11 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. 1 11 9 99 1 9 99 81 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. Therefore, if the square be pointed off in periods of two figures each, there will be as many periods as figures in the root. 121 ( 10 +1 10 x 10 = 100 100 10 x 2 = 20 21 ) 21 1 x 1= 1 21 121 = 10 x 10 = 100 144 1,44 ( 10 + 2 100. 22 ) 44 44: a + b a=10, then 12 = a + b 10 + 2 b= 2, 12 10 + 2 100+20 20+4 ab 62 a + 2ab+ba+b 100+40+4(10 + 2 100 40+4 a 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 (32) 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)", and its breadth is b, which is the term wanting in the root. ac Let a + b + c 100+20+3 a2 + ab + ac ab + 32 + bc Jablo ac+bc+c? a 2ab +62 2a +26+0) 2ac+2bc+c? 2ac+2bc +62 100+20+3 123 100+20+3 123 10000 + 2000 + 300 369 246 123 10000+4000+400+600+120+9 1,51,29 (100+20+3 100 00 20 44 00 3 729 1 COR.-In order to extract the square root of any number, point it off in periods of two figures each, beginning with units, the left hand period may have but one figure; find the greatest figure whose square will either be equal or less than the number in the left-hand period regarded as units, place this figure in the root and subtract its square from the left-hand period, and to the remainder, if there is any, annex the next period ; if there is no remainder, the next period itself will be the dividend; and for a trial divisor, double the root already found, regarding it as tens, as we are now considering it as with a figure annexed to which it will hold the tens' place; find how often the trial divisor is contained in the dividend and annex that figure to the root already found, and add this last figure of the root to the trial divisor, which makes the divisor complete ; perform the division and if there are other periods bring them down in like manner, and repeat the above method until the whole root is found. EXAMPLES. 1. Extract the square root of 9801. As a fraction is squared by multiplying it by itself, thus, 4x4 = it, so its root is extracted by extracting the roots of both its terms; hence, VH=#, V=d, = upon By a proposition 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 the base and perpendicular; thus, Let ABC be a rightangled triangle, AB the base, AC the perpendicular, and BC the hypothe 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 ABP and AC? |В nuse. 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. EXAMPLES. 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? = = P = 22 + p = h2 h2 72.. 25 x 25 = 625 72 15 x 15 = 225 b = v h2 – po 4,00 ( 20 = p. 4 ) 00 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. 12 x 12 144 A B 3. What is the length of the diagonal of a rectangle whose sides are 45 and 60 ? |