product of their roots. As an illustration, take 9 and 16. Their product is 144, of which the square root is 12; that is, the root of their product is 12. The product of their roots is the same, for the square of 9 is 3, and of 16 is 4; and 4×3 is 12. The same is true of the cube roots, or of any roots whatever. Take the numbers 8 and 27. 8=2, √27=3, and 3×2=6, 3 3 the product of their roots. Again, 27×8=216, and √216=6, the root of their products. EXTRACTION OF THE SQUARE ROOT. The Square Root of any number, is that number which being multiplied into itself once, will produce the number given. The following table exhibits the square of all numbers from 1 to 12: Roots, |1|2|3| 4 5 6 17 8 9 10 11 12 190° FIG. 1st. 8 ft. 900 A square is a figure bounded by four equal sides, and has all its angles right angles, or angles of 90 degrees. This may be seen in figure 1st. Now the area of a square, that is, the number of square feet, or rods, &c., it contains, is found by multiplying the length of any two sides together; or since the sides are all equal, by multiplying the length of any one side into itself; or by squaring it. As each of the sides of the annexed figure is 8 feet in length, it is obvious they may each be divided into 8 ft. 90° 90° 8 ft. 8 ft. 8 equal parts, each of which will be one foot in length. Let each side be thus divided and the points of division united, aş =64. Therefore, the area of a square is obtained by multiply FIG. 2d. a 1 2 3 4 5 6 7 8 a b9|10|11|12|13|14|15|16|6 c17 18 19 20 21|22|23|24|c d25 26 27 28 29 30 31 32 d e33 34 35 36 37 38 39 40 e f41 42 43 44 45|46|47|48|ƒ 49 50|51|52|53|54|55|56g h57 58 59 60 61 62 63 64 h ing the length of the side by itself; or, in other words, by squaring it. Now to apply the above remarks to Evolution, suppose we have an area equal to what is given above, but placed in a different form. Suppose it to consist of a board one foot wide and 64 feet long; and that it is required to determine how large a square floor it will exactly cover. V648 feet, is the length of the side of the required floor. Hence, the area of a square is found by squaring one of its sides; and the length of its sides is found by extracting the square root of the given area. The following is the rule for extracting the square root: RULE 1st. Separate the given number into periods of two figures each, by placing a point or dot over the place of units, another over the place of hundreds, and another over the place of tens of thousands, &c. 2d. Find by trial the greatest square root of the left hand period, and place it for the first figure of the root, after the manner of the quotient in division. 3d. Subtract the square of this root from the left hand period, and to the remainder bring down the next period of two figures for a dividend. 4th. Double the root figure already obtained, for a divisor; then, omitting the right hand figure of the dividend, divide as in Simple Division, and place the figure obtained, as the second figure of the root or quotient; and also on the right hand of the divisor. 5th. Multiply the divisor thus increased, by the figure last placed in the root, and place the product under the dividend, as in Division; then subtract, and to the remainder bring down the next period of two figures. with 6th. Double the root already found for a new divisor, which divide as before. Continue the operation till the periods are all brought down: the number obtained will be the root required. Note 1st.-If the number, the root of which is required, consist in part of a decimal, place the first point over the unit figure, as already directed, and point off both ways from that figure, allowing two figures to each point. If the decimal consist of an odd number of figures, annex one cypher to complete the last period. 2d. If after all the periods have been brought down, there be a remainder, periods of two cyphers each may be annexed and the operation continued. The root figures obtained by thus annexing cyphers will be decimals, and must be so marked. 3d. The number of dots employed in pointing off the given number, always determines the number of figures in the required root. Ex. 1. What is the square root of 1296? 1296 pointed according to the rule is 1296. Hence we know that the root will consist of two figures. The next step is to determine the root of 12, the left hand period. This is done by trial. If 4 be taken, it will be found too large, since 4 x4=16. We will, therefore, take 3. 3x3 9, and since 9 is less than 12, it is not too large, and yet it is the greatest integral quantity, whose square is less than 12, and is therefore the root we want. OPERATION CONTINUED. 1296(3 Root squared=9 3 9 6 remainder increased by the next period. Now since we are to have another figure in the root, the 3 already obtained is 3 tens or 30, the square of which is 30× 30-900; the 12 is also 1200. After subtracting the 9, therefore, there remains 3, or 300, and the next period being brought down, we obtain the number 396. The next step is to find a divisor, which by the rule is 3+3 =6; therefore, 1296(36 9 6) 3 In dividing, the right hand figure, viz. 6, is omitted. Again, 12 9 6 (3 6 9 66) 396 000 The last root figure is placed on the right of the divisor, making it 66, and the whole is multiplied by the root figure, that is, 66×6=396. I then subtract and nothing remains. Therefore, 36 is the root required. The operation is proved by squaring the root; thus, 36 × 36 =1296. If the given number had consisted of three or four periods instead of two, the operation would have been continued by bringing down another period, and then doubling the root 36 for a new divisor Explanation. We will suppose the 1296 in the preceding operation to be so many feet of boards one foot in breadth; and that it is required to know how large a floor, exactly square, they will cover. FIG. 3d. 30 ft. As has already been said, the 12 in the number 1296 is 1200, and the root 3 is so many tens, or 30. This, therefore, is the length in feet of one side of a square floor, which 1200 feet of the boards will cover, and leave a remainder of 3 or 300. Now to find the area of a square, (see fig. 3d,) we multiply the length of any two sides together, or square the length of one side. Therefore, 30 x30-900. We have then disposed of 900 of 1296 feet given, and there remains 396 feet to be so added to this figure as to preserve its square form. This is done by making equal additions upon any two adjacent sides; and hence we see the obvious reason for doubling the root, (which is the length of one side of Area 900 square feet. 30 ft. 30 ft. 180 sqr. ft. FIG. 4th. 36 ft. 900 sqr. ft. the square,) for a divisor. But the root, figure 3d, being doubled, gives 6 for a divisor, and this is contained in the remaining figures, 396, 6 times, (the right hand figure, viz. 6, being omitted, agreeably to the rule.) Now this figure, 6, expresses the breadth of the addition which the remaining feet of boards are sufficient to make to the original square, as seen at fig. 3d. This addition is seen at fig. 4th. This diagram is not a perfect square; a corner remains to be filled up. We will, however, before completing it, ascertain how many of 396 feet, that remained after the first square of 900 feet was completed, are here disposed of. The length of each addition being 30 feet, and the breadth 6 feet, the area is 30×6=180 sq. feet; and the area of both, consequently, is 180 + 180 360. But 396-360-36. Therefore, 36 feet still remain to be added. The scholar, by reference to the preceding diagram, will perceive that the corner, which yet remains to be filled, to complete the square, is just 6 feet from corner to corner; therefore, 6×6=36. This addition just disposes of the remaining feet of boards, and completes the square, as may be seen at fig. 5th. The area of the original square, as seen at fig. = 3d, and also of the several additions made at figures 4th and 5th, disposes of the whole of the given quantity of board; for 900+180 +180+36=1296. But why, in dividing, is the right hand figure of the dividend omitted? The scholar will remember that the points placed over any number, determine the number of figures in its root. (See Note 3d.) Before performing the operation, we therefore know that the required root of the preceding num 6 ft. 6 ft. 180 sqr. ft. 30 ft. FIG. 5th. 36 ft. 800 sqr. ft. 36 ft. ber will consist of two figures. The 3, or left hand figure of |