Spring 2006
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“Babylonian” Translation Modern algebraic “equivalent “
sidi side x
mehr square (or area of square) x2
ush length x
sag breadth y
kush height z
asha area xy
sahar volume xyz
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Sample problems similar to some Babylonian problems but with “nice” numbers:
Suppose that the length and breadth of a rectangle added together is 20 and that twice the length added to five times the breadth is 55. Multiply 2 times 20; you get 40. Subtract 40 from 55; you get 15. Subtract 2 from 5; you get 3. Divide 15 by 3; you get 5. Subtract 5 from 20; you get 15. The length is 15 and the breadth is 5..
Suppose two numbers have a sum of 14 and a product of 45. Find the two numbers. First, halve 14; you get 7. Multiply 7 by 7; you get 49. Subtract 45 from the 49; you get 4. And 2 is the root. Add 2 to 7. Subtract 2 from 7. One number is 9 and the other is 5.
I have added up the area and sixteen sides of my square; 132. Write down 16, the coefficient. Break off half of 16; 8. 8 and 8 you multiply; 64. You add 64 to 132. The result is 196. This is the square of 14. From 14, subtract 8, which you squared above. 6 is the side of the square.
The problems below are transcriptions of problems from
actual Babylonian tablets. The
translations (which do not include the boldface) are from The History of mathematics : a reader, edited by John Fauvel and
Jeremy Gray, published by
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BM 13901
I have added up the area and the side of my square; 0;45. You write down 1, the coefficient. You break off half of 1. 0;30 and 0;30 you multiply: 0;15. You add 0;15 to 0;45: 1. This is the square of 1. From 1, you subtract 0;30, which you multiplied. 0;30 is the side of the square.
I have subtracted the side of my square from the area: 14,30. You write down 1, the coefficient. You break off half of 1. 0;30 and 0;30 you multiply. You add 0;15 to 14,30. Result 14,30;15. This is the square of 29;30. You add 0;30, which you multiplied to 29:30. Result 30, the side of the square.
I have added up seven times the side of my square and eleven times the area: 6;15. You write down 7 and 11. You multiply 6;15 by 11 : 1,8;45. You break off half of 7. 3;30 and 3;30 you multiply. 12;15 you add to 1,8;45. Result 1,21. This is the square of 9. You subtract 3;30, which you multiplied, from 9. Result 5;30. The reciprocal of 11 can not be found. By what must I multiply 11 to obtain 5;30? 0;30, the side of the square is 0;30.
The surfaces of my two square figures I have taken together: 21;15. The side of one is a seventh less than the other. You write down 7 and 6. 7 and 7 you multiply; 49. 6 and 6 you multiply. 36 and 49 you add: 1;25. The reciprocal of 1;25 cannot be found. By what must I multiply 1;25 to give me 21;15? 0;15. 0;30 the side. 0;30 to 7 you raise: 3;30 the first side. 0;30 to 6 you raise: 3 the second side.
YBC 6967
[Two numbers have a product of 1,0] The larger exceeds the smaller by 7. What are [the two numbers]? As for you, halve 7, by which the larger exceeds the smaller, and the result is 3;30. Multiply together 3;30 with 3;30, and the result is 12;15. To 12;15, which resulted for you, add 1,0, the product, and the result is 1,12;15. What is the square root of 1,12;15? 8;30. Lay down 8;30 and 8;30, its equal, and then subtract 3;30 from the one and add it to the other. One is 12, the other 5. [These are the two numbers.]