80+9 80+9 802+80×9 80×9+92 (80+9)=80+2×80X9+92 80+9 80+2×80 X9+80 × 92 80 X9+2X80X92+93 (80+9)=80+3x80x9+3x80x92+93 We will now illustrate geometrically the involution of the first example. How many cubic feet in a cube, each side of which is 89 feet? By reversing the above process, we obtain for extracting the cube root, the following GENERAL RULE. Commencing at units, separate the number into periods of three figures each. Then find the largest digit, the cube of which shall not exceed the left-hand period. Place this digit, which is called the first figure of the root, on the right, in the form of a quotient; also, on the left, for the first term of a first column, and its square for the first term of a second column, and from the left-hand period of the given number, subtract its cube. Then to the remainder, annex the next period, for the FIRST DIVIDEND. Now double the term in the first column, for its second term, and add its product into the root already found, to the first term of the second column, for the first TRIAL DIVISOR. Consider two ciphers annexed to the trial divisor, and write the number of times it is contained in the dividend, for the next figure of the root; also, annex it to the sum of the last term in the first column, and the first figure of the root; this will be the next term of the first column. Add the product of this term into the digit of the root last found, advancing it two places to the right, to the last term of the second column, for its next term; this will be the TRUE DIVISOR. From the DIVIDEND, subtract the product of the true divisor into the digit of the root found; and to the remainder annex the next period, for the second dividend. Proceed in a similar way until all the periods have been used. REMARK.-By carefully examining the foregoing involution, the pupil will be able to deduce other rules for the extraction of the cube root, some of which may perhaps, appear more plain than the one I have just given, as this is more readily deduced from Algebraic involutions. I have given this rule, as it will be less laborious to extract the cube root of large numbers by it, than by many other rules usually given; also, because it keeps distinct the three geometrical magnitudes-lines, surfaces, and solids. The first rule, however, is the most simple, and will be found of much im. po tance in reducing surd quantities to their simplest form, (as will hereafter be explained,) or in determining the roots of rational quantities. 1. What is the cube root of 704969? EXPLANATION. We first find the greatest cube contained in the left-hand period. more than 80, since 803 also, that it must be less is greater than 704969. We know that this number must be = 512000, which is less than 704969; than 900, since 903 = 729000, which Hence, the first, or left-hand figures of the root, is 8; whose cube is 512, which is the greatest cube contained in 704, the first or left-hand period. We first add the three square slab pieces AB, BC, and CD, whose length and breadth are each respectively 80 feet, (the side of the cube AD.) The area of the face of the first piece, AB, is 8026400 square feet. The length of the other two pieces, BC, and CD, is 80 + 80 feet, and their width 80 feet. Hence, their superficial contents is 160 x 80=12800 square feet, which added to 6400 square feet, the superficial contents of the piece, AB, gives 19200 square 160 B feet, the superficial contents of the three pieces, AB, BC, and CD. As these three square slabs make up by far the greatest amount of the whole increase, if we divide 192969, (the number of cubic feet remaining to be added,) by 192000, the number of square feet in the three pieces, AB, BC, and CD, (which may be called the trial divisor,) it will give their thickness; which we find to be 9 feet. Figure 2, represents the cube AD, with the three pieces AB, BC, and CD, added. We now add the three cornerpieces, EG, HF, and HX, whose lengths are respectively 80 feet, (the side of the cube AD,) and. whose width and thickness are each 9 feet respectively; also, the corner-piece AW, whose length, width, and thickness, are each 9 feet. Therefore, the length of the three pieces, EG, HF, and HX, is 240 feet; which being increased by the length of the corner-piece, AW, gives 249 feet for the length of the four pieces. Their width is 9 feet; therefore, 249 X 9 = 2241 square feet, is their superficial contents; which being increased by 19200 square feet, the superficial contents of the three pieces already added on, gives 21441 square feet, (the superficial contents of the seven pieces added on,) which being multiplied by 9, their thickness, gives 192969 cubic feet, their solid contents, which being subtracted from 192969 cubic feet, the quantity that remained to be added to the cube AD, leaves no remainder; therefore, а cube represented by Figure 3, whose side is 80 +9 89 feet, will cantain 704969 cubic feet. |