1 1 1 EVOLUTION. 5 25 4 16 21)21 21 1,21 (10+ 1 100 1090 6 36 12 13 144 169 The square of the highest digit has two places of figures, the square of the least one; the square of the least number of tens is hundreds, and when the tens reach 4 it has four places of figures, never more. 11 11 121 10 x 10 = 100 2 x 2 = 99 99 9801 An increase of one figure in the root makes an increase of two in the square, for the square of the least unit is a unit, and of the least units and tens is hundreds; the square of the largest digit is tens, and the square of the largest tens is thousands. 40 4 9 9 r 49 81 14 196 Therefore, if the square be pointed off in periods of two figures each, there will be as many periods as figures in the root. 15 225 8 64 a=10, then 12 = a + b b= 2, 12 a+b 144 a2 + ab a2 ab ab = +ab+b2 20+4 a2+2ab+b2(a+b 100+40+4(10+2 As in the multiplication the first term is multiplied into itself but once, so one of its equal factors multiplied by itself occupies the first square; the second number is twice multiplied into the first and once into itself; therefore the divisor must be twice the first plus the second. This is observable in the two rectangles, each of which has a for its length and b for its breadth; and one side of the little square (62) is b, and 2a+b make the full length of the rectangle, which together with the square (a) make the whole of the large square (a+b)2, and its breadth is b, which is the term wanting in the root. Let 100+20+3 a2 2a+b) 2ab+b2 a + b + c 10+ 2 100+20 = ab+b2+bc a2 2a+b)+2ab+b2 100 20+2) 40+4 ac+be+c2 a2+2ab+b2+2ac+2bc+c2 ( a +b+c 2ab+b2 2a+2b+c) 2ac+2bc+c2 123 300+60 +9 10000+4000+400+600+120+9 1,51,29 (100+20+3 1 00 00 200) 51 29 220 240) 729 243 In the square of the literal quantity, the first three terms are the same as above; hence (a+b) is the root of it, and the next divisor must be contained c times in the remainder of the square; that number is 2a+2b+c, which is found by doubling the root already found and adding to it the next figure c of the root. COR.-In order to extract the square root of any number, point it off in periods of two figures each, beginning at the units; then find the greatest root of the first period and place it in the root, square it and subtract the square from the period, bring down the next period, and for a trial divisor double the root already found, and see how often it is contained in the dividend, reserving the right-hand figure; as often as it is contained, place the figure in the root and in the divisor, to the right of the trial divisor. Continue this method each time until the whole root is found. EXAMPLES. 1. Extract the square root of 9801. 5. Extract the square root of 603729. As a fraction is squared by multiplying it by itself, thus, x=18, so its root is extracted by extracting the roots of both its terms; hence, √ √ = {, √18, etc. A By a propositíon in Geometry, it is proved that the square described on the hypothenuse of a right-angled triangle is equivalent to the sum of the squares described upon the base and perpendicular; thus, Let ABC be a rightangled triangle, AB the base, AC the perpendicular, and BC the hypothenuse. AB is 3 feet long, and contains 9 square feet; AC is 4 ft., and contains 16 sq. ft.; and BC is 5 ft., and contains 25 sq. ft., equal to the sum of AB and AC. B = This figure is exemplified by the walls of a house, which are always perpendicular to the surface of the earth or to the street. If the foot of a ladder rest on the ground some distance from a house, and the top of the ladder against the house, the distance of the foot of the ladder from the house is the base, the height of the house from the ground to the top of the ladder is the perpendicular, and the ladder is the hypothenuse. H b 1. A ladder 25 feet long, whose foot is 15 feet distant from the house, just reaches the top of the house. high is the house? How P = h2 b2 + p2 = h2 EXAMPLES. b = √h2 = p2 p = perpendicular. h = hypothenuse. b2 .. 25 × 25 = 625 be 15 × 15 = 225 12 × 12 2. What is the length of the diagonal of a square, each side of which is 12 feet? The diagonal of a square is the same as the hypothenuse, having base and perpendicular the same. = = 4,00 (20 = p. 4 ) 00 144 144 B A 3. What is the length of the diagonal of a rectangle whose sides are 45 and 60? |