We main, which we write down under the place of tens. then proceed to the hundreds. As we have borrowed 1 from the 9 hundreds, the 9 is too large by 1. We must therefore take the 6 (hundreds) from 8 hundreds and there will remain 2 (hundreds). We therefore write down the 2 in the place of hundreds. Or, because the 9 is too large by 1, we may add 1 to the 6, and say 7 from 9 and 2 will remain. Hence the following RULE. Place the less number under the greater; units under units, tens under tens, &c. Begin with the units, and if the lower figure be smaller than the one above it, write the difference below. But, if the upper figure be less than the lower, then add ten to the upper one, and write the difference between the sum thus obtained and the lower figure. Then carry or add one to the lower figure of the next column, and proceed as before, till all the numbers are subtracted, and the result will be the difference. NOTE. The upper number is called the Minuend, from the Latin word minuendum, signifying to be made less; and the lower one the Subtrahend, from subtrahendum, to be taken away. The result is the Remainder. PROOF. Add the remainder to the subtrahend, and, if their sum be like the minuend, the work may be considered correct. 44. Sir Isaac Newton was born in the year 1642, and he died in 1727; how old was he at the time of his decease? Ans. 85 years. 45. Gunpowder was invented in the year 1330; how long was this before the invention of printing, which was in 1441? Ans. 111 years. 46. The mariner's compass was invented in Europe in the year 1302; how long was this before the discovery of America by Columbus, which happened in 1492? Ans. 190 years. 47. What number is that, to which if 6956 be added, the sum will be one million? Ans. 993044. 48. A man bought an estate for seventeen thousand five hundred and sixty-five dollars, and sold it for twenty-nine thousand three hundred and seventy-five dollars. Did he gain or lose, and how much? Ans. Gained $11810. 49. Bought a pair of oxen for 85 dollars, a horse for 126 dollars, three cows at 25 dollars apiece; and sold the whole for three hundred dollars; how much did I gain? Ans. $14. 50. Bonaparte was declared emperor in 1804; how many years since? 51. The union of the government of England and Scotland was in the year 1603; how long was it from this period to the time of the declaration of the independence of the United States? Ans. 173 years. 52. Jerusalem was taken and destroyed by Titus in the year 70; how long was it from this period to the time of the first Crusade, which was in the year 1096 ? Ans. 1026 years. SECTION IV. MULTIPLICATION. MULTIPLICATION is the repetition of a number any proposed number of times. It consists of three parts, the Multiplicand, or number to be multiplied; the Multiplier, or number by which to multiply; and the result, which is called the Product. The. Multiplicand and Multiplier are called factors. RULE. Place the larger number uppermost for the multiplicand, and the smaller number under it for a multiplier, arranging units under units, tens under tens, &c. Then multiply each figure of the multiplicand by each figure of the multiplier, beginning with the right-hand figure, and carrying for every ten as in addition. If the multiplier consists of more than one figure, the right-hand figure of each product must be placed directly under the figure of the multiplier that produces it, which will cause the successive products to recede each one place to the left. The sum of the several products will be the whole product required. NOTE 1. When there are ciphers between the significant figures of the multiplier, pass over them in the operation, and multiply by the significant figures only, remembering to set the first figure of the product directly under the figure of the multiplier that produces it. See Ex. 15. NOTE. 2. If there are ciphers at the right hand either of the multiplier or multiplicand, or of both, they may be neglected to the close of the operation, when they must be annexed to the product. PROOF. The correctness of the result in Multiplication may be conveniently ascertained in three ways; viz., by Division, by Multiplication, or by casting out the nines. According to the first method,* divide the product by the multiplier; and, if the work is right, the quotient will be equal to the multiplicand. According to the second method, take the multiplier for the multiplicand and the multiplicand for the multiplier, and proceed according to the rule for multiplication; and, if the work be right, the product will be the same as by the former operation. According to the third method, begin at the left hand of the multiplicand, and add together its successive figures towards the right, till the sum obtained equals or exceeds the number 9. If it equals it, drop the nine, and begin to add again at this point, and proceed till you obtain a sum equal to or greater than nine. If it exceeds nine, drop the nine as before, and carry the excess to the next figure, and then continue the addition as before. Proceed in this way till you have added all the figures in the multiplicand and rejected all the nines contained in it, and write the final excess at the right hand of the multiplicand. Proceed in the same manner with the multiplier, and write the final excess under that of the multiplicand. Multiply these excesses together and place the excess of nines in their product under the other excesses. Then proceed to find the excess of nines in the product obtained by the original operation, and, if the work be right, the excess thus found will be equal to the excess contained in the product of the above excesses of the multiplicand and multiplier. See Example 15. NOTE. This method of proof, though perhaps sufficiently sure for common purposes, is not always a test of the correctness of an operation. Cases will sometimes occur in which the excesses above named will be equal, when the work is not right. * As the pupil is presumed not to be acquainted with Division, he will pass over this method of proof for the present. It is placed here as a method important to be known, and because there seems to be no better place for it, though it presupposes an acquaintance with a rule yet to be learned. TABLE OF PYTHAGORAS. 34 36 38 40 42 44 46 48 30 33 36 39 42 45 48 51 54 57 60 63 66 69 72 1 2 3 41 5 6 7 8 9 10 11 12 13 14 15 16 17 18 19 20 21 22 23 24 21 4 6 8 10 12 14 16 18 20 22 24 26 28 30 32 44 48 52 56 60 64 68 72 76 80 84 88 92 96 55 60 65 70 75 80 85 90 95 100|105|110|115|120| 72 78 84 90 96 102 108 114 120|126|132 138|144 70| 77 84 91 98|105|112|119|126|133140 147|154|161|168 72 80 88 96 104|112|120|128|136|144 152|160|168|176|184|192| 81 90|99|108117|126|135|144|153|162 171|180|189|198 207|216 90 100 110 120 130 140 150 160 170 180 190 200 210 220 230 240 88 99 110 121 132 143 154 165 176 187 198|209 220 231 242 253 264 841 96 108 120 132|144|156|168|180|192|204|216|228|240 252|264|276|288| 78| 91|104|117|130|143 156 169 182 195 208|221|234 247|260|273|286|299|312| 70 84 98|112|126|140|154|168|182|196|210|224|238|252|266 280 294|308|322|336| 60| 75 90 105 120 135 150 165 180 195 210 225 240 255|270|285 300 315 330 345 360 64 80|96|112|128|144|160|176|192|208|224|240 256 272 288|304|320 336 352 368 384 68 85 102|119|136|153|170|187|204|221|238|255|272|289|06|323 340 357 374 391 408 72| 90|108|126|144|162 180|198|216|234|252 270 288 306 324 342|360|378|396 414 432 76 95 114 133 152 171 190 209 228 247|266 285|304|323|342 361 380 399 418 437|456| 80 100 120 140 160 180 200 220 240 260 280 300 320 340 360 380 400 420 440 460 480 84|105 126 147|168|189|210|231 252 273 294 315 336 357 378 399 420 441 462 483 504 22 44 66 88 110 132 154 176 198 220 242 264 286 308 330 352 374 396 418 440 462 4841506 528 23 46 69 92 115 138 161|1841207|230|253|276 299 322|315 368 391|414 437 460 483 506 529 552 24 48 72 96 120 144|168|192|216|240 264 288 312 336 360 384|408 432 456 480 504 528 5521576 15 30 45 21 42 63 |