Early mathematical tablets (c. 1900–1700 BC) from Mesopotamia and Elam, with their geographical origin and diagrams of their content.
Mesopotamian mathematics refers to the mathematical practices of the ancient civilizations of Mesopotamia (modern-day Iraq and surrounding regions), including Sumer, Babylonia, and Assyria. It is one of the earliest highly-developed mathematical traditions in history, predating Greek mathematics by many centuries. The Mesopotamians developed a sexagesimal (base-60) place-value numeral system and recorded their calculations on clay tablets in cuneiform script. Hundreds of these tablets – especially from the Old Babylonian period (c. 1900–1600 BC) – have been unearthed, covering a wide range of topics from practical computations for trade and agriculture to advanced algebraic and geometric problems. For instance, ancient scribes drew up tables of multiplication and reciprocals, as well as tables of squares, cubes, and even exponential growth, and they could solve quadratic and cubic equations and calculate compound interestplus.maths.org. Such accomplishments underscore a dynamic mathematical culture that thrived in Mesopotamia for over three thousand years.
This long-lived mathematical heritage was not confined to Mesopotamia proper. In the neighbouring land of Elam (modern southwestern Iran), particularly at the city of Susa, archaeologists have discovered early mathematical tablets with similar content, indicating that advanced mathematical thinking spread across the region. In this bilingual article, we will explore some of the earliest known mathematical tablets – including Plimpton 322, the YBC 7289, YBC 7290, and YBC 11120 tablets from the Yale Babylonian Collection, and a notable tablet from Susa. Through these examples, we will examine the mathematical content of each tablet, their historical context and location, and how they demonstrate both the practical and abstract/theoretical mathematical understanding of the Mesopotamians.
The Plimpton 322 Tablet
Plimpton 322 is one of the most famous Mesopotamian mathematical tablets. It is a clay tablet from the Old Babylonian period (circa 1800 BC), believed to have originated from the city of Larsa in southern Mesopotamia. Roughly 13 cm wide and 9 cm tall (with part of the tablet broken, it is now preserved at Columbia University in New York (George A. Plimpton acquired it around 1922 and later bequeathed it to. The cuneiform text on Plimpton 322 is laid out as a table with four columns and fifteen rows. Each row corresponds to a set of numbers that form a Pythagorean triple—that is, integers (a, b, c) satisfying the equation a² + b² = c². In other words, the Babylonians clearly knew the relationship equivalent to the Pythagorean theorem long before Pythagoras, and they could find multiple examples of such integer solutions. The triples recorded on Plimpton 322 are neither few nor simple; in fact, the numbers involved are quite large, and there are far too many combinations to have been obtained by brute force trial-and-error, which implies that Babylonian scribes had systematic methods for generating Pythagorean triples. Modern research suggests they may have used algebraic techniques involving reciprocal pairs of numbers to produce these triples.
The exact purpose of Plimpton 322 is still debated. Some scholars argue it may have served as a teaching tool or reference for school instruction—for instance, a master scribe could use it to generate problems about right triangles for students, using each row’s numbers as a different example. Others have noted its similarity to administrative lists, but the prevailing view is that it was intended for mathematical work. In any case, Plimpton 322 demonstrates an exceptionally advanced and abstract understanding of mathematics. By around 1800 BC, Babylonian scholars were not only aware of the right-triangle relationship but had tabulated 15 distinct solutions for it—a clear indication of theoretical interest. Notably, this occurs at least a millennium before the time of Pythagoras, showing that the principle of the Pythagorean theorem was known and used in Mesopotamia long before classical Greek antiquity.
YBC 7289 Tablet (Square Root of 2)
Another impressive example of Babylonian mathematical prowess is found on the tablet YBC 7289, part of the Yale Babylonian Collection. This small, circular clay tablet from the Old Babylonian period (18th century BC) contains an extremely accurate approximation of the square root of 2. The tablet’s round shape and modest size suggest it was likely a student exercise in a scribal school – indeed, round tablets were commonly used by apprentice scribes for practice. On YBC 7289, a square is drawn, and numbers are inscribed relating to the square’s side and its diagonal. Along one side of the square is written “30” (presumably in some unit of length), and 1;24,51,10 is written along the diagonal. In Babylonian sexagesimal notation, 1;24,51,10 represents 1 + 24/60 + 51/60² + 10/60³, which in decimal equals approximately 1.41421296 – a remarkably close value for √2 (since √2 ≈ 1.41421356), correct to at least five decimal places. Below the diagonal, the product 42;25,35 is also written, which is exactly 30 × 1;24,51,10 (~42.4263888 in decimal). In this way, the scribe verified that for a square of side 30, the diagonal is ~42.43 in the same units, confirming the relationship 30 × 1;24,51,10 = 42;25,35. Essentially, the value 1;24,51,10 serves as the “diagonal coefficient” for the square – effectively the value of √2 in sexagesimal form. The approximation is so good that the square of 1;24,51,10 yields 1,59,59,59,38,1,40 in sexagesimal notation (nearly 2 exactly).
Beyond its numerical content, YBC 7289 highlights how Mesopotamians combined practical geometry with abstract number theory. On a practical level, finding a square’s diagonal given its side is useful for surveying land or constructing right angles in architecture. However, its appearance on a student tablet shows it was also used as a teaching example – instructors likely employed it to teach the concept of the diagonal and its non-integer value. Notably, the same numerical value (1;24,51,10) appears in other Babylonian sources, such as in coefficient lists for geometric computations. This suggests that the Babylonians had general tables of constants – in this case, they knew that for any square, the ratio of diagonal to side is ~1.4142. Thus, YBC 7289 is clear evidence of a sophisticated theoretical awareness: an understanding of an irrational number (√2 cannot be expressed as a finite fraction) and the ability to calculate it with high precision. As modern scholars have noted, achievements like this – along with the Pythagorean triples of Plimpton 322 – demonstrate the advanced level of Babylonian mathematical education by 1800 BC.
YBC 7290 Tablet (Trapezoid Area)
The tablet YBC 7290 highlights the practical side of Mesopotamian mathematics, as it contains an exercise in computing the area of a trapezoid – a problem directly related to land surveying and field measurement in antiquity. This clay tablet, also from the Old Babylonian period (c. 1800–1600 BC) and now in Yale’s collection, has on its obverse a drawn trapezoid with cuneiform notations specifying the lengths of the bases and the non-parallel sides. Specifically, one base is given as 2;20 (sexagesimal, i.e. 2 + 20/60 = 2.333... in some length unit), the other base as 2;00 (2 units), and both slanted side lengths are 2;20. The resulting area written within the figure is 5;03 20 (5 + 3/60 + 20/3600 in sexagesimal) square units. From these numbers, we deduce the scribe applied a formula equivalent to taking the average of the two side lengths and the average of the two bases, then multiplying those averages to find the area. In modern terms, this approximates the trapezoid area formula $A = \frac{(B_1+B_2)}{2} \times h$, except here the “height” $h$ was effectively estimated as the average of the slanted sides (a practical approach when perpendicular height was not directly measured). In essence, Babylonian surveyors used a rule of thumb for quadrilaterals: the area is the product of the average length of the sides and the average length of the bases. While this formula is not exact for every trapezoid unless it is isosceles, it provides a reasonable approximation in many cases and would have been sufficient for practical survey needs.
YBC 7290 reveals how ancient scribes computed land areas – a task of prime importance for taxation and agriculture. At the same time, it showcases an abstract generalization: the scribe did not treat each field uniquely but applied a general formula (an average) that could be used in multiple situations. This indicates a recognition of underlying patterns in geometry: essentially, an empirical mathematical formula had been formulated. The presence of a figure diagram (with no text) on the reverse side suggests that the student also practiced drawing the shape, not just the arithmetic – a detail that provides insight into the teaching method: combining geometric drawing with calculation. This tablet, roughly contemporary with the previous examples (~18th–17th century BC), underscores that Mesopotamian mathematics included geometric rules with practical applications. Such ancient surveying rules, like the trapezoid area method, are forerunners of later geometric formulas and demonstrate how Mesopotamian scholars were beginning to articulate general principles about space and measurement.
YBC 11120 Tablet (Circle and π)
The tablet YBC 11120 from the Yale collection takes us into the study of circular measurements – specifically, the area of a circle. This Old Babylonian tablet (c. 18th–17th century BC) shows how the ancients calculated the area of a circle given its circumference. On the tablet, a circle is drawn accompanied by numerical annotations: the circumference is given as 1;30 (in some unit of length), and the square of that circumference is noted as 2;15. Using these, the scribes computed the area by applying the formula $A = (;05) \times (\text{circumference})^2$, where “;05” in sexagesimal is 5/60 = 1/12. In other words, they assumed the area of a circle equals one-twelfth of the square of its circumference. Translating this to modern notation: $A = \frac{1}{12} C^2$. We know that actually $A = \pi r^2$ and $C = 2\pi r$, so $A = \frac{C^2}{4\pi}$. Comparing with the Babylonian formula $A = \frac{C^2}{12}$, we see this corresponds to taking $\pi = 3$ (since $4\pi$ in the denominator becomes 12 when $\pi=3$). Indeed, the Babylonians often assumed $\pi \approx 3$ in practical calculations – for example, they typically took the circumference as 3 times the diameter, using the convenient approximation of 3 in most cases. In YBC 11120, following this rule, with a circumference of 1;30 (which is 1.5 in decimal), they obtained an area of 0;11 15, which equals 0 + 11/60 + 15/3600 = 3/16 = 0.1875 in decimal. This is precisely the result given by $1/12 \times (1.5)^2$, confirming the use of the 1/12 constant (i.e. $\pi=3$). Notably, the scribes themselves indicate on the tablet that the factor 1/12 (written as ;05) was a “standard constant” employed in Babylonian computation.
Despite this simplified value of $\pi$, the important fact is that the Babylonians had a procedure for finding a circle’s area – they understood that the area is related to the square of the perimeter (or equivalently the square of the diameter). YBC 11120 shows that scribes could adapt their methods (used for squares and rectangles) to curved shapes by introducing a constant into the calculation. It also suggests that although they usually used 3 for $\pi$, they were aware this was an approximation. In another tablet – discussed next, from Susa – we find that in special cases they employed a more refined value, $\pi \approx 3.125$ (25/8), to achieve higher accuracy. Taken together, these circle computations reveal a blend of practicality and theoretical curiosity among Mesopotamian scholars: practical, because using 3 for $\pi$ was convenient for everyday purposes, but also theoretical, because there is evidence that learned scribes experimented with improving that constant when greater precision was desired.
The Susa Tablet (π ≈ 3.125)
The Susa tablet discovered in 1936 near Susa (in ancient Elam, modern southwest Iran) shows that Mesopotamian mathematicians did not always settle for π = 3, but sometimes pursued greater accuracy for π. This tablet – published by E. M. Bruins in 1950 and later fully in 1961 by Bruins & Rutten – dates to the Old Babylonian period (19th–17th century BC) and is interpreted to yield π approximately 3.125, or $25/8$. Specifically, the tablet’s text describes a geometric relationship: it states that the ratio of the perimeter of a regular hexagon to the circumference of the circumscribed circle equals a certain number. That number is given as 0;57 36 in sexagesimal (57/60 + 36/60²) – which in decimal is 0.96. This value essentially represents the fraction of the circle’s circumference relative to the hexagon’s perimeter: for a regular hexagon inscribed in a circle, the perimeter of the hexagon is 6r (where r is the radius), while the circle’s circumference is $2\pi r$. The ratio of these is $\frac{6r}{2\pi r} = \frac{3}{\pi}$. The tablet thus effectively asserts $\frac{3}{\pi} = 0.96$, which leads to $\pi = \frac{3}{0.96} = 3.125$. Indeed, $25/8 = 3.125$, a value that differs from the true π (~3.1416) by only about 0.5%. This improved approximation of π (3 + 1/8) is remarkable: although slightly low, it is significantly more accurate than the simple 3 that was normally used.
The Susa mathematical tablet underlines that there was a drive for theoretical exploration and precision when the context allowed it. While in everyday transactions or routine calculations the Babylonians deliberately used rounded values (like 3 for π) for simplicity, here we see a learned scribe engaging in a more nuanced geometric analysis. By examining a hexagon and a circle, he was essentially undertaking an early method to refine π via geometric comparison. This approach is reminiscent of similar efforts much later – for example, Archimedes in Greece (circa 3rd century BC) famously used inscribed and circumscribed polygons to approximate π. It appears, then, that the germ of such ideas was present in the ancient Near East over a millennium earlier.
We cannot be certain how, or if, such knowledge was directly transmitted to other cultures. However, Susa, being part of the Elamite and Babylonian cultural sphere, was later incorporated into the Persian Empire, which in turn interfaced with the Greek world. It is thus possible that the accumulated mathematical experience of Mesopotamia – including advanced notions of π and other constants – reached the Greeks indirectly through Persian rule and translations. Regardless of the transmission path, the very existence of this “π tablet” from Susa is evidence that the quest for mathematical accuracy and theoretical understanding had begun well before the classical era, within the civilizations of the ancient Near East.
Practical and Theoretical Knowledge in Mesopotamian Mathematics
The above examples show that Mesopotamian mathematics embodied both practical problem-solving and abstract theoretical exploration. On the one hand, tablets like YBC 7290 (trapezoid area) address immediate practical needs: measuring land, calculating areas for agriculture or construction, and handling economic computations (such as distribution of goods or interest on loans) were part of daily life, and mathematical methods were developed to serve these ends. Indeed, many Babylonian tablets are not “theorems” but tables and word problems related to commercial arithmetic, accounting, surveying, and engineering. On the other hand, we see tablets like Plimpton 322 and YBC 7289 delving into purely numerical or geometric ideas (e.g. Pythagorean triples, √2) with no obvious everyday application. The ancient scribes appeared comfortable moving between the real and the abstract: they could solve a concrete problem (such as a field’s area) and also engage with mathematical relationships for their own sake, showing a notable scientific curiosity. In fact, many of their “theoretical” problems were presented in the guise of practical riddles or story problems – for example, what we recognize today as quadratic equations appear in texts as problems about dividing plots of land or building projects – a didactic technique that allowed scribes to study abstract mathematics under the cover of realistic scenarios.
This blending of utility and theory is closely tied to the educational system of ancient Mesopotamia – the so-called “tablet house” (edubba) scribal schools. There, student scribes first learned basics: they memorized multiplication tables, reciprocal tables, and metrological (measurement) lists needed for bureaucracy and commerce. As they advanced, however, they tackled complex problems requiring ingenuity and generalization. For instance, from Old Babylonian tablets we know they developed general methods for solving equations (algebra) framed as narrative problems: e.g. “finding the sides of a rectangle given its area and sum of sides,” “distributing grain with certain ratios,” etc. In such problems, scribes used equivalent transformations and steps that are essentially forms of algebraic solutions. As the historian Jens Høyrup has noted, even a fundamental discovery like the rule of the right triangle (what we call the Pythagorean theorem) likely emerged from the practical environment of lay surveyors – perhaps a scribe trying to compute land boundaries discovered the relationship, sometime between 2300 and 1825 BC, and then the knowledge was generalized and entered the teaching texts. Once a new mathematical principle was found, it became part of the tradition and curriculum. This explains how we find the Pythagorean rule applied in seven different tablets from cities like Eshnunna, Sippar, and Susa – evidence that it had been incorporated as a broadly known result. In summary, the earliest mathematical tablets reveal a culture in which practical know-how coexisted with profound theoretical thought. The Mesopotamian mathematicians laid down foundational principles of number and measurement, achieving a level of abstraction and generalization that would (directly or indirectly) influence later developments in mathematics.
Legacy and Influence on Later Civilizations
The mathematical knowledge of Mesopotamia and Elam did not vanish with the passing of those civilizations – it influenced neighboring cultures and eventually became part of the broader stream of mathematical development. Several aspects of Babylonian mathematics appear to have been known or adopted by later peoples. One clear example is the division of the circle into 360 degrees and the use of the base-60 subdivision of time (minutes and seconds), which was taken up by the Greeks and remains in use to this day – a direct inheritance from Babylonian practice. During the Achaemenid Persian Empire (6th–4th century BC), which included Mesopotamia, and especially after Alexander the Great’s conquests (4th century BC), Babylonian astronomical and mathematical knowledge became accessible to Greek scholars. Greek astronomers such as Hipparchus and later Ptolemy explicitly utilized Babylonian records and sexagesimal computational methods for their calculations of planetary motions and celestial phenomena. It is well documented that Hellenistic astronomy heavily relied on the data and mathematical tools developed by the Babylonians (for example, lunar eclipse cycles and planetary position tables).
In the realm of pure mathematics, the transmission of knowledge is less directly attested, but there are suggestive hints and parallels. The famous Pythagorean theorem, for instance, was formally proved in Greek mathematics (in Euclid’s Elements during the 4th century BC) and attributed to Pythagoras (6th century BC), yet Babylonian tablets like Plimpton 322 show that the relationship $a^2 + b^2 = c^2$ was known and used over a thousand years earlier. It is quite possible that scholars such as Thales of Miletus or Pythagoras himself – who, according to tradition, traveled eastward – encountered advanced Babylonian mathematics during their visits to Babylon (or via Egypt, which was also influenced by the Near East). Ancient authors describe Babylon as a center of astronomy and learning – disciplines inherently tied to mathematics. The Greek historian Herodotus and later writers acknowledged that the Greeks borrowed certain mathematical ideas from older cultures like Egypt; by the same token, knowledge from Mesopotamia likely found its way into Greek thought through the interconnected networks of the ancient world.
One concrete pathway of influence was astronomy: Babylonian astral science, with its sophisticated mathematical models, deeply informed Greek astronomy in the Hellenistic period. Concepts such as the zodiac, accurate eclipse prediction, and mathematical tables of planetary positions in Greece were built upon centuries of Babylonian observations and calculations. Moreover, some technical terminology and methods (for example, the use of reciprocal tables or certain algebraic solution techniques) may have traveled via scholars in the Seleucid era or via translations in major centers (like the Library of Alexandria). While Greek mathematicians developed a different style – more geometric and axiomatic – the foundational ideas, such as place-value notation, general algebraic problem-solving, and certain numeric constants, were part of the cumulative knowledge of the Near East that preceded them.
In conclusion, Mesopotamian and Elamite mathematics provided an early blueprint for advanced mathematical thinking. This legacy was transmitted through time by both direct contact and the enduring utility of their innovations (for example, the sexagesimal system for measuring time and angles that we still use today). The ancient Greeks, and subsequently other civilizations, built upon this foundation, whether explicitly or implicitly. The clay tablets of Mesopotamia stand not only as archaeological artifacts but also as testimony to a seminal chapter in the global history of mathematics – one that set the stage for later mathematical achievements in the classical world and beyond.
References
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E. M. Bruins & M. Rutten, Textes mathématiques de Suse. Mémoires de la Mission Archéologique en Iran, vol. 34, Paris: P. Geuthner, 1961.
Eleanor Robson, Neither Sherlock Holmes nor Babylon: A Reassessment of Plimpton 322. Historia Mathematica 28 (2001): 167–206.
Jöran Friberg, Unexpected Links Between Egyptian and Babylonian Mathematics. World Scientific, 2005.
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