Another Rule, still shorter. 1. Find by trials, the nearest rational cube to the given number, and call it the affumed cube. 2. Then as twice the affumed cube added to the given number, is to twice the given number added to the affumed cube, fo is the root of the affumed cube to the root required nearly. 3. And by taking the cube of the root thus found, for the affumed cube, and repeating the operation, the root will be had to a ftill greater degree of exactness. Examples. 1. Let it be required to find the cube root of 12484? Here the nearest rational root is 23. and its cube 12167. Answer-23.198 the root required, which is true to the laft place of decimals. 2. Required the cube root of 2? Here the nearest rational root is 1, and its cube also 1. Whence, 1 X 2 2 4, and 2 X 2 + 1 = 5 + 4. Therefore, 4: 5: 1: 2/ 1.25 =root nearly. 3. What is the cube root of 34965783? NOTE. When the given number consists of a whole number and a decimal together, make the number of decimals consist of 3, 6, 9, &c. places, by adding cyphers thereto, to that there may a point fall on the units place of the whole number. 9. What is the cube root of 12.977875? Ans. 2.35 10. What is the cube root of 15926.97 504? Ans. 25.16+ 11. What is the cube root of 171.46776406? Ans. 5.555+ 12. What is the cube root of 36155.027576? Ans. 33.06+ 13. What is the cube root of .001906624? Ans. .124+ 14. What is the cube root of .053157376? Ans. 376+ RULE. To extract the Cube Root of a Vulgar Fraction. Reduce the fraction to its lowest terms; then extract the cube root of the numerator and denominator, for a new numerator and denominator; but should the fraction be a furd, reduce it to a decimal, and then extract the root from it. To extract the Cube Root of a mixed number. RULE. Reduce the fractional part to its lowest terms, and then the mixed number to an improper fraction; extract the cube roots of the numerator and denominator, for a new numerator and denominator; but if the mixed number be a furd, reduce the fractional part to decimals, annex it to the whole number, and extract the root therefrom. Practical Questions, and use of the Cube Root. 1. If a cubical piece of timber be 41 inches long, 41 inches broad, and 41 inches deep; how many cubical inches doth it contain? Anf. 68921 2. If a cellar be dug 12 feet every way, in length, breadth, and depth; how many folid feet of earth will there be taken out of it? Anf. 1728 3. Suppofe a piece of marble be of a cubic form to contain 474552 folid inches; what is the fuperficial content of one of its fides? Ans. 6084 inches. " To find two mean proportionals between any two given num bers. RULE 1.-Divide the greater by the lefs, and extract the cube root of the quotient. 2. Multiply the root thus found, by the leaft of the given numbers, and the product will be the leaft. 3. Multiply this product by the fame root and it will give the greatest. Examples. 4. What are the two mean proportionals between 4 and 108? Ans. 12. and 36. 5. What are the two mean proportionals between 6 and Ans. 30 and 150 750? NOTE The solid contents of similar figures are in proportion to each other, as the cubes of their similar sides or diameters. 6. If a bullet 6 inches diameter weigh 32 lb. what will a bullet of the fame metal weigh, whose diameter is 3 inches? 666-216. 3 3 3 27. 1 As .216 32: 27: 4 Answer. 7. If a globe of filver of 3 inches diameter, be worth £45. what is the value of another globe of a foot diameter ? To find the fide of a cube that shall be equal in solidity to any given solid, as a Globe, Cylinder, Prism, Coné, &c. RULE. The cube root of the folid content of any folid body given, is the side of a cube of equal folidity. Examples. 8. If the folid content of a globe be 10648; what is the side of a cube of equal solidity? Anf. 22 The fide of a cube being given, to find the side of that cube which fhall be double, triple, &c. in the quantity to the given cube. RULE.-Cube the side given, and multiply it by 2, 3, &c. the cube root of the product, will be the side sought. Examples. 9. If a cube of filver, whofe fide is 4 inches, be worth £50. I demand the side of a cube of the like filver, whofe value fhall be 4 times as much? Anf. 6.349 Inches. 10. There is a cubical veffel, whofe fide is 2 feet; I demand the fide of a veffel which fhall contain 3 times as much? Anf. 2.884 2 feet 103 inches, Extraction of the Biquadrate Root. RULE.-First extract the fquare root of the given number, and then extract the fquare root of that fquare root, and it will give the biquadrate root required. Anf. 5308416 Ans. 84934656 Ans. 48 Anf. 96 Ans. 27 2. What is the biquadrate root of 48? Roots of all Powers. A General Rule for Extracting the Roots of all Powers. 1. Prepare the given number for extraction, by pointing off from the unit's place, as the required root directs. 2. Find the first figure in the root by trial, or, by the table of powers, which fubtract from the given number. 3. Bring down the firft figure in the next point to the remainder, and call it the dividend. 4. Involve the root into the next inferior power to that which is given, multiply it by the given power, and call it the divisor. 5. Find a quotient figure by common divifion, and annex it to the root; then involve the whole root into the given power, and call that the fubtrahend. 6. Subtract that number from as many points of the given power as are brought down, beginning at the lowest place, and |