dividend, with which proceed as before, and fo on, till the whole be finished. EXAMPLE. Required the Cube root of 16194277 ? 16194277(253 Root. 8194 = Firft dividend. 124 6 126 60 150 125 7625 Triple fquare of 2. = First divifor. Triple fquare of 2 multiplied by 5. = Triple of 2 multiplied by the fquare of 5. = Cube of 5. First fubtrahend. Triple fquare of 25 multiplied by 3. Cube of 3. Second fubtrahend. FIRST METHOD by APPROXIMATION. RULE 1.Find by trial, a cube near to the given num ber, and call it the fuppofed cube. 2. Then, 2. Then, as twice the fuppofed cube, added to the given number, is to twice the given number, added to the fuppofed cube, fo is the root of the fuppofed cube, to the true root, or an approximation to it. 3. By taking the cube of the root, thus found, for the Juppofed cube, and repeating the operation, the root will be had to a greater degree of exactness. EXAM PLE. It is required to find the cube root of 54854153? Let 64000000 fuppofed cube, whofe root is 400; Then 64000000 54854153 182854153)69483321400(379=root nearly. Again, let 54439939=fuppofed cube, whofe root is 379. Then, 54439939 54854153 1 SECOND METHOD by APPROXIMATION. RULE 1.-Divide the refolvend by three times the affumed root, and referve the quotient. 2. Subtract one twelfth part of the fquare of the affumed root from the quotient. 3. Extract the fquare root of the remainder. 4. To this root add one half of the affumed root, and the fum will be the true root, or an approximation to it; --Take this approximation as the affumed root, and, by repeating the procefs, a root farther approximated will be found, which operation may be farther repeated as often. as neceffary, and the root discovered to any affigned exactnefs. Note. In order to find the value of the firft affumed root, in this or any other power, divide the refolvend into periods by beginning at the place of units, and including in each period, fo many figures as there are units in the exponent of the root; viz. 3 figures in the cube root ;-4 for the biquadrate, and fo on: Then by a table of powers, or otherwife, find a figure, which, (being involved to the power whofe exponent is the fame with that of the required root) is the nearest to the value of the first period of the refolvend at the left hand; and to that figure annex fo many cyphers as there are periods remaining in the integral part of the refolvend; this figure, with the cyphers annexed, will be the affumed root: And it is of no importance whether the figure thus chofen, be, when involved, greater or less than the left hand period. 1. What is the cube root of 436036824287 ? 8 21,000)436036324287 (20763658,2994 16680324,9661 = 4084,15 Add the affumed root,= 3500 And it gives the approximated root = 7584,15. For For the fecond operation, ufe the approximated root as the affumed one, and proceed as above. THIRD METHOD by APPROXIMATION. 1. Affume the root in the ufual way, then multiply the fquare of the affumed root by 3, and divide the refolvend by this product; to this quotient add of the affumed root, and the fum will be the true root, or an approximation to it. 2. For each fucceeding operation let the laft approximated root be the affumed root, and proceeding in this manner, the root may be extracted to any affigned exactnefs. What is the cube root of 7 ? Let the affumed root be 2. the divifor. Then 2 X2 X3=12 12)7,0(,583, to this add of 21,333, &c. that is, 583+1,3331,916 approximated root. Now affume 1,916 for the root, then, by the fecond 7 process, the root is—2 +× 1,916=1,9126, &e. 3 X 1,916 APPLICATION and USE of the CUBE ROOT. 1. To find two mean proportionals between any two given numbers. RULE 1. Divide the greater by the less, and extra& the cube root of the quotient. 2. Multiply the root, fo 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 greateft. U 2 EXAMPLES. EXAMPLES. 1. What are the two mean proportionals between 6 and 750? 3 750÷6=125, and 125=5. Then 5×6=30=leaft, and 30×5150 greatest. Anf. 30 and 150. Proof. As 6: 30:: 150: 750. 2. What are the two mean proportionals between 56 and 19208? Anf. 392 and 2744. Note The folid contents of fimilar figures are in proportion to each other, as the cubes of their fimilar fides or diameters. 3. If a bullet 6 inches diameter weigh 32, What will a bullet of the fame metal weigh, whose diameter is 3 inches? : 6×6×6=216. 3×3×3=27. As 216 32:: 27: 41b, Anf. 4. If a globe of filver of 3 inches diameter, be worth £45, what is the value of another globe of a foot diamę ter? 3X3X327. 12 X 12 X 12=1728. As 27:45: 1728: 2880 Anf. The fide of a cube being given, to find the fide of that cube which shall be double, triple, &c. in quantity to the given cube. RULE.-Cube your given fide, and multiply it by the given proportion between the given and required cube, and the cube root of the product will be the fide fought. 5. If a cube of filver, whofe fide is 4 inches, be worth £50, I demand the fide of a cube of the like filver, whose value fhall be 4 times as much. 3 4X4×4=64, & 64×4=256. √ 256±6,349+ inches, Anf. 6. There is a cubical veffel, whofe fide is 2 feet: I demand the fide of a veffel, which hall contain three times as much? 3 2 X 2 X 28, and 8x3=24. 242,8842 feet, 103 inches, Anf |