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The system now becomes: xyc = 210 x + yc = 29 and the solution is found by applying a famous Babylonian recipe. The arithmetic mean of x and y' is 14 1--- . 14 1--2- 2 – 210 is calculated, which is 1--- . The square root of 2 4 this number is 1--2- and according to the recipe, the correct values for x and y' are then 14 1--- + 1--- and 14 1--- – 1--2- , respectively. The final solution is x = 15, y = 12; this can be 2 2 2 seen on the pictured fragment under the line with 27 and 3,3, because the solution is shown first, and the method used is shown underneath.

During this process, a second-degree equation is solved in passing by complet------ . The final solution for the problem itself ing the square. This is followed by z = 21 11 ------------ . This number is the secis found by substituting this value in z + 2 and squaring: 1849 121 ond-largest square and must therefore be equal to 6x + 4 . ------------ this gives the following three numbers as a After finding the value of x = 1365 762 solution: 58 --------484 1878 -----------484 7338 -----------484 As shown by the final steps of his treatment, Diophantus no longer wants to use fractions in the numerator or denominator, but in this case he does use the common quadratic numerator, so that the fractions are not given in their simplest form.

DEJ GH ]KT. This represents 4, because the M with a circle above means (according to Diophantus) that it concerns units, Monaden; so here the 4 is just a number. The letterL next to T represents 10, the following letters stand for 20, 30, etc. This is followed by the nine hundreds. There were separate words for 1000 and 10,000. We can now read the ‘1’ and the ‘2’ units (monads). The sign in the front is new; this is the building block of Diophantus’ fame. Diophantus said that this was the unknown quantity of units to be found, the arithmos.

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