Babylonian Mathematics

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YBC 7289 shows the square root of two

J J O'Connor and E F Robertson wrote: “The Babylonians had an advanced number system, in some ways more advanced than our present systems. It was a positional system with a base of 60 rather than the system with base 10 in widespread use today. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

“The Babylonians divided the day into 24 hours, each hour into 60 minutes, each minute into 60 seconds. This form of counting has survived for 4000 years. To write 5h 25' 30", i.e. 5 hours, 25 minutes, 30 seconds, is just to write the sexagesimal fraction, 5 25/60 30/3600. We adopt the notation 5; 25, 30 for this sexagesimal number... As a base 10 fraction the sexagesimal number 5; 25, 30 is 5 4/10 2/100 5/1000 which is written as 5.425 in decimal notation. ==

“Perhaps the most amazing aspect of the Babylonian's calculating skills was their construction of tables to aid calculation. Two tablets found at Senkerah on the Euphrates in 1854 date from 2000 B.C.. 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 × 60 + 4 = 64 and so on up to 592 = 58, 1 (= 58 × 60 +1 = 3481). ==

“The Babylonians 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, simply taking the difference of the two squares that were looked up in the table then taking a quarter of the answer. ==

“Division is a harder process. The Babylonians did not have an algorithm for long division. Instead they based their method on the fact that a/b = a × (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 billion. Of course these tables are written in their numerals, but using the sexagesimal notation we introduced above, ==

“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 Babylonians could not compute 1/13, say. They would write 1/13 = 7/91 = 7 × (1/91) = (approx) 7 × (1/90) and these values, for example 1/90, were given in their tables. In fact there are fascinating glimpses of the Babylonians 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 [5]):-

Advanced Babylonian Mathematics

Babylonian numerals

J J O'Connor and E F Robertson wrote: “Babylonian mathematics went far beyond arithmetical calculations. In our article on Pythagoras's theorem in Babylonian mathematics we examine some of their geometrical ideas and also some basic ideas in number theory. In this article we now examine some algebra which the Babylonians developed, particularly problems which led to equations and their solution.[Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

“We noted above that the Babylonians were famed as constructors of tables. Now these could be used to solve equations. For example they constructed tables for n3 + n2 then, with the aid of these tables, certain cubic equations could be solved. For example, consider the equation ax3 + bx2 = c. Let us stress at once that we are using modern notation and nothing like a symbolic representation existed in Babylonian times. Nevertheless the Babylonians could handle numerical examples of such equations by using rules which indicate that they did have the concept of a typical problem of a given type and a typical method to solve it. For example in the above case they would (in our notation) multiply the equation by a2 and divide it by b3 to get (ax/b)3 + (ax/b)2 = ca2/b3. Putting y = ax/b this gives the equation y3 + y2 = ca2/b3 which could now be solved by looking up the n3 + n2 table for the value of n satisfying n3 + n2 = ca2/b3. When a solution was found for y then x was found by x = by/a. We stress again that all this was done without algebraic notation and showed a remarkable depth of understanding. ==

“Again a table would have been looked up to solve the linear equation ax = b. They would consult the 1/n table to find 1/a and then multiply the sexagesimal number given in the table by b. An example of a problem of this type is the following. Suppose, writes a scribe, 2/3 of 2/3 of a certain quantity of barley is taken, 100 units of barley are added and the original quantity recovered. The problem posed by the scribe is to find the quantity of barley. The solution given by the scribe is to compute 0; 40 times 0; 40 to get 0; 26, 40. Subtract this from 1; 00 to get 0; 33, 20. Look up the reciprocal of 0; 33, 20 in a table to get 1;48. Multiply 1;48 by 1,40 to get the answer 3,0. ==

“It is not that easy to understand these calculations by the scribe unless we translate them into modern algebraic notation. We have to solve 2/3× 2/3 x + 100 = x which is, as the scribe knew, equivalent to solving (1 - 4/9)x = 100. This is why the scribe computed 2/3 × 2/3 subtracted the answer from 1 to get (1 - 4/9), then looked up 1/(1 - 4/9) and so x was found from 1/(1 - 4/9) multiplied by 100 giving 180 (which is 1; 48 times 1, 40 to get 3, 0 in sexagesimal). ==

“To solve a quadratic equation the Babylonians essentially used the standard formula. They considered two types of quadratic equation, namely x2 + bx = c and x2 - bx = c where here b, c were positive but not necessarily integers. The form that their solutions took was, respectively x = v[(b/2)2 + c] - (b/2) and x = v[(b/2)2 + c] + (b/2). Notice that in each case this is the positive root from the two roots of the quadratic and the one which will make sense in solving "real" problems. For example problems which led the Babylonians to equations of this type often concerned the area of a rectangle. For example if the area is given and the amount by which the length exceeds the breadth is given, then the breadth satisfies a quadratic equation and then they would apply the first version of the formula above. ==

“A problem on a tablet from Old Babylonian times states that the area of a rectangle is 1, 0 and its length exceeds its breadth by 7. The equation x2 + 7x = 1, 0 is, of course, not given by the scribe who finds the answer as follows. Compute half of 7, namely 3; 30, square it to get 12; 15. To this the scribe adds 1, 0 to get 1; 12, 15. Take its square root (from a table of squares) to get 8; 30. From this subtract 3; 30 to give the answer 5 for the breadth of the triangle. Notice that the scribe has effectively solved an equation of the type x2 + bx = c by using x = v[(b/2)2 + c] - (b/2). ==

“In [10] Berriman gives 13 typical examples of problems leading to quadratic equations taken from Old Babylonian 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+ containing 36 problems of this type, is the earliest known attempt to set up and solve cubic equations. Hoyrup discusses this fascinating tablet in [26]. Of course the Babylonians did not reach a general formula for solving cubics. This would not be found for well over three thousand years.” ==

Babylonian Base-60 Numerals

J J O'Connor and E F Robertson wrote: “Certainly in terms of their number system the Babylonians inherited ideas from the Sumerians and from the Akkadians. From the number systems of these earlier peoples came the base of 60, that is the sexagesimal system. Yet neither the Sumerian nor the Akkadian system was a positional system and this advance by the Babylonians was undoubtedly their greatest achievement in terms of developing the number system. Some would argue that it was their biggest achievement in mathematics. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==] “Often when told that the Babylonian number system was base 60 people's first reaction is: what a lot of special number symbols they must have had to learn. Now of course this comment is based on knowledge of our own decimal system which is a positional system with nine special symbols and a zero symbol to denote an empty place. However, rather than have to learn 10 symbols as we do to use our decimal numbers, the Babylonians only had to learn two symbols to produce their base 60 positional system. Now although the Babylonian system was a positional base 60 system, it had some vestiges of a base 10 system within it. This is because the 59 numbers, which go into one of the places of the system, were built from a 'unit' symbol and a 'ten' symbol. ==

“Now given a positional system one needs a convention concerning which end of the number represents the units. For example the decimal 12345 represents 1 × 104 + 2 × 103 + 3 × 102 + 4 × 10 + 5. If one thinks about it this is perhaps illogical for we read from left to right so when we read the first digit we do not know its value until we have read the complete number to find out how many powers of 10 are associated with this first place. The Babylonian sexagesimal positional system places numbers with the same convention, so the right most position is for the units up to 59, the position one to the left is for 60 × n where 1 = n = 59, etc. Now we adopt a notation where we separate the numerals by commas so, for example, 1,57,46,40 represents the sexagesimal number 1 × 603 + 57 × 602 + 46 × 60 + 40 which, in decimal notation is 424000.” ==

Babylonian math textbook

Babylonian Numeral Problems

J J O'Connor and E F Robertson wrote: “Now there is a potential problem with the system. Since two is represented by two characters each representing one unit, and 61 is represented by the one character for a unit in the first place and a second identical character for a unit in the second place then the Babylonian sexagesimal numbers 1,1 and 2 have essentially the same representation. However, this was not really a problem since the spacing of the characters allowed one to tell the difference. In the symbol for 2 the two characters representing the unit touch each other and become a single symbol. In the number 1,1 there is a space between them. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

“A much more serious problem was the fact that there was no zero to put into an empty position. The numbers sexagesimal numbers 1 and 1,0, namely 1 and 60 in decimals, had exactly the same representation and now there was no way that spacing could help. The context made it clear, and in fact despite this appearing very unsatisfactory, it could not have been found so by the Babylonians. How do we know this? Well if they had really found that the system presented them with real ambiguities they would have solved the problem - there is little doubt that they had the skills to come up with a solution had the system been unworkable. Perhaps we should mention here that later Babylonian civilisations did invent a symbol to indicate an empty place so the lack of a zero could not have been totally satisfactory to them. ==

“An empty place in the middle of a number likewise gave them problems. Although not a very serious comment, perhaps it is worth remarking that if we assume that all our decimal digits are equally likely in a number then there is a one in ten chance of an empty place while for the Babylonians with their sexagesimal system there was a one in sixty chance. Returning to empty places in the middle of numbers we can look at actual examples where this happens. ==

“Here is an example from a cuneiform tablet (actually AO 17264 in the Louvre collection in Paris) in which the calculation to square 147 is carried out. In sexagesimal 147 = 2,27 and squaring gives the number 21609 = 6,0,9. Here is the Babylonian example of 2,27 squared. Perhaps the scribe left a little more space than usual between the 6 and the 9 than he would have done had he been representing 6,9.

Doing Fractions and Division with Babylonian Numbers

J J O'Connor and E F Robertson wrote: “Now if the empty space caused a problem with integers then there was an even bigger problem with Babylonian sexagesimal fractions. The Babylonians used a system of sexagesimal fractions similar to our decimal fractions. For example if we write 0.125 then this is 1/10 + 2/100 + 5/1000 = 1/8. Of course a fraction of the form a/b, in its lowest form, can be represented as a finite decimal fraction if and only if b has no prime divisors other than 2 or 5. So 1/3 has no finite decimal fraction. Similarly the Babylonian sexagesimal fraction 0;7,30 represented 7/60 + 30/3600 which again written in our notation is 1/8. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

“Since 60 is divisible by the primes 2, 3 and 5 then a number of the form a/b, in its lowest form, can be represented as a finite decimal fraction if and only if b has no prime divisors other than 2, 3 or 5. More fractions can therefore be represented as finite sexagesimal fractions than can as finite decimal fractions. Some historians think that this observation has a direct bearing on why the Babylonians developed the sexagesimal system, rather than the decimal system, but this seems a little unlikely. If this were the case why not have 30 as a base? We discuss this problem in some detail below. ==

“Now we have already suggested the notation that we will use to denote a sexagesimal number with fractional part. To illustrate 10,12,5;1,52,30 represents the number 10 × 602 + 12 × 60 + 5 + 1/60 + 52/602 + 30/603 which in our notation is 36725 1/32. This is fine but we have introduced the notation of the semicolon to show where the integer part ends and the fractional part begins. It is the "sexagesimal point" and plays an analogous role to a decimal point. However, the Babylonians has no notation to indicate where the integer part ended and the fractional part began. Hence there was a great deal of ambiguity introduced and "the context makes it clear" philosophy now seems pretty stretched. If I write 10,12,5,1,52,30 without having a notation for the "sexagesimal point" then it could mean any of: 0;10,12, 5, 1,52,30; 10;12, 5, 1,52,30; 10,12; 5, 1,52,30; 10,12, 5; 1,52,30; 10,12, 5, 1;52,30; 10,12, 5, 1,52;30; 10,12, 5, 1,52,30 in addition, of course, to 10, 12, 5, 1, 52, 30, 0 or 0 ; 0, 10, 12, 5, 1, 52, 30 etc.

Plimpton 322

Why Did the Babylonians Have a Base-60 Numeral System?

J J O'Connor and E F Robertson wrote: “Finally we should look at the question of why the Babylonians had a number system with a base of 60. The easy answer is that they inherited the base of 60 from the Sumerians but that is no answer at all. It only leads us to ask why the Sumerians used base 60. The first comment would be that we do not have to go back further for we can be fairly certain that the sexagesimal system originated with the Sumerians. The second point to make is that modern mathematicians were not the first to ask such questions. Theon of Alexandria tried to answer this question in the fourth century AD and many historians of mathematics have offered an opinion since then without any coming up with a really convincing answer. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

“Theon's answer was that 60 was the smallest number divisible by 1, 2, 3, 4, and 5 so the number of divisors was maximised. Although this is true it appears too scholarly a reason. A base of 12 would seem a more likely candidate if this were the reason, yet no major civilisation seems to have come up with that base. On the other hand many measures do involve 12, for example it occurs frequently in weights, money and length subdivisions. For example in old British measures there were twelve inches in a foot, twelve pennies in a shilling etc. ==

“Perhaps the most widely accepted theory proposes that the Sumerian civilisation must have come about through the joining of two peoples, one of whom had base 12 for their counting and the other having base 5. Although 5 is nothing like as common as 10 as a number base among ancient peoples, it is not uncommon and is clearly used by people who counted on the fingers of one hand and then started again. This theory then supposes that as the two peoples mixed and the two systems of counting were used by different members of the society trading with each other then base 60 would arise naturally as the system everyone understood. ==

“I have heard the same theory proposed but with the two peoples who mixed to produce the Sumerians having 10 and 6 as their number bases. This version has the advantage that there is a natural unit for 10 in the Babylonian system which one could argue was a remnant of the earlier decimal system. One of the nicest things about these theories is that it may be possible to find written evidence of the two mixing systems and thereby give what would essentially amount to a proof of the conjecture. Do not think of history as a dead subject. On the contrary our views are constantly changing as the latest research brings new evidence and new interpretations to light.” ==

Babylonian Base-60 System Based on Weights, Astronomy or Geometry?

J J O'Connor and E F Robertson wrote: “Neugebauer proposed a theory based on the weights and measures that the Sumerians used. His idea basically is that a decimal counting system was modified to base 60 to allow for dividing weights and measures into thirds. Certainly we know that the system of weights and measures of the Sumerians do use 1/3 and 2/3 as basic fractions. However although Neugebauer may be correct, the counter argument would be that the system of weights and measures was a consequence of the number system rather than visa versa. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==] “Several theories have been based on astronomical events. The suggestion that 60 is the product of the number of months in the year (moons per year) with the number of planets (Mercury, Venus, Mars, Jupiter, Saturn) again seems far fetched as a reason for base 60. That the year was thought to have 360 days was suggested as a reason for the number base of 60 by the historian of mathematics Moritz Cantor. Again the idea is not that convincing since the Sumerians certainly knew that the year was longer than 360 days. Another hypothesis concerns the fact that the sun moves through its diameter 720 times during a day and, with 12 Sumerian hours in a day, one can come up with 60. ==

“Some theories are based on geometry. For example one theory is that an equilateral triangle was considered the fundamental geometrical building block by the Sumerians. Now an angle of an equilateral triangle is 60° so if this were divided into 10, an angle of 6° would become the basic angular unit. Now there are sixty of these basic units in a circle so again we have the proposed reason for choosing 60 as a base. Notice this argument almost contradicts itself since it assumes 10 as the basic unit for division! ==

“I [EFR] feel that all of these reasons are really not worth considering seriously. Perhaps I've set up my own argument a little, but the phrase "choosing 60 as a base" which I just used is highly significant. I just do not believe that anyone ever chose a number base for any civilisation. Can you imagine the Sumerians setting set up a committee to decide on their number base - no things just did not happen in that way. The reason has to involve the way that counting arose in the Sumerian civilisation, just as 10 became a base in other civilisations who began counting on their fingers, and twenty became a base for those who counted on both their fingers and toes. ==

“Here is one way that it could have happened. One can count up to 60 using your two hands. On your left hand there are three parts on each of four fingers (excluding the thumb). The parts are divided from each other by the joints in the fingers. Now one can count up to 60 by pointing at one of the twelve parts of the fingers of the left hand with one of the five fingers of the right hand. This gives a way of finger counting up to 60 rather than to 10. Anyone convinced? A variant of this proposal has been made by others.” ==

Did Babylonians Develop Trigonometry, 3,700s Years Ago?

Kenneth Chang wrote in the New York Times: “Suppose that a ramp leading to the top of a ziggurat wall is 56 cubits long, and the vertical height of the ziggurat is 45 cubits. What is the distance x from the outside base of the ramp to the point directly below the top? (Ziggurats were terraced pyramids built in the ancient Middle East; a cubit is a length of measure equal to about 18 inches or 44 centimeters.) Could the Babylonians who lived in what is now Iraq more than 3,700 years ago solve a word problem like this? Two Australian mathematicians assert that an ancient clay tablet was a tool for working out trigonometry problems, possibly adding to the many techniques that Babylonian mathematicians had mastered. “It’s a trigonometric table, which is 3,000 years ahead of its time,” said Daniel F. Mansfield of the University of New South Wales. Dr. Mansfield and his colleague Norman J. Wildberger reported their findings in the journal Historia Mathematica.[Source: Kenneth Chang, New York Times August 29, 2017 ^]

“The tablet, known as Plimpton 322, was discovered in the early 1900s in southern Iraq and has long been of interest to scholars. It contains 60 numbers organized into 15 rows and four columns inscribed on a piece of clay about 5 inches wide and 3.5 inches tall. It eventually entered the collection of George Arthur Plimpton, an American publisher, who later donated his collection to Columbia University. With all the publicity, the tablet has been put on display at the university’s Rare Book & Manuscript Library. Based on the style of cuneiform script used for the numbers, Plimpton 322 has been dated to between 1822 and 1762 B.C.

“One of the columns on Plimpton 322 is just a numbering of the rows from 1 to 15. The other three columns are much more intriguing. In the 1940s, Otto E. Neugebauer and Abraham J. Sachs, mathematics historians, pointed out that the other three columns were essentially Pythagorean triples — sets of integers, or whole numbers, that satisfy the equation a2 + b2 = c2. This equation also represents a fundamental property of right triangles — that the square of the longest side, or hypotenuse, is the sum of the squares of the other two shorter sides. That by itself was remarkable given that the Greek mathematician Pythagoras, for whom the triples were named, would not be born for another thousand years. ^

Solution to the Problem Above: “A Babylonian faced with the ziggurat word problem may have found it easy to set up: a right triangle with the long side, or hypotenuse, 56 cubits long, and one of the shorter sides 45 cubits. Next, the problem solver could have calculated the ratio 56/45, or about 1.244 and then looked up the closest entry on the table, which is line 11, which lists the ratio 1.25. From that line, it is then a straightforward calculation to produce an answer of 33.6 cubits. In their paper, Dr. Mansfield and Dr. Wildberger show that this is better than what would be calculated using a trigonometric table from the Indian mathematician Madhava 3,000 years later. These days, someone with a calculator can quickly come up with a bit more accurate answer: 33.3317.

Debate Over Babylonian Possible Development of Trigonometry

Kenneth Chang wrote in the New York Times: “Why the Babylonians compiled the triples and wrote them down has remained a matter of debate. One interpretation was that it helped teachers generate and check problems for students. Dr. Mansfield, who was searching for examples of ancient mathematics to intrigue his students, came across Plimpton 322 and found the previous explanations unsatisfying. “None of them really seemed to nail it,” he said. Other researchers have postulated that the tablet originally had additional columns listing ratios of the sides. (There’s a break along the left side of the tablet.) [Source: Kenneth Chang, New York Times August 29, 2017 ^]

“But what is conspicuously missing is the notion of angle, the central concept impressed upon students learning trigonometry today. Dr. Wildberger, down the hall from Dr. Mansfield, had a decade earlier proposed teaching trigonometry in terms of ratios rather than angles, and the two wondered that Babylonians took a similar angle-less approach to trigonometry. “I think the interpretation is possible,” said Alexander R. Jones, director of the Institute for the Study of the Ancient World at New York University, who was not involved with the research, “but we don’t have much in the way of contexts of use from any Babylonian tablets that would confirm such an intention, so it remains rather speculative.” ^

“Eleanor Robson, a Mesopotamia expert now at University College London who proposed the idea of the tablet as a teacher’s guide, is not convinced. Although she declined interviews, she wrote on Twitter that the trigonometry interpretation ignores the historical context. Perhaps the strongest argument in favor of the hypothesis of Dr. Mansfield and Dr. Wildberger is that the table works for trigonometric calculations, that someone had put in the effort to generate Pythagorean triples to describe right triangles at roughly one-degree increments. “You don’t make a trigonometric table by accident,” Dr. Mansfield said. “Just having a list of Pythagorean triples doesn’t help you much. That’s just a list of numbers. But when you arrange it in such a way so that you can use any known ratio of a triangle to find the other sides of a triangle, then it becomes trigonometry. That’s what we can use this fragment for.”“ ^

Babylonians Used Pythagorean Theorem 1,000 Years Before it Was 'Invented' in Ancient Greece

A 3,700-year-old clay tablet has revealed that the ancient Babylonians understood the Pythagorean theorem more than 1,000 years before the birth of the Greek philosopher Pythagoras, who is widely associated with the idea. Ben Turner wrote in Live Science: The tablet, known as Si.427, was used by ancient land surveyors to draw accurate boundaries and is engraved with cuneiform markings which form a mathematical table instructing the reader on how to make accurate right triangles. The tablet is the earliest known example of applied geometry. [Source Ben Turner, Live Science, September 22, 2022]

A French archeological expedition first excavated the tablet, which dates to between 1900 and 1600 B.C in what is now Iraq in 1894, and it is currently housed in the Istanbul Archeological Museum. But it is only just now that researchers have discovered the significance of its ancient markings. "It is generally accepted that trigonometry — the branch of maths that is concerned with the study of triangles — was developed by the ancient Greeks studying the night sky," in the second century B.C., Daniel Mansfield, a mathematician at the University of New South Wales in Australia and the discoverer of the tablet's meaning, said in a statement. "But the Babylonians developed their own alternative 'proto-trigonometry' to solve problems related to measuring the ground, not the sky."

According to Mansfield, Si.427 is the Old Babylonian period's only known example of a cadastral document, or a plan surveyors used to define land boundaries. "In this case, it tells us legal and geometric details about a field that's split after some of it was sold off," Mansfield said. The tablet details a marshy field with various structures, including a tower, built upon it. The tablet is engraved with three sets of Pythagorean triples: three whole numbers for which the sum of the squares of the first two equals the square of the third. The triples engraved on Si.427 are 3, 4, 5; 8, 15, 17; and 5, 12, 13. These were likely used to help determine the land's boundaries.

Though the tablet does not express the Pythagorean theorem in the familiar algebraic form it’s expressed in today, coming up with those triples would have required understanding the general principle that governs the relationship between length of the sides and the hypotenuse.

In 2017, Mansfield had discovered a tablet from the same period, named Plimpton 322 (See Below), which he identified as containing another trigonometric table. But it wasn't until he saw the triples on Si.427 that he was able to piece together that the ancient Babylonians were using rudimentary trigonometric theory to split up parcels of land. Si.427 is thought to pre-date Plimpton 322 — and may have even inspired it, Mansfield said. "There is a whole zoo of right triangles with different shapes. But only a very small handful can be used by Babylonian surveyors. Plimpton 322 is a systematic study of this zoo to discover the useful shapes," Mansfield said, referring to the fact that different types of right triangles can have different interior angles. "This is from a period where land is starting to become private — people started thinking about land in terms of 'my land and your land', wanting to establish a proper boundary to have positive neighbourly relationships. And this is what this tablet immediately says. It's a field being split, and new boundaries are made."

Although the reasons behind the calculations of land boundaries on the tablet aren’t entirely clear, Si.427 does mention a dispute over date-palms on the border between the properties of a prominent individual called Sin-bel-apli and a wealthy female landowner, according to Mansfield. "It is easy to see how accuracy was important in resolving disputes between such powerful individuals," he said.

What is surprising to Mansfield, however, is the level of theoretical sophistication the tablets reveal the ancient Babylonians to have had at such an early stage of human history. "Nobody expected that the Babylonians were using Pythagorean triples in this way," he said. "It is more akin to pure mathematics, inspired by the practical problems of the time."

Plimpton 322

J J O'Connor and E F Robertson wrote: Plimpton 322 “has four columns with 15 rows. The last column is the simplest to understand for it gives the row number and so contains 1, 2, 3, ... , 15. The remarkable fact which Neugebauer and Sachs pointed out in [5] is that in every row the square of the number c in column 3 minus the square of the number b in column 2 is a perfect square, say h. c2 - b2 = American So the table is a list of Pythagorean integer triples. Now this is not quite true since Neugebauer and Sachs believe that the scribe made four transcription errors, two in each column and this interpretation is required to make the rule work. The errors are readily seen to be genuine errors, however, for example 8,1 has been copied by the scribe as 9,1. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

“The first column is harder to understand, particularly since damage to the tablet means that part of it is missing. However, using the above notation, it is seen that the first column is just (c/h)2. Now so far so good, but if one were writing down Pythagorean triples one would find much easier ones than those which appear in the table. For example the Pythagorean triple 3, 4 , 5 does not appear neither does 5, 12, 13 and in fact the smallest Pythagorean triple which does appear is 45, 60, 75 (15 times 3, 4 , 5). Also the rows do not appear in any logical order except that the numbers in column 1 decrease regularly. The puzzle then is how the numbers were found and why are these particular Pythagorean triples are given in the table. ==

“Several historians have suggested that column 1 is connected with the secant function. Zeeman has made a fascinating observation. He has pointed out that if the Babylonians used the formulas h = 2mn, b = m2-n2, c = m2+n2 to generate Pythagorean triples then there are exactly 16 triples satisfying n = 60, 30° = t = 45°, and tan2t = h2/b2 having a finite sexagesimal expansion (which is equivalent to m, n, b having 2, 3, and 5 as their only prime divisors). Now 15 of the 16 Pythagorean triples satisfying Zeeman's conditions appear in Plimpton 322. Is it the earliest known mathematical classification theorem? Although I cannot believe that Zeeman has it quite right, I do feel that his explanation must be on the right track. ==

“To give a fair discussion of Plimpton 322 we should add that not all historians agree that this tablet concerns Pythagorean triples. For example Exarchakos, claims that the tablet is connected with the solution of quadratic equations and has nothing to do with Pythagorean triples: ‘we prove that in this tablet there is no evidence whatsoever that the Babylonians knew the Pythagorean theorem and the Pythagorean triads.’ I feel that the arguments are weak, particularly since there are numerous tablets which show that the Babylonians of this period had a good understanding of Pythagoras's theorem. Other authors, although accepting that Plimpton 322 is a collection of Pythagorean triples, have argued that they had, as Viola writes in a practical use in giving a: ‘general method for the approximate computation of areas of triangles.’” ==

Babylonians, Isosceles Triangles and Rectangle Areas

J J O'Connor and E F Robertson wrote:“The Susa tablet sets out a problem about an isosceles triangle with sides 50, 50 and 60. The problem is to find the radius of the circle through the three vertices. Here we have labelled the triangle A, B, C and the centre of the circle is O. The perpendicular AD is drawn from A to meet the side B.C.. Now the triangle ABD is a right angled triangle so, using Pythagoras's theorem AD2 = AB2 - BD2, so AD = 40. Let the radius of the circle by x. Then AO = OB = x and OD = 40 - x. Using Pythagoras's theorem again on the triangle OBD we have x2 = OD2 + DB2.So x2 = (40-x)2 + 302 giving x2 = 402 - 80x + x2 + 302 and so 80x = 2500 or, in sexagesimal, x = 31;15. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

“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. Now this to us is quite an easy exercise in solving equations. If the sides are x, y we have xy = 0.75 and x2 + y2 = (1.25)2. We would substitute y = 0.75/x into the second equation to obtain a quadratic in x2 which is easily solved. This however is not the method of solution given by the Babylonians and really that is not surprising since it rests heavily on our algebraic understanding of equations. The way the Tell Dhibayi tablet solves the problem is, I would suggest, actually much more interesting than the modern method. ==

“Here is the method from the Tell Dhibayi tablet. 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). Compute 2xy = 1;30. Subtract from x2 + y2 = 1;33,45 to get x2 + y2 - 2xy = 0;3,45. 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. 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. Is this not a beautiful piece of mathematics! Remember that it is 3750 years old. We should be grateful to the Babylonians for recording this little masterpiece on tablets of clay for us to appreciate today.” ==

Babylonians and the Concept of Zero

J J O'Connor and E F Robertson wrote:“Numbers in early historical times were thought of much more concretely than the abstract concepts which are our numbers today. There are giant mental leaps from 5 horses to 5 "things" and then to the abstract idea of "five". If ancient peoples solved a problem about how many horses a farmer needed then the problem was not going to have 0 or -23 as an answer. [Source: J J O'Connor and E F Robertson, St. Andrews University, December 2000 ==]

“One might think that once a place-value number system came into existence then the 0 as an empty place indicator is a necessary idea, yet the Babylonians had a place-value number system without this feature for over 1000 years. Moreover there is absolutely no evidence that the Babylonians felt that there was any problem with the ambiguity which existed. Remarkably, original texts survive from the era of Babylonian mathematics. The Babylonians wrote on tablets of unbaked clay, using cuneiform writing. The symbols were pressed into soft clay tablets with the slanted edge of a stylus and so had a wedge-shaped appearance (and hence the name cuneiform). Many tablets from around 1700 B.C. survive and we can read the original texts. Of course their notation for numbers was quite different from ours (and not based on 10 but on 60) but to translate into our notation they would not distinguish between 2106 and 216 (the context would have to show which was intended). It was not until around 400 B.C. that the Babylonians put two wedge symbols into the place where we would put zero to indicate which was meant, 216 or 21 '' 6. ==

“The two wedges were not the only notation used, however, and on a tablet found at Kish, an ancient Mesopotamian city located east of Babylon in what is today south-central Iraq, a different notation is used. This tablet, thought to date from around 700 B.C., uses three hooks to denote an empty place in the positional notation. Other tablets dated from around the same time use a single hook for an empty place. There is one common feature to this use of different marks to denote an empty position. This is the fact that it never occured at the end of the digits but always between two digits. So although we find 21 '' 6 we never find 216 ''. One has to assume that the older feeling that the context was sufficient to indicate which was intended still applied in these cases. ==

“If this reference to context appears silly then it is worth noting that we still use context to interpret numbers today. If I take a bus to a nearby town and ask what the fare is then I know that the answer "It's three fifty" means three pounds fifty pence. Yet if the same answer is given to the question about the cost of a flight from Edinburgh to New York then I know that three hundred and fifty pounds is what is intended. We can see from this that the early use of zero to denote an empty place is not really the use of zero as a number at all, merely the use of some type of punctuation mark so that the numbers had the correct interpretation. ==

“Now the ancient Greeks began their contributions to mathematics around the time that zero as an empty place indicator was coming into use in Babylonian mathematics. The Greeks however did not adopt a positional number system. It is worth thinking just how significant this fact is. How could the brilliant mathematical advances of the Greeks not see them adopt a number system with all the advantages that the Babylonian place-value system possessed? The real answer to this question is more subtle than the simple answer that we are about to give, but basically the Greek mathematical achievements were based on geometry. Although Euclid's Elements contains a book on number theory, it is based on geometry. In other words Greek mathematicians did not need to name their numbers since they worked with numbers as lengths of lines. Numbers which required to be named for records were used by merchants, not mathematicians, and hence no clever notation was needed.” ==

Image Sources: Wikimedia Commons

Text Sources: Internet Ancient History Sourcebook: Mesopotamia sourcebooks.fordham.edu , National Geographic, Smithsonian magazine, especially Merle Severy, National Geographic, May 1991 and Marion Steinmann, Smithsonian, December 1988, New York Times, Washington Post, Los Angeles Times, Discover magazine, Times of London, Natural History magazine, Archaeology magazine, The New Yorker, BBC, Encyclopædia Britannica, Metropolitan Museum of Art, Time, Newsweek, Wikipedia, Reuters, Associated Press, The Guardian, AFP, Lonely Planet Guides, “World Religions” edited by Geoffrey Parrinder (Facts on File Publications, New York); “History of Warfare” by John Keegan (Vintage Books); “History of Art” by H.W. Janson Prentice Hall, Englewood Cliffs, N.J.), Compton’s Encyclopedia and various books and other publications.

Last updated July 2024