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

The geometric formulas found on the ancient Babylonian tablet force other tablets from that era to be revised, and the Australian scholar says that this indicates that "the ancient society reached a high level of mathematical sophistication."

An ancient Babylonian tablet, probably used for geodetic work, it uses Pythagorean triplets at least 1000 years before Pythagoras.

An Australian mathematician found perhaps the oldest known example of applied geometry on a 3,700-year-old Babylonian clay tablet.

The plate, known as Si.427, shows a plan of a field where the boundaries of a piece of land are measured.

The tablet dates back to the ancient Babylonian period between 1900 and 1600 BC and was discovered in the late 19th century on the territory of modern Iraq. It was kept in the Istanbul Archaeological Museum before Dr. Daniel Mansfield of the University of New South Wales tracked it down.

Mansfield and Norman Wildberger, assistant professor at the University of South Wales, previously determined that another Babylonian tablet contains the oldest and most accurate trigonometric table in the world. At the time, they suggested that the tablet probably had some kind of practical use, perhaps in surveying or construction.

This tablet, according to Plympton, described right-angled 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 just accidentally come to trigonometry, you usually do something practical," Mansfield said. Plympton pushed him to look for other tablets from the same period containing Pythagorean triplets, which eventually led him to Si.427.

"Si.427 is information about the land that is being sold," Mansfield said. The cuneiform script with characteristic wedge-shaped depressions on the tablet describes a field with swampy areas, as well as a threshing ground and a nearby tower. The rectangles depicting the field have opposite sides of the same length, which suggests that surveyors of the time came up with a way to create perpendicular lines more accurately than before, says Mansfield.

"Like today, individuals are trying to figure out where the boundaries of their lands are, and then a surveyor appears, but instead of using GPS equipment, he uses Pythagorean triplets."

Although Pythagorean triplets are used in Plimpton 322 and Si.427, they appeared more than 1000 years before the Greek mathematician Pythagoras.

"Once you understand what Pythagorean triplets are, your society has reached a certain level of mathematical sophistication," says Mansfield.

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

The Babylonians used a base 60 number system - much like we count time today - which made it difficult to work with prime numbers greater than five.

Tablet Si.427, described in a study published in the journal Foundations of Science, dates back to a period of increasing private ownership of land.

"Now that we know what problem the Babylonians were solving, it changes the meaning of all the mathematical tablets of this period," says Mansfield. "You see that mathematics is evolving to meet the needs of the day."

One thing that puzzles Mansfield in the Si.427 is the sexagesimal number "25:29" - similar to 25 minutes and 29 seconds - which is engraved in large print on the back of the nameplate.

"Is this part of the calculation they have done? Is this some information that I haven’t come across yet. Is this a measurement of something?" - he said. "It really annoys me because I don't understand so much about the information written on this ancient table. I gave up trying to figure out what it is."