History Weblecture for Unit 6
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The impulse that drove the citizens of Mesopotamia and Egypt to explain the rising of the rivers, the storms, and the daily motion of the sun as the acts of divine beings also drove them to try to explore these phenomena more carefully. Both civilizations kept careful records of solar positions against the background sky. Both civilizations dealt with the practicalities of reestablishing property boundaries after flooding, and with creating trade inventories, exchanging monies, and measuring the stone blocks required for temples and tombs for their kings. By the second millennia before Christ, both civilizations had established methods of writing, including writing numbers and carrying out exercises that we would now recognize as algebra and geometry.
In Babylon, a sharply pointed stick or stylus was used to make wedge-shaped marks on soft clay tablets. The tablets would be left to dry out, or baked and hardened, after which they could be stacked. Their cuneiform writing was originally a kind of pictograph, where small, stylized pictures called ideographs stood for individual words, much the way we might write I ♥ NYC. Unless broken by impact, the tablets with their cuneiform markings would last for centuries.
The early Babylonians used two symbols in writing numbers: an "up" wedge or ∇ that stood for one, and a sideways wedge or > that stood for ten. The number 23 would look like >>∇∇∇. Obviously, writing 999 would be somewhat cumbersome, so they developed a positional system that allowed them to compress a bit how they wrote large numbers. This system used as its base the number 60, the way that we now use the number 10. "Unit" marks (∇) had a different meaning, depending on where they appeared. Unit marks before a "tens" mark indicated "sixty" rather than "one". So ∇∇ >>> meant 2 * 60 + 3 * 10 = 150.
Using a base of 60 has some distinct advantages, especially when doing division. The number 60 can be evenly divided by 2, 3, 4, 5, 6, 10, 12, 15, 20, and 30. This makes it easier to express some fractions exactly. The Babylonians would write 3 and 2/15s (three and two fifteenths) as 3,8, meaning 3 units plus 8 60ths. Lest this seem really bizarre, I could point out that as I write this, my digital clock reads 3:16, or 3 hours and 16 "60ths of an hour". We just happen to call a 60th of an hour a "minute", and the fact that we keep time this way, with 60 seconds to the minute and 60 minutes to the hour, dates back to the number system the Babylonians invented.
With this number system, and their ability to write and calculate relatively easily, the Babylonians made some major strides into algebra. They had clay tablets that listed solutions to certain kinds of problems, like the squares and square roots of all numbers between 1 and 150, or the areas of rectangles of different sides, or the hypotenuse of right triangles, and for the solution of the quadratic equation.
From the last set of tablets, we can conclude that some of them, at least, knew and used the Pythagorean theorem, which states that the square of the hypotenuse (long side) of a right triangle will be the sum of the square of its sides: D2 = H2 * L2. They used the concept of exponents, the way we use 23 to mean 2 * 2 * 2, to calculate compound interest problems. In fact, on a tablet dating from around 2000 bce, there is a student exercise to calculate how long it would take for a sum of money to double itself at 20% interest.
Despite their advances in algebra-like formula, the Babylonians don't seem to have looked for general solutions to mathematical problems. Their tables were set up so that those without mathematical training could use the results of prior calculations to solve immediate practical problems, like figuring out the area of a square plot of land by looking up the sides.
Babylonian geometrical calculations
By comparison with the Egyptians, though, the Babylonians were amateurs at geometry. They used a value of 3 for π, and their formula for calculating a truncated pyramid, one with the point at the top missing, was an approximation much less exact than the formula the Egyptians used.
The Egyptians built a pyramid sometime before 2700 bce in which they laid to rest their king Khufu (called Cheops by the Greeks). The pyramid was 230 meters on the side and stood 140 meters high. The sides of the pyramid are exact to within 2 centimeters. If you want to get an idea of how precise this is, think of $230.00 give or take 2 cents. That's a variation of 1 part in 10000. The right angles at each corner are precise to 2 minutes of arc (1/30th of one degree). The tomb faces due north, aligned to the north celestial pole star (at the time) with the same degree of accuracy.
Section of temple wall in the British Museum. © 1986,2005 Christe A. McMenomy
Such precision gives evidence not only of a good understanding of geometry, but of a very practical ability to apply geometry to the problems of construction and land measurement. When we consider that the regular flooding of the Nile made it necessary to survey and mark the boundaries of property every year, it isn't very surprising that the Egyptians became interested in finding easy ways to solve their geometric problems.
One of the advantages the Egyptians had was the development of hieroglyphic forms of writing, which they were able to do using papyrus, a kind of flattened paper-like material made from the leaves of a plant that grows in the Nile River marshes. Their pictorial language covers the walls of their tombs and temples as well. In this section of a temple wall, you can easily pick out birds, fish, people and plants.
Over time, the symbols came to indicate sounds rather than specific concepts, so that the Egyptians used their pictures a bit like the way we use letters. This actually made the language much harder to interpret if one didn't know what sounds the symbols stood for. The information in many Egyptian inscriptions and papyri was undecipherable to later generations until the discovery of the Rosetta Stone in the early nineteenth century. This tablet had a section in Egyptian hieroglyphs at the top, nicely translated into Greek below. Using this as a key, archaeologists were able to "crack the code" of Egyptian writing.
Rosetta Stone, British Museum © Christe A McMenomy
Numbers from temple wall (Louvre) © 2015 Christe A. McMenomy
Can you find the numbers above on the wall to the left?
Like us, the Egyptians used a base 10 system. However, they didn't have symbols for 1, 2, 3, and so on. Instead, they had different symbols for 1 (|), 10 (^), 100 (?), and 1000 (a crescent-moon on a pole), and each symbol was repeated however many times it was necessary to write out a number. The numeral for 236 would be ??^^^||||||.
To do a multiplication problem, an Egyptian mathematician would double numbers as many times as necessary to get the right result. For example, to multiply 7 * 8, he would start with a table of multiplications of 8 and select the multiplications that added up to the 7:
|Choose||Number of 8s||Value of Doubled number|
|Sum of chosen rows||7||56|
While both their method of writing numerals and their methods of calculation were cumbersome, the Egyptians did manage to use this information to solve some interesting problems. The Rhind papyrus, composed around 1650 bce during the Middle Kingdom and finally translated in the 19th century, contains a problem in fractions:
Divide 100 loaves among five men in such a way that the shares received shall be in arithmetic progression and that 1/7 of the sum of the largest three shares shall be equal to the sum of the smallest two. What is the difference of the shares?
But the supreme achievement of the Egyptian mathematicians lies in their understanding and application of geometry. Not only did they use the formula of height * width for calculating rectangular area that was known to the Babylonians, they had a better approximation for the value of π: they used the formula
area of a circle = (16/9 d)2
This works out (as you can see below) to 256/81 times the radius squared. 256/81 is 3.16, not a bad approximation for π = 3.14159.....
Despite these accomplishments, neither the Babylonians nor the Egyptians had a symbol for "nothing in this place". We use a zero to show that a there are no tens in the number 103, just one hundred, and three ones. This kind of notation problem complicated calculations. Even with all their tables of specific solutions for particular problems, the Babylonians failed to make the leap to a general solution for situations like the solution of the Pythagorean theorem or the quadratic equation.
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