_a_ - (_b_ + _c_ - _d_ - _e_) = _a_ - _b_ - _c_ + _d_ + _e_. 2_a_ + 3_b_ - (_a_ - 2_b_) = 2_a_ + 3_b_ - _a_ + 2_b_ = _a_ + 5_b_. 4_x_ + _y_ - (17_x_ - 9_y_) = 4_x_ + _y_ - 17_x_ + 9_y_ = 10_y_ - 13_x_. 42. I want to find the difference of the numbers 57762 and 34631. Take these to pieces as in (29) and 57762 is 5 ten-th. 7 th. 7 hund. 6 tens and 2 units. 34631 is 3 ten-th. 4 th. 6 hund. 3 tens and 1 unit. Now 2 units exceed 1 unit by 1 unit. 6 tens 3 tens 3 tens. 7 hundreds 6 hundreds 1 hundred. 7 thousands 4 thousands 3 thousands. 5 ten-thousands 3 ten-thous. 2 ten-thous. 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. [7] Those numbers which have been altered are put in italics. 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.