scending from the minuend with a horizontal column extending to the right of the subtrahend. Thus the difference between 11 and 5 is 6. 23. To illustrate the principles given in Art. 21, let it be required to subtract 14235 from 27869. so that, as in Addition, units may fall under units, tens under tens, &c.; the subtraction is effected by taking the units of the lower line from those of the upper line, the tens from the tens, &c. Now 9-5-4-difference of the units; 2-1-1-difference of the ten thousands. The difference of the proposed numbers is therefore composed of 1 ten thousand, 3 thousands, 6 hundreds, 3 tens, and 4 units; that is, the difference is 13634. 2nd Ex. From 575 take 193. Proceeding as in the first example,― 5-3 2 difference of the units; 7-9, the subtraction cannot be effected. This method of finding the difference of two numbers leads apparently to the impossibility of taking a greater number from a less. Yet since 575 is greater than 193, it is evident that the subtraction of the whole number, 193 from 575, is possible. The difficulty is got over thus. Any figure on the left of another expresses units of the order immediately superior to the units of that other figure (Art. 3.). Consequently, in this example, 1 in the place of hundreds is equal to 10 in the place of tens. Let 1 be taken, or, as is commonly said, borrowed from the 5 hundreds, which are thus reduced to 4. Then the 1 hundred borrowed 10 tens; and 10 tens + 7 tens = 17 tens. Whence, resuming the subtraction, 17-9=8=difference of the tens; 4-1=3=difference of the hundreds. The partial remainders being 2 units, 8 tens, and 3 hundreds, To effect in this example the subtraction of 9 tens from 7 tens, 1 hundred, or 10 tens, are borrowed from 5, the next higher figure of the minuend; and the 5 is thus reduced to 4. But the absolute difference of two numbers is not changed by the addition of 1 to each of them; the result will, therefore, be the same whether the 1 borrowed be taken from the next higher figure of the minuend, and the corresponding figure of the subtrahend be left as it is, or the said figure of the minuend be left as it is, and I be added to the corresponding figure of the subtrahend; for this amounts simply to the increasing of each figure by 1. The second method, namely, that of carrying 1 for 10 borrowed, to the next higher figure of the subtrahend, is adopted in practice, because of its convenience. 3rd Ex. From 7967 take 3984. 7-4-3 difference of the units, = 6 being less than 8, ten are borrowed. 6+10=16, and 16-8=8=difference of the tens. Carrying to 9, the next figure of the subtrahend, 1 for 10 borrowed, 9+1=10. 9+10=19, and 19-10=9=difference of the hundreds. Carrying to 3, the next figure of the subtrahend, 1 for 10 borrowed, 3+1=4. And 7-4-3-difference of the thousands. Whence, if from 7967 there be taken 3984 the remainder is 3983 In the third partial subtraction of this example, it is found necessary to subtract 10 from 19, a contingency provided for in the table of subtraction. The detail of the operation may be abridged, as in the following example: 4th Ex. From 937040 take 878654. 10-4-6 units of the remainder; 5+1=6 and 14-6-8-tens of the remainder; 8+1=9 and 9-9-0-hundred thousands of the remainder. 24. In Subtraction, as in Addition, the operation is performed from right to left. In effect, if it were never necessary in Addition to carry, or in Subtraction to borrow, the calculation might be made either way indifferently. But when the units in the sum of a column exceed 9, the next figure to the left must be augmented; and when, in Subtraction, a figure of the subtrahend is greater than the corresponding figure of the minuend, one must be borrowed from the next higher figure of the latter. Proceeding from right to left, these operations fall into the work; but from left to right, they give rise to new additions and subtractions. 25. To find the difference of two unequal numbers: General Rule. Write the figures of the subtrahend under the figures of the same order of the minuend. Then if every figure of the subtrahend is less than the corresponding figure of the minuend, subtract the simple units of the subtrahend from those of the minuend, and write the remainder below the subtrahend in the place of simple units. Afterwards, in succession, subtract the tens of the subtrahend from the tens of the minuend, the hundreds from the hundreds, &c., and write the partial remainders in their proper places; the result thus obtained is the difference sought. But if a figure of the minuend is less than the corresponding figure of the subtrahend, add 10 to the units expressed by that figure of the minuend, subtract the figure of the subtrahend from this sum, carry 1 to the next figure of the subtrahend as an equivalent for the 10 borrowed, subtract the augmented figure of the subtrahend from the corresponding figure of the minuend, borrowing and carrying if necessary; and thus proceed till all the partial subtractions are effected. The result is the difference of the proposed numbers. 26. Examples. Find the difference between 1st. 9 and 5. 2nd. 18 and 7. 3rd. 13 and 10. 4th. 17 and 6. 5th. 19 and 9. 6th. 18 and 13. 7th. 25 and 14. 9th. 38 and 25. 13th. 19685 and 7420. 19th. 10 and 7. 20th. 16 and 9. 21st. 21 and 13. 22nd. 35 and 27. 23rd. 54 and 39. 24th. 135 and 97. 25th. 586 and 279. 26th. 974 and 596. 27th. 1376 and 897. 50th. 1000000000 and 123456789. 51st. How often can 53746 be subtracted from 214984 and the successive remainders? 52nd. How often can 32756 be subtracted from 196536 and the successive remainders ? 53rd. How often can 840750 be subtracted from 672600 and the successive remainders ? 54th. How often can 152708 be subtracted from 1159979 and the successive remainders? and what is the remainder left from the last subtraction? 27. If it be required to find the difference between one number and several other numbers, the most obvious course is to take one subtrahend from the minuend, a second from the first remainder, a third from the second remainder, &c., till all the numbers are subtracted. Since, by subtracting the same numbers in any order, the absolute number of simple units taken away is the same, it follows that the last remainder is independent of the order of subtraction. Also, since the last remainder is the difference between the minuend and all the simple units in the subtrahends, it follows that if the sum of all the subtrahends be at once taken from the minuend, the result is equal to that obtained by the method of repeated subtraction. 28. The arithmetical complement of a number is the difference between that number and 10 in the place of its highest figure. Thus the arithmetical complement of 6 is 4=10—6. In subtracting a number from 1 followed by as many zeros as that number contains figures, it is evident that ones for tens borrowed must either be carried to the figures of the subtrahend, or taken from the corresponding figures of the minuend. If the second method is adopted, the last significant figure of the subtrahend must be taken from 10, and the other figures each from 9. Whence the arithmetical complement of a number is obtained by taking, from left to right, the significant figures of the subtrahend each from 9, excepting the last figure, and this from 10. When in a calculation, some numbers are to be added, and others subtracted, it may be convenient to use the arithmetical complements of the latter; but it is chiefly in calculating with logarithms that the employment of arithmetical complements is advantageous. 29. The sum of any number and its arithmetical complement is equal to 10, in the place of the highest figure of the number. Hence, to find the difference of two numbers, let the arithmetical complement of the subtrahend be added to the minuend, |