On Addition

In regard to astronomy we have then the need of six types of operations¹³, of which the first is that referred to as numeration, ¹⁴ and then addition, subtraction, multiplication and division, while the sixth is the extraction of the square root¹⁵ of each (given) number. There has already been mention made of the representations, so let us turn our attention to the method of addition.

Addition is the combination of two or more numbers into a sum resulting in one number, for example when we add two and three we make five. It is performed as follows.

Write down the symbols in turn, as many and whichever ones you like, and again underneath these, write as many digits as you like, with the same number of places or more or less. Let them be placed so that the units¹⁶ are placed under the units, the tens under the tens and so on in turn. Then add each to each, that is, the first to the first, the second to the second and so on in similar fashion. Write the number resulting from the two above the first column, that is, the number formed from adding the first column above the first column, that resulting from the second column above the second column and so on in turn. If then the total of the numbers added is two or three or four and so on as far as nine, it is written above, as has been said. But if the total is ten then write a cipher, which indicates `nothing', take a one instead of ten and combine it with those numbers being next in line to be added. If the total is more than ten, write down that part of its excess over ten above the numbers added, as before, and take one instead of ten and combine it with those numbers about to be added.

To make the explanation clear by example, I give the following diagram.

We begin then with 3 and 7; 3 and 7 gives 10, write the cipher above, and carry¹⁷ the unit, which indicates ten.  Again, 4 and 8 gives 12, add in¹⁸ also the unit which you carried and we get¹⁹ 13.  We again take the unit which indicates ten and write the 3 above. Now 3 and 6 gives 9, add in the unit you carried and we get 10.  Write the cipher above and keep the unit; now say  2 and 5 makes 7, add in the unit you held and we get  8 and write this above. Thus the sum of five thousand six hundred and eighty seven and two thousand three hundred and forty three is eight thousand and thirty.

There follows a test²⁰ by which we can learn whether we have performed the addition correctly or not. Regard the values of the signs no longer as monadic, decadic and so on, but take them all as monadic and add (the numbers in) each of the rows individually and look at the number resulting from each. Subtract nine from each, look at what is left for each number, and place what is left at the end of the row which produced it. If the remaining numbers of the two rows add to less than nine, then it is not necessary to subtract (further) from them, but if what is left exceeds this, then again subtract nine and look at the remainder. If it equals the remaining number in the top third row, then one knows that the addition was effected correctly, but if they are not equal then the opposite is true.

For the sake of clarity, let us demonstrate on the previous example.  We say that 8 and 3 is 11, take away 9 leaves [2. Now  5 and  6 is 11, take away  9 leaves  2, 2 and 8 is 10, take away 9 leaves]²¹ 1. And  1 and  7 is  8 and so write  8 in turn in the middle row.   Again 2 and 3 is 5,  5 and  4 is 9, take away 9 and the resulting number 3 remains. Write this in turn after the numbers added together. Now 3 and 8 is 11, take away 9 and 2 is left over. You then have the number equal to the number from the previous check and so the addition is correct.

So much for addition.