should then take a 10 foot pole, every one of his marks would be one foot behind my marks. But, as he keeps on measuring with his 9 foot pole, his second mark will be 2 feet behind mine ; his third, 3 feet; his fourth, 4 feet; and so on. Thus, at every measurement, his mark falls a foot farther behind mine. Therefore, our marks will be always just as many feet apart, as the number of measurements. 3. From this illustration it will be very clear, that as 9 is one less than 10, so two 9s are 2 less than two 10s; three 9s, 3 less than three 10s; and so on. For one 10 consists of a 9 and a unit. And two 10s are one 10 and one 10. Then, if each of these 10s contain a 9 and a unit, both of them must contain two nines and two units. So, likewise, three 10s contain three 9s and 3 units ; four 10s, four 9s and 4 units; and any number of 10s contains just as many 98 as there are 108, and just as many units besides. 4. Therefore, if from one 10 I take 1 unit, one 9 remains; if from two 10s I take away 2 units, two 9s remain ; if from three 10s, 3 units, three Is remain; and, if from any number of 10s, I take the same number of units, the same number of 98 remains. That part of this idea which we wish to have distinctly remembered, may be concisely expressed as follows: If from ANY NUMBER OF TENS, the same number of UNITS be taken away, THE REMAINDER WILL CONSIST OP EVEN 9s. ART. 11. It appears (1: 4) that if from ten 10s, that is, from 100, I should take away 10 units, ten 9s would remain. Now these 10 units taken away, make one 10, which is equal to one 9 and 1 unit. Therefore, in the ten 10s or 100, there are ten 9s, and one 9 and 1 unit; that is, eleven 9s and 1 unit. Therefore, if from 100, I take away one unit, a number of even 9s, that is, eleven 9s, will remain. If, from 200 I should take 2 units, a number of even 9s would, likewise, remain. For 200 is 100 and 100; and if one unit be taken from each hundred, it will leave a number of even 9s in each case. But taking one unit from each, is taking 2 units from the whole. Therefore, if2 units be taken from 200, a number of even 9s remains; if 3 units be taken from 300, a number of even Is remains ; and, if any number of units be taken from the same number of hundreds, a number of even 9s remains. Hence, If from ANY NUMBER OF HUNDREDS, the same number of UNITS be taken away, THE REMAINDER WILL CONSIST OF EVEN 9s. ART. III. 1. It appears (11.) that ten hundreds, or 1,000 consist of even Is and 10 units. But these 10 units make one 10, which is equal to one 9 and 1 unit. Therefore, 1,000 consists of even 98, and 1 unit. Hence, if from 1,000, I should take away 1 unit, a num. ber of even 9s would remain. Then, as in hundreds, it may be shown, that, if 2 units be taken from 2,000, a number of even 9s will remain ; if 3 units from 3,000; 4 units from 4,000; or, if any number of units be taken from the same number of thousands, a num. ber of even 9s will remain. And, If from ANY NUMBER OF THOUSANDS, the SAME number of UNITS be taken away, the REMAINDER WILL CONSIST OF EVER 98, 2. And, by the same method of proof it may be shown, that, If from ANY NUMBER OF UNITS OF ANY ORDER, the same number of SIMPLE UNITS be taken away, THE REMAINDER WILL CONSIST OF EVEN 9s. ART. IV. 1. Let there be given any number, as 324. 20 4 Now it has been shown, (111. 3.) that, 300 contains even Is and 3 units, And 20 contains even 9s and 2 units, And 4 contains no 9s and 4 units. The whole number, 324, then, consists of even 9s, and 3 units, and 2 units, and 4 units. It is plain, then, that if 3 units and 2 units and 4 units, when added together, make 9, or any number of even 9s, the number 324, will consist of even 9s, with no remainder. 3 and 2 and 4 are 9. Therefore, 324 consists of even 9s, without remainder. But 3 and 2 and 4 are the figures which make 324. Therefore, IF THE SUM OF THE FIGURES WHICH COMPOSE A NUMBER, CONSISTS OF EVEN 9s, THE NUMBER ITSELF CONSISTS OF EVEN 9s. Also, IF THE SUM OF THE FIGURES WHICH COMPOSE A NUMBER, DOES NOT CONSIST OF EVEN 98, THE NUMBER ITSELF DOES NOT CONSIST OF EVEN 9s, BUT LEAVES THE SAME REMAINDER AS THAT LEFT BY THE SUM OF ITS FIGURES, Let there now be given an example in addition, as the following. Add 178 17 543 3 821 | 2 15423 After adding, draw a line up and down, on the right. Add the figures 1, 7 and 8, which compose the upper number. Their sum is 16=9+7. Reject the. 9, and set down the 7, opposite the upper number, and outside the line. Add the figures of the second num. ber, reject and set down, as before. Do the same with the third number. We know, therefore, that the upper number consists of even 9s and 7; the second, of even 98 and 3; and the third of even 98 and 2. Of course, if 7 and 3 and 2 added together, are even 98, the sum of the numbers ought to consist of even 9s. But 7+3+2 =12=9+3. Therefore, the sum of the numbers ought to be even 9s, and 3 over. Set down this 3 opposite the amount, 1542. It is plain, that if the example is correctly performed, this amount ought to contain just as many 9s as the numbers added, with the same excess, or remainder. What this remainder is, we shall find by adding the figures 1, 5, 4 and 2, which compose the amount, and rejecting the 9s from their sum. The sum is 12,=9+3. Reject the 9 and 3 remains. This remainder is exactly what it was shown above that it ought to be. Hence, we conclude, that the amount is correct, and that the addition has been correctly performed. From what has been said, we derive the following rule for the proof of addition. I. ADD THE FIGURÉS COMPOSING EACH OF THE GIVEN NUMBERS. II. REJECT OR CAST OUT THE 9s FROM EACU SUM, AND SET DOWN THE EXCESSES OPPOSITE THE SEVERAL NUMBERS. III. ADD THESE EXCESSES, REJECT THE 9s FROM THEIR SUM, AND PLACE THE REMAINDER OPPOSITE THE AMOUNT. IV. ADD THE FIGURES COMPOSING THE AMOUNT, REJECT THE 9s, AND, IF THE REMAINDER, THUS FOUND, AGREES WITH THE LAST, THE OPERATION MAY BE CONSIDERED AS CORRECTLY PERFORMED. Note. It will be found most convenient and expeditious, to reject the 98 WHILE ADDING, that is, as often as you obtain an amount of 9 or more, reject the 9, and proceed with the remainder, if any, is before. This rule may, possibly, make a false result appear correct. For, if we make one figure of the amount a unit or two too large, and another, as much too small, the sum of the figures, and of course, its excess of 9s, will remain the same. Or if there be any number of errors made, such, that half their amount shall be on the side of excess, and the other half on the side of deficiency, the excess of 9s will remain unaltered, and, of course, the result will seem correct. But it is very improbable tha errors, exactly counterbalancing each other, will be made, in the same operation; and hence, this mode of proof may, perhaps, be better depended upon than any other. It is, perhaps, needless to remark that errors may sometimes be made in proving, as well as in performing the first operation : so that no mode of proof is a certain test of the accuracy of a result. The property of the number 9 on which the above mode of proof depends, is not an essential property of the number ; that is, it is not a property which belongs to it from its nature, but it is one which results from its place in the scheme of notation. It would belong to 11, if we were to reckon by twelves; and to 29, if we were to reckon by thirties. In other words, it always belongs to the number, which is one less than the rudix of the system. The same property, also, belongs to the number 3; because 3 is con. tained in 9 an exact number of times. Of course, any number of 9s will be an exact number of 3s. We may, therefore, prove addition by casting out of 3s, in the same manner as by casting out of 9s. MULTIPLICATION. MENTAL EXERCISES. X. 1. A shoemaker can make one pair of shoes in a day. How many can he make in 2 days ? 2 times 1, or twice 1 are how many ? 2. A cabinet maker makes two tables in a day. How many can he make in two days ? 2 times 2, or twice 2 are how many ? 3. If one inkstand has 3 pens in it, how many pens have two inkstands ?. 2 times 3 are how many ? 4. If one camel has two humps on his back, how many humps have 4 camels ?-4 times 2 are how many ? 5. One boy has 3 apples. How many apples have 5 boys? 5 times 3 are how many? 6. If one chair has 4 legs, how many legs have 6 chairs? 6 times 4 are how many? 7. A window has 5 rows of panes, and 7 panes in each How many panes of glass in the whole window? 5 times 7 are how many ? 8. One boy has 6 marbles. How many marbles have 8 boys? 8 times 6 are how many ? 9. Three boys have 7 cents apiece. How many cents have all together? 3 times 7 are how many? 10. One boy has 5 school books. How many books have 4 boys? 4 times 5 are how many ? 11. In one peck are 8 quarts. How many quarts in 3 pecks? 3 times 8 are how many ? 12. One tree produced 6 bushels of apples. How many bushels would 5 trees produce? 5 times 6 are how 13. Two oranges cost 6 cents apiece. What did both cost? 2 times 6, or twice 6 are how many ? 14. At 12 cents a pound, how much will 3 pounds of raisins cost? 3 times 12 are how many ? 15. A chess board has 8 rows of squares, and 8 squares in a row. How many squares on the board ? : 8 times 8 are how many? 16. Five men drank à glass of ale apiece, and paid 3 cents a glass. How much was paid in all ? 3 times 5 are how many ? 17. If one lemon cost 4.cents, what will 2 cost? What will 3 cost? What will 4? 5? 6? 7? 8? 9? 10 ? 18. At 3 dollars a barrel, what will 2 barrels of cider cost? What will 3 ? 4? 5? 6? 7? 8? 9? 10 ? 11 ? 12? 13? 14? 15 ? 16 ? 19. At 5 cents apiece, what will 2 story books cost? 3? 4? 5? 6? 7 ? 8? 9? 10 ? 11 ? 12? 20. How many are 3 times 5 ? 4 times 5 ? 6?7? 8? 9? 21. Repeat the many ? MULTIPLICATION TABLE. 2 times ( are 015 times are 08 times are 011 times are 0 2 X 1 25 X 1 58 X 1 = 811 X 1 11 2. 45 2 108 2 1611 2 22 3 65 3 158 3 2411 3 33 4 85 4 2018 4 3211 4 44 105 5 258 5 40911 5 55 2 6 125 6 3013 6 4811 6 66 7 1415 7 3518 7. 5611 7 77 8 165 403 8 6411 8 88 9 185 9 -458 7211 9 99 10 2015 10 508 10 8011 10 110 11 11 121 12 245 12 608 12 9611 132 225 11 12 24 3 times () are 016 times are 09 times ( are 0 12 times are 0 3 X 1 316 X 1 619 X 1 9 12 X 1 =12 2 66 2 129 2 1812 3 3 96 3 189 3 2712 3 36 3 126 4 240 4 36 12 4 48 3 5 1516 5 309 5 45 12 5 60 3 6 186 6 369 6. 5412 6 72 3 7 21 6 7 429 7 63 12 my 84 3 8 246 8 489 7212 8 96 3 9 2716 9 549 9 8112 9 108 3 10 306 609 10 90 12 10 120 3 11 336 11 99 12 11 132 3 12 12 729 12 108 12 12 144 10 366 127 4 times are 0 7 times are 0 10 timnes() are 0 13 times are 0 4 x 1 47 X 1 = 710 X 1 = 10 13 x 1 13 2 87 2 14 10 2 2013 2 26 3 3 2110 3 3013 3 39 4 4 1617 4 2810 4 4013 4 52 4 5 2017 5 3510 5 50 13 5 65 4 6 247- 6 4210 6 60/13 6 78 4 2817 4910 7013 7 91 4 8 327 8 56/10 8 8013 8 104 4 9 367 9 6310 9 9013 9 117 10 407 10 70 10 10 10013 10 130 4 11 447 11 11013 11 143 4 12 487 12 8110 12 12013 12 156 |