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The Pythagorean Theorem
The theorem did not originate with Pythagoras. Archaeological evidence suggests that the Babylonians understood it a thousand years before the birth of Pythagoras, and even the ancient Chinese used it in their calculations. However Pythagoras developed a proof for the theory and taught it to his students. No writings or texts authored by Pythagoras have survived, but his disciples memorized his ideas and passed them along orally.
The Pythagorean Theorem applies only to right triangles; that is any triangle with a 90-degree angle and a ratio of sides of 3:4:5. As he studied the legs of the right triangle, he referred to the shorter sides in general terms as "a" and "b" and the longest side as "c" or the hypotenuse. The theorem states, "The area of the square built upon the hypotenuse of a right triangle is equal to the sum of the areas of the squares upon the remaining sides."
He used a writing stylus of metal, bone or ivory with a wax tablet to lay out his geometric constructions. The tablets had wax coats on wooden boards that were hinged together in such a way that the wax sides closed together with the boards on the outside. The wax could easily melt or dent, and the boards protected the written material on the inside. They were reusable by simply heating the wax or scraping off the marks. His tool kit also included a bronze compass and a straight edge marked with equal measurements to draw the constructions. The compass was actually a divider. It consisted of two collapsible arms connected with a central pivot hinge. Because of their flimsiness, it was impossible to transfer lengths from one drawing to another.
Tools for Surveying
Archaeological evidence shows that surveyors, architects and builders used the surveying tools of the Pythagorean Theorem in ancient times to build roads, design cities and lay out boundaries. The Egyptian engineers developed a way to lay out a right triangle with a rope. They tied 12 evenly spaced knots in the rope and pegged it to the ground using the dimensions of 3:4:5 and laid out a right triangle. These engineers who built the pyramids were called the "rope stretchers."
Pythagoras acknowledged the rope trick but never used it in his own work because of the difficulty of tying the knots at equal intervals. In Egypt, geometry developed from practical necessity and the need to measure land. The word "geometry" means earth measuring.
He used a straight edge to draw accurate lines. This tool was similar to a ruler without numerical marking; however, many straight edges displayed equidistant marks, which were accurately spaced. He might also have used string lined with chalk to lay out a consistently straight line on a large surface. The compass was the only other tool needed for geometric constructions. It used a center point and a radius measurement to lay out circles and arcs. The concept behind the mechanical handheld compass could be easily adapted to larger constructions by pinning a string to a center point and using the length of string (representing the radius) tied to a marker to sweep out the curve on the surface.
Numbers as Tools
Pythagoras taught that knowledge was the greatest purification; for his students, knowledge meant the numerical tools of mathematics. He declared that "Numbers rule the universe," and assigned numerical values with mystical and spiritual qualities to alphabet letters, ideas and some objects, thus developing a form of numerology. As he investigated the relationships in the 3:4:5 unit sides of the right triangle, he concluded that these numbers might reveal other number patterns in nature.
Mathematical and Philosophical Tools
The Pythagorean worldview revolved around the study of mathematics that represented the universe, the ultimate reality and God. Pythagoras taught that although we experience the material world with our senses, especially sight and touch, we can easily go astray from reality. Plato was a Pythagorean who reasoned that cave dwelling people can see shadows coming from the cave's entrance and assume that the shadows represent reality, but they ignore the objects that cast those shadows. The logical and rational tools of mathematics and philosophy reveal the truth of those objects themselves.