number, then place a period over tenths in decimals, and one over every third figure beyond it, counting to the right, and if the right hand period should not be complete, annex ciphers to complete the period; then extract the root the same as in whole numbers. The periods over whole numbers, show that the root must have so many figures in whole numbers; the rest will be decimals. 14. What is the cube root of 41421,736? 15. What is the cube root of 85766,121 ? 16. What is the cube root of 117,649 ? 17. What is the cube root of 84,604519? Ans. 34,6. Ans. 44,1. Ans. 4,9. Ans. 4,39. NOTE. If there be a remainder after all the periods are brought down, the operation may be continued, at pleasure, by annexing periods of ciphers. 18. What is the cube root of 22 19. What is the cube root of 1? Ans. 1,2599. Ans. 1,442. 20. What is the cube root of 3? Application and use of the Cube Root. CASE I. To find two mean proportionals between any two given numbers RULE. Divide the greater extreme by the less, and the cube root of the quotient, multiplied by the less extreme, gives the less mean; multiply the said cube root by the less mean, and the product will be the greater mean proportional. EXAMPLES. 1. What are the two mean proportionals, between 4 and 256? Ans. 16 the less, 64 the greater. NOTE. Sixteen and 64 are the two mean proportionals between 4 and 256; because 4:16:: 64: 256, and the product of the extremes, is equal to the product of the means. It will be seen that the extremes are the given numbers, and the means, the two numbers in the answer to the question. 2. What are the two mean proportionals between 8 and 4096? Ans. 64 less mean, 512 greater mean. CASE II.-To find the side of a cube that shall be equal in solidity to any given solid, as a globe, cylinder, prism, cone, &c. RULE.—Extract the cube root of the solid contents of the given body, and the said root will be the side of the cube required. EXAMPLES. 1. The statute bushel contains 2150,4252 cubick or solid inches; what must be the side of a eubick box, that shall contain the same quantity? Ans. 12,907 inches. 2. There is a certain cistern, 24 feet long, 18 feet wide, and 4 feet high; required the side of a cistern of a cubick form, that shall hold the same quantity. Ans. 12 feet. NOTE. The solid contents of similar Figures are in proportion to each other, as the cubes of their similar sides or diameters. 3. If a ball, 2 inches in diameter, weigh 3lb.; what will a ball of similar metal weigh, whose diameter is 4 inches? 2X2X2=8; 4X4X4 64. As 8:31b.:: 64: 24lb. Ans. 24lb. 4. If a globe of gold 1 inch in diameter be worth $100; what is the value of another globe 4 inches in diameter ? Ans. $6400. CASE III-The side of a cube being given, to find the side of another cube which shall be double, triple, &c. in quantity to the given cube: Or one fourth, one fifth, &c. of the given cube. RULE.-Cube the given side, and multiply it by the given proportion, and the cube root of the product will be the side sought. Or if required to be less; divide the cube of the given side, by the given proportion, and extract the cube root of the quotient; the root will be the side of the cube sought. EXAMPLES. 1. There is a cubical vessel whose side is 3 feet; required the side of another cubical vessel which shall contain 8 times as much. Ans. 6 feet. 2. There is a cubical vessel whose side is 6 feet; required the side of one that shall contain only one eighth as much. Extraction of the Biquadrate Root, Ans. 3 feet. Is finding a number, which being multiplied by its cube, or involved four times into itself, will be equal to the given number. RULE.-You may first extract the square root of the given number, and then the square root of that root, and the last root will be the required biquadrate root. EXAMPLES. 1. What is the biquadrate root of 33362176? Ans. 76. Ans. 275. A general rule for extracting the roots of all powers. 1. Prepare the given number for extracting, by pointing off from the unit's place, as the required root directs, that is, a period over every second figure, for the square root, and one over every third, for the cube root; one over every fourth, for the biquadrate, and one over every fifth, for the fifth root, and so on. 2. Find the first figure of the root by the table of powers, or by trial; subtract its power from the left hand period, and to the remainder bring down the first figure in the next period for an imperfect dividend. 3. Involve the root to the next inferiour power to that which is given, and multiply it by the number denoting the given power, for a divisor; by which find a second figure of the root. 4. Involve the whole ascertained root to the given power; and sub 1 tract it from the first and second periods. Bring down the first figure of the next period to the remainder, for a new dividend; to which find a new divisor as before, and so proceed till the work is finished. NOTE. The roots of the 4th, 6th, 8th, 9th, and 12th powers may be obtained more readily, thus: For the 4th root, extract the square root of the square root. 1. What is the fifth root of 130691232? * We call 2829 an imperfect dividend, because we make no other use of it than determining the next figure in the root, after which we make use of the periods in our given number. NOTE.-When a number is given to find its root, the given number may be considered as the power of the root required; thus, when the square root is required, the given number is the second power of the required root; when the cube root is required, the given number is the third power, and when the biquadrate root is required, the given number is the 4th power of the required root, and so on. 2. What is the biquadrate root of 5308416? Ans. 48, the root. QUESTIONS ON THE CUBE ROOT. What is the extraction of the cube root? A. It is finding a number which being multiplied by its square, will produce the given number. How is a number prepared for extracting the root? A. By pointing it off into periods of three figures each, beginning at the place of units. Why is the given number pointed off into periods of three figures each? A. Because the cube of one figure never exceeds three figures. How is the first figure of the root found? A. By the table of powers or by trial; the greatest cube that shall not exceed the left hand period, is placed directly below it, and its root in the quotient. What does the first figure of the root show, taken in its local value? A. It expresses the side of a cube formed from what has been subtracted from the given number. What does the first divisor express? A. It expresses, like all other divisors, the area or suriace of three sides of the cube which has already been formed. What is the product when this divi sor is multiplied by the last quotient figure? A. It expresses the solidity contained in the three pieces, or additions, made to the three sides of the cube. What do the other parts of the subtrahend express? A. The solid contents of the pieces required to fill the spaces, in order to complete the cube. What does the root show at all stages of the work? A. The length of one side of a cube, formed from what has been subtracted from the given number. How is the cube root proved? A. By cubing the quotient or root, or by adding together the solidity of the several parts of the cube. When the solid contents of a cube are given, how do you find the length of one of the sides? A. By extracting the cube root. When the side of a cube is given, how do you find its solidity? A. By cubing the side. MENSURATION OF SUPERFICIES, Teaches how to find the area or surface of any square, parallelogram, circle, triangle, &c. Every magnitude is measured by some magnitude of the same kind; thus, a line is measured by a lineal inch, foot, yard, &c.; superficies, or surface, by the square inch, square foot, square yard, &c.; and a solid, by the cubick inch, cubick foot, cubick yard, &c. The superficial inch, foot, yard, &c., is one inch, foot, yard, &c., in length and breadth; and because 12 inches make one foot in long measure, 144 inches make one superficial foot, thus 12X12=144 inches, and 9 feet a superficial yard, thus, 3×3=9. ► Το To find the area or surface of a Square having equal sides. RULE.-Multiply the side of a square into itself, and the product will be the area or the superficies. 24 feet. 24 feet. EXAMPLES. 1. How many square feet of boards will lay a floor which is 24 feet square? 2. How many acres does a square lot contain, measuring 40 rods on each side? Ans. 10 acres. To find the area of a parallelogram, or long square. RULE.-Multiply the length by the breadth, and the product will be the area, or superficial content, in the same name with the length of the sides. 2. How many acres are there in a lot of land 320 rods long, and 160 rods wide? Ans. 320 acres To find the area of a right angled triangle. RULE.-Multiply the base of the triangle by half its perpendicular, or multiply half the base by the whole perpendicular, and the product will be the area. 20 rods. 12 rods. EXAMPLES. 1. Required the area of a triangle, the base of which is 20 rods, and the perpendicular 12 rods. 20X6=120rds. Ans. 120 sq. rds. DEM.-The reason of multiplying the base by half the perpendicular, is plain, because the triangle is half of a parallelogram, one side of which is 20 rods, and the other 12 rods. 2. How many acres are contained in a triangle whose base is 320 rods, and perpendicular 40 rods? Ans. 40 acres. To find the area of a triangle having the three sides given. RULE.-Add the three sides together, then take the half of that sum, and out of it subtract each side severally, and multiply the half of the sum and these remainders, and the square root of this product will be the area of the triangle. EXAMPLES. 1. Required the area of an oblique triangle, the 3 sides of which are 13, 14, and 15 rods. Ans. 84 square rods. 21 half the sum. 6 15 8 7 6 126 2. Required the area of an oblique triangle, the 3 sides of which are 80, 120, and 160 rods. Ans. 29 acres, 7 poles. |