Mathematics and Astronomy in Mesopotamia

views updated






















B. L. van der Waerden

MATHEMATICS. The Sources. In 1935-1937, Otto Neugebauer published all mathematical cuneiform texts then known in three volumes entitled Mathematische Keilschriftterte (MKT). They contain photographs, transcriptions, and German translations of all texts, with a very good commentary. A later publication by F. Thureau-Dangin contains new transcriptions and French translations of most texts published in Neugebauer’s MKT, with a new, valuable commentary.

In 1945, Neugebauer and Sachs published another collection. Mathematical Cuneiform Texts (MCT), supplementing Neugebauer’s MKT. It contains newly discovered texts and corrections to earlier publications. More recently E. M. Bruins and M. Rutten produced a volume entitled Textes mathématiques de Suse (1960, in which cuneiform texts from Susa were published and discussed.

All our knowledge concerning Babylonian mathematics is ultimately derived from these sources. The majority of the texts are either Old Babylonian (dynasty of Hammurapi. probably 1830–1531 B.C.) or Hellenistic (Seleucid era, beginning 311 B.C .).

Babylonian arithmetic, algebra, and geometry were already fully developed under the dynasty of Hammurapi. Mathematical astronomy came into existence much later. Therefore. Babylonian mathematics can be studied without any attention to astronomy.


E.M. Bruins and M. Rutten, Textes mathématiques de Suse, in the series Mémoires de la Mission Archéologique en Iran (Paris. 1961).

O. Neugebauer, Mathematische Keilschrifttexte, 3 vols., which is Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. A, 3 (Berlin. 1935–1937). Cited as MKT.

O. Neugebauer and A. Sachs, Mathematical Cuneiform Texts, which is American Oriental Series. 29 (New Haven, 1945). Cited as MCT.

F. Thureau-Dangin. Textes mathématiques babyloniens (Leiden, 1938).

The Sexagesimal System , Before 2000 B.C. the Sumerians. the inventors of cuneiform script, who lived in southern Mesopotamia, used a positional system with base 60 for writing integers and fractions. “Positional” means that the value of a numeral depends on where it stands. The symbol for I, a vertical wedge, can denote any power of 60 or any power of 1/60. Integers up to 59 were written in the decimal system. Thus, 21 would be written as , the symbols for 10 and 1 being repeated as often as necessary. Beyond 59, integers were written as sums of multiples of powers of 60. Thus, 80 = 60 + 20 would be written as .

The normal way of writing fractions was to express them as multiples of 1/60, or of 1/602, and so on. The denominator 60n was not written: it had to be inferred from the context. Thus, 1 1/2 would be written as 1,30; and the reader would have to determine from the context whether I 1/2 or 90 was meant, or even 90 X 60n or 1 1/2 X 60-n.

After 2000 B.C. the Babylonian and Assyrian scribes inherited this system from the Sumerians. together with cuneiform script, which they adapted to their Semitic languages.

In transcribing Sumerian or Babylonian numerals, I shall follow the method introduced by O. Neugebauer, separating successive sexagesimal places by a comma:

1,30 = 60 + 30 = 90.

If it is known that the first unit has the value 1, integers and fractions will he separated by a semicolon:

On the other hand, if it is known that the first unit has the value 602. I shall write



O. Neugebauer, “Zur Entstehung des Sexagesimalsystems.” in Abhandlungen der Akademie der Wissenschaften zu Göttingen, n.s. 13 (1927), 1–55.

F. Thureau-Dangin , Esquisse d’une histoire du système sexagésimal (Paris. 1932).

Methods of Calculation . The Sumerians and Babylonians performed such arithmetical operations as multiplication, division, squaring, and extracting square and cube roots by means of tables. The Sumerians already had multiplication tables, each table containing the multiples of a given “head number,” For instance, if the head number is 1,30. the table would contain the items

1,30 times 1

times 2


times 19

times 20

times 30

times 40

times 50

All other multiples of 1,30 were obtained by adding items found in the table.

Other tables contain reciprocals. The reciprocal of an integer is a finite sexagesimal fraction, provided the integer contains no prime factors other than 2, 3, and 5. Such integers may be called “regular.” A normal table of reciprocals would give the reciprocals of all regular integers from 2 to 1,21 as follows:

1:2 = 30

3 20


1,20 45

1,21 44,26,40.

From the Hellenistic era we have large tables giving the reciprocals of regular numbers up to seven sexagesimal places (see Neugebauer, MKT, 1, 16).

The usual procedure to divide b by c was to take c-1 from a table of reciprocals and to calculate b c-1by means of a multiplication table. There are multiplication tables headed by numbers that are the reciprocals c-1 of simple regular integers. Other tables contain approximate reciprocals of irregular numbers. We also have tables of squares, of cubes. and of square roots and cube roots. Thus, even extensive calculations presented no difficulties for the Sumerian and Babylonian scribes.


The structure of the system of tables for multiplication, division. and squaring was first described in a series of papers by O. Neugebauer: “Sexagesimalsystem und babylonische Bruchrechnung I-IV,” in Quellen und Studien zur Geschichte der Mathematik. Astronomie und Physik. Abt. B, 1 and 2 (1930–1932). For further details see MKT and MCT.

Babylonian Algebra . Babylonian algebra was concerned mainly with the solution of equations. The methods for solving equations were developed in the Old Babylonian period, and the same methods were still used in the Hellenistic age.

Equations and methods of solution were always formulated in words. An unknown x was often called “the side” and its square, “the square.” When two unknowns occurred, they were usually called “length” and “breadth,” and their product, “area.” Three unknowns were sometimes called “length,” “breadth,” and “height,” and their product, “volume.”

Despite this geometric terminology, the Babylonians did not hesitate to subtract a side from an area. Thus, the quadratic equation

(1) x2-x=14,30

was formulated as follows in the text BM (British Museum) 13901; “I have subtracted the side of the square from the area, and the result is 14,30.” (See Neugebauer, MKT, III,6, prob, 2.)

The rules for solving equations were always explained as operations on definite numerical coefficients, but in such a way that the general rule is easy to recognize. For example, equation (1) was solved as follows:

Take 1, the coefficient (of the linear term).
Take one-half of l (result, 0;30),
Multiply 0;30 by 0;30 (result, 0;15).
Add 0;15 to 14.30. Result, 14.30;15.
The square root of this is 29;30.
Take the 0;30, which you have multiplied by itself, and add it to 29;30.
Result, 30; this is the [required side of the] square.

Of course, the procedure is equivalent to the modern formula for solving the equation

(2) x2-ax=b,



A complete discussion of all types of equations and all methods of solution occurring in Babylonian texts was given by H. Goetsch. My account is based mainly on his paper.

Six types of equations were solved.

First, there were linear equations in one unknown.

Second, there were systems of linear equations in two or three unknowns. The usual method was to solve one of the equations for one of the unknowns and substitute it into the other equations. In some cases, however, a different method was used. Given the equation

(4) x + y = a

and another (linear or nonlinear) equation for x and y. If x and y were equal, each of them would be a/2. Now suppose .x exceeds a/2 by an unknown amount, s:


Then for y we must take


Substituting these two expressions into equation (4), one obtains an equation for s, which can be solved by Babylonian methods if it is linear or quadratic. Let us call this method the “plus and minus” method. Diophantus often used it if a sum of two unknown numbers x + y was given.

A good example of the “plus and minus” method is found in the Old Babylonian text VAT 8389 (see Neugebauer. MKT, I, 323). In this text a pair of equations of the form

x + y = a

bx -cy = d

is not solved by eliminating one of the two unknowns, but by the “plus and minus” method. The unknowns .x and y are calculated simultaneously according to equations (5) and (6), while s is found as the solution of a linear equation.

In a linear and a quadratic equation, the following standard types occur frequently:

(I) x + y = a, xy = b

(II) x - y = a, xy = b

(III)x + y = a, x2 + y2 = b

(IV)x - y = a, x2y2 = b

All four types could have been solved by the method of substitution, solving .x or y from the linear equation and substituting it into the quadratic equation. This method would first have yielded one of the two unknowns, and the other would be found by a formula such as y = ax y = xa. No case is known in which such formulas were used. It appears that in cases (1) and (III), in which x + y is known, the “plus and minus” method was used, for x and y are regularly found as x = a/2 +y = a/2— s, while s is calculated as a square root.

In cases (II) and (IV). where xy = a is given, a similar method can be used. One can write



where s is a new unknown. In case (II) as well as in case (IV), one finds a pure quadratic equation for s, from which .s can be obtained as a square root. The Babylonian expressions for.x and y always have the forms of (7) and (8). Diophantus also used this method, as S. Gandz has noted.

Very often, other types of equations were reduced to the standard forms (I, II, III. IV). Thus, in the text AO 8862 (Neugebauer, MKT, I. 113) the pair of equations

xy + (xy) = 3,3

x + y = 27

was reduced to the standard form (I) by adding the two equations and writing

y = y-2.

(For other examples of nonlinear systems of equations, see the paper by Goetsch. 126–141.)

Among single quadratic equations, two types occur frequently:

(9) x2ax = b

(10) x2 + ax = b

The positive root of such an equation is always found by a procedure equivalent to our solution formula, well known since al–Khwārizmī. Most probably the Babylonians found their solution by adding (a/2)2 to both sides of the equation and applying the formulas

with which the Babylonians were just as familiar as the Greeks and Arabs.

The third type used by al–Khwārizmī,

x2 + b = ax,

is not found in Babylonian texts.

Pairs of quadratic equations are discussed by Goetsch, who presents two examples (103, 141; for the first example, see also Neugebauer, MKT, I, 486; and van der Waerden, Science Awakening, I, 70–71), The two equations in this example can be written as

0;20(x+y)-0;1(x-y)2 = 15

xy =10,0.

Cubic equations occur in the following types:

(12) x3 = a

(13) x2(x + 1) = a

(14) x(10 - x) = a

The Babylonians had tables of cube roots, by means of which they could solve equations of type (12). They also had a table giving numbers n2(n +1) in the first column and their “roots” n (from I to 60) in the second column (Neugebauer, MKT, I. 76), By means of such a table they could solve equations of type (13). We do not know how they solved (14), but they did give the solution

x-=6for a = 2;48

(see Goetsch, 149).

Numbers of the form n2(n - 1) with their “roots” also were tabulated (see Thureau-Dangin, Textes mathématiquesbabyloniens, 123). The neo-Pythagorean Nicomachus of Gerasa (first century) had a special name for the numbers n2(n +1) and n2(n -1): arithmoi paramekepipedoi. (See O. Becker’s paper in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. B, 4 [19381. 181.)

The Babylonians certainly were familiar with such rules of elementary algebra as

(15) (a+b) (a-b)=a2-b2

(16) (a+b)2 =a2+2ab+b2

(17) (a-b)2 =a2- 2ab+b2

The Greeks and Arabs proved such formulas by drawings of rectangles and squares, and it is possible that the Babylonians also derived these formulas by means of drawings,


Cuneiform texts dealing with algebraic equations were first discussed by O. Neugebauer, H. Waschow. F, ThureLiu-Dangin. K. Vogel, and others between 1929 and 1936. Full quotations of these pioneer papers can be found in Neugebauer, MKT.

Later books and papers include the following:

S. Gandz. “The Origin and Development of Quadratic Equations in Babylonian, Greek and Early Arab Algebra,” in Osiris, 3 (1937), 405–543.

H. Goetsch, “Die Algebra der Babylonier,” in Archive for History of Exact Sciences, 5 (1968), 79– 153.

O. Neugebauer, The Exact Sciences in Antiquity, 2nd ed. (Providence, R. I., 1957).

O. Neugebauer, Vorgriechische Mathematik (Berlin, 1934; repr. Heidelberg. 1969).

Babylonian Arithmetic . From several texts we know that the Babylonians were able to calculate the sum of an arithmetical progression. The problem discussed in these texts is to distribute a sum of money among a number of men according to an arithmetical progression.

In the text AO 6484 (Neugebauer, MKT, 1, 96– 107) we find the summation of a geometrical progression the ratio of which is 2:

1+2+4+…+ 29 = 29 + (29-l).

In the same text the sum of the squares of the integers from 1 to 10 is computed according to the formula


See the publications mentioned in the bibliography to “The Sources.”

Babylonian Geometry . Areas of triangles and trapeziums were calculated by the correct formulas


respectively. The area of a circle with radius r was calculated as 3r2, and its perimeter as 6r. The volumes of prisms and cylinders were determined by multiplying the area of the base by the height.

In several Old Babylonian texts, volumes of frustums of cones and pyramids were calculated by the incorrect formula

in which h is the height and A and B are the areas of bottom and top. It is remarkable that at about the same time, the Egyptians used the correct formula for calculating the volume of a pyramid frustum on a square base:

If a trapezium is divided into two parts of equal areas by a line parallel to the base, the length x of the dividing line is given by the formula


in which a and h are the two parallel sides. The Babylonians knew this formula (see Neugebauer, MKT, I, 131).

In the text VAT 8512 (Neugebauer. MKT, I, 342), one of the problems is accompanied by a drawing showing a triangle that is divided into a triangle and a trapezium by a line parallel to the base. The problem solved in the text concerns the areas of the two parts. Neugebauer was able to explain the solution, assuming that the Babylonians knew about the proportionality of the sides of the two similar triangles. Another, more geometrical explanation of the same text, based upon formula (18), was given by P. Huber.

In the texts from Susa published by E. M. Bruins and M. Rutten, one finds approximate calculations of the sides of regular polygons.


In addition to the publications mentioned in the bibliography to “The Sources.” see P. Huber, “Zu einem mathematischen Keilschrifttext (VAT 8512)” in Isis, 46 (1955). 104–106.

On the texts from Susa, see E. M. Bruins and M Rutten (cited under “The Sources”) and E. M. Bruins. “Quelques textes mathématiques de la mission de Suse,” in Proceedings K. Nederlandse akademie van wetenschüppen, 53 (1950), 1025–1033.

On the Egyptian pyramid formula, see W. W. Struve, “Mathematischer Papyrus des… Museums… in Moskau,” which is Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik, Abt. A, I (1930), 134–145.

The “Theorem of Pythagoras.” ; Several texts, from the Old Babylonian as well as from the Hellenistic period, show clearly that the Babylonian mathematicians knew the formula

(19) d2 = h2+b2

for the sides of a right triangle. They calculated b as the square root of d2 -b2. (For examples see Neugebauer, MAT, II, 53; and III, 22.)

The Old Babylonian text Plimpton 322, published by Neugebauer and Sachs (MCT, 38–41), contains an extensive table of “Pythagorean triples—of rational triples h, b, d satisfying equation (19).


E. M. Bruins, “Reciprocals and Pythagorean Triads.” in Physis, 9 (1967), 373–392.

P. Huber, “Bemerkungen über mathematische Keilschrifttexte,” in Enseignement mathématique, 2nd ser., 3 (1957), 19–27.

D. Price, “The Babylonian ‘Pythagorean Triangle’ Tablet,” in Centaurus, 10 (1964), 219–231.

B. L. van der Waerden, Science awakening, I (Groningen, 1954; New York, 1961), 78–80.

Babylonian Influence on Greek Mathematics . We have already seen that the Babylonians had tables of numbers n2(n ± 1) and that the neo-Pythagorean Nicomachus had a special name for these numbers, We also have seen that the “plus and minus” method for solving equations in two unknowns x and y. of which the sum x +y or difference x - y is given, was used by the Babylonians, as well as by Diophantus of Alexandria.

The four standard types of pairs of equations in two unknowns recur in Euclid’s Elements. Theorems 5–6 and 9–10 of book II are the theorems needed for the geometrical solution of the Babylonian standard types. As Neugebauer was the first to see, the single steps in the Greek geometrical solutions are exactly the same as the steps in the algebraic solutions of the Babylonians. Whenever the Babylonians say “Take the square root of A,” the Greeks say “Take the side of the square of area equal to A” and so on. It is known that theorems 5–6 and 9–10 of book II of the Elements are due to the Pythagoreans.

In Pythagorean geometry a fundamental theorem is the “theorem of Pythagoras.” The Babylonians also knew and used this theorem. They had methods to calculate “Pythagorean triples” —triples of integers ,x, y, z satisfying the equation

z2 = x2 + y2.

The Greeks also had methods for calculating such triples. The most general method, regularly used by Diophantus. is given by the equations

x = p2 - q2

y = 2 pq

z = p2 + q2

According to Neugebauer and Sachs (MCT, 38–41). it is quite possible that the Babylonians used the same formulas.

All these facts seem to indicate a Babylonian influence in Greek geometry and arithmetic. According to a Greek tradition. Pythagoras went to Babylon and learned the science of numbers, the science of music, and the other sciences from the magi. This may well be true.


The first to draw attention to the relations between Babylonian and Greek mathematics was Otto Neugebauer in the series of papers “Studien zur Geschichte der antiken Algebra.” in Quellen und Studien zur Geschichte der Mathematik, Astronomie und Physik. Abt, B. 2 and 3 (1933–1936). See also B. L. van der Waerden, Science Awakening. 1.66–101. 116–127.

For the spread of Babylonian algebra and geometry to Egypt, see R. A. Parker, Demotic Mathematical Papyri, Brown Egyptological Studies, Vll (Providence. R.I.London, 1972).

ASTRONOMY. Four Stages of Babylonian Astronomy . The history of Babylonian astronomy is much clearer today than it was about 1930. This is due mainly to O. Neugebauer’s thorough analysis of Babylonian mathematical astronomy and to the publication and interpretation of observational texts by A. Sachs. We are now in a position to recognize four distinct periods:

The first, or Old Babylonian, period coincides with the reign of the dynasty of Hammurapi (most probably 1830– 1531 B.C.).

The second period encompassed the Kassite reign, followed by Assyrian domination (1530–612). It ended with the destruction of Nineveh and its great library in 612 B.C.

The third, or Neo-Babylonian, period spanned the Chaldaean dynasty (611–540).

The fourth period saw the highest development of observational and mathematical astronomy under the Persian, Seleucid, and Arsacid reigns (539 B.C.-A.D. 75).

Each of these four clearly separated periods produced a particular type of astronomy connected with a particular type of astrology.


The first important publications on Babylonian astronomy were the following:

J. Epping. Astronomisches aus Babylon (Freiburg im Breisgau, 1889).

F. X. Kugler. Babylonische Mondrechnung (Freiburg im Breisgau, 1900).

F. X. Kugler. Sternkunde und Sterndienst in Babel, 2 vols., 3 supps. (Münster. 1907– 1935), 3rd supp. written by J. Schaumberger.

Indispensable for all serious students of the subject are three standard publications:

O. Neugebauer, Astronomical Cuneiform Texts. 3 vols. (London, 1955).

A. Sachs, Late Babylonian Astronomical and Related Texts (Providence. R.I., 1955).

O. Neugebauer. A History of Ancient Mathematical Astronomy, 3 vols. (Heidelberg-New York. 1975).

A summary of the entire subject is in B. I., van der Waerden, Sciene e Awakening, 11, The Birth of Astronomy (Leiden-New York, 1974).

Astronomy and Astrology in the Old Babylonian Period . From the Old Babylonian period we have just one astrological and one astronomical (observational) text. The astrological text, published by V. Shileyko in 1927, contains predictions based on the state of the sky on the day when the crescent just becomes visible, at the beginning of a new year. The predictions are of the following kind:

  1. If the sky is dark, the year will be bad.
  2. If the face of the sky is bright when the new moon appears and [it is greeted] with joy, the year will be good.
  3. If the north wind blows across the face of the sky before the new moon, the corn will grow abundantly.

The astronomical text, of which several copies exist, is a list of dates for the first appearance and disappearance of Venus as an evening star or morning star during the twenty-one years of the reign of King Ammizaduga. This reign began just 160 years after that of Hammurapi. The observations were connected with astrological predictions. I shall quote one example from the tenth year: “If on the tenth of Arahsamna. Venus disappeared in the east, remaining absent two months and six days, and was seen on the sixteenth of Tebitu in the West, the harvest of the land will be successful.”

If the astrological predictions (which may or may not be later additions) are ignored, we are left with a sequence of observations of the appearance and disappearance of Venus during twenty-one years. These observations may be used to date the reign of Ammizaduga. Within reasonable limits, four chronological hypotheses are possible: one “long chronology,” two “middle chronologies, and one “short chronology.” According to P. Huber’s careful investigation of all dates, the “short chronology” is much more probable than the others: Hammurapi reigned from 1728 to 1686. Ammizaduga from 1582 to 1562, and the dynasty lasted from 1830 to 1531 B.C.

In Babylonian astronomical texts, most names of stars and constellations are Sumerian. For example:

mulgu an na(bull of the sky) = Taurus
mulur gu la (lion or lioness) = Leo
mulgir tab(scorpion) = Scorpio.

According to Willy Hartner one may conclude from this fact that the Sumerians already regarded these constellations as representing a bull, a lion. and a scorpion respectively.

Hartner also has drawn attention to the numerous pictorial representations of a Lion-Bull combat in ancient Near Eastern art. These pictures show a remarkable constancy from the fourth millennium B.C. (Elamite and Sumerian pictures) through the Achaemenid period up to a Persian miniature from the Mogul period (see the photographs in Hartner’s paper). In many of these pictures, stars and other celestial symbols are visible, which seems to indicate that the lion and the bull were meant to represent not just animals, but the constellations Leo and Taurus. Thus, the Sumerian bull attacked by a lion in Hartner’s Figure 2 carries a star rosette in his horns.

For the geographical latitude of Persepolis (30°) and for 4000 B.C. Hartner has calculated the dates of heliacal rising or Morningfirst of the Pleiades (in Taurus), of Regulus (the main star of Leo), of Antares (the main star of Scorpio), and of β Pegasi (the first rising star of the Pegasus quadrangle). These four easily recognizable stars are nearly 90° apart in the zodiacal circle. Hartner found that these four heliacal rising dates coincide exactly with the spring equinox, the summer solstice, the autumn equinox, and the winter solstice for 4000 B.C. This would explain why the Sumerians regarded Taurus, Leo, and Scorpio as important constellations and why pictorial representations often show two or three of these animals.

Hartner explains the Lion-Bull fight as representing the situation of the sky at nightfall on the date of Eveninglast (heliacal setting) of the Pleiades, when the sun is just 20° below the horizon. About 4000 B.C. this date was 10 February (Gregorian), and in the time of the Achaemenids (500 B.C.) it was 28 March. In any case Leo was standing in the zenith and thus was displaying its greatest power, while the Bull was just disappearing below the horizon.


W. Hartner, “The Earliest History of the Constellations in the Near East and the Motif of the Lion-Bull Combat.” in Journal of Near Eastern Studies, 24 (1965), 1– 16, and pi. I-XVI,

On the astrology of the Old Babylonian period see the following:

T. Bauer. “Eine Sammlung von Himmelsvorzeichem,” in Zeitschrift für Assyríologie. 43 (1936). 308–314.

V. Shileyko, “Mondlaufprognosen aus der Zeit der ersten babylonischen Dynastie.” in Doklady Akademii nauk SSSR(1927),ser. B, 125–128.

The Venus tablets of Ammizaduga are discussed in the following:

P. Huber, “Early Cuneiform Evidence for the Planet Venus.” paper presented to the annual meeting of the American Association fur the Advancement of Science, San Francisco. Feb. 1974.

S. Langdon. J. K. Fotheringham, and C. Schoch, The Venus Tablets of Ammizaduga (Oxford, 1928).

B. L. van der Waerden, Die Anfänge der Astronomie (Basel. 1968).

J. D. Weir. The Venus Tablets of Ammizaduga (Istanbul—Leiden, 1972).

Second Period. We know much more of the second period because many reports and letters from the Assyrian court astrologers to their kings (written 722 – 612 B.C.) and several other texts from the same period are preserved. Among these texts are lists of fixed stars, circular diagrams of the sky, and many parts of the great omen collection “Enūma Anu Enlil.”


On the letters and reports of the Assyrian court astrologers see the following:

R. F. Harper, Assyrian and Babylonian Letters, 14 vols. (Chicago, 1892–1914).

S. Parpola, Letters From Assyrian Scholars, I (Kevelaer, Germany, 1970).

R. C. Thompson, The Reports of the Magicians and Astrologers…(London, 1900).

The Omen Collection “Enūma Anu Enlil.” ; An omen is a statement concerning the astrological signification of a phenomenon observed in the sky. An example of a very old omen is “If Venus appears in the east in the month Airu and the Great and Small Twins surround her, all four of them, and she is dark, then will the king of Elam fall sick and die” (see J. Schaumberger, supp. 3 to Kugler’s Sternkunde, 344).

The omen collection “Enūma Anu Enlil” consisted of seventy or more tablets, altogether containing some 7,000 omina. It probably was composed before 900 B.C. but was quoted extensively long afterward. If, for example, an Assyrian king wished to know whether the stars were favorable, the royal astronomer would observe the sky and consult the old omen series. He would then report his observation in a letter to the king, quote an appropriate omen, and explain its application to the observed situation.

Omen astrology, which prevailed in the first and second periods, differs in several respects from horoscope astronomy, which came into use in the fourth period. It is concerned mainly with matters of general interest, such as good or bad harvests, peace and war, and the health of the king: whereas the chief aim of horoscope astronomy is to predict the fate of individuals from the aspect of the sky at the moment of their birth or conception. Also, omen astrology does not use the division of the zodiac into twelve signs. In fact, in the numerous extant texts from the second period, zodiacal signs are never mentioned. The phenomena mentioned in the great omen collection are always visible phenomena that everyone can observe in the sky without any knowledge of zodiacal signs.

Between the old and the new astrology there is an intermediate stage, in which the zodiacal signs appear but birth horoscopes are not yet cast. In this “primitive zodiacal astrology” predictions are based either on the zodiacal sign in which Jupiter dwells or on the zodiacal sign in which the moon stays on the day of first visibility of Sirius. These predictions can be found in Greek texts ascribed to Orpheus or to Zoroaster. In Science Awakening, II, 180, I have tentatively given some arguments in favor of the hypothesis that this “primitive zodiacal astrology” originated in the third, or Neo-Babylonian, period.


On the series “Enūma Anu Enlil.” see the Following;

C. Virolleaud, L’astrologie chaldêenne (Paris. 1905–1912).

E, F. Weidner, “Die astrologische Serie Enuma Anu Enlil,” in Archiv fur orientforsckung, 14 (1943). 172–195.

On the astrological texts ascribed to Orpheus and Zoroaster see the following:

J. Bidez and F. Cumont. Lei mages hellénisés, 2 vols, (Paris, 1938), I, 107–127, and II, O37–O52.

B, L. van der Waerden, Science Awakening, II (Leiden-New York, 1974), 177–180.

The “Three Stars Each.” The modern, ill-chosen name “astrolabes” denotes a class of texts in which three stars are listed for every month of the year. The Babylonians had a better name for these lists: “the three stars each.” The thirty-six stars were divided into three sets of twelve: the “stars of Ea,” or northern stars: the “stars of Anu,” near the equator; and the “stars of Enlil,” or southern stars.

The texts state that the stars become visible only in the months to which they are assigned, but this is not always true. Three of the stars —Venus, Mars, and Jupiter —are actually planets.

There are astrolabes in circular and rectangular form (see Figures ] and 2). The circular form is probably older (see B. L. van der Waerden, Science Awakening, II, 67). The astrolabes were very popular in Babylonia and Assyria. Today we have astrolabes from Assur, Nineveh, Uruk (Erech), and Babylon. The oldest known, from Assur, was written about 1100 B.C.

Numbers are written on some of the astrolabes. In the outer ring of the circular astrolabe the numbers increase from 2, 0 to 4, 0 by equal increments of 20 and then decrease by equal amounts. The numbers in the middle ring are half, and those in the inner ring a quarter, of those in the outer rings. We have here an early example of a “linear zigzag function,” a function that increases linearly to a maximum and then decreases linearly to a minimum. In the mathematical astronomy of the fourth period, many astronomical quantities were represented by linear zigzag functions.

The numbers on the astrolabes reach their maximum in the summer and their minimum in the winter. They were supposed to be proportional to the duration of the day. Day and night were divided into three “watches” each, and the duration of the watches was regulated by means of water clocks. A certain amount of water was poured into a vessel, and its becoming empty determined the end of the watch. The numbers in the outer ring of the astrolabe were supposed to determine the weight of the water to be poured into the water clock. If this explanation, given by O. Neugebauer, is accepted, the numbers in the middle ring must refer to half-watches, and those in the inner ring to quarter-watches (twelfth parts of the light day).

This division of the day into twelve hours of varying duration confirms the statement of Herodotus (Histories. II, 109): “Polos and Gnomon and the twelve parts of the day did the Greeks learn from the Babylonians.”


O. Neugebauer. “The Water Clock in Babylonian Astronomy,” in Isis, 37 (1947). 37–43.

A. Schott, “Das Werden der babylonisch-assyrischen Positions-Astronomic” in Zeitschrift der Deutschen morgenländischen Gesellschaft. 88 (1934). 302–337.

The Compendium MUL. AP1N . About 700 B.C. or earlier, an astronomical compendium was composed, consisting of at least three tablets, of which several copies are preserved. One of these, found in Assur, is dated 687 B.C. Another copy bears the remark “copy from Babylon.” The series, which is named after its opening words, MUL.APIN, seems to be a compilation of all or nearly all astronomical knowledge of the period before 700 B.C.

The first tablet begins with a list of thirty-three stars of Enlil, twenty-three stars of Anu, and fifteen stars of Ea, with indications of their relative positions. With few exceptions, all stars of Anu lie in a zone between +17° and —17° declination. The “path of Enlil” is north of this zone, and the “path of Ea” is south of it.

Next comes a list of dates of first visibility of thirty-six fixed stars and constellations. In a separate list the date differences are given. Another list states that certain constellations rise while others set in their daily motion. Still another section names stars that culminate while others rise. Using these lists, C. Bezold, F. X. Kugler, and B. L. van der Waerden have identified many Babylonian constellations.

Another section contains a list of differences between the times of culmination of fixed stars. From a letter written by an astrologer we know that culminations were used to determine the exact time of eclipses. See J. Schaumberger. Zeitschrift für Assyriologie, 47 (1941), 127, and 50 (1942), 42.

The division of the zodiacal circle into twelve “zodiacal signs” of 30° each, which is fundamental for all texts of the fourth period, does not occur in MUL.APIN. Yet the author knows that the sun, the moon, and the five planets are always in a zone that he calls “the path of the moon,” which coincides with our “zodiacal belt.” The text enumerates some eighteen constellations and single stars in the path of the moon. Among these are twelve constellations bearing the same names as the zodiacal signs in later texts. For instance, the constellation Aries appears in MUL.APIN as LU-HUN-GA, and the star Spica is called ABSIN. In texts of the fourth period, LU HUN GA (often abbreviated to HUN) usually denotes the zodiacal sign Aries, a segment of 30° in the ecliptic; and AB-SIN denotes the sign Virgo as well as the star Spica. In a drawing from the Seleucid period, Virgo is represented as a virgin bearing an ear of corn (in Latin. spica.= ear of corn). On another drawing, URGULA (Leo) is drawn as a lion,

According to MUL.APIN, the sun dwells for three months in the “path of Enlil” in the summer, then for three months in the “path of Anu,” three months in the “path of Ea” in the winter, and again three months in the “path of Anu.” Since the three paths are bounded by circles parallel to the equator and since the orbit of the sun was supposed to cross all three paths, it follows that the author was aware of the obliquity of the zodiac.

The text also gives rules for calculating the shadow length of a vertical bar at different times of the year and of the day, and for calculating the times of rising and setting of the moon. Although crude and inaccurate, these rules represent a first attempt toward the calculation of celestial phenomena by means of rising and falling arithmetical sequences.

We find next to nothing in MUL.APIN on the course and periods of the planets. The text merely states that all planets move within the “path of the moon.”


B. L. van der Waerden, “The Thirty-Six Stars,” in Journal of Near Eastern Studies, 8 (1949). 6–26.

B. L. van der Waerden, Science Awakening, II (Leiden-New York, 19741,70–86.

E. F. Weidner. “Ein babylonisches Kompendium der Himmelskunde,” in American Journal of Semitic Languages and Literatures, 40 (1924), 186– 208.

Diaries and Eclipse Records . According to A. Sachs, cuneiform texts that contain records of daily observations are called “diaries.” There are more than 1,200 fragments of astronomical diaries, most of them at the British Museum. Usually a diary contains a record of lunar and planetary observations made during half of a Babylonian year. Not all entries in the diaries are records of actual observations. Some are accompanied by remarks like NU PAP (“not watched for”) or DIR NU PAP (“cloudy, not watched for”). Such notations often occur with predicted eclipses.

Most diaries are from the period 400–50 B.C., but a few go back to the Neo-Baby Ionian or even to the Assyrian period. The oldest datable fragment contains observations made in 652 B.C.

Among the planetary phenomena recorded in the diaries are conjunctions of planets with the moon and fixed stars, usually with an indication of the northern or southern distance, and dates of first and last visibility of planets. Among the lunar phenomena are, of course, lunar and solar eclipses. Regular observations also were made of six time intervals that A. Sachs calls the “lunar six.” In our texts these intervals are denoted by standard cuneiform signs, which may be read as

na, Šu, me, na, mi (or ge), kur.

The first of these was observed just after the new moon, on the evening of first visibility of the crescent. The next four were observed just before and after the full moon, and kur was observed on the day of last visibility of the moon in the morning. The meanings of the terms are given below:

na = time between sunset and moonset
Šu = time between moonset and sunrise
we = time between moon rise and sunset
na =time between sunrise and moonset
mi= time between sunset and moonrise
kur= time between moonrise and sunrise.

If bad weather prevented the observation, calculated values were inserted; we do not know how they were made.

Other items are the first and last visibility of Sirius, and equinoxes and solstices. These dates were nearly always calculated, not observed. (On the method of calculation see the paper by O. Neugebauer cited in the bibliography.) All sorts of meteorological events are reported in the diaries: rainbows. halos, thunder, rain, clouds, storms. Also mentioned are the prices of barley, dates, and wool in Babylon.

In other texts, observations of only one planet for several years or long sequences of eclipse records are assembled. These texts probably are compiled from diaries. The oldest texts of this kind contain detailed reports of lunar eclipses ranging over long periods: one text, for example, records eclipses from 731 to 317 B.C.: another goes back to the reign of the Babylonian king Nabonassar (747–734 B.C.).

Ptolemy informs us in book III of the Almagest that records of eclipse observations beginning with the reign of Nabonassar were still available to him. This statement is confirmed by the cuneiform texts.

The reports of the Assyrian court astrologers contain not only records of observed eclipses but also predictions of eclipses (see R. C. Thompson, The Reports of the Magicians). Report no, 273 reads: “On the fourteenth an eclipse of the moon will take place: woe for Elam and Amurru, good for my Lord, the King. Let the heart of my Lord the King rejoice.… An eclipse will take place. , . .” Report no. 274F seems to contain confirmation of the prediction: “… To my Lord the King I have written: An eclipse will take place. Now it has not gone by, it has taken place. In the occurrence of this eclipse lies happiness for my Lord the King.…”

We do not know by what method these predictions were made. (For other examples and for a discussion of possible methods, see B. L. van der Waerden, Science Awakening. II, 115– 122.)


A. Sachs, Late Babylonian Astronomical and Related Texts (Providence, R. 1.. 1955).

A. Sachs, “Babylonian Observational Astronomy,” in F. R. Hodson, ed.. The Place of Astronomy in the Ancient World (London. 1974), 43 –50

R. C. Thompson, The Reports of the Magicians and Astrologers (London, 1900).

On the computation of Sirius dates and equinoxes and solstices, see O. Neugebauer, “Solstices and Equinoxes in Babylonian Astronomy,” in Journal of Cuneiform Studies, 2(1948), 209–222.

The Neo-Babylonian Period . The most important astronomical text from this period is a diary for year 37 of Nebuchadnezzar II (568 B.C.). The text shows that more and more attention was paid to the course of the moon and the planets. Conjunctions of the moon and the planets with fixed stars were regularly noted, as were the dates of first and last visibility of planets and the “lunar six.”

For the date 4 July 568 B.C. an “eclipse of the moon that failed to occur” was recorded. The eclipse actually took place during the day, when the moon was invisible. We may conclude that the eclipse was predicted.

We have seen that in the MUL.APIN the zodiacal circle was divided into four parts corresponding to the four seasons and that the sun was supposed to dwell in each for three months. If each part is further divided into three parts corresponding to the months of the schematic year, one obtains a division of the circle into twelve zodiacal signs. Since every month of the schematic year has thirty days, it is natural to divide every zodiacal sign into thirty degrees. Thus, it is only a small step from the zodiacal schema of MUL.APIN to the division of the zodiac into twelve signs of thirty degrees each. There are reasons to suppose that this division was known and used for astrological purposes in the Neo-Babylonian period (see B. L. van der Waerden, Science Awakening II, 180).


P. V. Neugebauer and E. F. Weidner, “Ein astronomischer Beobachtungstext aus dem 37. Jahre Nebukadnezars II,” in Berichte, Sächsische Gesellschaft der Wissenschaften, phil.-hist. Kl.. 67 (1915). 29–89.

Babylonian Horoscopes . The greatest period of Babylonian astronomy was the Persian reign (539–333 B.C.), during which mathematical theories of the motion of the sun, the moon, and the five classical planets were developed. Historically, these theories are closely connected with horoscope astrology, because astrologers need methods for calculating solar, lunar, and planetary positions. As long as such methods did not exist, astrologers could not cast horoscopes, except in the very rare cases where they could observe the sky immediately after the birth of the child.

The oldest surviving cuneiform horoscope was cast for a date in 410 B.C. At that time lunar and planetary theories were already fully developed. Other Babylonian horoscopes were cast for 263, 258, 235, 230. and 142 B.C.


A. Sachs, “Babylonian Horoscopes.” in Journal of Cuneiform Studies, 6 (1952), 49–75.

Chaldaeans in the Persian Era and Later . During and after the Persian reign. Babylonian astronomers and astrologers were called Chaldaeans. According to Herodotus (Histories. I. 181), the Chaldaeans were priests in the temple of the highest god, Bel (=Marduk, or, as Herodotus calls him. Zeus-Belos), at Babylon.

When Alexander came to Babylon, two processions of priests met him: first the magi and then the Chaldaeans (Quintus Curtius Rufus l. Historia Alexandri Magni, 22). According to Diodorus, the Chaldaeans made astrological predictions for Alexander and his successors Antigonus and Seleucus.

The geographer Strabo makes a distinction between the Chaldaean people, who lived near the Persian Gulf, and the astronomers, who were called Chaldaeans and lived in a special quarter of Babylon. There were also Chaldaean astronomers at Uruk (Erech) and Borsippa. As far as Babylon and Uruk are concerned, Strabo’s statements are confirmed by cuneiform texts, for we have hundreds of astronomical cuneiform texts from Babylon and Uruk. Those from Babylon cover the period from 748 B.C. to A.D. 75, and those from Uruk range from ca. 230 to 160 B.C.

Strabo also mentions the names of famous Chaldaean astronomers: Kidenas. Naburianus, Sudines, and Seleucus of Seleucia. The first two are also known from cuneiform texts as Kidinnu and Naburimannu. According to B. L. van der Waerden (Science Awakening, II. 247–248. 281–283). Naburimannu probably lived under Cambyses (530–522) and Darius I (522–486). and Kidinnu under Artaxerxes I (465–424). In the texts mentioned above, the name Naburimannu is associated with the method of calculation of system A of lunar theory and Kidinnu with system B, (These two systems wilt be explained below.)

The other two astronomers mentioned by Strabo lived in the Hellenistic age. Sudines was astrologer at the court of Attalus I at Pergamum (241– 197), and Seleucus of Seleucia was a follower of the Greek astronomer Aristarchus of Samos (ca. 280 B.C.), who first proposed the heliocentric system.


B. L. van der Waerden, “The Date of Invention of Babylonian Planetary Theory.” in Archive for History of Exact Sciences, 5 (1968), 70–78.

B. L. van der Waerden, “Die ‘ Aegypter ’ und die ‘ Chaldaer, ’” in Sitzungsberichte der Heidelberger Akademie der Wissenschaften, math.-naturwiss. Kl. (1972), 197–227.

Systems A and B . For calculating positions of the sun, the moon, and the five planets the Babylonians had two kinds of theories, which Neugebauer calls system A and system B. To explain the characteristic differences between them, let us first consider the motion of the sun.

The fundamental unit of time in all lunar and planetary tables is the mean synodic month (slightly more than 29½12 days). According to system A, the motion of the sun in a certain part of the zodiacal circle was 30° per month, and in the remaining part 28°7’ 30” per month. In system B.on the other hand, the monthly motion of the sun was supposed to be a function of time, increasing by constant differences up to a maximum M and then decreasing in the same manner to a minimum m. Such a function is called a linear zigzag function (see Figure 3).

In both systems the velocity of the moon was supposed to be a linear zigzag function of lime. For the lunar latitude and the duration of daylight, more complicated piecewise linear functions were adopted, the graphs of which roughly approximate sine waves. Methods were developed that enabled the scribes to calculate, by using purely arithmetical operations, the longitude and latitude of the moon at any time, the time of the new moon and of the full moon, eclipse magnitudes, and times of rising and setting of the sun and the moon. These quantities were regularly calculated from month to month, some of them even from day to day.

In planetary theory the methods were quite similar. Planetary tables that I have called cardinal tables (Neugebauer calls them ephemerides) serve to calculate the longitudes and dates of certain characteristic points in the orbits of the planets called cardinal points. These are of three kinds:

1. The points of first and last visibility of the planets (to be denoted as Morningfirst, Eveninglast, and so on),

2. The stationary points, where the retrograde motion begins or ends.

3. The opposition to the sun (for outer planets).

The first element to be calculated was always the synodic arc, the increase in longitude of any planet from one cardinal point to the next of the same kind (for instance, from one Morningfirst to the next). In system A this synodic arc was supposed to have

different values in different parts of the zodiacal circle. For Jupiter the zodiacal circle was divided into two or four parts, and for Mars into six parts. In the case of Mars, each of the six parts consisted of two adjacent zodiacal signs: Pisces and Aries, Taurus and Gemini, and so on. Exactly the same division was used in Egyptian planetary tables of the first two centuries A.D. and in Hindu texts of the sixth century.

For computing the dates of cardinal points, the Babylonians applied a principle that I have called the sun-distance principle. It states that at each of the cardinal points, the planet has a fixed elongation from the sun. This is obvious for the oppositions, where the elongation is 180°. The Babylonians also assumed fixed elongations for the other cardinal points. For instance, the first and last visibilities of Mars were supposed to take place at elongations of 15°, and the first stationary point at 120°. (For details see B. L. van der Waerden. Science Awakening, II, ch. 6.)

For the motion of planets between cardinal points, the Babylonians, as well as the Egyptians and Hindus, used velocity schemata. A velocity schema for Jupiter on the “fast arc” of the zodiacal circle, drawn from a procedure text for system A, reads as follows:

After Morningfirst 30 days’ velocity 15’ per day
3 months to first stationary point 8’ per day
4 months retrograde 5’ per day
3 months after second stationary point 7’ 40” per day
30days to Eveninglast 15’ per day
30 days to Morningfirst 15’ per day

By means of such schemata, the Babylonians were able to calculate dates of entrance of planets into zodiacal signs. These dates were compiled, together with other calculated data, in so-called almanacs (see below) that probably were used for casting horoscopes.

In one text discussed by P. Huber. longitudes of Jupiter between Morningfirst in 165 B.C. and the next Eveninglast are calculated as terms of a third-order arithmetical progression (their third differences are constant).


The first to explain the methods of calculation of lunar tables were J. Epping and J. N. Strassmaier, followed by F. X. Kugler, who also explained planetary tables. All lunar and planetary tables known in 1955 were assembled and explained in the three-volume ACT by O. Neugebauer. (See the bibliography 10 “Four Stages of Babylonian Astronomy.”)

The earliest known lunar texts (system A) were published by A. Aaboe and A. Sachs. “Two Lunar Texts From the Achaemenid Period From Babylon,” in Centuurus. 14 (1969). 1–22.

Probably the oldest planetary tables are the dateless cardinal tables for Saturn, Jupiter, and Mars published by A. Aaboe and A. Sachs, “Some Dateless Computed Lists of Longitudes of Characteristic Planetary Phenomena…,” in Journal of Cuneiform Studies, 20 (1966), 1–33.

These dateless tables belong to system A. In B. L. van der Waerden’s “The Dale of Invention of Babylonian Planetary Theory,” in Archive for History of Exact Sciences, 5 (1968), 70–78, it was shown that the most probable dates for the first lines of the tables for Saturn and Mars are 511 and 499 B.C. Both dates are in the reign of Darius I.

The following papers have contributed to a better understanding of Babylonian planetary theory:

A. Aaboe, “On Period Relations in Babylonian Astronomy,” in Centaurus, [0 (1964), 213–231.

P. Huber, “Zur täglichen Bewegung des Jupiter,” in Zeitschrift für Assyriologie. n.s. 18(1957), 265–303.

B. L. van der Waerden, “Babylonische Planetenrechnung,” in Vierteljahresschrift der Naturforschenden Gesellschaft in Zürich. 102 (1957), 39–60,

A very good summary of the principles of Babylonian mathematical astronomy is given by A. Aaboe, “Scientific Astronomy in Antiquity,” in F. R. Hodson. ed., I he Place of Astronomy in the Ancient World (London, 1974), 21–42.

For a more extensive treatment, see O. Neugebauer, A History of Ancient Mathematical Astronomy. 3 vols. (Heidelberg-New York, 1975).

For the relation of Babylonian theories to Greek, Egyptian, and Hindu astronomy, see the following:

O, Neugebauer, “The Survival of Babylonian Methods in the Exact Sciences of Antiquity and Middle Ages,” in Proceedings of the American Philosophical Society, 107 (1963),528–535.

D. Pingree, “The Mesopotamian Origin of Early Indian Astronomy,” in Journal for the History of Astronomy, 4 (1973). 1– 12.

B. L. van der Waerden, Science Awakening, 11 (Leiden-New York, 1974), ch. 8,

Goal-Year Texts and Almanacs . The Babylonian scribes knew that the phenomena of the planet Venus are repeated after eight years —or. more precisely, after ninety-nine lunar months minus four days. Thus, if one has observed a Morninglast of Venus in the year x — 8 on the seventh day of a certain month, one may expect a Morninglast in the year x near the third day of a month, ninety-nine months minus 4 days after the month in which the earlier observation was made. Phenomena of Saturn are repeated after fifty-nine years, and so on. These facts were used by the Babylonians to obtain predictions of lunar and planetary phenomena.

In goal-year texts observational data from earlier years are assembled in order to obtain predictions for a certain “goal year.” By “observational data” 1 mean data of the kind found in diaries, without claiming that all these data were actually observed. The data include conjunctions of planets with fixed stars, the “lunar six,” and dates of first and last visibility of planets.

In a goal-year text for the year X, we find the following data:

for Jupiter; years x - 71 and x -83
for Venus: Year x -8
for Mercury: year x -46
for Saturn: years x -59
for Mars: years x -79 and x -47
for the moon: x -18 and x- 19

We have goal-year texts for many years between 240 and 20 B.C. There is also a text in which corrections are indicated, such as the four days to be subtracted from a Venus date in order to obtain the correct date in the goal year.

After having applied the necessary corrections, the Babylonian scribes obtained predictions for the most important lunar and planetary phenomena. These predictions were assembled, together with information from other sources, in an almanac for the goal year. Among the data from other sources are the dates of first and last visibility of Sirius and the equinoxes and solstices, which were calculated from year to year by simple rules.

Two kinds of almanacs are known, which A. Sachs calls normal-star almanacs and almanacs in the narrower sense. 1 n the former, conjunctions of planets with “normal stars” in the zodiac are mentioned, whereas the other almanacs note dates of entrance of planets into zodiacal signs. These dates, which were needed for casting horoscopes, probably were computed by means of velocity schemata.

Apart from these conjunctions and dates of entrance. the two types of almanacs are concerned with the same phenomena: the “lunar six,” cardinal points of planets, solstices and equinoxes, appearance and disappearance of Sirius, and predictions of possible eclipses.


A. Sachs, “A Classification of the Babylonian Astronomical Tablets of the Seleucid Period,” in Journal of Cuneiform Studies, 2 (1948), 271 – 290.

About this article

Mathematics and Astronomy in Mesopotamia

Updated About content Print Article


Mathematics and Astronomy in Mesopotamia