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Mathematics is a logical method ... Mathematical propositions express no thoughts. In life it is never a mathematical proposition which we need, but we use mathematical propositions only in order to infer from propositions which do not belong to mathematics to others which equally do not belong to mathematics.
Tractatus Logico Philosophicus, New York, 1922, p. 169.
The Great Calculation According to the Indians, of Maximus Planudes
On Subtraction or Taking Away
Subtraction from a number, means to take one number from another and look at what is left over. We always either take the lesser number from the greater, so that there is some remainder, for example three from five leaves two, or we subtract equal from equal, where the remainder is nothing; for example when we subtract three from three. It is not possible to take a greater from a lesser number, for it is not possible to take away what is not there.
Write the digits in turn as many and whichever you like. Below it write the same number of digits or less, but not more. If the number below which is being subtracted has fewer digits then no other condition need be stated. But if it has the same number of digits as the top number then make sure the last digit of the top row is greater than the last corresponding digit of the bottom row. As stated previously, when I speak of the `last digit', I mean the one on our left hand side. We need this digit to be greater than the bottom one to ensure that the whole number (on top) is greater than the whole number (on the bottom). This being so, even if the rest of the digits in the bottom number are greater than the digits in the top number, taken one at a time, nevertheless the top number is greater and in no way smaller than the bottom one, that is, than the number being subtracted.
Suppose then that the numbers are written with digits corresponding, units with units, tens with tens and so on in turn. If the first digit22 on the bottom row is less than the first digit of the top, then take the lesser from the greater and write the number remaining above the first digit on the first row. If it is equal, then subtract equal from equal and, as stated above, write the remaining zero above. If it is greater, since we cannot take the greater from the lesser, borrow a unit which signifies ten, from the next digit after it, that is from the second digit of the bottom row. (This digit after the first place occupies the second decadic position.) Having added this ten to the smaller number in the top row, subtract the larger from that total and write the remainder again above the smaller digit. If you can take the second digit on the bottom line, after adding the unit to it - for this unit is again thought of as a unit in respect to the number in the corresponding column, but as ten in respect to the preceeding column, since every number is taken to be ten times the number in the preceeding column, viz; ten compared to the unit, a hundred to ten, a thousand to a hundred and so on - if then you can take away this second digit, after adding to it the unit, then take it away and write the remainder, if there is any, above that second digit on the top row, but if not then write 0. Again, if the second digit on the bottom row, with the unit, is greater than the second digit on the top, again borrow a unit, that is a ten, from the third column on the bottom and add the ten to the second digit on the top and take away the second digit on the bottom, with the unit, and again write the remainder above the second digit on the top. Proceeding thus to the end, you will have the desired result. For the number remaining after you take the whole of the lesser number from the whole of the greater is precisely the number written above the top row.
5 4 6 1 2
1 8 7 6 9
5 4 6 1 2
3 5 8 4 3
1 1 1 1
I take from 11 the 4 with the unit, that is 5, leaving 6 and I write this above the 1. Using this method I reach the last column and since I can subtract 3 with a unit from 5, I subtract from it 4, that is 3 with a unit, leaving a unit. This number I write above the 5.
With this in mind another example is given, in order to show that if the remainder is zero, it is still written. Beginning in the third place in the accompanying example,
after adding the unit to the digit thus making 2, I can subtract this completely from 2 and nothing is left. I write 0 above the two. I continue in the same way to the end of the diagram.
6 4 5 4 3 2
6 3 8 6 7 4
6 4 5 4 3 2
6 7 5 8
1 1 1 1 1
Going to the 4th place, since I am unable to take the 6 with the unit added, that is 7, from the 5 I write a unit next to the units under the 4 and I regard this as 10. Having added this to the 5, I proceed as previously explained. After this I take the unit from the 4 leaving 3 and I write this above the 4. If there are two, three or more digits less in the bottom than the top26 a unit is not written below those digits from there to the end along the top row, but the digits are written as they are, each one written above itself in turn after those digits written above and included with the remainders.27