcasiddhāntikā by Varāhamihira (505 CE) approximates the method for determination of the meridian direction from any three positions of the shadow using a gnomon. By the time of Aryabhata the motion of planets was treated to be elliptical rather than circular. Other topics included definitions of different units of time, eccentric models of planetary motion, epicyclic models of planetary motion, and planetary longitude corrections for various terrestrial locations.

A page from the Hindu calendar 1871–72.

Calendars

The divisions of the year were on the basis of religious rites and seasons (Rtu). The duration from mid March—Mid May was taken to be spring (vasanta), mid May—mid July: summer ("grishma"), mid July—mid September: rains (varsha), mid September—mid November: autumn, mid November—mid January: winter, mid January—mid March: dew (śiśira).

In the Vedānga Jyotiṣa, the year begins with the winter solstice. Hindu calendars have several eras:

• The Hindu calendar, counting from the start of the Kali Yuga, has its epoch on 18 February 3102 BCE Julian (23 January 3102 BCE Gregorian).
• The Vikrama Samvat calendar, introduced about the 12th century, counts from 56–57 BCE.
• The "Saka Era", used in some Hindu calendars and in the Indian national calendar, has its epoch near the vernal equinox of year 78.
• The Saptarshi calendar traditionally has its epoch at 3076 BCE.

J.A.B. van Buitenen (2008) reports on the calendars in India:

The oldest system, in many respects the basis of the classical one, is known from texts of about 1000 BCE. It divides an approximate solar year of 360 days into 12 lunar months of 27 (according to the early Vedic text Taittirīya Saṃhitā 4.4.10.1–3) or 28 (according to the Atharvaveda, the fourth of the Vedas, 19.7.1.) days. The resulting discrepancy was resolved by the intercalation of a leap month every 60 months. Time was reckoned by the position marked off in constellations on the ecliptic in which the Moon rises daily in the course of one lunation (the period from New Moon to New Moon) and the Sun rises monthly in the course of one year. These constellations (nakṣatra) each measure an arc of 13° 20′ of the ecliptic circle. The positions of the Moon were directly observable, and those of the Sun inferred from the Moon's position at Full Moon, when the Sun is on the opposite side of the Moon. The position of the Sun at midnight was calculated from the nakṣatra that culminated on the meridian at that time, the Sun then being in opposition to that nakṣatra.

Astronomers

Name	Year	Contributions
Lagadha	1st millennium BCE	The earliest astronomical text—named Vedānga Jyotiṣa details several astronomical attributes generally applied for timing social and religious events. The Vedānga Jyotiṣa also details astronomical calculations, calendrical studies, and establishes rules for empirical observation. Since the texts written by 1200 BCE were largely religious compositions the Vedānga Jyotiṣa has connections with Indian astrology and details several important aspects of the time and seasons, including lunar months, solar months, and their adjustment by a lunar leap month of Adhimāsa. Ritus are also described as ((yugams)). Tripathi (2008) holds that ' Twenty-seven constellations, eclipses, seven planets, and twelve signs of the zodiac were also known at that time.'
Aryabhata	476–550 CE	Aryabhata was the author of the Āryabhatīya and the Aryabhatasiddhanta, which, according to Hayashi (2008): 'circulated mainly in the northwest of India and, through the Sāsānian dynasty (224–651) of Iran, had a profound influence on the development of Islamic astronomy. Its contents are preserved to some extent in the works of Varahamihira (flourished c. 550), Bhaskara I (flourished c. 629), Brahmagupta (598–c. 665), and others. It is one of the earliest astronomical works to assign the start of each day to midnight.' Aryabhata explicitly mentioned that the earth rotates about its axis, thereby causing what appears to be an apparent westward motion of the stars. In his book, Aryabhatiya, he suggested that the Earth was sphere, containing a circumference of 24,835 miles (39,967 km). Aryabhata also mentioned that reflected sunlight is the cause behind the shining of the moon. Aryabhata's followers were particularly strong in South India, where his principles of the diurnal rotation of the earth, among others, were followed and a number of secondary works were based on them.
Brahmagupta	598–668 CE	Brahmasphuta-siddhanta (Correctly Established Doctrine of Brahma, 628 CE) dealt with both Indian mathematics and astronomy. Hayashi (2008) writes: 'It was translated into Arabic in Baghdad about 771 and had a major impact on Islamic mathematics and astronomy.' In Khandakhadyaka (A Piece Eatable, 665 CE) Brahmagupta reinforced Aryabhata's idea of another day beginning at midnight. Bahmagupta also calculated the instantaneous motion of a planet, gave correct equations for parallax, and some information related to the computation of eclipses. His works introduced Indian concept of mathematics based astronomy into the Arab world. He also theorized that all bodies with mass are attracted to the earth.
Varāhamihira	505 CE	Varāhamihira was an astronomer and mathematician who studied and Indian astronomy as well as the many principles of Greek, Egyptian, and Roman astronomical sciences. His Pañcasiddhāntikā is a treatise and compendium drawing from several knowledge systems.
Bhāskara I	629 CE	Authored the astronomical works Mahabhaskariya (Great Book of Bhaskara), Laghubhaskariya (Small Book of Bhaskara), and the Aryabhatiyabhashya (629 CE)—a commentary on the Āryabhatīya written by Aryabhata. Hayashi (2008) writes 'Planetary longitudes, heliacal rising and setting of the planets, conjunctions among the planets and stars, solar and lunar eclipses, and the phases of the Moon are among the topics Bhaskara discusses in his astronomical treatises.' Baskara I's works were followed by Vateśvara (880 CE), who in his eight chapter Vateśvarasiddhānta devised methods for determining the parallax in longitude directly, the motion of the equinoxes and the solstices, and the quadrant of the sun at any given time.
Lalla	8th century CE	Author of the Śisyadhīvrddhida (Treatise Which Expands the Intellect of Students), which corrects several assumptions of Āryabhata. The Śisyadhīvrddhida of Lalla itself is divided into two parts:Grahādhyāya and Golādhyāya. Grahādhyāya (Chapter I-XIII) deals with planetary calculations, determination of the mean and true planets, three problems pertaining to diurnal motion of Earth, eclipses, rising and setting of the planets, the various cusps of the moon, planetary and astral conjunctions, and complementary situations of the sun and the moon. The second part—titled Golādhyāya (chapter XIV–XXII)—deals with graphical representation of planetary motion, astronomical instruments, spherics, and emphasizes on corrections and rejection of flawed principles. Lalla shows influence of Āryabhata, Brahmagupta, and Bhāskara I. His works were followed by later astronomers Śrīpati, Vateśvara, and Bhāskara II. Lalla also authored the Siddhāntatilaka.
Bhāskara II	1114 CE	Authored Siddhāntaśiromaṇi (Head Jewel of Accuracy) and Karaṇakutūhala (Calculation of Astronomical Wonders) and reported on his observations of planetary positions, conjunctions, eclipses, cosmography, geography, mathematics, and astronomical equipment used in his research at the observatory in Ujjain, which he headed.
Śrīpati	1045 CE	Śrīpati was an astronomer and mathematician who followed the Brhmagupta school and authored the Siddhāntaśekhara (The Crest of Established Doctrines) in 20 chapters, thereby introducing several new concepts, including moon's second inequality.
Mahendra Suri	14th century CE	Mahendra Suri authored the Yantra-rāja (The King of Instruments, written in 1370 CE)—a Sanskrit work on the astrolabe, itself introduced in India during the reign of the 14th century Tughlaq dynasty ruler Firuz Shah Tughluq (1351–1388 CE). Suri seems to have been a Jain astronomer in the service of Firuz Shah Tughluq. The 182 verse Yantra-rāja mentions the astrolabe from the first chapter onwards, and also presents a fundamental formula along with a numerical table for drawing an astrolabe although the proof itself has not been detailed. Longitudes of 32 stars as well as their latitudes have also been mentioned. Mahendra Suri also explained the Gnomon, equatorial co-ordinates, and elliptical co-ordinates. The works of Mahendra Suri may have influenced later astronomers like Padmanābha (1423 CE)—author of the Yantra-rāja-adhikāra, the first chapter of his Yantra-kirnāvali.
Nilakanthan Somayaji	1444–1544 CE	In 1500, Nilakanthan Somayaji of the Kerala school of astronomy and mathematics, in his Tantrasangraha, revised Aryabhata's model for the planets Mercury and Venus. His equation of the centre for these planets remained the most accurate until the time of Johannes Kepler in the 17th century. Nilakanthan Somayaji, in his Aryabhatiyabhasya, a commentary on Aryabhata's Aryabhatiya, developed his own computational system for a partially heliocentric planetary model, in which Mercury, Venus, Mars, Jupiter and Saturn orbit the Sun, which in turn orbits the Earth, similar to the Tychonic system later proposed by Tycho Brahe in the late 16th century. Nilakantha's system, however, was mathematically more efficient than the Tychonic system, due to correctly taking into account the equation of the centre and latitudinal motion of Mercury and Venus. Most astronomers of the Kerala school of astronomy and mathematics who followed him accepted his planetary model. He also authored a treatise titled Jyotirmimamsa stressing the necessity and importance of astronomical observations to obtain correct parameters for computations.
Acyuta Pisārati	1550–1621 CE	Sphutanirnaya (Determination of True Planets) details an elliptical correction to existing notions. Sphutanirnaya was later expanded to Rāśigolasphutānīti (True Longitude Computation of the Sphere of the Zodiac). Another work, Karanottama deals with eclipses, complementary relationship between the sun and the moon, and 'the derivation of the mean and true planets'. In Uparāgakriyākrama (Method of Computing Eclipses), Acyuta Pisārati suggests improvements in methods of calculation of eclipses.

Instruments used

Sawai Jai Singh (1688–1743 CE) initiated the construction of several observatories. Shown here is the Jantar Mantar (Jaipur) observatory.

 

Yantra Mandir (completed by 1743 CE), Delhi.

 

Astronomical instrument with graduated scale and notation in Hindu-Arabic numerals.

 

Detail of an instrument in the Jaipur observatory.

Among the devices used for astronomy was gnomon, known as Sanku, in which the shadow of a vertical rod is applied on a horizontal plane in order to ascertain the cardinal directions, the latitude of the point of observation, and the time of observation. This device finds mention in the works of Varāhamihira, Āryabhata, Bhāskara, Brahmagupta, among others. The Cross-staff, known as Yasti-yantra, was used by the time of Bhaskara II (1114–1185 CE). This device could vary from a simple stick to V-shaped staffs designed specifically for determining angles with the help of a calibrated scale. The clepsydra (Ghatī-yantra) was used in India for astronomical purposes until recent times. Ōhashi (2008) notes that: "Several astronomers also described water-driven instruments such as the model of fighting sheep."

The armillary sphere was used for observation in India since early times, and finds mention in the works of Āryabhata (476 CE). The Goladīpikā—a detailed treatise dealing with globes and the armillary sphere was composed between 1380–1460 CE by Parameśvara. On the subject of the usage of the armillary sphere in India, Ōhashi (2008) writes: "The Indian armillary sphere (gola-yantra) was based on equatorial coordinates, unlike the Greek armillary sphere, which was based on ecliptical coordinates, although the Indian armillary sphere also had an ecliptical hoop. Probably, the celestial coordinates of the junction stars of the lunar mansions were determined by the armillary sphere since the seventh century or so. There was also a celestial globe rotated by flowing water."

An instrument invented by the mathematician and astronomer Bhaskara II (1114–1185 CE) consisted of a rectangular board with a pin and an index arm. This device—called the Phalaka-yantra—was used to determine time from the sun's altitude. The Kapālayantra was an equatorial sundial instrument used to determine the sun's azimuth. Kartarī-yantra combined two semicircular board instruments to give rise to a 'scissors instrument'. Introduced from the Islamic world and first finding mention in the works of Mahendra Sūri—the court astronomer of Firuz Shah Tughluq (1309–1388 CE)—the astrolabe was further mentioned by Padmanābha (1423 CE) and Rāmacandra (1428 CE) as its use grew in India.

Invented by Padmanābha, a nocturnal polar rotation instrument consisted of a rectangular board with a slit and a set of pointers with concentric graduated circles. Time and other astronomical quantities could be calculated by adjusting the slit to the directions of α and β Ursa Minor. Ōhashi (2008) further explains that: "Its backside was made as a quadrant with a plumb and an index arm. Thirty parallel lines were drawn inside the quadrant, and trigonometrical calculations were done graphically. After determining the sun's altitude with the help of the plumb, time was calculated graphically with the help of the index arm."

Ōhashi (2008) reports on the observatories constructed by Jai Singh II of Amber:

The Mahārāja of Jaipur, Sawai Jai Singh (1688–1743 CE), constructed five astronomical observatories at the beginning of the eighteenth century. The observatory in Mathura is not extant, but those in Delhi, Jaipur, Ujjain, and Banaras are. There are several huge instruments based on Hindu and Islamic astronomy. For example, the samrāt.-yantra (emperor instrument) is a huge sundial which consists of a triangular gnomon wall and a pair of quadrants toward the east and west of the gnomon wall. Time has been graduated on the quadrants.

The seamless celestial globe invented in Mughal India, specifically Lahore and Kashmir, is considered to be one of the most impressive astronomical instruments and remarkable feats in metallurgy and engineering. All globes before and after this were seamed, and in the 20th century, it was believed by metallurgists to be technically impossible to create a metal globe without any seams, even with modern technology. It was in the 1980s, however, that Emilie Savage-Smith discovered several celestial globes without any seams in Lahore and Kashmir. The earliest was invented in Kashmir by Ali Kashmiri ibn Luqman in 1589–90 CE during Akbar the Great's reign; another was produced in 1659–60 CE by Muhammad Salih Tahtawi with Arabic and Sanskrit inscriptions; and the last was produced in Lahore by a Hindu metallurgist Lala Balhumal Lahuri in 1842 during Jagatjit Singh Bahadur's reign. 21 such globes were produced, and these remain the only examples of seamless metal globes. These Mughal metallurgists developed the method of lost-wax casting in order to produce these globes.

Global discourse

Greek equatorial sun dial, Ai-Khanoum, Afghanistan 3rd–2nd century BCE.

Indian and Greek astronomy

The earliest known Indian astronomical work (though it is restricted to calendrical discussions) is the Vedanga Jyotisha of Lagadha, which is dated to 1400–1200 BCE (with the extant form possibly from 700–600 BCE). According to Pingree, there are a number of Indian astronomical texts that are dated to the sixth century CE or later with a high degree of certainty. There is substantial similarity between these and pre-Ptolomaic Greek astronomy. Pingree believes that these similarities suggest a Greek origin for certain aspects of Indian astronomy.

The Pingree – van der Waerden hypothesis in the history of astronomy holds that Indian texts from the 7th century reflect Greek astronomy of the first century. These texts represent information not available in Western libraries. Naturally the hypothesis is rejected by historians attributing originality to the Indian authors of these texts. First proposed by Bartel van der Waerden, it has been thoroughly argued by David Pingree.

With the rise of Greek culture in the east, Hellenistic astronomy filtered eastwards to India, where it profoundly influenced the local astronomical tradition. For example, Hellenistic astronomy is known to have been practiced near India in the Greco-Bactrian city of Ai-Khanoum from the 3rd century BCE. Various sun-dials, including an equatorial sundial adjusted to the latitude of Ujjain have been found in archaeological excavations there. Numerous interactions with the Mauryan Empire, and the later expansion of the Indo-Greeks into India suggest that transmission of Greek astronomical ideas to India occurred during this period. The Greek concept of a spherical earth surrounded by the spheres of planets, further influenced the astronomers like Varahamihira and Brahmagupta.

Several Greco-Roman astrological treatises are also known to have been exported to India during the first few centuries of our era. The Yavanajataka was a Sanskrit text of the 3rd century CE on Greek horoscopy and mathematical astronomy. Rudradaman's capital at Ujjain "became the Greenwich of Indian astronomers and the Arin of the Arabic and Latin astronomical treatises; for it was he and his successors who encouraged the introduction of Greek horoscopy and astronomy into India."

Later in the 6th century, the Romaka Siddhanta ("Doctrine of the Romans"), and the Paulisa Siddhanta ("Doctrine of Paul") were considered as two of the five main astrological treatises, which were compiled by Varāhamihira in his Pañca-siddhāntikā ("Five Treatises"), a compendium of Greek, Egyptian, Roman and Indian astronomy. Varāhamihira goes on to state that "The Greeks, indeed, are foreigners, but with them this science (astronomy) is in a flourishing state." Another Indian text, the Gargi-Samhita, also similarly compliments the Yavanas (Greeks) noting that the Yavanas though barbarians must be respected as seers for their introduction of astronomy in India.

Indian and Chinese astronomy

Indian astronomy reached China with the expansion of Buddhism during the Later Han (25–220 CE). Further translation of Indian works on astronomy was completed in China by the Three Kingdoms era (220–265 CE). However, the most detailed incorporation of Indian astronomy occurred only during the Tang Dynasty (618–907 CE) when a number of Chinese scholars—such as Yi Xing— were versed both in Indian and Chinese astronomy. A system of Indian astronomy was recorded in China as Jiuzhi-li (718 CE), the author of which was an Indian by the name of Qutan Xida—a translation of Devanagari Gotama Siddha—the director of the Tang dynasty's national astronomical observatory.

Fragments of texts during this period indicate that Arabs adopted the sine function (inherited from Indian mathematics) instead of the chords of arc used in Hellenistic mathematics. Another Indian influence was an approximate formula used for timekeeping by Muslim astronomers. Through Islamic astronomy, Indian astronomy had an influence on European astronomy via Arabic translations. During the Latin translations of the 12th century, Muhammad al-Fazari's Great Sindhind (based on the Surya Siddhanta and the works of Brahmagupta), was translated into Latin in 1126 and was influential at the time.

Indian and Islamic astronomy

In the 17th century, the Mughal Empire saw a synthesis between Islamic and Hindu astronomy, where Islamic observational instruments were combined with Hindu computational techniques. While there appears to have been little concern for planetary theory, Muslim and Hindu astronomers in India continued to make advances in observational astronomy and produced nearly a hundred Zij treatises.  Humayun built a personal observatory near Delhi, while Jahangir and Shah Jahan were also intending to build observatories but were unable to do so. After the decline of the Mughal Empire, it was a Hindu king, Jai Singh II of Amber, who attempted to revive both the Islamic and Hindu traditions of astronomy which were stagnating in his time. In the early 18th century, he built several large observatories called Yantra Mandirs in order to rival Ulugh Beg's Samarkand observatory and in order to improve on the earlier Hindu computations in the Siddhantas and Islamic observations in Zij-i-Sultani. The instruments he used were influenced by Islamic astronomy, while the computational techniques were derived from Hindu astronomy.

Indian astronomy and Europe

Some scholars have suggested that knowledge of the results of the Kerala school of astronomy and mathematics may have been transmitted to Europe through the trade route from Kerala by traders and Jesuit missionaries. Kerala was in continuous contact with China, Arabia and Europe. The existence of circumstantial evidence such as communication routes and a suitable chronology certainly make such a transmission a possibility. However, there is no direct evidence by way of relevant manuscripts that such a transmission took place.

In the early 18th century, Jai Singh II of Amber invited European Jesuit astronomers to one of his Yantra Mandir observatories, who had bought back the astronomical tables compiled by Philippe de La Hire in 1702. After examining La Hire's work, Jai Singh concluded that the observational techniques and instruments used in European astronomy were inferior to those used in India at the time - it is uncertain whether he was aware of the Copernican Revolution via the Jesuits. He did, however, employ the use of telescopes. In his Zij-i Muhammad Shahi, he states: "telescopes were constructed in my kingdom and using them a number of observations were carried out".

Following the arrival of the British East India Company in the 18th century, the Hindu and Islamic traditions were slowly displaced by European astronomy, though there were attempts at harmonising these traditions. The Indian scholar Mir Muhammad Hussain had travelled to England in 1774 to study Western science and, on his return to India in 1777, he wrote a Persian treatise on astronomy. He wrote about the heliocentric model, and argued that there exists an infinite number of universes (awalim), each with their own planets and stars, and that this demonstrates the omnipotence of God, who is not confined to a single universe. Hussain's idea of a universe resembles the modern concept of a galaxy, thus his view corresponds to the modern view that the universe consists of billions of galaxies, each one consisting of billions of stars. The last known Zij treatise was the Zij-i Bahadurkhani, written in 1838 by the Indian astronomer Ghulam Hussain Jaunpuri (1760–1862) and printed in 1855, dedicated to Bahadur Khan. The treatise incorporated the heliocentric system into the Zij tradition.

Abacus

From Wikipedia, the free encyclopedia

A Chinese abacus, Suanpan

 

Calculating-Table by Gregor Reisch: Margarita Philosophica, 1503. The woodcut shows Arithmetica instructing an algorist and an abacist (inaccurately represented as Boethius and Pythagoras). There was keen competition between the two from the introduction of the Algebra into Europe in the 12th century until its triumph in the 16th.

The abacus (plural abaci or abacuses), also called a counting frame, is a calculating tool that was in use in Europe, China and Russia, centuries before the adoption of the written Hindu–Arabic numeral system. The exact origin of the abacus is still unknown. Today, abacuses are often constructed as a bamboo frame with beads sliding on wires, but originally they were beans or stones moved in grooves in sand or on tablets of wood, stone, or metal.

Abacuses come in different designs. Some designs, like the bead frame consisting of beads divided into tens, are used mainly to teach arithmetic, although they remain popular in the post-Soviet states as a tool. Other designs, such as the Japanese soroban, have been used for practical calculations even involving several digits. For any particular abacus design, there usually are numerous different methods to perform a certain type of calculation, which may include basic operations like addition and multiplication, or even more complex ones, such as calculating square roots. Some of these methods may work with non-natural numbers (numbers such as 1.5 and ​³⁄₄).

Although today many use calculators and computers instead of abacuses to calculate, abacuses still remain in common use in some countries. Merchants, traders and clerks in some parts of Eastern Europe, Russia, China and Africa use abacuses, and they are still used to teach arithmetic to children. Some people who are unable to use a calculator because of visual impairment may use an abacus.

Etymology

The use of the word abacus dates before 1387 AD, when a Middle English work borrowed the word from Latin to describe a sandboard abacus. The Latin word came from Greek ἄβαξ abax which means something without base, and improperly, any piece of rectangular board or plank. 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 (for the use of mathematics)" (the exact shape of the Latin perhaps reflects the genitive form of the Greek word, ἄβακoς abakos). Whereas the table strewn with dust definition is popular, there are those that do not place credence in this at all and in fact state that it is not proven. Greek ἄβαξ itself is probably a borrowing of a Northwest Semitic, perhaps Phoenician, word akin to Hebrew ʾābāq (אבק), "dust" (or in post-Biblical sense meaning "sand used as a writing surface").

The preferred plural of abacus is a subject of disagreement, with both abacuses and abaci (hard "c") in use. The user of an abacus is called an abacist.

History

Mesopotamian

The period 2700–2300 BC 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.

Some scholars point to a character from the Babylonian cuneiform which may have been derived from a representation of the abacus. It is the belief of Old Babylonian scholars such as Carruccio that Old 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".

Egyptian

The use of the abacus in Ancient Egypt is mentioned by the Greek historian Herodotus, who writes 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, wall depictions of this instrument have not been discovered.

Persian

During the Achaemenid Empire, around 600 BC the Persians first began to use the abacus. 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, when it is thought to have been exported to other countries.

Greek

An early photograph of the Salamis Tablet, 1899. The original is marble and is held by the National Museum of Epigraphy, in Athens.

The earliest archaeological evidence for the use of the Greek abacus dates to the 5th century BC. Also Demosthenes (384 BC–322 BC) talked of the need to use pebbles for calculations too difficult for your head. A play by Alexis from the 4th century BC mentions an abacus and pebbles for accounting, and both Diogenes and Polybius mention 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 saw 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 AD (the Salamis Tablet), dates back to 300 BC, making it the oldest counting board discovered so far. It is a slab of white marble 149 cm (59 in) long, 75 cm (30 in) wide, and 4.5 cm (2 in) 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 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.

Chinese

A Chinese abacus (suanpan) (the number represented in the picture is 6,302,715,408)

The earliest known written documentation of the Chinese abacus dates to the 2nd century BC.

The Chinese abacus, known as the suanpan (算盤, lit. "calculating tray"), is typically 20 cm (8 in) tall and 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. The beads are usually rounded and made of a 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. 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.

Suanpan 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. There are currently schools teaching students how to use it.

In the long scroll Along the River During the Qingming Festival painted by Zhang Zeduan during the Song dynasty (960–1297), a suanpan is clearly visible beside an account book and doctor's prescriptions on the counter of an apothecary's (Feibao).

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 abacuses may be coincidental, both ultimately arising from counting with five fingers per hand. Where the Roman model (like most modern Korean and Japanese) has 4 plus 1 bead per decimal place, the standard suanpan has 5 plus 2. (Incidentally, this allows use with a hexadecimal numeral system, which was used for traditional Chinese measures of weight.) Instead of running on wires as in the Chinese, Korean, and Japanese models, the beads of Roman model run in grooves, presumably making arithmetic calculations much slower.

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, allowing them to acquire the concept of zero and the decimal point from Indian merchants and mathematicians.

Roman

Copy of a 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. Due to Pope Sylvester II's reintroduction of the abacus with modifications, it became widely used in Europe once again during the 11th century This abacus used beads on wires, unlike the traditional Roman counting boards, which meant the abacus could be used much faster.

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 here 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 –five units, five tens etc., essentially in 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" (i.e. fractions).

Indian

The decimal number system invented in India replaced the abacus in Western Europe.

The Abhidharmakośabhāṣya of Vasubandhu (316-396), a Sanskrit work on Buddhist philosophy, says that the second-century CE philosopher Vasumitra said that "placing a wick (Sanskrit vartikā) on the number one (ekāṅka) means it is a one, while placing the wick on the number hundred means it is called a hundred, and on the number one thousand means it is a thousand". It is unclear exactly what this arrangement may have been. Around the 5th century, Indian clerks were already finding new ways of recording the contents of the Abacus. Hindu texts used the term śūnya (zero) to indicate the empty column on the abacus.

Japanese

Japanese soroban

In Japanese, the abacus is called soroban (算盤, そろばん, lit. "Counting tray"), imported from China in the 14th century. It was probably in use by the working class a century or more before the ruling class started, as the class structure did not allow for devices used by the lower class to be adopted or used by the ruling class. The 1/4 abacus, which is suited to decimal calculation, appeared circa 1930, and became widespread as the Japanese abandoned hexadecimal weight calculation which was still common in China. The abacus is still manufactured in Japan today even with the proliferation, practicality, and affordability of pocket electronic calculators. The use of the soroban is still taught in Japanese primary schools as part of mathematics, primarily as an aid to faster mental calculation. Using visual imagery of a soroban, one can arrive at the answer in the same time as, or even faster than, is possible with a physical instrument.

Korean

The Chinese abacus migrated from China to Korea around 1400 AD. Koreans call it jupan (주판), supan (수판) or jusan (주산).

Native American

Representation of an Inca quipu

 

A yupana as used by the Incas.

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. The word Nepōhualtzintzin [nepoːwaɬˈt͡sint͡sin] comes from Nahuatl and it is formed by the roots; Ne – personal -; pōhual or pōhualli [ˈpoːwalːi] – the account -; and tzintzin [ˈt͡sint͡sin] – small similar elements. Its complete meaning was taken as: counting with small similar elements by somebody. Its use was taught in the Calmecac to the temalpouhqueh [temaɬˈpoʍkeʔ], who were students dedicated to take the accounts of skies, from childhood.

The Nepōhualtzintzin was divided in two main parts separated by a bar or intermediate cord. In the left part there were four beads, which in the first row have unitary values (1, 2, 3, and 4), and in the right side there are three beads with values of 5, 10, and 15 respectively. In order to know the value of the respective beads of the upper rows, it is enough to multiply by 20 (by each row), the value of the corresponding account in the first row.

Altogether, there were 13 rows with 7 beads in each one, which made up 91 beads in each Nepōhualtzintzin. This was a basic number to understand, 7 times 13, a close relation conceived between natural phenomena, the underworld and the cycles of the heavens. One Nepōhualtzintzin (91) represented the number of days that a season of the year lasts, two Nepōhualtzitzin (182) is the number of days of the corn's cycle, from its sowing to its harvest, three Nepōhualtzintzin (273) is the number of days of a baby's gestation, and four Nepōhualtzintzin (364) completed a cycle and approximate a year (11/4 days short). When translated into modern computer arithmetic, the Nepōhualtzintzin amounted to the rank from 10 to the 18 in floating point, which calculated stellar as well as infinitesimal amounts with absolute precision, meant that no round off was allowed.

The rediscovery of the Nepōhualtzintzin was due to the Mexican engineer David Esparza Hidalgo, who in his wanderings throughout Mexico found diverse engravings and paintings of this instrument and reconstructed several of them made in gold, jade, encrustations of shell, etc. There have also been found very old Nepōhualtzintzin attributed to the Olmec culture, and even some bracelets of Mayan origin, as well as a diversity of forms and materials in other cultures.

George I. Sanchez, "Arithmetic in Maya", Austin-Texas, 1961 found another base 5, base 4 abacus in the Yucatán Peninsula that also computed calendar data. This was a finger abacus, on one hand 0, 1, 2, 3, and 4 were used; and on the other hand 0, 1, 2 and 3 were used. Note the use of zero at the beginning and end of the two cycles. Sanchez worked with Sylvanus Morley, a noted Mayanist.

The quipu of the Incas was a system of colored 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 Italian mathematician Nicolino De Pasquale. 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 a minimum.

Russian

Russian abacus

The Russian abacus, the schoty (счёты), usually has a single slanted deck, with ten beads on each wire (except one wire, usually positioned near the user, with four beads for quarter-ruble fractions). 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, 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 are of a different color from the other eight beads. Likewise, the left bead of the thousands wire (and the million wire, if present) may have a different color.

As a simple, cheap and reliable device, 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 until the 1990s. Even the 1874 invention of mechanical calculator, Odhner arithmometer, had not replaced them in Russia and likewise the mass production of Felix arithmometers since 1924 did not significantly reduce their use in the Soviet Union. The Russian abacus began to lose popularity only after the mass production of microcalculators had started in the Soviet Union in 1974. Today it is regarded as an archaism and replaced by the handheld calculator.

The Russian abacus was brought to France around 1820 by the mathematician Jean-Victor Poncelet, who served in Napoleon's army and had been a prisoner of war in Russia. The abacus had fallen out of use in western Europe in the 16th century with the rise of decimal notation and algorismic methods. 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 also used abacuses similar to the Russian schoty. It was named a coulba by the Turks and a choreb by the Armenians.

School abacus

Early 20th century abacus used in Danish elementary school.

 

A twenty bead rekenrek

Around the world, abacuses 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.

The wire frame may be used either with positional notation like other abacuses (thus the 10-wire version may represent numbers up to 9,999,999,999), or each bead may represent one unit (so that e.g. 74 can be represented by shifting all beads on 7 wires and 4 beads on the 8th wire, so numbers up to 100 may be represented). In the bead frame shown, the gap between the 5th and 6th wire, corresponding to the color change between the 5th and the 6th bead on each wire, suggests the latter use.

The red-and-white abacus is used in contemporary primary schools for a wide range of number-related lessons. The twenty bead version, referred to by its Dutch name rekenrek ("calculating frame"), is often used, sometimes on a string of beads, sometimes on a rigid framework.

Renaissance abacuses gallery

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Uses by the blind

An adapted abacus, invented by Tim Cranmer, 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 cube root.

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 mathematical skills that can never be replaced with talking calculators and is an important learning tool for blind students. Blind students also complete mathematical assignments using a braille-writer and Nemeth code (a type of braille code for mathematics) but large multiplication and long division problems can be long and difficult. The abacus gives blind and visually impaired students a tool to compute mathematical 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.

Binary abacus

Two binary abacuses constructed by Dr. Robert C. Good, Jr., made from two Chinese abaci

The binary abacus is used to explain how computers manipulate numbers. The abacus shows how numbers, letters, and signs can be stored in a binary system on a computer, or via ASCII. The device consists of a series of beads on parallel wires arranged in three separate rows. The beads represent a switch on the computer in either an "on" or "off" position.