9. The wall of a town is 25 feet high, which is surrounded by a moat of 30 feet in breadth, I defire 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 hypothen ufe and perpendicular given to find the base. Rule. The square root of the difference of the squares of the hypothenuse and perpendicular is the length of the base. The base and hypothenufe given to find the perpendicular. Rule. The square root of the difference of the hypothenuse and base is the heighth of the perpendicular. N. B. The two last questions may be varied for examples to the two last propositions. Any number of men being given to form them 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 cither in rank or file. An army consisting of 331776 men, I desire to know how many in rank and file ? 11. A certain square pavement contains 48841 square stones, all of the same size, I demand how many are contained in one of the fides. Ans. 221. 10. Aní. 576. EXTRACTION OF THE CUBE ROOT. TO extract the Cube Root is to find out a number which being multiplied into itself, and then into that product, produceth the given number. RULE. Point every third figure of the cube given, beginning 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 divifor by multiplying the square of the quotient by 3. See how often it is contained in the resolvend, rejecting the units and put the answer in the quotient. 3. To find the fubtrahend. 1. Cube the last figure in the quo. tient. 2. Multiply all the figures in the quotient by 3, except the laft, and that product by the square of the last. 3. Multiply the divifor by the last figure. Add these products together, gives the fubtrahend, which lubtract from the resolvend ; to the remainder bring down the next point and proceed as before. Roots. 3. 4. 5. 6. 7. 9. tens, and 2. 1. 1. 99252847(463 64=Cube of 4. Divisor. 216=Cube of 6. 33336 Subtrahend. Divisor. 27=Cube of 3. a Another new and more concise method of extracting the Cube Root. RULE. 1. Point every third figure of the cube given, beginning at the unit's place, then find the nearest cube to the first point, and subtract it therefrom, put the root in the quotient, bring down the figures in the next point to the remainder for a resolvend. 2. Square the quotient and triple the square for a divisoras 4X4X3=48. Find how often it is contained in the resolvend, reječting units and tens, and put the answer in the quotient. 3. Square the last figure in the quotient, and put it on the right hand of the divisor : As 6x6=36 put to the divisor 48=4836. 4. Triple the last figure in the quotient, and multiply by the former, put it under the other, units under the tens, add them together, and multiply the sum by the last figure in the quotient, subtract that product from the resolvend, bring down the next point and proceed as before. Anf. 73. EXAMPLES. Square of 4 x3=48 Divisor, 99252847(463 64 35252 5556 X 6 = 33336 Square of 46=216x3=6348 Divisor Square of 3=9 put to 6348=*631809 1916847 3X3 X 46=414 638949 X3=1916847 2. What is the cube root of 389017 ? 3. What is the cube root of 5735339 ? 4 What is the cube root of 32461759 ? s. What is the cube root of 84604519 ? 6. What is the cube root of 259694072 ? Ans. 638. 7. What is the cube root of 48228544 ? Anf. 364 8. What is the cube root of 27054036008 ? 9. What is the cube root of 22069810125 ? Anf. 2805 10. What is the cube root of 12261 5327232 ? Anf. 4968. 11. What is the cube root of 219365327791 ? Ans. 6031. What is the cube root of 673373097125? Anf. 8765. When the given number consists of a whole number and decimal together, make the number of decimals to consist of 3, &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 ? 14. What is the cube root of 36155:027576 ? Anf. 33,06+ 15. What is the cube root of .001906624? 16. What is the cube root of 33-230979637 ? Anf. 3,215+ 17 What is the cube root of 1 15926,972504 ? Anf. 25,19+ 18. What is the cube root of ,053258279 ? Anf. ,376 + Anf. 3002. 12. Anf. 2.35 Anf. ,124 To extra&t the Cube Root of a Vulgar Fraction. Rule. Reduce the fraction to its lowest terms, then extract the cube root of the numerator and denominator for a new numerator and denominator, but if the fraction be a surd, reduce it to a decimal, and then extract the root from it. EXAMPLES. 19. What is the cube root of thi? Anf. . 20. Ani .. • When the quotient is 2 or 3 there must be a cypher put to supply the place of tens, Ans. 35. 2 To extraet the Cube Root of a mixt number. Rule. Reduce the fractional part to its lowest terms, and then the mixt number to an improper fraction, extract the cube roots of the numerator and denominator for a new numerator and denominator ; but if the mixt number given be a surd, reduce the fractional part to a decimal, annex it to the whole number, and extract the root there. from. EXAMPLES. 25. What is the cube root of 12 į? ? Anf. 21. 26. What is the cube root of 31 335? 27. What is the cube root of 405125 ? SURDS. 28. What is the cube root of 7}? Ans. 1.93+ 29. What is the cube root of 91? Anf. 2,092+ 30. What is the cube root of 8 Ž? Anl. 2,057+ THE APPLICATION. If a cubical piece of timber be 47 inches long, 47 inches broad, and 47 inches deep, how many cubical inches doth it contain ? Anf. 103823 There is a cellar dug that is 12 feet every way, in length, breadth and depth, how many folid feet of earth were taken out of it ? Anf. 1728. 3. There is a stone of a cubic form, which contains 389017 solid feet, what is the superficial content of one of its fides ? 2. a a Anf. 5329. Between two numbers given, to find two mean proportionals. RULE. Divide the greater extreme by the lesser, and the cube root of the quotient multiplied by the lesser extreme, gives the lesser mean ; multiply the said cube root by the lesser mean, and the produet will be the greater mean proportional. Dd EXAMPLES. 4. What are the two mean Proportionals between 6 and 162 ? Anf. 18 and 54. 5. What are the two mean Proportionals between 4 and 108 ? Anf. 12 and 36. To find the side of a cube that shall be equal in folidity to any givets Solid, as a globe, cylinder, prism, cone, &c. Rule. The cube root of the solid content of any solid body given is the side of the cube of equal folidity. EXAMPLE. 6. If the solid content of a globe is 10648, what is the side of a cube of equal solidity ? Ans, 22, a The side of the cube being given, to find the side of that cube, that shall be double, treble, &c. in quantity to the given cube. Rule. Cube the fide given, and multiply it by 2,3, &c. the cube root of the product is the fide fought. EXAMPLE, 7. There is a cubical vessel, whose fide is 12 Inches, and it is required to find the side of another vessel, that is to contain 3 times as much ? Anf. 17,306. a EXTRACTION 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. Anf. 531441 2. 3. 4 5 6. EXAMPLES. 33362176 What is the biquadrate of 275 P 5719140625 What is the biquadrate root of 531441 ? 27 What is the biquadrate root of 33362176 ? 76 What is the biquadrate root of 5719940625 ? 275 |