together or into each other,—one of which is the number repeated, or multiplicand, and the other, the number which shows how many times it is repeated, or multiplier. Hence, any number is the product of its generic unit multiplied by that number. Thus 1 x 24 24. 58. If, from a number which contains another a certain number of times, we subtract that other until nothing remains, the number from which we subtract, or dividend, is said to be divided by the number subtracted, or divisor: the number of times it is subtracted being the quotient. Hence, any number is the quotient, found in dividing that 24 number by its generic unit. Thus 1 Wherefore a number multiplied or divided by its generic unit, remains the same. 59. Of the two numbers forming a product, either may be the number repeated, or multiplicand; the other being the number of times it is repeated, or multiplier. For example, if we write separately the units of the figure 6, and repeat the number separated, 4 times, in rows under each other thus : For 3 rows, so on. we see, that for every unit in the row repeated, we have a column containing a number of units, which agrees with the number of times the row is repeated. Thus, for_2 rows, we have 6 columns of 2 units each, or 6 times 2. 6 times 3. For 4 rows, 6 times 4, as here exemplified, and Hence, we see that the product is formed by a repetition of either the multiplicand or multiplier, which two numbers are, therefore, called the two factors (makers, facere, Lat. to do or make) of the product; because the product is the sum obtained by the addition of the series formed by repeating either factor as often as there is a unit in the other. Thus, 6+6+6+6=4+4+4+4+4+4= 24; or, 6 X 434 X 6=24; that is, 4 times 6 equals 6 times 4: and it is the same with any two numbers multiplied. 60. Again : either factor, in showing how often the other is contained in the product, shows how often that other could be subtracted, and is, consequently, the quotient in dividing the product by that other factor. Wherefore, if we divide the product of two numbers by either of them, the quotient 24 24 will be the other. Thus, 4 and - 6. 6 4 Hence, also, a number multiplied and divided by the same 6 X 4 number is still the same. Thus, 6; that is, multi 4 plication and division destroy the effect of each other; also, addition and subtraction destroy the effect of each other ; which properties are the natural result of the opposite attributes of quantity. SECTION III. ADDITION. 61. Addition is the method by which we classify and combine the orders of several numbers so as, of the whole, to form but one number, called the sum. Note. The word Sum should not be used for Question. By rendering the mind familiar with the combination of each of the digits with each of the others, as in the following Addition Table : 2 + 2 4 3 + 3 6 4+4= 8 5+5=10 2 + 3 5 3+ 7 4+5 9 5+6=11 2 +4 6 3+ 8 +6 10 5+7=12 2+5 7 3 +6 9 4+7 - 11 5+8 13 2 + 6 8 3 +7 10 4+8 : 12 5 + 9 14 2 +7 9 3 + 8 11 4 +9 : 13 2+8 10 3 + 9 : 12 6 + 6 =12 2 +9 11 7+7=14 6 + 7 13 8 + 8 16 7+8=15 6 + 8 14 9+9=18 8 +9=17 7 +9 = 16 6 +9=15 the addition of the longest line of figures is performed with expedition and certainty. For example, if we know that 8 and 5 is 13, we shall easily perceive that 28 and 5 is 33; that 38 and 5 is 43; that 48 and 5 is 53, &c., seeing that the figure 3 remains the same, and that it is easy to keep account of the additional unit in the place of tens. 62. The correctness of an addition is proved by adding the figures, a second time, in an inverse or contrary order : which second addition must, if the work be right, give the same sum as the first; consequently, when a different sum is found, the work must be repeated till we find the same. For example, in the following addition, 2+3+4+1+5=15 beginning at the left, we say, 2, 5, 9, 10, 15: which last we write as the sum. We then begin at the right, and say, 5, 6, 10, 13, 15: which sum, being the same as before, wu consider correct. 63. The following Additions are (57) Multiplications, and should be remembered for future use. For practice in addition, and to imprint them, as multiplications, on the memory, they may (beginning with 2 and proceeding regularly throughout) be repeated thus : 2 and 2 is 4; twice 2 is 4; 3 and 3 is 6; twice 3 is 6; and (59) inversely, 3 times 2 is 6: and so on for the rest. Again: for the combination of 3 figures, we say, 3 and 3 is 6 and 3 is 9; 3 times 3 is 9: 4 and 4 is 8 and 4 is 12; 3 times 4 is 12; and inversely 4 times 3 is 12, &c. For 4 figures : 4 and 4 is 8, and 4 is 12, and 4 is 16; 4 times 4 is 16 : 5 and 5 is 10, and 5 is 15, and 5 is 20; 4 times 5 is 20: also, inversely, (which, throughout the whole, should not be forgotten,) 5 times 4 is 20, &c. In the same manner repeat to 9 figures inclusive. 2+2 = 4=2 X 2 3 +3 +3 9=3 X 3 4+4 8= 4+4+4=12 3 x 4 5 + 5 10 = 5+5+5=15 3 x 6+6 12 = 6+6+6=18= 3 X 6 7+7=14=2 7+7+7=21=3 X 7 8+8=16=2 2 X 8 8 + 8 +8=24=3 X 8 9+9=18=2 X 9 9+9+9=27= 3 X 9 4+4+4+4=1634X4 5+5+5+5=20=4X5 5+5+5+5+5=25=5X5 6+6+6+6=244x6 6+6+6+6+6=30=5X6 7+7+7+7=28=4x7 7+7+7+7+7=353587 8+8+8+8=32=4X 8 8+8+8+8+8=40=5X8 9+9+9+9=36=479 9+9+9+9+9=45=579 +8 6+6+6+6+6+6 36 = 6 x 6 :48 = 6 X 8 9+9+9+9+9+9+9=63=7X9 9+9+9+9+9+9+9+9=72=8X9 9+9+9+9+9+9+9+9+9=81=9 X 9 In this last combination, observe, that, as the number 9 requires but one unit to constitute 10, the unit figure of the number added to it is diminished, and that in the place of tens increased, each by a unit. Thus : in adding 9 to 9 we have 18; then, in adding 9 to 18, the 8 becomes 1 unit less, and the 1, one unit greater: hence, we have 27 : and SO, in succession, we have 36; 45; 54; 63; 72; 81. 64. When, in adding, we meet with two digits, the sum of which is 10, we comprehend both at once, in simply reading our last result, as having one more unit in the place of tens. Thus, in the following addition, 2+3+9+7+3+9+6+4+9+8+2+9=71 beginning at the left, we say: 2 and 3 is 5; and 9 is 14; and 10 (taking 7+3) is 24; and 9 is 33; and 10 (6+4) is 43; and 9 is 52; and 10 (8+2) is 62; and 9 is 71. Then, beginning at the right, we say : 9 and 10 is 19; and 9 is 28; and 10 is 38; and 9 is 47; and 10 is 57; and 9 is 66; and 3 is 69; and 2 is 71. Let the following examples be proved by adding both ways: 1. 2+2+1= 5 11. 7+3+5= 15 7 13. 8 +4+5=17 17. 9+ 8 = 24 8. 6+4+2=12 18. 9+8+8=25 9. 8+2+3=13 19. 9 + 9 +8= 26 10. 8 +3+3 14 20. 9 + 9 + 9 -27 Let the results of the following be written and proved, as in the foregoing: 1. 5 +4+4+3 6. 9+7 +8+8= 2. 5+6+5 +6 7. 7+9+5+6 3. 7+4+3 + 8 8. 4+9+9+3+8 4. 8 +5+8+ 6 9. 5+6+8+9+7 5. 8 +6+ +9 10. 7+4+9+8+8= 11. 7+3+5+8+9+6= 12. 5+6+8+7+8+9 13. 9+9+5+7+4+7+3 14. 6 + 6 +9+7+9+5 +6. 15. 4 +8+6 +3+9+7 +8+ 6 16. 3 +9.+7+5+8+ +9+5 17. 5+5+8+3+7+5+9+6+9 18. 8+7+6+4+7+3+8+2+9 19. 9+7+3+8+5+2+5+7+9 20. 9+9+8+8+4+7+ 6 +5+8= 65. All the units contained in many given finite or limited numbers, however great, can be expressed by a single finite number. For, as the Scale of Natural Numbers is (26) infinite, progressing by a unit at a time, it is evident that in continuing its formation, we shall, at length, arrive at a number expressing all the units contained in the given numbers. 66. Some learners perform the addition of large numbers, by separating the units composing each figure in those numbers and adding them one at a time, which slow enumeration very much retards their progress. The necessity for this injurious practice may be obviated by a frequent repetition of the addition table. 67. From the above, it is plain that the only difference between the formation of the sum of an addition and the regular formation of numbers, as in numeration, is, that in adding, instead of progressing by a unit at a time, we comprehend, at each step, all the units contained in any digit which may present itself to our view. 68. In forming the sum of large numbers, care must be taken to combine with each other only those figures which express units of the same order. For, as it is by ten units of the same order that we form one unit of the next order towards the left, it is plain that, if we add units of different orders to each other, the sum, not being of any order, cannot be expressed, neither can it fórm new orders. |