 Very few examples of Greek computational methods have survived. In fact, our best examples come from Theon of Alexandria, who lived in the 4th century AD, and Eutokias, who lived in the 6th century AD almost a thousand years after the main developments in Greek mathematics. The examples suggest algorithms for multiplication and division very similar to our own, but worked with Greek numerals. An example of a Greek multiplication comes to us from Eutokias in the 6th century, who computed the product as… Now translating these numbers into more recognizable forms, this is 1351 times 1351, and the partial products are… Where we use the comma to separate the ten thousands place from the thousands, so there are in fact four zeros after the comma. So note that the first row of the partial products is 110,000, 30,000, 510,000, and 1,000. And what's important here is to note that if you were to multiply 1,000 times 1,351, you would get this number, but Eutokias doesn't write it that way. And the fact that Eutokias wrote it the way that he did indicates that he multiplied the digits one at a time. In other words, we found 1,000 times 1,000 to give us 110,000s, 1,000 times 300, that's 30,000s, 1,000 times 50, 510,000s, 1,000 times 1, 1,000. And a similar pattern prevails on the second, third, and fourth partial products. And this suggests that the process of multiplication is to multiply each digit of the upper factor by the first digit of the lower factor, repeat with the remaining digits, and then add the partial products. So we might try Eutokias's method to multiply 75 by 38. So 75, that's 70 and 5, and 38 is 30 and 8. So for the first row, we'll multiply the top row by 30, 30 times 70 is 2100, 30 by 5 is 150. For the second row, we'll multiply the top row by 8, 8 by 70 is 560, 8 by 5 is 40, and then we'll add the partial products. And you can find products of three digit numbers as well, 217 by 195, that's 210 and 7 by 190 and 5, and first we'll multiply 210 and 7 each by 100, then by 90, then by 5, and add the partial products.