Babylonian mathematics

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...

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 scarcity 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 ${\sqrt {2}}$ accurate to nearly six decimal places.

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 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).

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.

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.

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

$ab={\frac {(a+b)^{2}-a^{2}-b^{2}}{2}}$

$ab={\frac {(a+b)^{2}-(a-b)^{2}}{4}}$

to simplify multiplication.

The Babylonians did not have an algorithm for long division. Instead they based their method on the fact that

${\frac {a}{b}}=a\times {\frac {1}{b}}$

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.

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

${\frac {1}{13}}={\frac {7}{91}}=7\times {\frac {1}{91}}\approx 7\times {\frac {1}{90}}=7\times {\frac {40}{3600}}.$

Algebra

As well as arithmetical calculations, Babylonian mathematicians also developed algebraic methods of solving equations. Once again, these were based on pre-calculated tables.

To solve a quadratic equation the Babylonians essentially used the standard quadratic formula. They considered quadratic equations of the form

$\ x^{2}+bx=c$

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

$x=-{\frac {b}{2}}+{\sqrt {\left({\frac {b}{2}}\right)^{2}+c}}$

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

$\ ax^{3}+bx^{2}=c.$

Multiplying the equation by a² and dividing by b³ gives

$\left({\frac {ax}{b}}\right)^{3}+\left({\frac {ax}{b}}\right)^{2}={\frac {ca^{2}}{b^{3}}}.$

Substituting y = ax/b gives

$y^{3}+y^{2}={\frac {ca^{2}}{b^{3}}}$

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.

The ancient Babylonians had known of theorems on the ratios of the sides of similar triangles for many centuries, but they lacked the concept of an angle measure and consequently, studied the sides of triangles instead.

The Babylonian astronomers kept detailed records on the rising and setting of stars, the motion of the planets, and the solar and lunar eclipses, all of which required familiarity with angular distances measured on the celestial sphere.

Plimpton 322

Main article: Plimpton 322

In p and q are two coprime numbers, then $(p^{2}-q^{2},\,2pq,\,p^{2}+q^{2})$ 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.

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 would nowadays describe as quadratic equations of the form $x-{\tfrac {1}{x}}=c$, 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.

Influence

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, i.e., 26 February 747 BC.

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):

• 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 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.

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.

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.

See also

• Babylonia
• History of mathematics
• Babylonian astronomy

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". {{cite journal}}: Cite journal requires |journal= (help)
• 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.
• Eleanor Robson, Words and pictures: New light on Plimpton 322, The American Mathematical Monthly. Washington: Feb 2002. Vol. 109, Iss. 2; pg. 105
• Toomer, G. J., Hipparchus and Babylonian Astronomy, (1981).

External links

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