the length of the side of the square, the area of which the square number represents; that is, in each square, the square root or length of the side, is seen in either of the opposite angles, through which the diameter does not pass. Note. The figure 1, as a root, is the side of the square or length of 1 inch. As a square it is the area or square inch; the generic unit by which the whole surface of the Table or that of any parallelogram or square in it is measured. 110. The unit, to which we compare, and by which we measure, any surface, is always a square. Thus the surface of the globe we inhabit, or of any country, kingdom, state, township, &c., is generally measured by the square mile . Most kinds of artificers' work, such as bricklaying, flooring, plastering, painting, glazing, &c. is measured by the square foot or square yard. Also the work, in all the branches, is generally in the form of a parallelogram.' So that, in multiplying any two numbers, the scholar may suppose that he is finding the content or area of a piece of artificer's work in the form of a parallelogram; the product being a number of square feet or square yards, according to the measure used in taking its length and breadth. Suppose the above Table to represent a square township, each side of which is 9 miles : then, it is plain that the township contains 81 square miles. Again, suppose the Table a window, each side of which is 9 feet long; then, the glazing work contains 81 square feet. Again, suppose it a court-yard, the side of which is 9 yards long; then the paying work consists of 81 square yards. 111. Though the Pythagorean Table is sufficient for the multiplication of all numbers, an acquaintance with the following extension of it will, in many cases, be found advantageous. The student may, however, make use of it or not, as he pleases. We shall only remark that arithmeticians always multiply by 11 or 12, as by a single figure. As the multiplication of a number by 10, consists in merely placing 0 on the right, this number, in the Table, is omitted as unnecessary. To multiply by 11 any number expressed by two figures, the sum of which does not exceed 9, we have only to separate the figures and place their sum in the middle. Thus, to multiply 23 by 11, we separate the figures 2 and 3, and placing their sum 5 in the middle, we have 253 for the product. Also 32 X 11 = 352. 11 22 33 44 55 66 77 88 99 12/132 143 154 165 176 187 198 209 220 12 24 36 48 60 72 84 96 108 132 14 156 168 180 192 204 216 228 240 1326 39 52 65 78 91 104 117 143 156 100/182 195 208 221 234 247 260 14 28 42 56 70 84 98 112 126 154 168 182 196 210 224 238 252 266 280 15 30 45 60 75 90 105 120135 165 180 195 210 285 240 255 270 285 300 16 32 48 64 80 96 112 128 144 176 192208 224 240 256 272 288 304 320 |17|34 51 68 85 102 119 136 153 187 204 221 238 255 272289 306 323 340 18 36 54 72 90 108 126 144 162 198 216234 252 270288 306 324 342 360 19 38 57 76 95 114 133 152 171 209 228 247 266 285 304 323 342 361 380 20 40 60 80 100 120 140 160 180 220 240 260 280 300 320 340 360 380 400 112. When a number subtracted from another number as often as possible leaves no remainder, it is called a measure of that other number. Also the number which contains another a certain number of times is called its multiple. Thus, 6 is a measure of 24, and 24 a multiple of 6. A number which measures each of several numbers is called their common measure. Also a number which can be measured by each of several numbers is called their common multiple. Thus, 2 is a common measure of all even numbers 2, 4, 6, 8, 10, 12, &e. Also 24 is a common multiple of 2, 3, 4, 6, 8, and 12: hence, these numbers are all factors of 24. Now, if one of these be assumed as one of two factors of 24, the other is arbitrary, being the number which shows how many times the assumed number can be subtracted. Thus if we assume 12, the other factor is 2; if we assume 8, the other is 3. 113. To facilitate the acquisition of the above Table, as well as to acquaint the scholar with some useful properties of numbers, let us observe : 1. That any number multipled by 2 or any other even pumber is an even number, and, consequently, terminates with 2, 4, 6, 8 or 0. Hence, in the Table, the numbers in all the columns, except the last, under the even numbers, terminate with these figures, which succeed each other in some regular order. 2. That, as every alt unit is composed of nines and a unit, and, (as 9 is divisible by 3, also, of threes and a unit, it is evident that an alt unit divided by 3 or by 9 will always leave a unit for remainder. Wherefore, each unit being considered a remainder, if the sum of the figures, in any number, be divisible by 3, the number also is divisible by 3 : if the sum be divisible by 9, the number is divisible by 9; consequently, when any number is multiplied by 3 or by 9, the sum of the figures of the product is a multiple of 3 or of 9 respectively. Hence, the sum of the figures of any number in the column under 3, is a multiple of 3 : under 9, the sum is 9 or its multiple. As we descend in this column, the unit's figure is diminished, and the tens figure increased by a unit at a time, (63.) Under 19, the unit figure diminishes in like manner, and the tens figure (on account of the ten on the left of 9) increases by 2 units at a time. 3. That, as 5 X 2=10, five times any even number is a number of tens, and terminates with 0. Also 5 times any odd number must terminate with 5; because, 5 times its even part is a number of tens, and 5 times the unit which renders it odd is 5. Hence the numbers under 5 and 15 terminate with 5 and 0 alternately. 4. That, under 11, the unit figure of each number agrees with the unit figure of the number multiplied by 11. That the tens figure is one unit greater, and that the left-hand figure is 1, except in the two lower numbers. 5. That, under 7 and 17, the unit figure, excepting 0, 1, and 2, diminishes by 3 units at a time, (because with the additional 7, this constitutes 10,) and under 13 increases The oblique line ab passes through the square numbers, which, omitting 9, are Squares 121 144 169 196 225 256 289 324 361 400 Square Roots 11 12 13 14 15 16 17 18 19 20 114. When the multiplicand consists of several figures, and the multiplier of a single figure, we place the multiplier under the unit figure of the multiplicand; and, having drawn a line underneath, we multiply, successively, all the figures of the by 3. then say, 2: multiplicand by the multiplier, beginning with the units. The unit figure of each product we write under the figure which gave it, and the tens we retain, as in addition, to add to the next product. Having multiplied the last or left-hand figure of the multiplicand, we write the whole product underneath, which completes the operation, the number under the line being the product of the two factors—that is, the sum which would arise from the addition of either as often as the other contains a unit. 1. To multiply 6342 by 6, we write the number thus : 6342 multiplicand. 6 multiplier. 38052 product. 221 and, beginning with the units, we say : 6 times 2 is 12. Here, as in ordinary addition, we write 2 and carry 1: we 6 times 4 is 24, and 1 is 25; we write 5 and carry 2: then, 6 times 3 is 18, and 2 is 20; we write 0 and carry lastly, 6 times 6 is 36, and 2 is 38, the whole of which we write underneath, and have 38052 for the product. If we write the number 6342 six times, thus, 6342 6342 6342, 6342 6342 6342 38052 221 and perform the addition, we bave the same result. Hence, it is evident that multiplication is a species of addition; consequently, we begin with the units, proceed towards the left, carry one for every ten, &c., as in ordinary addition, and for the same reasons. Also, that if we add two or more unequal numbers, the operation is an ordinary or common addition. But if we add a number which is repeated any number of times, the operation is properly a multiplication, and in this consists the only essential difference in the nature of the operations. 115. If any number be multiplied by 0, nought, the product will be nought; because it is evident that nought added to itself ever so many times can nerer produce any thing. Therefore, whenever, in multiplying, we meet with in the multiplicand, and have nothing to carry from the product of the preceding figure, we write 0 in the product. Also, in multiplying the figure which precedes 0, if its product consis of two figures, we write it as we should that of the last figure of the multiplicand. If its product consist of three figures, we carry only the left-hand figure, after we have written the two others underneath. 2. 860142 x 5 = 4300710 860142 5 4300710 Here, we say, 5 times 2 is 10; 0 and 1; five times 4 is 20, and 1 is 21; 1 and go 2; 5 times 1 is 5, and 2 is 7; the next figure being 0, we write 0; then, 5 times 6 is 30; O and go 3; 5 times 8 is 40, and 3 is 43. 3. 6753 x 2 13506 7. 48706 x 6 4. 8492 X 3 8. 87965 X 7 5. 5467 X 4 9. 76009 X 8 6. 390835 X 5 = 10. 93584 x 9 11. Multiply 5384679 by each of the numbers 2, 3, 4, 5, 6, 7, 8, 9, and find the sum of the several products. Answer. Two hundred and thirty-six millions, nine hundred and twenty-five thousand eight hundred and seventy-six. 12. Multiply 93578864 by each of the numbers 2, 3, 4, 5, 6, 7, 8, 9, and find the difference between the sum of the products and one trillion. Answer. Nine hundred and ninety-five billions, eight hundred and eighty-two millions, five hundred and twenty-nine thousand nine hundred and eighty-four. 116. When the multiplier, as well as the multiplicand, consists of several figures, having placed the multiplier under the multiplicand and drawn a line underneath, first multiply all the figures of the multiplicand by the unit figure of the multiplier, as above. Then multiply by the tens in the same manner, placing the product under the first product, so that its unit figure may stand under the tens of the first product. Continue to multiply successively by the figures of the multiplier, always placing the first figure of each product under that |