Ancient architects and surveyors needed to be able to make right angles in the field as needed. The method used by the Egyptians earned them the name “rope pullers” in Greece, apparently because they used ropes to organize their construction instructions. One way they could use rope to form right triangles was to mark a looped rope with knots so that when the knots were held and pulled tight, the rope should form a right triangle. The easiest way to do the trick is to take a rope 12 units long, tie a knot 3 units from one end and 5 units from the other end, and then tie the ends together to form a loop. However, Egyptian scribes have left us no instructions on these procedures, much less any indication that they knew how to generalize them to obtain the Pythagorean theorem: the square of the line opposite the right angle. is equal to the sum of the squares on two. Sides Similarly, the Vedic scriptures of ancient India include sections on the Salva Sutra, or “Rope Rule,” for the correct positioning of sacrificial altars. The required right angles were constructed from the marked chords to give the triads (3, 4, 5) and (5, 12, 13).
In Babylonian clay tablets (c. 1700-1500 BC) modern historians have discovered problems whose solutions show that the Pythagorean theorem and certain triads were known more than a thousand years before Euclid. A randomly generated right triangle, however, is unlikely to have all its sides measured in the same unit—that is, each side is a whole number multiple of some common unit of measure. This fact, which came as a shock when discovered by the Pythagoreans, gave rise to the concept and theory of incommensurability.
Assessing wealth
A Babylonian cuneiform tablet written about 3,500 years ago treats problems of dams, wells, water clocks and excavations. It also has an exercise on circular enclosures with an implicit value of π = 3. King Solomon’s swimming pool contractor, who built a pool 10 cubits and 30 cubits around (1 Kings 7:23), uses the same value. However, the Hebrews must have taken their π from the Egyptians before crossing the Red Sea, for the Rhind Papyrus (c. 2000 bce; our original source for ancient Egyptian mathematics) implies π = 3.1605.
Knowledge of the area of a circle was of practical importance to the officials who kept track of the pharaoh’s tribute, as well as to builders of altars and swimming pools. Ahmadus, the author who copied and annotated the Rhind Papyrus (c. 1650 BCE), says much about cylindrical grains and pyramids, whole and truncated. He could calculate their volume, and, as shown by his taking the Egyptian secad, the horizontal distance associated with the vertical increment of a cubit, as the defining quantity for the slope of the pyramids, he could make similar triangles. knew something about
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