Although the theorem has been attributed to and named after Pythagoras and his community of scholars, it is believed that the concepts behind the theorem were known long before the Pythagoreans proved it. Among historians, an ongoing debate ensues about the possibilities that the ideas behind the Pythagorean Theorem were independently discovered by different groups at different times. A wide variety of theories exist, but there is substantial evidence that various civilizations used the Pythagorean Theorem, or were at least aware of the main principles of the theorem, to find the side lengths of right triangles.
In 1800 BCE, more than a thousand years before Pythagoras founded his school, a group of people living in Mesopotamia (located in present-day Iraq) already understood the relationship between the side lengths of a right triangle. These people, called Babylonians, were the first known group to demonstrate a conceptual understanding of the Pythagorean Theorem.
Historians have gained an understanding of the Babylonians from studying the ancient clay tablets they have left behind. These tablets were used throughout Mesopotamia to record a variety of information about commerce, culture, and daily life. Two of these clay tablets have particular relevance to the Pythagorean Theorem. On one of these tablets, which has been named YCB (short for Yale Babylonian Collection) 7289 since its discovery, there is an illustration of a tilted square with its two diagonals drawn in. In their own numeration system, Babylonians labeled the sides of the square as having a length equivalent to the value of 1 in our number system and the diagonal with a length equivalent to 1.414213. This decimal is a miraculously accurate approximation of 2√ , which proves that the Babylonians had very refined methods of calculation.
The Pythagorean Theorem was never explicitly written on any of the recovered clay tablets, but the engravings on tablet YBC 7289 display an early understanding of the Pythagorean Theorem because the diagonal of the square can be thought of as the hypotenuse of a right triangle. The legs of this right triangle, which are simply the sides of the square, each have a length of 1. By the Pythagorean Theorem, of which the Babylonians must have had some understanding, the diagonal must have a length of 12+12−−−−−−√ (because (leg1)2+(leg2)2=(hypotenuse)2). This is simply 2√ or, as the Babylonians approximated, 1.414213.
Plimpton 322 tablet with engravings of Pythagorean triples.
A second tablet (shown in Figure above), named Plimpton 322 after the collection to which it belongs, reveals the Babylonians’ advanced understanding of right triangles. Inscribed in this tablet is a table of Pythagorean triples, which are sets of three positive integers (a, b, c) that would satisfy the Pythagorean Theorem (a2+b2=c2). One example of a Pythagorean triple is the set (3, 4, 5), as seen in Example 1. We will explore Pythagorean triples more fully in the chapter “Applying the Pythagorean Theorem.”
Like Mesopotamia, Egypt was a great ancient civilization whose inhabitants were very commercially and culturally advanced. The Egyptians never explicitly expressed the Pythagorean Theorem as we know it today, but they must have used it in constructing their pyramids. It is known that, when building the pyramids, Egyptians used a knotted rope as an aid in making right angles. This rope had twelve evenly spaced knots (similar to the diagram below) that could be formed into a 3-4-5 right triangle with one angle of 90∘. The ropes were used as a model for the much larger right triangles used in the pyramids, which were built during a period of 1,500 years as a way to honor the pharaohs.
Try what the Egyptians did yourself! Cut a 12-inch piece of yarn and mark every inch by tying a knot. Next, try to construct a right triangle with this piece of yarn so that each side has an integer length, just like the Egyptians did.
Other Ancient Civilizations
It is believed that other ancient civilizations, such as China and India, also understood the Pythagorean Theorem before Pythagoras himself proved it. The debate about whether the theorem was discovered in one place at one time or in many places at different times still lingers today.