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

An Australian mathematician has discovered what may be the oldest known example of applied geometry, on a 3,700-year-old Babylonian clay tablet.

Known as Si.427, the tablet bears a ground plan measuring the boundaries of certain lands.

The tablet dates to 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 kept in the Istanbul Archaeological Museum before Dr Daniel Mansfield of the University of New South Wales found it.

Mansfield and Norman Wildberger, an associate professor at UNSW, had previously identified another Babylonian tablet as containing the oldest and most accurate trigonometric table in the world. At the time, they speculated that the tablet would likely have had a practical use, perhaps 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. Photography: UNSW Sydney

This tablet, Plimpton 322, described right triangles using Pythagorean triples: three integers in which the sum of the squares of the first two is equal to the square of the third – for example, 32 + 42 = 52.

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

“Si.427 relates to land that is sold,” Mansfield said. In cuneiform script, with its characteristic wedge-shaped indentations, the tablet describes a field containing marshy areas, as well as a threshing floor and an adjoining tower.

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

“Just like we would today, you have individuals trying to figure out where their land boundaries are, and the surveyor comes out but instead of using GPS equipment, they use Pythagorean triples.”

Although 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 it difficult to work with prime numbers greater than five.

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, it recolors all the mathematical tablets from that period,” Mansfield said. “You see mathematics developing to meet the needs of the times.”

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

“Is this part of a calculation they made?” Is this an area I haven’t discovered yet? Is it a measure of something? he said. “It’s really boring to me because there’s so much on the tablet that I understand. I’ve given up trying to figure out what it is.

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