Ancient Tablet May Show Earliest Use of This Advanced Math

From National Geographic:

For nearly 100 years, the mysterious tablet has been referred to as Plimpton 322. It was first discovered in Iraq in the early 1900s by Edgar Banks, the American archaeologist on which the character Indiana Jones is thought to have been largely based. It was later bought by George Arthur Plimpton in 1922 and has been called the Plimpton 322 tablet ever since.
Now researchers from the University of New South Wales are calling it one of the oldest and possibly most accurate trigonometric tables of the ancient world.
Findings published in the journal Historia Mathematica, the official journal for the International Commission on the History of Math, reveal how researchers dated the ancient clay tablet and came to conclusions about its use. 
The tablet is arranged in a series of 15 rows intersected by four columns. According to the UNSW researchers the tablet uses a base number of 60, which may have been used to allow ancient Babylonians to derive integers instead of fractions.
Norman Wildberger, explained that the research team reached their conclusions that the tablet was used for the study of triangles by findings based on ratios, not angles. In the top row of the tablet, said Wildberger, relatively equal ratios create a near equilateral triangle. Descending down the tablet, the ratios decrease the triangle's inclination, creating narrower triangles.
"It is a fascinating mathematical work that demonstrates undoubted genius," said University of New South Wales researcher Daniel Mansfield in a press release.
The researchers speculate the tablet could have been used to survey fields or construct buildings. For example, knowing the height and width of a building, ancient builders would have been able to calculate the exact measurements need to build pyramid slopes.


The Greek astronomer Hipparchus has widely been considered the father of trigonometry. During his life, roughly dating to 120 B.C., he famously created a table of chords drawn from the center of a circle that resulted in angles from which he derived trigonometric formulas.
Does this study dethrone him? Not quite, say two experts on ancient mathematics.
Despite being in top condition for a tablet likely created around 1762 B.C., the left-hand edge of the artifact is broken. (Glue residue found on the side suggest the break was recent.) The team used previous research on Plimpton 322 to speculate that it was originally built with six columns and 38 rows.
Duncan Melville is a professor of mathematics at St. Lawrence University who specializes in Mesopotamian mathematics.
"Apart from the column headings, the tablet just consists of columns of numbers, and this invites a great deal of purely mathematical speculation," said Melville in an emailed statement to National Geographic. "Some of the different interpretations for construction of the tablet are mathematically equivalent and so just having the output on the tablet does not tell you much about the process used to generate that output."
Melville stated that to accept the study's results would in a sense redefine trigonometry, but Wildberger, who has previously argued for new theories of trigonometry, argued adopting a new mindset to understand how ancient Babylonians may have worked is essential.
Donald Allen, a mathematics professor at Texas A&M University, is also skeptical that the researchers have proven Plimpton 322 was used for trigonometry.
"It is old and accurate, but the interpretation of it as a trig table is conjecture, as it is broken, and the telling part would be contained with the part broken off, and never found," he said in an emailed statement. 
Allen noted the most important finding from the tablet is the evidence of Pythagorean triples, indicating that Babylonians were seemingly aware of the Pythagorean theorem—years before Pythagorus. If the UNSW study does show how the tablet was used to find approximate solutions to equations involving triangles, only speculative historical context can determine exactly how the tablet was applied in day-to-day life said Wildberger.
If the Babylonians were the originators of trigonometry, say Allen and Melville, it was drastically improved in efficiency and accuracy by the Greeks nearly a thousand years later.
"Bottom line is this," says Allen, "if interpreted as a trig table, it would be the oldest known. Some of their computations were very accurate. Babylonian arithmetic was rather clumsy, but then so were Egyptian and Greek variations."
He noted that mathematicians in the ancient world heavily borrowed from one another, making it difficult to track their origins.


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