EXAMPLE. Suppose a triangular piece of land, whose sides are 24, 16, and 12 rods; what is the area? 24+16+12-52-2-26 for half. 26-24 2 then 26×2×10×14=7280, the square 26-16-10 root of which is 85,32+ rods. 26-12-14 Multiplying the longest side by half of the nearest distance, to its opposite angle; or, multiplying the longest side by the nearest distance to its opposite angle and taking half of the product, gives the area. EXAMPLE. If the three sides of a piece of land in form of a triangle be 15, 14, and 13 rods, required the area? The nearest distance would be 11,2 rods. Or, 11,2 15 2)168,0 If the three sides be 120,5, 112,6 and 90,3 rods, required the area? Ans. 4832,7 rods 30 acres and 327 perches. Any irregular four-sided piece of land may be divided into two triangles by a diagonal line, and a five-sided piece into three triangles by two diagonals. If the length of the sides be agreed on, there can be no dispute on the admeasurement, as all who are acquainted with the rule will agree in the result. 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. 1. 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 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 resolvend; to the remainder bring down the next point, and proceed as before. 8. 9. Roots. 1. 2. 3. 4. 5. 6. 7. What is the cube root of 99252847? 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 divisor -as, 4x4x3=48. Find how often it is contained in the resolvend, rejecting 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 6×6=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. 1916847 Square of 46 2116×3=6343 div. Square of 3-9 put to 6348=*634809 3X3X46= 414 638949X3=1916847 2. What is the cube root of 389017? 3. What is the cube root of 5735339? Ans. 73 Ans. 179. 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 ciphers thereto, so that there may be a point fall on the unit's place of the whole number. 4. What is the cube root of 12,977875? Ans. 2,35. 5. What is the cube root of 36155,027576? Ans. 33,06+ To extract 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. *When the quotient is 1, 2, or 3, there must be a cipher put to supply the place of tens. 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. 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. 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. The cube root of the solid content of any solid body given is the side of the cube of equal solidity. EXAMPLE. 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 given, to find the side of that cube, that shall be double, treble, &c. in quantity to the given cube. RULE. Cube the side given, and multiply it by 2, 3, &c. the cube root of the product is the side sought. EXAMPLE. There is a cubical vessel, whose side is 12 inches, and it is required to find the side of another vessel that is to contain three times as much? Ans. 17,306. 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, then extract the square root of that square root, and it will give the biquadrate root required. EXAMPLES. 1. What is the biquadrate of 27? 2. What is the biquadrate root of 531441? Ans. 531441. A GENERAL RULE FOR EXTRACTING THE ROOTS OF ALL POWERS. 1. Prepare the number given for extraction, by 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 are brought down, beginning at the lowest place, and to 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. EXAMPLES. 1. What is the square root of 141376? 2. What is the cube root of 53157376? Ans. 376. Ans. 376. 3. What is the biquadrate root of 19987173376? Ans. 376, |