Forgotten Algebra of the Babylonians

The trilingual Rosetta stone is justly famous for enabling the decipherment of Egyptian hieroglyphics. But its story is far from unique to the 19th century. Hundreds of thousands of clay tablets, written with cuneiform scripts in languages such as Hittite and Persian, also became readable for the first time in over a thousand years, giving scholars a wealth of knowledge about the culture of the ancient Middle East. And numerous Akkadian tablets from the Old Babylonian Empire—contemporary with the Biblical patriarchs—revealed something unexpected: algebra. 

The Babylonians did problems with actual numbers much more than the Greeks would, thanks to their uniformly-structured number system. This system was a mixture of base 10—writing numbers in terms of powers of 10—and base 60. As shown above, it expressed numbers from 1 to 59 using combinations of the two symbols 𒑰 and 𒌋, which respectively signify 1 and 10. For larger numbers, it used these to express powers of 60, just as the decimal digit 7 can symbolize both 7 and 700. For instance, the Babylonians would write 197, equal to 60*3+17, as 𒐈 followed by 𒌋𒑂. 8000, equal to (60²)*2 +60*3+20, would be 𒐖 followed by 𒐈 followed by 𒎙. 7 ¹⁄₄, equal to 7+(60^-1)*15, would be 𒑂 followed by 𒌋𒐊.

Scholars transcribe these numbers with Arabic numerals, separating them with commas to avoid a blur of digits. They also put a semicolon between whole numbers and fractions. But this clarity hides a genuine obscurity: the Babylonians put numbers right next to each other regardless of their place value. For instance, 67 and 1 ⁷⁄₆₀, easily distinguishable in modern rewriting as 1,7 and 1;7, look exactly the same on the tablets. So, too, would 3608 and 68. Context is thus indispensable to interpreting these numerals, and recognizable equations in which they appear are an important type of such context. 

The Babylonians concentrated on obtaining the solutions to equations; their lack of concern for proving correctness greatly contrasts with the Greek attachment to careful logic. They had no formulas either (their lack of imaginary numbers made the quadratic formula impossible anyway). Instead, they used step-by-step methods such as completing the square, one of the most familiar second-degree (quadratic) techniques. A transcription of one of these tablets, however, might at first seem odd:

I have accumulated the surface and my confrontation; it is equal to 0;45. You posit 1 as the projection. You break 1 in half, and make 0;30 and 0;30 hold each other. You append 0;15 to 0;45, and 1 is the square. Tear out 0;30 from what you made hold to get 1, and 0;30 is the confrontation.

Like the medieval Arab algebraists, the Babylonians used concretely geometric language. “Confrontation” refers here to the side length of a square, and the “surface” is the square itself. The Babylonians distinguished between addition of values and combining one thing with another of the same type; “accumulation” is the first of these. Thus the first sentence is equivalent to x²+x=³⁄₄.

The “projection” gives x a width of one, turning an abstract addition of two numbers into the combined areas of two rectangles. The next two sentences, as the diagram shows, literally complete the square; “holding” 1 is multiplication and “appending” is the other addition. In modern algebraic terminology, one adds the square of half the x-coefficient to both sides of the equation, producing x²+x+¹⁄₄=³⁄₄, or (x+¹⁄₂)²=³⁄₄. Finally, “tearing out” is subtraction from the square root.

This problem is typical in its terminology and precise shifts between first and second person. Such simplicity of computation, however, is far from ubiquitous. The Babylonians often dealt with quadratics such as ²⁄₃x²+¹⁄₃x=¹⁄₃. Numerous tables of square and even cube roots survive, often accompanied by less sophisticated copies which students made for memorization. Reciprocals are the subject of many more tables; while the Babylonians had no direct division besides simple “breaking” into halves or thirds, they could easily “hold” by a number’s reciprocal. (These tables, of course, fail for numbers such as ¹⁄₇, which is impossible to write in base 60. Sometimes the Babylonians used approximations, which they clearly recognized as such. Other times, when division came up in a specific problem, they simply stated the answer, testable by multiplication.) 

Problems survive which use vaguely real-life scenarios as well as the standard geometric language. The Babylonians had, for instance, some simple third-degree equations about the relations between a cubical hole’s depth and the dirt dug out of it. These were solved similarly to quadratics, though with more care about conversion factors between units of length and volume. The Babylonians also had problems about building walls, selling oil, and measuring fields.

This last type, a broken reed problem, could get very complicated. One typical example involves a surveyor measuring the perimeter of a trapezoidal field with a very fragile reed. Though this reed breaks three times in the course of the measurement, the surveyor keeps going, only repairing one of the breaks for the last side. The problem is to determine the reed’s original length, given the area, the amounts of the reed broken off, and the measurements. The solution begins by assuming the reed begins with length 1, computing the side lengths and area, then adjusting the length to make the area match. Sometimes known as the “rule of false,” this was a common technique in the Middle Ages. 

All these “applications,” however, are as contrived as any recreational math problem. Though Babylonian algebra was far from a mere pastime—scribes regularly learned it in school—its lack of true application would bring it to an end. The Old Babylonian Empire fell around 1600 BC, and the Kassite conquerors put little value on such purely intellectual activity. Algebraic cuneiform tablets vanished from the record, briefly reemerged in the Chaldean Empire around 400 BC, then disappeared altogether. Only in the Middle Ages would algebra again reach that level.

Eventually, though, scholars rediscovered the tablets. And despite the remoteness of time and culture, they recognized the mathematics they saw.

Notes:

1. Often confused with an even stranger Akkadian term meaning “eating.”

Image Credit:

https://archeologie.culture.gouv.fr/orient-cuneiforme/en/mesopotamian-mathematics

https://commons.wikimedia.org/wiki/File:Babylone_1.PNG

https://commons.wikimedia.org/wiki/File:Babylonian_numerals.svg

https://www.mathsisfun.com/algebra/completing-square.html (edited by Hudson Green)

https://www.britishmuseum.org/collection/object/W_1896-0402-1

Sources:

https://www.wikipedia.org/

Aaboe, Asger. Episodes from the early history of mathematics. New York: Random House, Inc., 1964.

Høyrup, Jens. Lengths, widths, surfaces: a portrait of old Babylonian algebra and its kin. New York: Springer-Verlag New York, Inc., 2002.

Joseph, George Gheverghese. The crest of the peacock: non-European roots of mathematics. Penguin Books, 1992.

Muroi, Kazuo. “Cubic equations in Babylonian mathematics.” May 2019. https://arxiv.org/ftp/arxiv/papers/1905/1905.08034.pdf