A Mathematician Deciphered The Code of a 3,700-Year-Old Babylonian Clay Tablet!
In ancient discoveries, a Babylonian Clay Tablet known as ‘Plimpton 322‘ has captivated mathematicians and historians alike. It was found in the ruins of the ancient city of Larsa, now part of modern-day Iraq. This artifact dates back approximately 3,700 years to the Babylonian era around 1800 BCE. Its significance, however, remained hidden for centuries. That is until Dr. Daniel Mansfield, an Australian mathematician from the University of New South Wales, decoded it. His breakthrough revealed that the tablet contains a highly advanced form of trigonometry. It was far older and different from the Greek system we traditionally associate with the discipline.
The tablet has four columns and 15 rows of numbers written in cuneiform script. It initially appeared to be a list of numbers without much mathematical context. However, upon deeper examination, Dr. Mansfield realized it held Pythagorean triples, a mathematical principle used for calculating right angles. These findings highlighted that the Babylonians were using a base-60 number system to create a more accurate form of trigonometry. It is a trigonometric table. They designed it specifically for practical tasks, such as surveying and land measurement. Let us now find out more exciting details about this clay tablet.
Unveiling the Origins and Age
Archeologists discovered Plimpton 322 in 1894, during an excavation of the ancient city of Larsa, situated along the Euphrates River. They found the clay tablet amidst a trove of other Babylonian artifacts. But it was unique due to the mathematical markings it carried. Historians date the tablet to approximately 1800 BCE, making it one of the oldest recorded examples of applied mathematics. After its discovery, the tablet was stored in the Istanbul Archaeological Museum, where it remained largely unnoticed. “It’s the only known example of a cadastral document from the Old Babylonian period, which is a plan used by surveyors to define land boundaries”. Says Dr. Mansfield
Dr. Mansfield Decoded the Babylonian Tablet
The new study was conducted by Dr Mansfield and UNSW Associate Professor Norman Wildberger. The findings were published in Historia Mathematica, the official journal of the International Commission on the History of Mathematics.
Dr Mansfield came across the Plimpton 322 while preparing study material for UNSW’s first-year mathematics students. He and Dr. Wildberger decided to study Babylonian mathematics and examine the different historical interpretations of the tablet’s meaning after realizing that it parallels the rational trigonometry of Dr. Wildberger’s book Divine Proportions: Rational Trigonometry to Universal Geometry.
The 15 rows on the clay tablet detail a sequence of 15 right-angle triangles, which are continuously decreasing in inclination. Babylonians wrote it in cuneiform text. Upon closer examination, he identified that the numbers inscribed on the tablet followed patterns associated with Pythagorean triples, which are sets of three whole numbers that satisfy the equation a²+ b²=c². This is the rule that describes the relationship between the sides of right-angled triangles. The integers 3, 4, and 5 are a well-known example of a Pythagorean triple. But the values on Plimpton 322 are often considerably larger with, for example, the first row referencing the triple 119, 120, and 169. So this was the oldest trigonometric table with practical use that we know.
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 BCE. But the Babylonians developed their own alternative ‘proto-trigonometry’ to solve problems related to measuring the ground, not the sky. Babylonians developed their own alternative ‘proto-trigonometry’ to solve problems related to measuring the ground, not the sky
Dr. Mansfield
The Accuracy of the Babylonian Method
The Babylonians’ use of a base-60 number system made this tablet stand out, as opposed to our modern base-10 system. This sexagesimal system, commonly associated with the way we measure time today, allowed for more accurate calculations in certain contexts. Dr. Mansfield explained in an interview that “the base-60 system allowed for far more accurate calculations than our decimal-based system,” which tends to round off numbers.
The Babylonian Clay Tablet method, based on exact ratios, is more precise in certain practical tasks than modern trigonometry, which often involves approximations. For example, the ratio-based approach avoids errors that arise from irrational numbers such as the value for pi. Using whole numbers (like the 3-4-5 triangle), the Babylonians were able to construct perfectly accurate right angles without the need for sophisticated tools or calculations.
Dr. Mansfield emphasized the significance of these methods, stating that the Babylonians “may not have been interested in angles at all, which makes their approach to trigonometry vastly different but no less effective.” This practical mindset, which focused on solving real-world problems, allowed them to create accurate land divisions and construction plans without the mathematical abstractions that would come later in Greek trigonometry.
Experts long regarded the Greek astronomer Hipparchus, who lived about 120 years BC, as the father of trigonometry, with his “table of chords” on a circle considered the oldest trigonometric table.“Plimpton 322 predates Hipparchus by more than 1,000 years,” says Dr. Wildberger.
Conclusion
The decoding of the Babylonian Clay Tablet by Dr. Daniel Mansfield has reshaped our understanding of early mathematics. The Babylonians’ method, based on ratios between triangle sides, provides a more practical and accurate form of trigonometry for specific applications, such as land measurement and construction. Their use of the base-60 number system enabled them to achieve a level of precision that surpasses modern trigonometry in some contexts. As Dr. Mansfield has revealed, the mathematical prowess of the Babylonians was far more advanced than previously believed, offering new insights into the development of mathematical thought in the ancient world.
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