Pythagorean Knowledge In Ancient Babylonia

In an effort to broaden our understanding of ancient Egyptian mathematics, I briefly review what is known from some the ancient Babylonian texts. As I have shown, Mesopotamia had a broad influence in Egypt. See my papers:



Knowledge of this influence should provide us with more penetrating insight into the cross fertilization between the two cultures.

This map shows the general area of ancient Babylonia. It covered what is now mostly the modern state of Iraq. I do not mean to imply that influence from this region was limited to ancient Babylon. The cultural influence came from a broader Mesopotamian geographical area.

This region was the center of the Sumerian civilization from before 3500 BC. This advanced civilization built cities with irrigation systems, a legal system, a complex administration, and even a postal service. The development of writing predates our earliest knowledge. We do not know when it developed, likely before the people of the region resorted to cuneiform on clay tablets. As we know, counting was based on a sexagesimal system, that is to say base 60, that provided our modern sexagesimal time-keeping and geographical co-ordinate systems. Around 2300 BC, toward the end of the Old Kingdom in Egypt, Semitic Akkadians invaded the area and mixed with the more advanced culture of the Sumerians. During the 3rd Millennium BC, the Sumerians and the Semitic Akkadians lived peacefully together to create a common high civilization.

The Old Babylonian civilization replaced that of the Sumerians around 2000 BC. This was the period in Egypt that saw an interregnum of administration before establishment of the Middle Kingdom. Refer to my paper on the Mysterious Habiru:


This was also the period when the Semite Abraham moved from Ur to Canaan. Since Abraham was an educated man we should expect that he was aware of these mathematical developments in Mesopotamia and Egypt.

We should note the connection between the Akkadian and Babylonian (Chaldean) languages; they were mild dialects of the same Northwest Semitic tongue. The ancient name Akkadian is derived from the city-state of Akkad. Akkadia was founded by Sargon I who reigned from 2334 to 2279 BC. During those fifty-five years he created the world's first empire. The location of Akkadia is uncertain, although mentioned prevalently in the old texts. During his reign, the Akkadian language became the lingua franca of the region. The Biblical Shinar, home of the tribe of Terach, father of Abraham, about 2400 BC, was ancient Akkadia. The capital city of Babylon was Byblos, located about 80 km south of the present day Bagdhad.  The word babel comes originally from the Akkadian Bab-ilu meaning "gate of God".

Akkadian (or Babylonian-Assyrian) is the collective name for the spoken languages of the culture in Mesopotamia. Akkadian is the oldest known member of the Northwest family of Semitic languages. The principal subdivisions of this group are Canaanite, Ugaritic, and Aramaic. Canaanite included Phoenician, Moabite, and Hebrew.

The language is also preserved in inscriptions from ancient Phoenician colonies, especially Carthage, whose language was a variant of Phoenician known as Punic. The existence of Moabite is known from a single inscription in that language dating from about the 9th century BC, from proper names that occur in the Old Testament, and from the inscriptions of other peoples. The Ugaritic language was first encountered in 1929 at Ras Shamra, Syria, a village where ancient clay tablets with writing in this tongue were found. Since Ras Shamra, which flourished before the 12th century BC, was called Ugarit in antiquity, the language discovered there was named after that ancient city. The Ugaritic language has variously been regarded as an early form of Hebrew, an early form of Phoenician, an early dialect of Canaanite, and an independent dialect of Northwest Semitic. To Semitic scholars its classification is still unresolved. One can easily see why, since it was nearly the same as Hebrew and Phoenician.

To show how pervasive this language was consider that Akhnaton's capital was in Tell el Amarna. About 400 tablets with inscriptions in Akkadian cuneiform were found there in 1887. They constitute correspondence between Amenhotep III and Akhnaton and the governors of the cities in Palestine and Syria. Clearly the rulers of Egypt were multilingual and understood cuneiform writing. The book written by James E. Koch, Semitic Words in Egyptian Texts of the New Kingdom and Third Intermediate Period, Princeton University Press, 1994, attest to the pervasiveness of the Semitic influence in Egypt.

Many of the clay tablets concern topics which, although not deeply mathematical, nevertheless are fascinating to our understanding of that culture. For example K. Muroi writes about the expressions of zero and of squaring in the Semitic mathematical text VAT 7537, Historia Sci. (2) 1 (1) (1991), 59-62.

It was an important task for the rulers of Mesopotamia to dig canals and to maintain them, because canals were not only necessary for irrigation but also useful for the transport of goods and armies. The rulers or high government officials must have ordered Babylonian (Semitic) mathematicians to calculate the number of workers and days necessary for the building of a canal, and to calculate the total expenses of wages of the workers.

There are several of these Semitic mathematical texts in which various quantities concerning the digging of a canal are asked for. They are YBC 4666, 7164, and VAT 7528, all of which are written in Sumerian . . ., and YBC 9874 and BM 85196, No. 15, which are written in Akkadian . . . . From the mathematical point of view these problems are comparatively simple . . .

These Semites had an advanced number system, in some ways more advanced than our present decimal system. It was a positional system with a base of 60 rather than the system with base 10 in widespread use today.

Keep in mind that Abraham and his family had these skills available to them. They were not primitive nomads.

Perhaps the most amazing aspect of the Semitic mathematical skills was their construction of tables to aid calculation. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 BC. They give squares of the numbers up to 59 and cubes of the numbers up to 32. The table gives 82 = 1,4 which stands for

82 = 1, 4 = (1 x 60) + 4 = 64

and so on up to 592 = 58, 1 = ( 58 cross 60) +1 = 3481.

These Semite mathematicians used the formula

ab = [(a + b)2 - a2 - b2]/2

to make multiplication easier. Even better is their formula

ab = [(a + b)2 - (a - b)2]/4

which shows that a table of squares is all that is necessary to multiply numbers. They simply took the difference of the two squares by looking them up in the table and then taking a quarter of the answer.

Division is a harder process. The Semites did not have an algorithm for long division. Instead they based their method on the fact that

a/b = a cross (1/b)

so all that was necessary was a table of reciprocals. We still have their reciprocal tables going up to the reciprocals of numbers up to several million! Of course these tables are written in their numerals, but using the sexagesimal notation the beginning of one of their tables would look like this. Each number is 1/2, and so on.


0; 30


0; 20


0; 15


0; 12


0; 10


0; 7, 30


0; 6, 40


0; 6


0; 5


0; 4


0; 3, 45


0; 3, 20


0; 3


0; 2, 30


0; 2, 24


0; 2, 13, 20


Now the table had gaps in it since 1/7, 1/11, 1/13, etc. are not finite base 60 fractions. This did not mean that the Semites could not compute 1/13, say. They would write

1/13 = 7/91 = 7 cross (1/91) = (approx) 7 cross (1/90)

and these values, for example 1/90, were given in their tables. In fact there are fascinating glimpses of the Semites coming to terms with the fact that division by 7 would lead to an infinite sexagesimal fraction. A scribe would give a number close to 1/7 and then write statements such as (see for example G. G. Joseph, The Crest of the Peacock, London, 1991): . . . an approximation is given since 7 does not divide. Obviously he understood that he was making an approximation.


J. J. O'Connor and E. F. Robertson go on to discuss the complex mathematics of these Old Semitic people. See their articles at http://www-history.mcs.st-andrews.ac.uk/Indexes/Babylonians.html. They are from the School of Mathematics and Statistics at St. Andrews University, the oldest school in Scotland.

As stated:

Geometric considerations play a very secondary role in Babylonian algebra, even though geometric terminology may be used. Areas and lengths are freely added, something that would not be possible in Greek mathematics. Overall, the role of geometry is diminished in comparison with algebraic and numerical methods. Questions about solvability or insolvability are absent. The concept of proof is unclear and uncertain. Overall, there is no sense of abstraction. In sum, Babylonian mathematics, like that of the Egyptians, is mostly utilitarian, but apparently more advanced.

As I have noted elsewhere, these utilitarian methods depended upon a knowledge of mathematics that predated their practical use. By 2000 BC insights into mathematical developments had been lost, and we see only the remnants of those works. For example, how did they know that they could construct tables for n3 + n2 and then, with the aid of those tables, certain cubic equations could be solved? As O'Connor and Robertson noted: We stress again that all this was done without algebraic notation and showed a remarkable depth of understanding. A. E. Berriman discussed The Babylonian Quadratic Equation in Mathematical Gazateer, Vol 40, (1956), 185-192. He gives 13 typical examples of problems leading to quadratic equations taken from Old Semitic tablets. If problems involving the area of rectangles lead to quadratic equations, then problems involving the volume of rectangular excavation (a "cellar") lead to cubic equations. The clay tablet BM 85200+ contains 36 problems of this type, is the earliest known attempt to set up and solve cubic equations. J. Hoyrup analyzes these in "The Babylonian Cellar Text BM 85200+ VAT 6599; Retranslation and Analysis," Amphora (Basel, 1992), 315-358. As O'Connor and Robertson stated, "Of course the Babylonians did not reach a general formula for solving cubics. This would not be found for well over three thousand years." They fail to understand that mathematic knowledge was degrading because of the losses with time, and the original inspiration had died.


As examples of the range of concept of the Old Semitic scholars consider these Tablets from the Sch°yen Collection. The Sch°yen Collection comprises most types of manuscripts from the whole world spanning over 5000 years. It is the largest private manuscript collection formed in the 20th century. See


MS 3048

Table with data for solving cubic equations, in the Old Semitic sexagesimal system.

MS on clay, Babylonia, 19th c. BC, 1 tablet, 7,6x4,4x2,3 cm, 3 columns, 30 lines in cuneiform script.

Context: The only similar text known before is a Late Babylonian table text, where the numbers m at left take the values nxnx(n+1). Problems of the mentioned type are known from a large Old Babylonian clay tablet (BM 85200+VAT 6599).

Commentary: Every line of the table says, "m has the root n". The numbers n at right take the values 1 to 30. The numbers m at left take the corresponding values nx(n+1)x(n+2). In the 6th line, for instance, n = 6 and m = 6x7x8 = 336 = 5x60 + 36. The table was probably used to set up a series of problems leading to cubic equations guaranteed to have integers as solutions. The problems would have been of the form "An excavated room. Its length equals its width plus 1 cubit. Its height equals its length. Its volume plus its bottom area is ... (a given number)."

MS 2351

Extremely large 15-place sexigesimal number,

13 22 50 54 59 09 29 58 26 43 17 31 51 06 40,

equaling the 20th power of 20, which is 104,857,600,000,000,000,000,000

MS on clay, Babylonia, 19th c. BC, 1 tablet, 4.5x11.7x2.8 cm, single column, 2 lines in cuneiform script.

Commentary: The number 104 quintillions, 857 quadrillions and 600 trillions is so large that it occupies 2 lines on the obverse and continues on the reverse, being one of the largest numbers recorded on a cuneiform tablet.


Examples of Pythagorean Knowledge, circa 1800 BC

I shall now illustrate three examples from the Old Semitic repertoire on mathematics that shows their knowledge of the Pythagorean relationship.

First Example

This diagram shows a translation of the notation on the clay tablet. It has a diagram of a square with 30 on one side. The diagonals are drawn in and near the centre is written 1,24,51,10 and 42,25,35. Now the Babylonian numbers are always ambiguous and no indication occurs as to where the integer part ends and the fractional part begins. Assuming that the first number is 1: 24,51,10 then converting this to a decimal gives 1.414212956 while √2 = 1.414213562. Calculating 30 cross [1: 24,51,10] gives 42: 25,35 which is the second number. In other worlds, the diagonal of a square of side 30 is found by multiplying 30 by the approximation to √2.



I shall show how this was done. In decimal notation, and assuming that the Old Semitic mathematician and his audience understood the positional meaning of the numbers, we have:


Decimal Calculation Sexagesimal Calculation
X 30
1 1 1 1 30


24/60 0.4 24/60 (x1) 12
51 51/3,600 0.01416666 51/3,600 (x60) 25 +
10 10/216,000 0.000046296 10/216,000 (x3600) 30 + 5
Sum 1.414212956   42: 25, 35


Now consider the error tolerated by the scribe. If he had used only the third notational position, (1+24+51) his error would have been only 4.7 parts per 100,000. But he carried this another step (1+24+51+10). This made his error - 6 parts in 10,000,000. If he had used 9 instead of 10 for the last position, his error would have been  - 5.3 parts in 1,000,000. If he had used 11 instead of 10 for the last position, his error would have been + 4 parts in 1,000,000. This shows the refinement in the calculation of the Semitic scribe.


√2 1.414213562 Difference from Ideal
1 + 0.4 + 0.0141666 1.4141666 - 0.000046962
+ 9/216,000 1.414208267 - 0.000005295
+ 10/216,000 1.414212956 - 0.000000606
+ 11/216,000 1.414217526 + 0.000003963

He knew the value of √2 to better than 1 part per ten million.

Two important considerations apply: First, that he knew the Pythagorean relationship, and second, that he could carry it with such precision.

To allow the Pythagorean relationship he had to know that the two sides were at right angles to one another. This is a basic condition for the Pythagorean relationship to apply. Then he had to know that the square of each side, when added together, formed the square of the diagonal, a2 + b2 = c2 , where, in this case, a = b. But he also had to know that if 12 + 12 = 22 then the actual value of the diagonal had to be √2 X one side. But this was an irrational number. This was not a simple 3-4-5 relationship, where each side has an integral value, (a Pythagorean triple). This means he had to know the Pythagorean relationship in general, other than Pythagorean triples: 302 + 302  = 1800, (in our decimal notation), and that the square root of this number, √1800, would give the correct result.

The question before us is this: Was he carrying this process from some very ancient knowledge repertoire by rote, or did he have insight into the process? It seems hardly possible that his generation would have inherited this knowledge with such keen precision merely by rote. Certainly, those people would have lost the significance of the √2 if that were so. If we take the social model that he had inherited this knowledge from some ancient source, then we could estimate how much of that ancient knowledge was preserved by that social process, the form of it, and where we might find voids. But this would require us to investigate the gamut of the ancient Semitic or Sumerian knowledge, something I am unprepared to do. The work would require a major life commitment.

The point here is that we can look at this evidence as merely the beginning of NEW mathematical knowledge, the position of our modern scholarly world, or we can look upon it as the remnants of something lost from the more ancient past. But we can clarify our concern by noting that these mathematical methods were lost to historic record until Pythagoras revived them in the Greek age. This shows the general downward trend of our knowledge from that more ancient past.

A Second Example

A Susa tablet of 2000 BC requires the reader to find the radius of a circle in which is inscribed an isosceles triangle of sides 50, 50, and 60.

This is an analytical problem, not a numerical one as in the previous example. Here I have labeled the triangle A, B, C and the center of the circle as O. The perpendicular AD is drawn from A to meet the side BC = 60.

The triangle ABD is a right triangle. Using Pythagoras's theorem AB2 - BD2  = AD2. 502 - 302 = 402 ; hence AD = 40.


Let the radius of the circle be given by x. Then AO = OB = x and OD = 40 - x. Using Pythagoras's theorem again on the triangle OBD we have x2 = OD2 + DB2. Hence x2 = (40-x)2 + 302 . Multiplying out we have x2 = 402 - 80x + x2 + 302 . This reduces to 80x = 2500. Therefore, in decimal, x = 31.25,or, in sexagesimal, x = 31;15.

A Third Example

Finally consider the problem from the Tell Dhibayi tablet. It asks for the sides of a rectangle whose area is 0:45 and whose diagonal is 1:15. The equivalent decimal numbers are 0.75 for the area, and 1.25 for the diagonal.

In decimal solution this is quite an easy exercise. If we make the sides x, y we have

xy = 0.75 and

x2 + y2 = (1.25)2.

To solve we substitute y = 0.75/x into the second equation to obtain a quadratic in x2:

x2 + (0.5625 /x2 ) = 1.5625

x4  + 0.5625 = 1.5625 x2

x4 - 1.5625 x2  + 0.5625 = 0 

The solution is x = 1 and y = 0.75

We should note that this solution rests heavily on our algebraic understanding of equations.

However this is not the method of solution given by the Semitic scribe.

His solution is much more interesting than the modern method, and offers insight into the ancient mind.

We preserve the modern notation x and y as each step for clarity but we do the calculations in sexagesimal notation (as of course does the tablet). This method relies on the use of the squares of numbers to multiply, which the ancient Semitic scribe could easily consult. Refer to the above discussion.

Compute 2xy = 1;30. (xy was given.)

Subtract 2xy from x2 + y2 where x2 + y2 = 1:33,45 (Remember that x2 + y2 is the diagonal squared, 1:152 is 1:33,45. Here the scribe resorts to his knowledge of the Pythagorean rule.)

This give x2 + y2 - 2xy = 0:3,45.

Note that x2 - 2xy + y2 can be expressed as (x - y)2. (The Semitic scribe resorted to his knowledge of quadratics.)

Take the square root to obtain x - y = 0:15.

Divide by 2 to get (x - y)/2 = 0:7,30.

Divide x2 + y2 - 2xy = 0:3,45 by 4 to get x2/4 + y2/4 - xy/2 = 0:0,56,15.

Add xy = 0:45 to get x2/4 + y2/4 + xy/2 = 0:45,56,15. (x2 + xy/2 + y2)

Take the square root to obtain (x + y)/2 = 0:52,30.

Add (x + y)/2 = 0:52,30 to (x - y)/2 = 0:7,30 to get x = 1.

Subtract (x - y)/2 = 0:7,30 from (x + y)/2 = 0:52,30 to get y = 0:45.

Hence the rectangle has sides x = 1 and y = 0:45.

Remember that this is 3750 years old. We should be grateful to the Semitic scribe for recording this little masterpiece on tablets of clay for us to appreciate today. Clearly he was familiar with quadratics as well as the Pythagorean rule.