a + b a=10, then 12 = a + b 10 + 2 b=2, 12 10 + 2 144 a2 + ab 100+20 + ab +32 20+4 62 a + 2ab+ba+b 100+40+4(10+ 100 40+4 ab 8 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)?, and its breadth is b, which is the term wanting in the root. = ac bcc2 ab a? Let a + b + c 100+20+-3 ab + b + bc ac+bc+ca. a2 2ab +32 2ac+2bc + 2 100+20+3 123 100+20+3 123 10000 + 2000+300 369 2000+400 + 60 246 300+60 +9 123 10000+4000+400+600 +120+9 1,51,29 (100+20+3 100 00 20 44 00 3 729 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 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, fxf = it, so its root is extracted by extracting the roots of both its terms; hence, VH=4, V=), V = 4, etc. 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 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?. 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. How high is the house ? Let 6 р base. b = 72 + pa ha 72 15 x 15 = 225 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 V 288 = А B 3. What is the length of the diagonal of a rectangle whose sides are 45 and 60 ? 4. A ladder 75 feet long being placed with its foot in the street reaches a window on one side 45 feet high, and on the other side 60 feet high. How wide is the street? Ans. Street, 105. Angle 60 Angle 45 When a number is both integral and decimal, as 455.742, the integers must be pointed off as if there were no decimals, and the decimals as if there were no integers; thus, 4'55.74'20'. Add more ciphers if necessary. To extract the square root of a fraction whose denomi'nator is not a perfect square, multiply both terms of the fraction by the denominator; thus, to extract the root of z. $ x 5 = H. The root of the denominator is now 5; of the numerator, 15.0000 ( 3.873 9 68 ) 600 44 767 ) 5600 5369 774 ) 23100 The result = 3.873 = .7746 nearly. CUBE ROOT. The Cube is a solid having for its base a square and every side equal to the base all equal squares; it has therefore three dimensions, all equal; and as 1 foot is 12 |