Babylonian clay tablet YBC 7289 with annotations. The diagonal displays an approximation of the square root of 2 in four sexagesimal figures, which is about six decimal figures.
1 + 24/60 + 51/60² + 10/60³ = 1.41421296...
(Image by Bill Casselman)

Babylonian mathematics refers to any mathematics of the peoples of Mesopotamia (situated in present day Iraq), from the days of the early Sumerians to the fall of Babylon in 539 BC. In contrast to the sparsity of sources in Egyptian mathematics, our knowledge of Babylonian mathematics is derived from some 400 clay tablets unearthed since the 1850s. Written in Cuneiform script, tablets were inscribed whilst the clay was moist, and baked hard in an oven or by the heat of the sun. The majority of recovered clay tablets date from 1800 to 1600 BC, and cover topics which include fractions, algebra, quadratic and cubic equations, the Pythagorean theorem, and the calculation of Pythagorean triples and possibly trigonometric functions (see Plimpton 322). The Babylonian tablet YBC 7289 gives an approximation to accurate to nearly six decimal places. Image File history File links No higher resolution available. ... Image File history File links No higher resolution available. ... The square root of 2 is equal to the length of the hypotenuse of a right triangle with legs of length 1. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... For other uses, see Decimal (disambiguation). ... For other uses, see Mesopotamia (disambiguation). ... Sumer (or Å umer) was the earliest known civilization of the ancient Near East, located in lower Mesopotamia (modern Iraq) from the time of the earliest records in the mid 4th millennium BC until the rise of Babylonia in the late 3rd millennium BC. The term Sumerian applies to all speakers... For other uses, see Babylon (disambiguation). ... This article or section is in need of attention from an expert on the subject. ... Babylonia was a state in southern Mesopotamia, in modern Iraq, combining the territories of Sumer and Akkad. ... “Cuneiform” redirects here. ... For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... This article is about the branch of mathematics. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ... Graph of a cubic polynomial: y = x3/4 + 3x2/4 âˆ’ 3x/2 âˆ’ 2 = (1/4)(x + 4)(x + 1)(x âˆ’ 2) In mathematics, a cubic equation is a polynomial equation in which the highest occurring power of the unknown is the third power. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ... The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nature. ...

Babylonian numerals

Main article: Babylonian numerals

The Babylonian system of mathematics was sexagesimal (base-60) numeral system. From this we derive the modern day usage of 60 seconds in a minute, 60 minutes in an hour, and 360 (60×6) degrees in a circle. The Babylonians were able to make great advances in mathematics for two reasons. Firstly, the number 60 is a Highly composite number, having divisors 1, 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30, facilitating calculations with fractions. Additionally, unlike the Egyptians and Romans, the Babylonians and Indians had a true place-value system, where digits written in the left column represented larger values (much as in our base ten system: 734 = 7×100 + 3×10 + 4×1). Babylonian numerals were written in cuneiform, using a wedge-tipped reed stylus to make a mark on a soft clay tablet which would be exposed in the sun to harden to create a permanent record. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ... A numeral is a symbol or group of symbols, or a word in a natural language that represents a number. ... A highly composite number is a positive integer which has more divisors than any positive integer below it. ... For other meanings of the word fraction, see fraction (disambiguation) A cake with one quarter removed. ... The place value system is a method of writing numbers with a base 10 numerical system. ...

Sumerian mathematics (3000-2300 BC)

The earliest evidence of written mathematics dates back to the ancient Sumerians, who built the earliest civilization in Mesopotamia. They developed a complex system of metrology from 3000 BC. From 2600 BC onwards, the Sumerians wrote multiplication tables on clay tablets and dealt with geometrical exercises and division problems. The earliest traces of the Babylonian numerals also date back to this period.^[1] Sumer (or Šumer) was the earliest known civilization of the ancient Near East, located in lower Mesopotamia (modern Iraq) from the time of the earliest records in the mid 4th millennium BC until the rise of Babylonia in the late 3rd millennium BC. The term Sumerian applies to all speakers... Metrology (from Greek metron (measure), and -logy) is the science of measurement. ... In mathematics, a multiplication table is a mathematical table used to define a multiplication operation for an algebraic system. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In mathematics, especially in elementary arithmetic, division is an arithmetic operation which is the inverse of multiplication. ...

Old Babylonian mathematics (2000-1600 BC)

The Old Babylonian period is the period to which most of the clay tablets on Babylonian mathematics belong, which is why the mathematics of Mesopotamia is commonly known as Babylonian mathematics. Some clay tablets contain mathematical lists and tables, others contain problems and worked solutions. The term Old Babylonian is a period in Mesopotamian history that refers, roughly, to the period between the end of the Third Dynasty of Ur (c. ...

Arithmetic

The Babylonians made extensive use of pre-calculated tables to assist with arithmetic. For example, two tablets found at Senkerah on the Euphrates in 1854, dating from 2000 BC, give lists of the squares of numbers up to 59 and the cubes of numbers up to 32. The Babylonians used the lists of squares together with the formulas 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. ... Surfer Rosa The Euphrates (IPA: /juːˈfreɪtiːz/; Greek: EuphrátÄ“s; Akkadian: Pu-rat-tu; Hebrew: פְּרָת PÄ•rāth; Syriac: Prâth; Arabic: الفرات Al-Furāt; Turkish: Fırat; Kurdish: فرهات, Firhat, Ferhat, Azeri: FÉ™rat) is the western of the two great rivers that define Mesopotamia (the other... 1854 (MDCCCLIV) was a common year starting on Sunday (see link for calendar). ... (Redirected from 2000 BC) (21st century BC - 20th century BC - 19th century BC - other centuries) (3rd millennium BC - 2nd millennium BC - 1st millennium BC) Events 2064 - 1986 BC -- Twin Dynasty wars in Egypt 2000 BC -- Farmers and herders travel south from Ethiopia and settle in Kenya. ... In mathematics, a square number, sometimes also called a perfect square, is an integer that can be written as the square of some other integer; in other words, it is the product of some integer with itself. ... In arithmetic and algebra, the cube of a number n is its third power — the result of multiplying it by itself two times: n3 = n × n × n. ...

to simplify multiplication.

The Babylonians did not have an algorithm for long division. Instead they based their method on the fact that In arithmetic, long division is a procedure for calculating the division of one integer, called the dividend, by another integer called the divisor, to produce a result called the quotient. ...

together with a table of reciprocals. Numbers whose only prime factors are 2, 3 or 5 (known as 5-smooth or regular numbers) have finite reciprocals in sexagesimal notation, and tables with extensive lists of these reciprocals have been found. The reciprocal function: y = 1/x. ... In number theory, the prime factors of a positive integer are the prime numbers that divide into that integer exactly, without leaving a remainder. ... In number theory, a positive integer m is called B-smooth if all prime factors of m are such that . For example, 22335654 is 5-smooth since none of its prime factors are greater than 5. ... A Hasse diagram of divisibility relationships among the regular numbers up to 400. ... The reciprocal function: y = 1/x. ...

Reciprocals such as 1/7, 1/11, 1/13, etc. do not have finite representations in sexagesimal notation. To compute 1/13 or to divide a number by 13 the Babylonians would use an approximation such as

Algebra

As well as arithmetical calculations, Babylonian mathematicians also developed algebraic methods of solving equations. Once again, these were based on pre-calculated tables. Elementary algebra is a fundamental and relatively basic form of algebra taught to students who are presumed to have little or no formal knowledge of mathematics beyond arithmetic. ... An equation is a mathematical statement, in symbols, that two things are the same (or equivalent). ...

To solve a quadratic equation the Babylonians essentially used the standard quadratic formula. They considered quadratic equations of the form In mathematics, a quadratic equation is a polynomial equation of the second degree. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

where here b and c were not necessarily integers, but c was always positive. They knew that a solution to this form of equation is

and they would use their tables of squares in reverse to find square roots. They always used the positive root because this made sense when solving "real" problems. Problems of this type included finding the dimensions of a rectangle given its area and the amount by which the length exceeds the width.

Tables of values of n³+n² were used to solve certain cubic equations. For example, consider the equation Cubic can mean several things: cubic polynomial, a polynomial with a degree of at most three. ...

'''Multiplying the equation by''' a² and dividing by b³ gives

Substituting y = ax/b gives

which could now be solved by looking up the n³+n² table to find the value closest to the right hand side. The Babylonians accomplished this without algebraic notation, showing a remarkable depth of understanding. However, they did not have a method for solving the general cubic equation.

Geometry

The Babylonians may have known the general rules for measuring areas and volumes. They measured the circumference of a circle as three times the diameter and the area as one-twelfth the square of the circumference, which would be correct if π is estimated as 3. The volume of a cylinder was taken as the product of the base and the height, however, the volume of the frustum of a cone or a square pyramid was incorrectly taken as the product of the height and half the sum of the bases. The Pythagorean theorem was also known to the Babylonians. Also, there was a recent discovery in which a tablet used π as 3 and 1/8. The Babylonians are also known for the Babylonian mile, which was a measure of distance equal to about seven miles today. This measurement for distances eventually was converted to a time-mile used for measuring the travel of the Sun, therefore, representing time.^[2] Lower-case pi The mathematical constant Ï€ is a real number which may be defined as the ratio of a circles circumference (Greek περιφέρεια, periphery) to its diameter in Euclidean geometry, and which is in common use in mathematics, physics, and engineering. ... In mathematics, the Pythagorean theorem (AmE) or Pythagoras theorem (BrE) is a relation in Euclidean geometry among the three sides of a right triangle. ...

Trigonometry

There is also evidence that the Babylonians first used trigonometric functions, based on a table of numbers written on the Babylonian cuneiform tablet, Plimpton 322 (circa 1900 BC), which can be interpreted as a table of secants.^[3] In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ... “Cuneiform” redirects here. ... Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nature. ...

Plimpton 322

Main article: Plimpton 322

In each row of the Plimpton 322 tablet, the number in the second column can be interpreted as the shortest side s of a right triangle, and the number in the third column can be interpreted as the hypotenuse d of the triangle. The number in the first column is either the fraction or , where l denotes the longest side of the same right triangle. However, scholars differ on how these numbers were generated and why the Babylonians would have been interested in such tables. Of the approximately half million clay tablets excavated at the beginning of the 19th century, about 400 are of a mathematical nature. ... A right triangle and its hypotenuse, h, along with catheti, c1 and c2. ...

Neugebauer (1951) argued for a number-theoretic interpretation, pointing out that this table provides a list of (pairs of numbers from) Pythagorean triples. For instance, line 11 of the table can be interpreted as describing a triangle with short side 3/4 and hypotenuse 5/4, forming the side:hypotenuse ratio of the familiar (3,4,5) right triangle. If p and q are two coprime numbers, then form a Pythagorean triple, and all Pythagorean triples can be formed in this way. For instance, line 11 can be generated by this formula with p = 1 and q = 1/2. As Neugebauer argues, each line of the tablet can be generated by a pair (p,q) that are both regular numbers, integer divisors of a power of 60. This property of p and q being regular leads to a denominator that is regular, and therefore to a finite sexagesimal representation for the fraction in the first column. Neugebauer's explanation is the one followed e.g. by Conway and Guy (1996). However, as Robson points out, Neugebauer's theory fails to explain how the values of p and q were chosen: there are 92 pairs of coprime regular numbers up to 60, and only 15 entries in the table. In addition, it does not explain why the table entries are in the order they are listed in, nor what the numbers in the first column were used for. Otto E. Neugebauer (May 26, 1899 – February 19, 1990) was an Austrian-American mathematician and historian of science who became known for his research on the history of astronomy and the other exact (i. ... Number theory is the branch of pure mathematics concerned with the properties of numbers in general, and integers in particular, as well as the wider classes of problems that arise from their study. ... The Pythagorean theorem: a2 + b2 = c2 A Pythagorean triple consists of three positive integers a, b, and c, such that a2 + b2 = c2. ... Two types of special right triangles appear commonly in geometry, the angle based and the side based triangles. ... Coprime - Wikipedia /**/ @import /skins-1. ... This article is being considered for deletion in accordance with Wikipedias deletion policy. ...

Joyce (1995) provides a trigonometric explanation: the values of the first column can be interpreted as the squared cosine or tangent (depending on the missing digit) of the angle opposite the short side of the right triangle described by each row, and the rows are sorted by these angles in roughly one-degree increments. However, Robson argues on linguistic grounds that this theory is "conceptually anachronistic": it depends on too many other ideas not present in the record of Babylonian mathematics from that time. Wikibooks has a book on the topic of Trigonometry All of the trigonometric functions of an angle θ can be constructed geometrically in terms of a unit circle centered at O. Trigonometry (from Greek trigōnon triangle + metron measure[1]), informally called trig, is a branch of mathematics that deals with... In mathematics, the trigonometric functions (also called circular functions) are functions of an angle. ...

Robson (2001,2002), based on prior work by Bruins (1949,1955) and others, instead takes an approach that in modern terms would be characterized as algebraic, though she describes it in concrete geometric terms and argues that the Babylonians would also have interpreted this approach geometrically. Robson bases her interpretation on another tablet, YBC 6967, from roughly the same time and place.^[4] This tablet describes a method for solving what we would nowadays describe as quadratic equations of the form , by steps (described in geometric terms) in which the solver calculates a sequence of intermediate values v₁ = c/2, v₂ = v₁², v₃ = 1 + v₂, and v₄ = v₃^1/2, from which one can calculate x = v₄ + v₁ and 1/x = v₄ - v₁. Robson argues that the columns of Plimpton 322 can be interpreted as the following values, for regular number values of x and 1/x in numerical order: v₃ in the first column, v₁ = (x - 1/x)/2 in the second column, and v₄ = (x + 1/x)/2 in the third column. In this interpretation, x and 1/x would have appeared on the tablet in the broken-off portion to the left of the first column. For instance, row 11 of Plimpton 322 can be generated in this way for x = 2. Thus, the tablet can be interpreted as giving a sequence of worked-out exercises of the type solved by the method from tablet YBC 6967. It could, Robson suggests, have been used by a teacher as a problem set to assign to students. This article is about the branch of mathematics. ... Calabi-Yau manifold Geometry (Greek γεωμετρία; geo = earth, metria = measure) is a part of mathematics concerned with questions of size, shape, and relative position of figures and with properties of space. ... In mathematics, a quadratic equation is a polynomial equation of the second degree. ...

Neo-Babylonian mathematics (626-539 BC)

Further information: Babylonian astronomy

The Neo-Babylonian Empire flourished during the Chaldean period of Mesopotamia, which marked the second flowering of Babylon as a capital city and center of study. This period provides the second source of Babylonian mathematics, though somewhat more vague than the Old Babylonian mathematics. Babylonian astronomy refers to the astronomy that developed in Mesopotamia, the land between the rivers Tigris and Euphrates, where the ancient kingdoms of Sumer, Assyria, Babylonia and Chaldea were located. ... Through the centuries of Assyrian domination, Babylonia enjoyed a prominent status, or revolting at the slightest indication that it did not. ... For other uses, see Chaldean. ... For other uses, see Babylon (disambiguation). ...

Since the rediscovery of the Babylonian civilization, it has become apparent that Greek and Hellenistic mathematicians and astronomers, and in particular Hipparchus, borrowed a lot from the Chaldeans. The term Hellenistic (established by the German historian Johann Gustav Droysen) in the history of the ancient world is used to refer to the shift from a culture dominated by ethnic Greeks, however scattered geographically, to a culture dominated by Greek-speakers of whatever ethnicity, and from the political dominance... For the Athenian tyrant, see Hipparchus (son of Pisistratus). ... Look up Chaldean in Wiktionary, the free dictionary. ...

Franz Xaver Kugler demonstrated in his book Die Babylonische Mondrechnung ("The Babylonian lunar computation", Freiburg im Breisgau, 1900) the following: Ptolemy had stated in his Almagest IV.2 that Hipparchus improved the values for the Moon's periods known to him from "even more ancient astronomers" by comparing eclipse observations made earlier by "the Chaldeans", and by himself. However Kugler found that the periods that Ptolemy attributes to Hipparchus had already been used in Babylonian ephemerides, specifically the collection of texts nowadays called "System B" (sometimes attributed to Kidinnu). Apparently Hipparchus only confirmed the validity of the periods he learned from the Chaldeans by his newer observations. Franz Xaver Kugler (born 1862 in Königsbach, Germany–died 1929 in Lucerne, Switzerland) was a chemist, mathematician, Assyriologist, and Jesuit priest. ... An ephemeris (plural: ephemerides) (from the Greek word ephemeros= daily) was, traditionally, a table providing the positions (given in a Cartesian coordinate system, or in right ascension and declination or, for astrologers, in longitude along the zodiacal ecliptic), of the Sun, the Moon, and the planets in the sky at... Kidinnu (also Kidunnu) (circa 400 BC – possibly 14 August 330 BC) was a Chaldean astronomer and mathematician. ...

It is clear that Hipparchus (and Ptolemy after him) had an essentially complete list of eclipse observations covering many centuries. Most likely these had been compiled from the "diary" tablets: these are clay tablets recording all relevant observations that the Chaldeans routinely made. Preserved examples date from 652 BC to AD 130, but probably the records went back as far as the reign of the Babylonian king Nabonassar: Ptolemy starts his chronology with the first day in the Egyptian calendar of the first year of Nabonassar, i.e., 26 February 747 BC. Centuries: 8th century BC - 7th century BC - 6th century BC Decades: 700s BC 690s BC 680s BC 670s BC 660s BC - 650s BC - 640s BC 630s BC 620s BC 610s BC 600s BC Events and Trends Occupation begins at Maya site of Piedras Negras, Guatemala 657 BC - Cypselus becomes the... For other uses, see number 130. ... Nabonassar (also Nabonasser, Nabu-nasir, Nebo-adon-Assur or Nabo-n-assar) was a king of Assyria, who founded the Chaldean and Babylonian kingdom. ... is the 57th day of the year in the Gregorian calendar. ... Centuries: 9th century BC - 8th century BC - 7th century BC Decades: 790s BC 780s BC 770s BC 760s BC 750s BC - 740s BC - 730s BC 720s BC 710s BC 700s BC 690s BC Events and Trends February 26 747 BC - Nabonassar becomes king of Assyria 747 BC - Meles becomes king...

This raw material by itself must have been hard to use, and no doubt the Chaldeans themselves compiled extracts of e.g., all observed eclipses (some tablets with a list of all eclipses in a period of time covering a saros have been found). This allowed them to recognise periodic recurrences of events. Among others they used in System B (cf. Almagest IV.2): A Saros cycle is a period of 6585 + 1/3 days (approximately 18 years 10 days and 8 hours) which can be used to predict eclipses of the sun and the moon. ...

• 223 (synodic) months = 239 returns in anomaly (anomalistic month) = 242 returns in latitude (draconic month). This is now known as the saros period which is very useful for predicting eclipses.
• 251 (synodic) months = 269 returns in anomaly
• 5458 (synodic) months = 5923 returns in latitude
• 1 synodic month = 29;31:50:08:20 days (sexagesimal; 29.53059413… days in decimals = 29 days 12 hours 44 min 3⅓ s)

The Babylonians expressed all periods in synodic months, probably because they used a lunisolar calendar. Various relations with yearly phenomena led to different values for the length of the year. The orbital period is the time it takes a planet (or another object) to make one full orbit. ... In Egyptian mythology, Month is an alternate spelling for Menthu. ... In Egyptian mythology, Month is an alternate spelling for Menthu. ... A Saros cycle is a period of 6585 + 1/3 days (approximately 18 years 10 days and 8 hours) which can be used to predict eclipses of the sun and the moon. ... This article is about astronomical eclipses. ... In Egyptian mythology, Month is an alternate spelling for Menthu. ... Look up Month in Wiktionary, the free dictionary. ... A lunisolar calendar is a calendar whose date indicates both the moon phase and the time of the solar year. ...

Similarly various relations between the periods of the planets were known. The relations that Ptolemy attributes to Hipparchus in Almagest IX.3 had all already been used in predictions found on Babylonian clay tablets. The eight planets and three dwarf planets of the Solar System. ...

All this knowledge was transferred to the Greeks probably shortly after the conquest by Alexander the Great (331 BC). According to the late classical philosopher Simplicius (early 6th century AD), Alexander ordered the translation of the historical astronomical records under supervision of his chronicler Callisthenes of Olynthus, who sent it to his uncle Aristotle. It is worth mentioning here that although Simplicius is a very late source, his account may be reliable. He spent some time in exile at the Sassanid (Persian) court, and may have accessed sources otherwise lost in the West. It is striking that he mentions the title tèresis (Greek: guard) which is an odd name for a historical work, but is in fact an adequate translation of the Babylonian title massartu meaning "guarding" but also "observing". Anyway, Aristotle's pupil Callippus of Cyzicus introduced his 76-year cycle, which improved upon the 19-year Metonic cycle, about that time. He had the first year of his first cycle start at the summer solstice of 28 June 330 BC (Julian proleptic date), but later he seems to have counted lunar months from the first month after Alexander's decisive battle at Gaugamela in fall 331 BC. So Callippus may have obtained his data from Babylonian sources and his calendar may have been anticipated by Kidinnu. Also it is known that the Babylonian priest known as Berossus wrote around 281 BC a book in Greek on the (rather mythological) history of Babylonia, the Babyloniaca, for the new ruler Antiochus I; it is said that later he founded a school of astrology on the Greek island of Kos. Another candidate for teaching the Greeks about Babylonian astronomy/astrology was Sudines who was at the court of Attalus I Soter late in the 3rd century BC. For the film of the same name, see Alexander the Great (1956 film). ... Centuries: 5th century BC - 4th century BC - 3rd century BC Decades: 380s BC 370s BC 360s BC 350s BC 340s BC - 330s BC - 320s BC 310s BC 300s BC 290s BC 280s BC Years: 336 BC 335 BC 334 BC 333 BC 332 BC - 331 BC - 330 BC 329 BC... Simplicius, a native of Cilicia, a disciple of Ammonius and of Damascius, was one of the last of the Neoplatonists. ... (5th century — 6th century — 7th century — other centuries) Events The first academy of the east the Academy of Gundeshapur founded in Persia by the Persian Shah Khosrau I. Irish colonists and invaders, the Scots, began migrating to Caledonia (later known as Scotland) Glendalough monastery, Wicklow Ireland founded... Callisthenes, or Kallisthenes, ( in Greek) of Olynthus (c. ... Aristotle (Greek: AristotélÄ“s) (384 BC – 322 BC) was a Greek philosopher, a student of Plato and teacher of Alexander the Great. ... Sassanid Empire at its greatest extent The Sassanid dynasty (also Sassanian) was the name given to the kings of Persia during the era of the second Persian Empire, from 224 until 651, when the last Sassanid shah, Yazdegerd III, lost a 14-year struggle to drive out the Umayyad Caliphate... Calippus of Syracuse Callippus (or Calippus) (ca. ... The Metonic cycle or Enneadecaeteris in astronomy and calendar studies is a particular approximate common multiple of the year (specifically, the seasonal tropical year) and the synodic month. ... is the 179th day of the year (180th in leap years) in the Gregorian calendar. ... Centuries: 5th century BC - 4th century BC - 3rd century BC Decades: 380s BC 370s BC 360s BC 350s BC 340s BC - 330s BC - 320s BC 310s BC 300s BC 290s BC 280s BC 335 BC 334 BC 333 BC 332 BC 331 BC - 330 BC - 329 BC 328 BC 327... The Julian calendar was introduced in 46 BC by Julius Caesar and came into force in 45 BC (709 ab urbe condita). ... A Proleptic calendar or era is that calendar extrapolated to dates prior the its first adoption. ... In the Battle of Gaugamela in 331 BC Alexander the Great of Macedonia defeated Darius III of Persia. ... Centuries: 5th century BC - 4th century BC - 3rd century BC Decades: 380s BC 370s BC 360s BC 350s BC 340s BC - 330s BC - 320s BC 310s BC 300s BC 290s BC 280s BC Years: 336 BC 335 BC 334 BC 333 BC 332 BC - 331 BC - 330 BC 329 BC... This article cites its sources but does not provide page references. ... Centuries: 4th century BC - 3rd century BC - 2nd century BC Decades: 330s BC 320s BC 310s BC 300s BC 290s BC - 280s BC - 270s BC 260s BC 250s BC 240s BC 230s BC 286 BC 285 BC 284 BC 283 BC 282 BC 281 BC 280 BC 279 BC 278... Silver coin of Antiochus I Antiochus I Soter ( 324/323_262/261 BC reigned 281 BC - 261 BC) was half Persian, his mother Apame being one of those eastern princesses whom Alexander had given as wives to his generals in 324 BC. On the assassination of his father Seleucus I in... Hand-coloured version of the anonymous Flammarion woodcut (1888). ... Port and city view of Kos town on the island Kos. ... For other uses, see Astronomy (disambiguation). ... Hand-coloured version of the anonymous Flammarion woodcut (1888). ... Sudines (Greek: Σουδινες) ca. ... Bust of Attalus I, circa 200 BCE (Pergamon Museum, Berlin) Attalus I Soter (Greek: Savior; 269 BC – 197 BC)[1] ruled Pergamon, a Greek polis in what is now Turkey, from 241 BC to 197 BC. He was the second cousin and the adoptive son of Eumenes I,[2] whom... The 3rd century BC started the first day of 300 BC and ended the last day of 201 BC. It is considered part of the Classical era, epoch, or historical period. ...

In any case, the translation of the astronomical records required profound knowledge of the cuneiform script, the language, and the procedures, so it seems likely that it was done by some unidentified Chaldeans. Now, the Babylonians dated their observations in their lunisolar calendar, in which months and years have varying lengths (29 or 30 days; 12 or 13 months respectively). At the time they did not use a regular calendar (such as based on the Metonic cycle like they did later), but started a new month based on observations of the New Moon. This made it very tedious to compute the time interval between events. “Cuneiform” redirects here. ... The Metonic cycle or Enneadecaeteris in astronomy and calendar studies is a particular approximate common multiple of the year (specifically, the seasonal tropical year) and the synodic month. ... The lunar phase depends on the Moons position in orbit around Earth. ...

What Hipparchus may have done is transform these records to the Egyptian calendar, which uses a fixed year of always 365 days (consisting of 12 months of 30 days and 5 extra days): this makes computing time intervals much easier. Ptolemy dated all observations in this calendar. He also writes that "All that he (=Hipparchus) did was to make a compilation of the planetary observations arranged in a more useful way" (Almagest IX.2). Pliny states (Naturalis Historia II.IX(53)) on eclipse predictions: "After their time (=Thales) the courses of both stars (=Sun and Moon) for 600 years were prophesied by Hipparchus, …". This seems to imply that Hipparchus predicted eclipses for a period of 600 years, but considering the enormous amount of computation required, this is very unlikely. Rather, Hipparchus would have made a list of all eclipses from Nabonasser's time to his own. The ancient civil Egyptian Calendar, known as the Annus Vagus or Wandering Year, had a year that was 365 days long, consisting of 12 months of 30 days each, plus 5 extra days at the end of the year. ... Thales of Miletos (, ca. ...

Other traces of Babylonian practice in Hipparchus' work are:

• first Greek known to divide the circle in 360 degrees of 60 arc minutes.
• first consistent use of the sexagesimal number system.
• the use of the unit pechus ("cubit") of about 2° or 2½°.
• use of a short period of 248 days = 9 anomalistic months.

This article describes the unit of angle. ... A minute of arc, arcminute, or MOA is a unit of angular measurement, equal to one sixtieth (1/60) of one degree. ... The sexagesimal (base-sixty) is a numeral system with sixty as the base. ...

Babylonian mathematics in Alexandria

Main articles: Greek mathematics and Diophantus

Further information: Babylonian influence on Greek astronomy

Greek mathematics, as that term is used in this article, is the mathematics written in Greek, developed from the 6th century BC to the 5th century AD around the Eastern shores of the Mediterranean. ... Title page of the 1621 edition of Diophantus Arithmetica, translated into Latin by Claude Gaspard Bachet de Méziriac. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

Islamic mathematics in Mesopotamia

Main article: Islamic mathematics

Further information: List of Iraqis

Islamic mathematics is the profession of Muslim Mathematicians. ... This is a list of Iraqis or people from Iraq who have been famous and/or have an article. ...

Notes

St. ...

References

• Berriman, A. E., The Babylonian quadratic equation (1956).
• Boyer, C. B., A History of Mathematics, 2nd ed. rev. by Uta C. Merzbach. New York: Wiley, (1989) ISBN 0-471-09763-2 (1991 pbk ed. ISBN 0-471-54397-7).
• Joseph, G. G., The Crest of the Peacock, Princeton University Press (October 15, 2000), ISBN 0-691-00659-8.
• Joyce, David E. (1995). "Plimpton 322". 
• Neugebauer, O., "Exact Sciences of Antiquity", Dover (1969).
• O'Connor, J. J. and Robertson, E. F., "An overview of Babylonian mathematics", MacTutor History of Mathematics, (December 2000).
• Robson, Eleanor (2001). "Neither Sherlock Holmes nor Babylon: a reassessment of Plimpton 322". Historia Math. 28 (3): 167–206. DOI:10.1006/hmat.2001.2317. MR1849797. 
• Toomer, G. J., Hipparchus and Babylonian Astronomy, (1981).

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See also

• Babylonia
• History of mathematics
• Babylonian influence on Greek astronomy

Babylonia was a state in southern Mesopotamia, in modern Iraq, combining the territories of Sumer and Akkad. ... For a timeline of events in mathematics, see timeline of mathematics. ... To meet Wikipedias quality standards, this article or section may require cleanup. ...

External links

• Babylonian Mathematics, with particular emphasis on Pythagorean triples.
• Photographs of YBC 7289, taken by Bill Casselman at the Yale Babylonian Collection