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Science & technologyAug 7th 2021 edition

Ancient geometry
The Babylonians used Pythagorean ideas long before Pythagoras

Surveyors employed them to measure out land

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Aug 5th 2021

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MOST READERS will have encountered Pythagoras's theorem about
right-angled triangles--that the square on the hypotenuse is equal to
the sum of the squares on the other two sides--at school. But the
less-mathematically inclined might have been tempted to ask when such
knowledge would ever be useful in real life. One answer, predating
Pythagoras by over 1,000 years, is in land surveying.

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Lurking in a museum in Istanbul is a 3,700-year-old clay tablet known
as Si.427. It has been there since it was dug up in the 19th century
in Sippar, an ancient Babylonian city in what is now Iraq. Only this
year, however, has its significance been revealed--by Daniel
Mansfield, a mathematician at the University of New South Wales, in
Australia. As he describes in Foundations of Science, Dr Mansfield
has shown that Si.427 is inscribed with the world's oldest known
example of applied geometry.

Si.427 seems to have been created by a surveyor defining land
boundaries after part of a field was sold. This is, of itself,
unremarkable. What is remarkable is that its text describes the use
of what are now called Pythagorean triples to draw accurate right
angles.

Pythagorean triples are special cases of Pythagoras's theorem. They
are sets of whole numbers that correspond to the lengths of the sides
of particular right-angled triangles. Three, four and five for
example, or five, 12 and 13. Another tablet, Plimpton 322, showed
many decades ago that the Babylonians knew about these triples. What
researchers could not agree on was why they were interested.

Dr Mansfield's answer is surveying, for this was a period when
agriculture was developing, and with it the idea of land ownership.
One way to measure the size of a field is to divide it into
rectangles and right-angled triangles, both of which have easily
calculated areas. Knowing Pythagorean triples makes drawing the right
angles needed to construct these figures easier, and so is useful
information. QED.

An early version of this article was published online on August 4th
2021

This article appeared in the Science & technology section of the
print edition under the headline "No need for a protractor"

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