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The Great Calculation According to the Indians, of Maximus Planudes - On Multiplication

Author(s): 
Peter G. Brown

Multiplication takes place when we take one number a certain number of times, and from this produce a new number. For example, twice three is six. We took three, as many times as there are units in two, or vice versa, which produces six. Multiplication is performed as follows:

Write down the symbols in turn, as many and whichever ones you like, and again underneath these, another row of digits, with the same number of places or more or less. Do this however in an orderly fashion so that unit is under unit, the tens under the tens and so on in turn. Then multiply the first digit on the top row with the first digit of the bottom row. If the resulting number is less than ten29 then write it above the first, but if a multiple of ten30, such as ten or twenty or thirty and so on, write zero31 and carry as many units as there are tens in the product, but if (the product) is made up of both kinds, that is, units and tens, such as fifteen or twenty four and the like, then write the number of units above; for example five or four, but in regard to the tens, carry as many units as there were tens. Now multiply the first digit of the top line with the second digit of the bottom line and again the first digit of the bottom line with the second digit of the top and take the sum of these products. Add on the monads which you carried, whether two or more, and again, if a number less than 10 results, write it down above the second digit, or if a multiple of ten or mixed, proceed as previously shown. Now if each row has only two digits, there remains only to multiply the second digit by the second digit and then add to this any units that might be carried and write the result after the numbers previously written. If there are three digits, multiply the first by the third and again the first by the third chiastically32 and also the second by the second and adding these up write it down as you were instructed.33 Then multiply the second by the third and the second by the third chiastically, record it and then multiply the third by the third and again record it.

To make the explanation clear by example, I give firstly the following diagram having two rows each with two digits.\[\begin{array}{|ccc|}\hline 8 & 4 & 0 \\ \hline & 2 & 4 \\ & 3 & 5 \\ \hline \end{array}\]

We say four-times five is twenty and write zero34 above the 4 since twenty is a multiple of ten, and carry the two. We then say four-times three is twelve and five-times two is ten and together they make twenty two. To this add the two units we carried and it becomes 24. We write the 4 above the 2 and carry the two units. Again we say twice three is six and add on the two units and it becomes eight. I then write 8 in turn next to the previous digits and this number is the product of twenty four by thirty five.

Now consider another diagram where the lines have three digits. \[\begin{array}{|cccccc|}\hline 1 & 1 & 4 & 0 & 4 & 8 \\ \hline & & & 4 & 3 & 2 \\ & & & 2 & 6 & 4 \\ \hline \end{array}\]

We then say twice 4 is eight and write this above the 2. Twice 6 is 12 [and four-times 3 is 12]35 and together they make 24. We write 4 above the 3 and carry 2. Again twice 2 is 4, four-times 4 is 16, and thrice 6 is 18 and together they make 38. We add the two units to this to give 40. Write zero above the 4 and carry 4. Then we say thrice 2 is 6, six-times 4 is 24, total 30. Add on 4 gives 34. Write 4 next to the zero and carry 3. Now we say four-times 2 is 8 and add  3 making 11, and write this in turn after the 4.

This is how the multiplication proceeds if there is an equal number of digits in each of the rows, but if one exceeds the other, fill up the row with the smaller number of digits with zeros and repeat the method outlined. To make this clear by example, we illustrate as follows: \[\begin{array}{|ccccc|}\hline 7 & 6 & 8 & 4 & 2 \\ \hline & 1 & 4 & 2 & 3 \\ & 0 & 0 & 5 & 4 \\ \hline \end{array}\]

Thrice 4 is 12. Write 3 above the 2 and carry 1. Also thrice 5 is 15 and four-times 2 is 8, together giving 23. We add on the unit, total 24 and write 4 above the 2 and carry 2. Now thrice zero is zero, four-times 4 is 16, twice 5 is 10 , a total of 26. Add on the two giving 28. We write the 8 above the 4 and carry 2. Also thrice zero is zero, four-times 1 is 4, twice nothing is nothing and five-times 4 is 20, giving 24 . Add on 2 to give 26 . Write 6 above the the 1 and carry 2. Now twice nothing is nothing, five-times 1 is 5, four-times 0 is 0 making 5, and add on 2 gives 7. We write this in turn next to the 6. Then four-times 0 is 0, nothing-times 1 is nothing and (so) we do not write anything, also once 0 is 0 and again I do not write anything.

Observe that when you have multiplied the first digit (on the top row) by the last digit (on the bottom row), you should make some sign on the first digit to indicate that it has been multiplied by all the digits in turn and must not be multiplied again.

Peter G. Brown, "The Great Calculation According to the Indians, of Maximus Planudes - On Multiplication," Loci (March 2012)

Dummy View - NOT TO BE DELETED