2. What difference will there be in the number of vibrations made by a pendulum of 6 inches long, and another of 12 inches long, in an hour's time? Answ. 2695.08 the difference. 3. What difference is there in the length of two pendulums, the one swinging 30 times, the other 100 times in an hour? Answ. Difference 42806.4 feet. 4. Give the length of a pendulum that will swing once in a third, once in a minute, once in an hour, once in a day? Answ. 10108 inches the length of that which vibrates thirds. 2.5 miles, the length of a pendulum which vibrates once in a minute. 8018 miles, the length of a pendulum which vibrates once in an hour. 461818819 miles, the length of a pendulum which A THE CUBE ROOT. Number being multiplied into itself, and the Product again multiplied by the same, produceth a Cube Number, and the number multiplied is the Cube Root of the Product, as 4 x 4 x 4 produceth 64, which 64 is the Cube of 4, and 4 the Cube Root of 64. Or if a Square number be multiplied by its Root, the product is the Cube of that Root, as 16 × 464 the Cube of 4. From either of which definitions of a Cube, it will be easy to find the Cube of every single figure, according to the following TABLE. Roots 12 3 | 4 | 5 | 6 | 7 | 8 | 9 Application. Let it be required to extract the cube root of 157464? 157'461'(34 125 (1) The given number being pointed over the first and fourth figures, (2) the last 7500)32464 Resolvend. period is 157, and the next 30000 2400 64 cube, is 125, whose root 3 I put in the quotient, and the remainder, after 125 is taken from 157, is 32; to which bringing down the next pe 32464 Subtrahend riod 464, I get the resolvend 32464. (3) the square of 5 is 25, which being multiplied by 300 makes 7500 for a divisor. (4) Seeking how often this divisor will go in the resolvend, I find four times, which I put for the next quotient figure. (5) 4×7500=80000; Next the square of 4 (16) X5, the other figure of the quotient makes 80, which multiplied by 30 produces 2400, and the cube of 4 makes 64, which three numbers being added make the subtrahend 32464 the resolvend: So then the given number is a perfect cube whose root is 54 Again, let it be required to find the cube root of 164566592? 164'566'592 (548 125 The said number being pointed into periods of three figures, and proceeding as 39566 and its divisor 7500 which will be contained 7500,39566 1st Resolv. before, I get the resolvend 30000 2400 64 32464 Subtraliend therein times; but then there is not a surplus in the resolvend equal to the other two numbers to be brought in, for if I take the quotient $74800)7102502 2nd Resol. figure 5, and form a sub 6998400 103680 512 7102592 trahend therewith, d th Rule, it will result 41475, which is greater than the resolvend, solvend; and therefore cannot be subtracted from it, as the Rule directs, so I must take a lesser number as 4, and then I find the subtrahend 32464 (as before) which I subtract from the resolvend and there remains 7102, to which I bring down the next period, and get a new resolvend 7102592, with which I proceed as with the first, viz. I square 54 which makes 2916, this x 300 produces 874800 for a divisor to said resolvend in which it is found 8 times; and 8 × 874800=6998400; the square of Ɛ, viz. 64X54 × 30103680, and the cube of 8512; these 3 numbers added make the subtrahend 7102592 the resol vend, and every period is brought down. So 548 is the cube root of the given number. QUESTIONS. Quest. What is a cube number? Answ. A square number multiplied by its root. 2. What are the cubes of the single figures? A. The cube of 1 is 1; the cube of 2, 8; of 3, 27, &c. **. By just 300 times the square Which Which done, if ought remain there shall 2. How is the work proved? A. Multiply the root or quotient into itself, and then the product by the same root, adding the remainder, (if any to the last product: so shall we get the given number if the work is right. Bring the proposed fraction to its least terms, and extract the cube root of the numerator for the numerator, and the cube root of the denominator for the denominator of the root. Application. Let it be required to extract the cube root of? To extract the Cube Root of Surd Numbers. Extract the root of the given surd number the Rule before delivered, but when the work is done there will be a remainder; to the remainder prefix three cyphers, and repeat the process, and so to every succeeding remainder, until a root be got sufficiently near, (for it cannot be found exactly,) and all the figures arising after the prefixing of cyphers are a decimal, Example |