Did the Babylonians Predict the Pythagorean Theorem Before Pythagoras?
The Pythagorean Theorem, a cornerstone of ancient mathematics, was famously associated with the Greek philosopher Pythagoras in the 6th century BCE. However, evidence suggests the Babylonians had knowledge of a similar relationship long before Pythagorasrsquo; formulation. This article explores the remarkable discovery of the Plimpton 322 tablet and its implications for our understanding of early mathematical advancements.
The Plimpton 322 Tablet
One notable example of Babylonian mathematical prowess is the Plimpton 322 tablet, which dates back to around 1800 BCE. Found in modern-day Iraq, this clay tablet contains a list of Pythagorean triplesset of three positive integers (a, b, c) that satisfy the equation (a^2 b^2 c^2). This evidence suggests the Babylonians had a sophisticated understanding of the theorem and its practical applications, such as land measurement and construction.
Understanding the Plimpton 322 Tablet
The Plimpton 322 tablet is a rectangular sheet of clay inscribed with cuneiform script. It contains 15 rows of numbers, with each row representing a different Pythagorean triple. Each entry is composed of three sexagesimal numbers, corresponding to the side lengths of a right triangle.
The Old Babylonians interpreted these triples as the sides and diagonal of a rectangle, just as the Pythagoreans did. They recognized that there is no natural number triple where the two sides are equal to a square, a concept that predates our modern understanding.
The First Entry: A Near-Perfect Square
The first entry in the tablet is the 119-120-169 Pythagorean triple, illustrating a near-square rectangle. The equation for this entry is:
(119^2 120^2 169^2)
This entry is particularly interesting as it approximates a square, with the sides differing by only 1 unit.
Successive Entries and Regular Numbers
The tablet lists subsequent entries that gradually deviate more from near-square shapes. The first column of the tablet encodes an entire Pythagorean triple as a single sexagesimal number. The heading of the first column translates as:
ldquo;The takiltum of the diagonal from which 1 is subtracted and that of the width comes up.rdquo;
The term ldquo;takiltumrdquo; refers to the normalized square. By normalizing the long leg to 1, the Babylonians effectively expressed the following relationship:
(left(frac{c}{b}right)^2 1 left(frac{a}{b}right)^2)
Subtracting 1 from the value in the first column provides the normalized squared width, a relationship that can be expressed as:
(left(frac{c}{b}right)^2 - 1 left(frac{a}{b}right)^2)
The Case of the Missing Number
The first row of the tablet misses the number 120, theorized to be part of a column that has broken off. Each subsequent entry in the first column is a regular number (a number with only factors of 2, 3, and 5), indicating that when divided by a regular number, the quotient is a terminating finite-length number in their base 60 system.
An Example Calculation
Letrsquo;s consider the fourth row of the tablet, which corresponds to the sexagesimal number:
1.53.10.29.32.52.16
In base 60, this number is:
(1 cdot 60^6 53 cdot 60^5 10 cdot 60^4 29 cdot 60^3 32 cdot 60^2 52 cdot 60 16)
As a base 60 integer in base 10, it is:
(88004782336)
This represents a number between 1 and 2, equivalent to a decimal fraction in base 60 over three millennia before Stevin. Subtracting 1 from this value gives:
(left(frac{18541}{13500}right)^2)
Thus, we can deduce the triple:
(12709^2 13500^2 18541^2)
This remarkable calculation, done in 1800 BCE, demonstrates the advanced mathematical abilities of the Babylonians.
Conclusion
The Plimpton 322 tablet provides invaluable insights into the mathematical knowledge of ancient Babylonian civilization. Despite not having a formal proof equivalent to Pythagorasrsquo; development, the Babylonians had a practical understanding of the Pythagorean Theorem and its applications. Their ability to work with Pythagorean triples and their innovative methods of calculation showcase the sophistication of early mathematics.