Everything about Babylonian Mathematics totally explained
Babylonian mathematics refers to any mathematics of the peoples of
Mesopotamia (Ancient
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 while 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
, 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.
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's regular, and therefore to a finite sexagesimal representation for the fraction in the first column. Neugebauer's explanation is the one followed for example 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 doesn't explain why the table entries are in the order they're listed in, nor what the numbers in the first column were used for.
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.
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. This tablet describes a method for solving what we'd 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 =
v31/2, 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.
Neo-Babylonian mathematics (626-539 BC)
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.
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.
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.
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, for example,
26 February 747 BC.
This raw material by itself must have been hard to use, and no doubt the Chaldeans themselves compiled extracts of for example, 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):
- 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.
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.
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's 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's 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.
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 didn't 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.
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.
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.
Babylonian mathematics in Alexandria
Islamic mathematics in MesopotamiaFurther Information
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