35. The wall of a town is 25 feet high, which is surrounded by a moat of 30 feet in breadth: I desire to know the length of a ladder that will reach from the outside of the moat to the top of the wall ? Ans. 39,05 feet. The Base and Hypothenuse given to find the Perpendiculur. RULE. The square root of the difference of the squares of the hypothenuse and base is the height of the perpendicular. N. B. The two last Questions may be varied for Examples to the two lust Propositions. Any number of men being given to form thern into a square battle, or to find the number of ranks and files. Rule. The'square root of the number of men given, is the number of men either in rank or file. 36. An army consisting of 331776 men, I desire to know how many rank and file. Ans. 576. 87. A certain square pavement contains 48841 square stones, all of the same size, I demand how many are contained in one of the sides ? Ans, 221 EXTRACTION OF THE CUBE ROOT. being multiplied into itself, and then into that product, prom duceth the given number. Rule 1. Point every third figure of the cube given, begins ning at the unit's place ; seek the greatest cube to the first point, and subtract it therefrom; put the root in the quotient, and bring down the figures in the next point to the remainder for a RESOLVEND. 2. Find a Divisor by multiplying the square of the quotient by 3. See how often it is contained in the resolvend, rejecting the units and tens, and put the answer in the quotient. 3. To find the SUBTRAHEND. 1. Cube the last figure in the quotient. 2. Multiply all the figures in the quotient by 3, except the last, and that product by the square of the last. 3. Multiply the divisor by the last figure. · Add these products together, gives the subtrahend, which subtract from the resol vend ; to the remainder bring down the next point, and proreed as before. Roots. 1. 2. 3. 4. 5. 7. 9. CUBES. 1. 8. 27. 64. 125. 216. 343. 512. 729. 6 8. EXAMPLES. 1. What is the cube root of 99252847 ? 99252847. (463 64=cube of 4 216=cube of 6. 33336 subtrahend. Divisor. Square of 46X3=6348(1916847 resolvend. 27 =cube of 3. 1916847 subtrahend. 2. What is the cube root of 389017 ? Ans. 73. 8. What is the cuhe root of 5735339 ? Ans. 179. 4. What is the cube root of 32461759 ? Ans. 319. 5. What is the cube root of 8160.1519? Ans, 439. 6. What is the cube root of 259694072 ? Ans. 639. 7. What is the cube root of 48228541? Ans, 364. 8. What is the cube root of 27054036008? Ans. 3009. 9. What is the cube root of 22069810125 ? A17, 2805. 10. What is the cube root of 122615327232 ? Ans. 4968. 11. What is the cube root of 219365327791 ? Ans. 6031. 12. What is the cube root of 673373097125 ? Ans. 8765. When the given number consists of a whole number and decimal together, make the number of decimals to consist of 3, 6, 9, &c. places by adding cyphers thereto, so that there may be a point fall on the unit's place of the whole number. 13. What is the cube root of 12,977875 ? Ans. 2,35. 14. What is the cube root of 36155,027576 ? Ans. 33,06+ 15. What is the cube root of ,001906624? Ans. ,124. 16. What is the cube root of 33,230979637 ? Ans. 3,215+. 17. What is the cube root of 15926,972504 ? Ans. 25,16+. 18. What is the cube root of ,053157376 ? Ans. ,376. To extract the Cube Root of a Vulgar Fraction. RULE. Reduce the fraction to its lowest terms, then extract the cube root of its numerator and denominator, for a new nuse merator and denominator ; but if the fraction be a surd, reduce it to a decimal, and then extract the root from it, a EXAMPLES. 19. What is the cube root of 250 ? 20. What is the cube root of 38? 21. What is the cube root of 16zo Ans. SURDS. 22. What is the cube root of ;? Ans. ,829+ 23. What is the cube root of ? Ans. ,822+ 24. What is the cube root of ? Ans. ,873+. 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 given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root therefrom. LXAMPLES. SURDS. 95. What is the cube root of 1979 ? Ans. 25 26. What is the cube root of 311? Ans. 3. 27. What is the cube root of 405,1 ? Ans. 73. 28. What is the cube root of 7...? A18. 1,93 + 29. What is the cube root of gi? Ans. 2,092+: 30. What is the cube root of 8 ? Ans. 2,057 +. THE APPLICATION. 1. If a cubical piece of timber, be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches does it contain ? Ans. 103823. 2. There is a cellar dug, that is 12 feet every way, in length, breadth, and depth, how many solid feet of earth were taken out of it? Ans. 1728. 3. There is a stone of cubic form, which contains 389017 solid feet, what is the superficial contents of one of its sides? Ans, 5320. Between two Numbers given, to find two mean Proportionals. Rule. Divide the greater extreme by the less, and the cube root of the quotient multiplied by the less extreme gives the less inean; multiply the said cube root by the less mean, and the product will be the greater mean proportional. EXAMPLIS. 4. What are the two mean proportionals between 6 and 162? Ans, 18 and 64. 6. What are the two mean proportionals between 4 and 108? Ans. 12 and 36. To find the Side of a Cube that shall be equal in Solidity to any given Solid, as a Glob, Cylinder, Prism, Cone, gic. Rule. The cube root of the solici contents of any solid body given, is the side of the cube with equal solidity. 6. If the solid content of a globe is 10648, what is the side of a cube of equal solidity ? Ans, 22. The Side of the Cube being zicen, ta find the Side of the Cube that shall be double, treble, &c. in Qüantity to the Cube given. Rule. Cube the side given, and multiply it by 2, 3, &c. the cube root of the product is the side sought. EXAMPLE EXAMPLE. .7. There is a cubical vessel, whose side is 19 inches, and it is required to find the side of another vessel, that is to contain three times as much ? Ans. 17,306. EXTRACTING OF THE BIQUADRATE ROOT. To extract the Biquadrate Root is to find out a number, which being involved four times into itself, will produce the given number. Rule. First extract the square root of the given number, and then extract the square root of that square root, and it will give the biquadrate root required. 1. What is the biquadrate of 27 ? Ans. 531441, 2. What is the biquadrate of 76 ? Ans. 33362176. 3. What is the biquadrate of 275 ? Ans. 5719140625. 4. What is the biquadrate root of 531441 ? Ans. 27. 5. What is the biquadrate root of 33362176? Ans, 76. 6. What is the biquadrate root of 5719140625? Ans. 275. EXAMPI.ES, A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. 1. PREPARE the number given for extraction, hy pointing off from the unit's place as the root required directs. 2. Find the first figure in the root by the table of powers, which subtract from the given number. 3. Bring down the first 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 division, and annex it to the root; then involve the whole root into the given power, and call that the subtrahend, 6. Subtract that number from as many points of the given power, as is brought down, beginning at the lower place, and to EXAMPLES. the remainder bring down the first figure of the next point for a new dividend. 7. Find a new divisor, and proceed in all respects as before. 1. What is the square root of 141376 ? 141976(376 9 3 X 2=6 divisor. 37 X 37=1369 subtrahend. 6)51 dividend. 37 X 2=74 divisor. 376 x 376=141376 subtrahend. 1369 subtrahend. 53167376 subtrahend, 3 x 3 x 3527 divisor. 37 X 37 x 37=50653 subtrahend. 37 X 37 X 354107 divisor. 19987173376(376 108)1188 dividend. 1874161 subtrahend. 902612)1245563 dividend. 19987173376 subtrahend. 3 X 3 x 3 x 4 =108 divisor. 31 x 37 X 37 X 37=1874161 subtrahenda 9 Х 4–202612 divisor. 376 x 376 x 376 x 376–19987173376 subtrahona 37 x 37 X |