Therefore, by (40, Principle II.) all the first column together exceeds all the second column by all the third column, that is, by
2 ten-th. 3 th. 1 hund. 3 tens and 1 unit,
which is 23131. Therefore the difference of 57762 and 34631 is 23131, or 57762-34631 = 23131.
43. Suppose I want to find the difference between 61274 and 39628. Write them at length, and
61274 is 6 ten-th. 1 th. 2 hund. 7 tens and 4 units.
39628 is 3 ten-th. 9 th. 6 hund. 2 tens and 8 units.
If we attempt to do the same as in the last article, there is a difficulty immediately, since 8, being greater than 4, cannot be taken from it. But from (40) it appears that we shall not alter the difference of two numbers if we add the same number to both of them. Add ten to the first number, that is, let there be 14 units instead of four, and add ten also to the second number, but instead of adding ten to the number of units, add one to the number of tens, which is the same thing. The numbers will then stand thus,
6 ten-thous. 1 thous. 2 hund. 7 tens and 14 units.[7] 3 ten-thous. 9 thous. 6 hund. 3 tens and 8 units.
You now see that the units and tens in the lower can be subtracted from those in the upper line, but that the hundreds cannot. To remedy this, add one thousand or 10 hundred to both numbers, which will not alter their difference, and remember to increase the hundreds in the upper line by 10, and the thousands in the lower line by 1, which are the same things. And since the thousands in the lower cannot be subtracted from the thousands in the upper line, add 1 ten thousand or 10 thousand to both numbers, and increase the thousands in the upper line by 10, and the ten thousands in the lower line by 1, which are the same things; and at the close the numbers which we get will be,
6 ten-thous. 11 thous. 12 hund. 7 tens and 14 units.
4 ten-thous. 10 thous. 6 hund. 3 tens and 8 units.
These numbers are not, it is true, the same as those given at the beginning of this article, but their difference is the same, by (40). With the last-mentioned numbers proceed in the same way as in (42), which will give, as their difference,