# Australian mathematician discovers applied geometry engraved on 3,700-year-old tablet

An Australian mathematician has discovered what may be the oldest known example of applied geometry, on a 3,700-year-old Babylonian clay tablet. 3,700 year old Babylonian clay tablet revealing that they were doing more accurate trigonometry nearly 1,500 years before the Greeks.

Known as Si.427, the tablet bears a field plan measuring the boundaries of some land.

The tablet dates from the Old Babylonian period between 1900 and 1600 BCE and was discovered in the late 19th century in what is now Iraq. It had been housed in the Istanbul Archaeological Museum before Dr Daniel Mansfield from the University of New South Wales tracked it down.

Mansfield and Norman Wildberger, an associate professor at UNSW, had previously identified another Babylonian tablet as containing the world’s oldest and most accurate trigonometric table. At the time, they speculated the tablet was likely to have had some practical use, possibly in surveying or construction.

The ancient clay tablet was engraved with a stylus to describe a field containing marshy areas, as well as a threshing floor and nearby tower. Photograph: UNSW Sydney

That tablet, Plimpton 322, described right-angle triangles using Pythagorean triples: three whole numbers in which the sum of the squares of the first two equals the square of the third – for example, 32 + 42 = 52.

“You don’t just accidentally come up with trigonometry, you’re usually doing something practical,” Mansfield said. Plimpton 322 set him on a quest to find other tablets from the same period that contained Pythagorean triples, eventually leading him to Si.427.

“Si.427 is about a piece of land that’s being sold,” Mansfield said. In the cuneiform script, with its characteristic wedge-shaped indentations, the tablet describes a field containing marshy areas, as well as a threshing floor and nearby tower.

The rectangles depicting the field have opposite sides of equal length, suggesting surveyors of that time period had devised a way to create perpendicular lines more accurately than before, according to Mansfield.

“Much like we would today, you’ve got private individuals trying to figure out where their land boundaries are, and the surveyor comes out but instead of using a piece of GPS equipment, they use Pythagorean triples.”

Though Plimpton 322 and Si.427 both use Pythagorean triples, they predate the Greek mathematician Pythagoras by more than 1,000 years.

“Once you understand what Pythagorean triples are, your society has reached a particular level of mathematical sophistication,” Mansfield said.

Si.427 contains three Pythagorean triples: 3, 4, 5; 8, 15, 17; and 5, 12, 13.

The Babylonians used a base 60 number system – similar to how we keep time today – which made working with prime numbers larger than five difficult.

Si.427, described in a study in the journal Foundations of Science, dates from a period of increasing private land ownership. “Now that we know what problem the Babylonians were solving, that recolours all the mathematical tablets from this period,” Mansfield said. “You see mathematics being developed to address the needs of the time.”

One thing that puzzles Mansfield about Si.427 is the sexagesimal number “25:29” – similar to 25 minutes and 29 seconds – that is etched in large font on the back of the tablet.

“Is it part of a calculation that they performed? Is it an area that I haven’t come across yet? Is it a measurement of something?” he said. “It’s really annoying to me because there’s so much about the tablet that I understand. I’ve given up trying to figure out what that one is.”

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