(10) Eighty-six ; five hundred and ninety-four ; seventeen; two hundred and eight. L. (1) Five thousand eight hundred and sixtyseven ; seven hundred and ninety-four; eight thousand three hundred and five ; sixty ; eight hundred and fifty-four; three thousand and seventy-nine. (2) Six hundred and forty-eight ; three t'ousand nine hundred and seventy-five ; eighty-seven ; four hundred and nine ; six thousand five hundred and thirty ; seven hundred and sixty-eight. (3) Five thousand three hundred and eighty; seventy-nine ; four thousand and eight ; six hundred and fifty-seven ; ninety ; nine thousand four hundred and eight. (4) Eighty-seven; four thousand and sixty-nine ; three hundred and eight; five thousand seven hundred; four hundred and eight; thirty-nine. (6) Five hundred; six thousand and eighty-seven ; ninety-four ; one hundred and eleven ; seven thousand five hundred and six; four hundred and nine. (6) Three thousand eight hundred and seven ; four hundred and sixty ; seventy-eight thousand and eighty-five; nine hundred and two; one hundred and ten; thirty-six thousand and nine. (7) Four thousand one hundred and twenty-seven; twelve thousand and eight; five hundred and nine; sixty-seven ; one hundred and forty-nine ; eightyfour thousand and sixty-eight. (8) Six thousand three hundred and fifty-nine ; nine hundred and ninety-nine; ninety-five thousand and eighty-seven; nine hundred and nine; eighteen thousand and seventy-eight; ten thousand eight hundred and ten. (0) Nineteen thousand nine hundred and nine; eighty-nine; seven hundred and six; eighty thousand and forty-eight; nine hundred and eleven ; twentyfive thousand and nine. (10) Sixty ; fifteen thousand and eight; nine hundred and seventy-six ; seventy-nine thousand one hundred and eight; eighty-nine thousand and eighteen ; seven hundred and six. Further exercises in addition may be formed from the following table, where the numbers in every row, whether reckoned upright, or from left to right, or from corner to corner, when added together, will give the same number for their sum. 2452 1971 1490 1009 528 47 4450 3969 3488 3007 2526 248924151934 1453 972 491 417 44133932 3451 2970 2933 28592378 1897 1416 935 454 380 4376 38953414 3377 2896 28222341 1860 1379 898 824 34343393858 3821 3340 3266278523041823 1342 861 787 306 4302 4265 3784 3303 3229 2748 2267 1786 13051231 750269 232 4228 3747 36733192 27112230 1749 12681194 713 676 195|4191 371036363155 2674 21931712 1638 1157 1120 639 15841544080 3599 3118 2637 2156 1675 1601 1564 1083 602 121 411740433562 3081 2600 2119 2045 2008 15271046 565 84448740063525 30442563'2082 : CHAPTER III. SÚBTRACTION----SECTION 1. WHEN one number is taken from 'another, the number which is left is called the remainder or difference, and the process by which it is found is called subtraction. If from a bag of marbles, we take a certain number of marbles, we perform a subtraction. Subtraction is the converse of addition, for addition consists in uniting with, subtraction in withdrawing from. To subtract" 3 from 5, we may hold up three fingers and count downwards from 5 three steps, thus four, three, two; the remainder is 2. But in adding 3 to 5 we count upward three steps—thus six, seven, eight. Examples.--Find the difference between- 7 2 and 8. 5 and 2. 5 and 9. 3 and 6. 8 and 2. 7 and 9. 7 (18) 3 and 1o. 7 and 5. 9 and 2. 3. 9 and 15. 5 and 9. The difference of any two numbers may be found by the process explained above, but the pupil is recommended to learn the subtraction table, he will then be able to perform the operation with much less labour. i SECTION II. To find the difference between numbers of two or more figures. Find the difference between 86 and 54. (1) (8) (16) (3) (17) (10) (11) and 12. (5) 6 and 3 (12) (19) 6 and 12 (6) (13) (20) 9 and (7) 8 and 5. Place the less number under the greater, units under units and tens under tens, as in addition, and subtract the figures in the lower line from those above them beginning with the units' figure thus :86 4 units from 6 units leaves 2 units; and, 5 tens from 8 tens leaves 3 tens. 54 Thus the remainder is 3 tens 2 units, or 32, 32 In this example each figure in the lower line is less than the one above, but if we have to find the difference between 37 and 65 and we attempt to subtract as before, we meet with a difficulty immediately, because 7, being greater than 5, cannot be taken from it. Now the difference between two numbers is not altered by adding the same number to both of them; for if John has 4 marbles more than James and we give each of them 2 marbles, John will still have 4 marbles more than James. We may therefore add any number to 65 and 37 without altering their difference; let us add 10 to each of them. Then 65 or 6 tens 5 units will become 6 tens 15 units, and 37 will become 3 tens 17 units or 4 tens 7 units. Subtracting we get 2 tens 8 units or 28 for the remainder. In practice the operation is performed as follows: 65 Since 7, the units' figure in the lower line is 37 greater than the figure over it, add ten to 5 28 mentally, or, which is the same thing, imagine I to be placed before the 5, this makes 15; subtract 7 from 15 and set down the remainder 8; next add mentally, i ten to the 3 tens in the lower line, or as it is called, carry i to the next figure, and say 4 from 6 leaves 2. From the preceding observations we deduce the following rule for finding the difference between any numbers whatever. Place the less number under the greater, units under units, tens under tens, and so on ; begin at the units' place and subtract each figure in the lower line from that in the upper, increased by 10 if necessary, taking care, when ío has been added to the upper line, to add 1 to the next figure in the lower line. Proof.-Add together the remainder and the less of the given numbers, the sum will be equal to the greater number, if the operations be correctly performed. Example.-4 taken from 6 leaves 2, From 6 Take and this remainder added to 4, the less of 4 the given numbers, gives 6 the greater number. Signs. The operation of subtraction is indicated by a straight line, thus -, which is read minus, or less by. This sign was probably adopted because the distance between any two points is less by the straight line than by any other. The expression 8-3=5 is read 8 minus 3 equals 5. Examples : (3) '54 (3) 64 (4776 26 2 (1) (5) 86 (6) 79 (7) 89 |