appropriate sign --, implies their addition; with a contrary sign, their subtraction. 99. To find the value of an expression in which there are several positive and several negative numbers, add all the positive numbers together : then add all the negative numbers together, the difference between the two sums, preceded by the sign of the greater, is the value of the expression. 1. 569 + 19 -- 34 + 17 - 413-9=149 569 34 19 413 17 9 605 Sum of positive. - 456 Sum of negative. Then 605 456 = 149. The greater sum being positive, the remainder 149, which is the value of the expression, is positive, and is placed on the right of the sign =. Or, as it is of no consequence in wbat order the numbers are placed, we may proceed thus : 569+19–34+-17-413–9=569+19+17-34-413-9 -605_156=149 In this case the work may be performed without placing the numbers under each other. 2. 999 – 2014 + 164 13 - 79 + 40 = - 903 999 2014 164 13 2106 Sum of neg. num. 40 79 1203 Sum of pos. num. 1203 - 2106 903 diff. or required val. In this example, the sum of the negative numbers being the greater, the value of the expression is negative. 3. 254 + 619 46 – 75 + 843 - 997 6. 303 — 27 + 124 + 500 – 809 100. The ar. comp. may here be of great use. Ascertain by the eye whether the result is likely to be positive or negative, writing the ar. comps. of the numbers of contrary sign instead of the number : also taking care (90) to diminish by the alt unit for every complement used. 120 = 99 = 1. 5846 +2359 — 426851 - 4567 423213 -406851 4567 7641 ar. comp. of 2359 423213 Here, as we easily see that the result will be negative, we write the ar. comps. of the two positive numbers, the alts of which being 4th alt, we diminish the largest negative number by 2 units of this order. The scholar may prove the work by pursuing the ordinary method: 2. 532946 - 171 3436 + 546 - 27 = 529858 521846 829 ar. comp. of 171 6564 ar. comp. of 3436 546 73 ar. comp. of 27 529858 The result being evidently positive, we take the ar. comp. of the three negative numbers : and, as the alts are 2d, 3d, and 4th, we diminish the largest positive number by 1 unit of each of these orders. The scholar may prove as before. 3. 54927 9999 + 99 9 4. 483216 — 45— 259 + 354 5. 5436729 29346 + 9999 6. 64837 99 + 999 — 7254 = 7. 999 + 58476 356742 - 187 8. 9473 9999 519 + 64 + 999 9. 624 — 569 + 1940 — 2835 + 943 10. 854327 — 3745 + 9254 473 — 49 = 11. 747599 - 274 + 9999 + 999 —— 8451 747597 + 726 + 1549 749872 (90 and 93) 12. – 43560 — 999 — 99999 + 547 +57 + 9999 = SECTION V. MULTIPLICATION. 101. The word multiplication is derived from two Latin words—multus, many, and plicare, to fold. As an operation, it is the method of finding, with greater facility than by either Numeration or Addition, the sum produced by the combination of all the units expressed by writing a number two or more times. (See Art. 59.) 102. The multiplication of numbers, however great, requires nothing more than the multiplication of a single figure by a single figure. Hence, the following Table, which contains all the products arising from the multiplication of any two of the figures 2, 3, 4, 5, 6, 7, 8, 9, will prepare the scholar to perform any multiplication whatever. To make the exercise complete, he must perform each multiplication, where the factors are not alike, both ways. Thus, for the expression 2 X3=6, he must say, twice 3 is 6; three times 2 is 6. For the expression 2 X4=8, he must say, twice 4 is 8; four times 2 is 8, and so on. = 4 XXX = Multiplication Table. 3 X 3 9 4 X4=16 x 5= 25 2 X 3 6 3 x 4= 12 4 X 5 = 20 5 X 6= 30 2 8 3 x 5 - 15 4 x 6 = 24 5 X7= 35 2 X 5 10 3 X 6=18 4 X 7=28 5 X 8=40 2 X 6 12 3 x 7=21 4 X 8= 32 5 X 9 45 2 X 7=14 3 X 8=24 4 X 9= 36 2 x 8 16 Х 27 6 x 6= 36 X9 18 7 X 7 - 49 6 X 7 : 42 8 X 8=64 7 x 8 = 56 6 X 8 48 9 X 9=81 | 8 X 9=72 7 X9=63 6 X 9=54 103. The numbers produced by the continual addition of each of the digits form the following Table, which is attri. buted to Pythagoras. The first line is formed by adding 1 to itself, and continuing the addition till we have 9 times 1, The second line, by adding 2 in the same manner. The third, by adding 3, and so on for the others. ! C. 9 18 27 36 45 54 63 72 81 6 .When two straight lines cut each N other, so as to make the four angles at the point of intersection all equal, the lines are said to be perpendicular to each other, and the angles are called right angles. W Thus, the four angles, which the straight 9 lines N S and W E make at the point g, being equal, are right angles, and the lines are perpendicular to each other. S 104. The surface or outside of any thing is called its superficies, which is considered as having only length and breadth. A plane is that superficies in which, if any two points be taken, the straight line between them lies wholly in that superficies. 105. A geometrical figure is the space enclosed by one or more boundaries. The quantity of space enclosed is called the area of the figure. Figures enclosed by straight lines are called rectilineal figures : if by three straight lines, trilateral figures or triangles : if by four, quadrilateral : if by many, multilateral figures or polygons. 106. A square is a quadrilateral figure, having all its sides equal and all its angles right angles. A right-angled parallelogram, which is also called a rectangle, is a quadrilateral figure, having all its angles right angles, but only its opposite sides equal. 107. The Pythagorean Table, which has all its sides equal, and all its angles right angles, is a square. As the divisions of its sides are also equal, suppose each to be 1 inch long : then, as the small spaces in which the numbers are placed are formed by straight lines, one inch apart, which cut each other at right angles, they are all square inches. Now, as each row contains 9 square inches, the figure 9 in the upper row, on the right, shows the area of the rectangle formed by that row : and it is plain, that as we descend in the right-hand column, we shall have as many times 9 as there are units in the number in the left-hand column, (59 ;) that is, the number on the right expresses the area of a parallelogram, consisting of as many rows of 9 square inches as there are units in the lefthand figure; which figure is the breadth of the parallelogram. Wherefore, to find the area of any parallelogram, we multiply its length by its breadth, referring both dimensions to the same generic unit. 108. Hence it is obvious, that the product of any two integral numbers represents the area of a parallelogram-one number being its length and the other its breadth. Also, when the two numbers are alike, their product is the area of a square, and is, therefore, called a square number. Therefore any quantity multiplied into itself is the square of that quantity. Thus, a Xa is the square of a, usually written a a or a. The quantity itself, or length of one side, is called the square root. The small figure 2 on the right is called the index, and shows how many times a is factor. 109. The oblique line ab, which joins two opposite angles of the Table, is thence called diagonal : sometimes it is called diameter, because it metes or divides the square into two equal parts. This diameter passes through all the square numbers in the Table, which, taken from a to b, succeed each other, thus : Squares 1° 4 9 16 25 36 49 64 81 Square Roots 1 2 3 4 5 6 7 8 9 At the top of the column, and on the left in the row in which any square number is found, we find its root, which is |