egypt

Geometry Problem 50 of the Ahmes papyrus uses these methods to calculate the area of a circle, according to a rule that the area is equal to the square of 8/9 of the circle's diameter (so 1/9 is subtracted from the diameter, and the resulting figure is multiplied by itself, using the doubling method). This assumes that π is 4×(8/9)² (or 3.160493...), with an error of slightly over 0.63 percent. This value was slightly less accurate than the calculations of the Babylonians (25/8 = 3.125, within 0.53 percent), but was not otherwise surpassed until Archimedes' approximation of 211875/67441 = 3.14163, which had an error of just over 1 in 10,000). Interestingly, Ahmes knew of the modern 22/7 as an approximation for pi, and used it to split a hekat, hekat x 22/x x 7/22 = hekat; however, Ahmes continued to use the traditional 256/81 value for pi for computing his hekat volume found in a cylinder.
Problem 48 involved using a square with side 9 units. This square was cut into a 3x3 grid. The diagonal of the corner squares were used to make an irregular octagon with an area of 63 units. This gave a second value for π of 3.111...
The two problems together indicate a range of values for Pi between 3.11 and 3.16.
A problem in the Moscow Mathematical Papyrus considered finding the volume of a truncated pyramid with sides of 2 and 4 units and a height of 6: "Add together this 16 with this 8 and this 4. You get 28. Compute a third of 6. You get 2. Multiply 28 by 2. You get 56. Behold: it is 56. You have found right.

Water clocks are possibly the oldest time-measuring instruments, with the only exceptions being the gnomon and day-counting tally sticks. Given their great antiquity, where and when they first existed are not known and perhaps unknowable. The simplest form of water clocks, the bowl-shaped outflow type, are known to have existed in Babylon and in Egypt around the 16th century B.C. Other regions of the world, including India and China, also have early evidence of water clocks but the earliest dates are less certain. It is possible that the earliest water clocks existed as early as 3000 BC, given the advancement of civilizations during that time.
Some water clock designs were developed independently and some knowledge was transferred through the spread of trade. It is important to point out that the need for the common person to 'know what time it is' largely did not exist until the Industrial Revolution, when it became important to keep track of hours worked. In the earliest of time, however, the purpose for using a water clock was for astronomical and astrological reasons. These early water clocks were calibrated with a sundial. Through the centuries, water clocks were used for timing lawyer's speeches during a trial, labors of prostitutes, night watches of guards, sermons and Masses in church, to name only a few. While never reaching the level of accuracy based on today's standards of timekeeping, the water clock was the most accurate and commonly used timekeeping device for millennia, until it was replaced by the more accurate pendulum clock in the 17th century.

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