What Have the Ancient Babylonians Ever Done for Us?
Benjamin Skuse
In a classic comedy sketch from Monty Python’s Life of Brian, ‘Reg,’ planning a rebellion against his Roman occupiers, asks his followers what the Romans have ever given them, to which he receives a long list of huge benefits that the Romans had brought to their lives. After an exasperated pause, Reg responds: “Apart from the sanitation, the medicine, education, wine, public order, irrigation, roads, the freshwater system and public health … what have the Romans ever done for us?!”
What happens if we ask the same question of even earlier civilisations: the ancient Babylonians and their forebears in Mesopotamia? The Babylonian Empire stretched across present-day Iraq and parts of Kuwait, Syria and Iran between roughly 1800 BC and 539 BC. The people of ancient Babylon were highly advanced for their time, recording a wide range of aspects of life on clay tablets. From these, we know that they developed laws, made meticulous and accurate observations of the night sky, and devised innovative irrigation systems that optimised farming and even allowed the Hanging Gardens of Babylon to flourish; by all accounts, an astonishing sight now lost, but regarded as one of the Seven Wonders of the Ancient World.
Babylonian Mathematics
Though these achievements are remarkable and certainly worthy of an ancient Babylonian version of the list ‘Reg’ begrudgingly quoted, one area worth exploring in more detail is their contribution to mathematics.
Perhaps the biggest influence from ancient Babylonian mathematics today is their popularisation of the sexagesimal (base 60) numeral system (that they inherited and further developed from the earlier ancient Sumerians), which still divides the circle into 360 degrees (6 \(\times\) 60) and counts our minutes and seconds in groups of 60 to make a 24-hour day.
For this sexagesimal system, they used an abstract form of writing based on just two wedge-shaped (cuneiform) symbols to create 59 symbols that represent 1–59. Bizarrely, this system was built around a 1 symbol and a 10 symbol, so it also had elements of what we would regard as a conventional base 10 numeral system.
For numbers beyond 59, the Ancient Babylonians’ system was positional – meaning the number in the rightmost position represented units up to 59, next on the left would be \( 60 \times n \), where \( 1 \leq n \leq 59 \), then \(60^{2} \times n \), etc – and relied on accurate tables of squares, and even tables of cubes, to aid calculation. This provided an efficient way of representing very large numbers. For example, if we translate the cuneiform into conventional units and separate numbers by commas, this system writes 999,999 as ‘4,37,46,39’, i.e.
\[ (4 \times 60^{3}) + (37 \times 60^{2}) + (46 \times 60^{1}) + (39 \times 60^{0}).\]
Arithmetic, Squares and Fractions
Now, unlike our conventional numerical system, the ancient Babylonians had no equivalent of a decimal point to indicate where the integer part ended and the fractional part began of a given number. Far from being a hindrance, this feature – combined with the base 60 system and the use of multiplication tables – simplified calculations for ancient Babylonian mathematicians considerably, and led to important advances.
Significant among these is that it allowed for a simple way of representing fractions; ancient Babylon was therefore one of the first, if not the first civilisation capable of this feat. As there was no symbolic distinction between whole numbers and fractions, fractions could be represented by finite sexagesimal expansions just like whole numbers, and their fractional nature inferred from the context of the calculation. As an example, 1/128 would be written ‘28,7,30’, i.e.
\[28 \times 60^{-2} + 7 \times 60^{-3} + 30 \times 60^{-4}.\]
These could even be mixed. The number 2.5, say, could be written ‘2,30’ \((2 + (30 \times {60^{-1}})).\) Tablets of reciprocals have been found that were used like look-up tables to calculate large fractions, in the same way as their tablets of squares and cubes.
Pythagoras Before Pythagoras
As every secondary schoolchild knows, Pythagoras’ theorem is the equation that connects the lengths of all three sides of a right-angled triangle: \( a^{2} + b^{2} = c^{2} \), where \(a\) and \(b\) are the sides that form the right angle, and \(c\) is the side opposite, the hypotenuse. Schoolchildren are also taught that this theorem was devised by Greek philosopher Pythagoras of Samos, who lived from 570 to 495 BC. Only the former is true. Across a number of tablets is left evidence that the ancient Babylonians had a deep understanding of Pythagoras’ theorem, even if never stated explicitly.
The Yale tablet YBC 7289, held at Yale University, is a prime example, detailing how to find a square’s diagonal given its side. Although this would have been useful for ancient Babylonians surveying land or building structures, YBC 7289 is not thought to be a practical tablet, but instead a teaching aid. It illustrates a geometric square with intersecting diagonals. Translated into conventional numbers, one length (top-left) is labelled ‘30’, with ‘1,24,51,10’ written along the horizontal diagonal, and ‘42,25,35’ below that.
Using our understanding of the Babylonian sexagesimal system, ‘1,24,51,10’ is:
\[ (1 \times 60^{3}) + (24 \times 60^{2}) + (51 \times 60^{1}) + (10 \times 60^{0}) = 305,470. \]
So far, so confusing. But remembering that we have no idea of the absolute values of the numbers presented, it could equally mean 1 followed by ‘24,51,10’:
\[ 1 + (24 \times 60^{-1}) + (51 \times 60^{-2}) + (10 \times 60^{-3}) = \textbf{1.414212962}. \]
Similarly, for ‘42,25,35’, the meaning could be:
\[ (42 \times 60^{0}) + (25 \times 60^{-1}) + (35 \times 60^{-2}) = \textbf{42.426389}. \]
From a basic use of Pythagoras’ theorem, we know that the length of the diagonal \(d \) is:
\[ d = 30 \times \sqrt{2} = 30 \times\ \textbf{1.414213562} = \textbf{42.4264068}.\]
The tablet appears to be teaching the student that the diagonal of a square of side 30 is found by multiplying 30 by a close approximation to \(\sqrt{2}\). Not only does this show a strong understanding of Pythagoras’ theorem, but also an awareness of and ability to use irrational numbers, such as \(\sqrt{2}\).
Could the evidence given in the Yale tablet be a trick of biased interpretation or sheer luck? Several other tablets have been uncovered that add credence to the idea that the ancient Babylonians were familiar with and used Pythagoras’ theorem extensively. Plimpton 322, housed in Columbia University, USA, is particularly convincing, appearing to give a list of Pythagorean integer triples, and not just simple ones like 3, 4, 5; the smallest quoted is 45, 60, 75.
More Advanced Mathematics
Though Pythagoras’ theorem is a quadratic equation, it is a particularly simple one. Did the ancient Babylonians attempt more advanced problems that today would involve more challenging equations or geometrical concepts? There are many examples in recovered tablets posing and answering more difficult problems. For example, a tablet might ask: “If I add to the area of a square twice its side, I get 48. What is the side?” Writing \(x\) for the side, this translates to the equation \( x^{2} + 2x = 48\). This is readily solved using the general quadratic formula today, but would be tackled using more algorithmic brute force methods by the ancient Babylonians.
Another well-known example is the Susa 194 tablet found about 350 km from the ancient city of Babylon and today housed in the Louvre in Paris, France. It illustrates the use of arithmetic constants to calculate the areas of regular six- and seven-sided figures when their perimeters are identical to the circumference of a circle. One of these constants, 24/25, relates the ratio of a perimeter of a hexagon with side length \(r\), inscribed in a circle of radius \(r\), to the circumference of that circle. 24/25 is then \(6r/(2\pi r)\), which when rearranged yields a value of \(\pi = 25/8\) or 3.125, reasonably close to the true value 3.14159… and far better than 3, which is what most civilisations used as an approximation before ancient Greek philosopher Archimedes made a serious attempt at calculating \(\pi\).
Even more advanced problems are seen in BM 85200 + VAT 6599, known as the Babylonian Cellar Text, a tablet whose two parts are housed in the UK and Germany, respectively. This tablet cites 30 problems involving the volume of rectangular excavation (a ‘cellar’), and which would be solved largely by means of quadratic or cubic equations today.
Tablets like these not only address immediate practical needs, like measuring land, calculating areas for agriculture or construction, and handling trade, but also a type of abstraction, generalising problems and answers so they could be applied in these different contexts.
Astrology for Astronomy
Though also practical in a general sense in marking the seasons for farming and aiding navigation for travellers, mathematics applied to astronomy stands out in ancient Babylonian culture as a particularly important pursuit. For ancient Babylonians, astrology was central to society, with the movement of celestial bodies thought to predict everything from the price of grain to plagues. As such, accurate astronomical observations and predictions were critical.
Of particular significance is a set of later era tablets detailing how ancient Babylonian astronomers calculated and predicted Jupiter’s position in the sky. They describe the apparent decreasing velocity of Jupiter from the planet’s first appearance along the horizon to 60 and then 120 days later. If plotted on a graph, the relationship forms the shape of two conjoined trapezoids, with sides representing the speeds and elapsed time. The area of each trapezoid describes Jupiter’s total displacement measured in degrees along the ecliptic, or the path of the Sun. Remarkably, nothing similar appears until 14th Century English mathematicians devise the mean speed theorem to understand the velocity and displacement of an object over time.
Though many of the ancient Babylonian tablets use mathematics to solve very practical problems, these examples – particularly the latter, which describes a highly abstract application of geometry – underline how ahead of their time the ancient Babylonians were. So, in the extremely unlikely event that someone asks you “What have the ancient Babylonians ever done for us?”, you can add one more item to your answer: They laid the foundations for abstract mathematical reasoning.
The post What Have the Ancient Babylonians Ever Done for Us? originally appeared on the HLFF SciLogs blog.