A. Any Number multiplied by its Square, produces a Cube. Q. What is the extraction of the Cube-Root ? A. If a Cube be given to find out a Number, which, being multiplied into its Square, produceth the number given; this is called the extraction of the Cube-Root Q. How is the given Cube to be prepared for Extraction A. By pointing off at every three Figures, both ways, from the unit's place, for a resolvend. Q. What is a Surd? A. It is an imperfect Cube, or such a number, whose Cube-Root can never be exactly found. Q. What is the rule for extracting the Cube-Root of a Number. A. This the first figure sought is the Root of the greatest Cube contained in the first member, and it is eal'ed a; then 3aa3+ a is the Divisor, which finds a new figure called e; then 3ane+coa+eee is the Subtrahend or Number to be subducted; which operation is to be continued to every resolvend. Note. This rule being somewhat dark, I shall, by way of illustration, subjoin the operation at large for extracting the Cube-Root of any number. What is the Cube-Root of 444194.947? (1) Let the given Number be pointed as before directed; thus: 444194.947 (2) The first member, which contains the greatest Cube, is 444; and the nearest Root, whose Cube is not greater than it, is 7, which set thus: 444194.947(7 (3) The Cube of 7 is 343, which set down and subtract, annexing the next three figures, or member, viz. 194 for a resolvend; thus: 444194.947(7 343 101194 Resolvend. (4) The number 7, in the Root is called ; then by the Rule 3aa+3a is the Divisor; thus, (5) The next figure in the Root, viz. 6 (found by common Division) is called e; then by the rule 3aae+3eea+eee is the Subtrahend, or Number to be subducted; thus, (6) When the next number is brought down, viz. 947 as before, both figures in the Root, viz. 76 must be called a; then to find a Divisor to this last Resolvend, say, as before, 3aa+3a; thus, 17328 Saa 173508(5218.947 Resolvend. 228=3a Divis. 173508=3aa+3a (7) The next figure in the Root, viz. 3, found as before is also called e; then again Saae + Seea+eee is the other Subtrahend, or number to be subducted; thus, 1 What is the Cube of 6.4 ? Answer 262. 144 5 What is the Cube of .007 ? Answer .000000343 Answ. 19.67+ Answ. 196.71+ Answ. 39.41+ Answ. 19.238+ 10 What is the Cube Root Answ. 22.89+. of 12000.812161 ? 11 What is the Cube Root of .121861281 12 What is the Cube Root Answ..495 of .0069761218? Answ. .19107+ 13 If a cubical piece of timber be 41 inches long, 41 inches broad, and 41 inches deep, how many cubical inches doth it contain? Ans. 68921 cubical inches. 14 Suppose a Cellar to be dug, that shall be 12 feet every way.in length, breadth and depth; how many solid feet of earth must be taken out to complete the same? Ans. 1728. 15 Suppose a stone of a cubic form to contain 474552 solid inches; what is the superficial content of one of its sides? Ans. 6084 inches. OF THE CUBE-ROOT OF A VULGAR FRACTION. Q. How do you extract the Cube Root of a Vulgar Fraction? A. 1 Reduce the Fraction to its lowest terms. 2 Extract the Cube Roots of the Numerator and Denominator for a new Numerator and Denominator. 3 If the Fraction be a Surd reduce it to a Decimal, and then extract the Cube Root from it. 4 The Decimal Fraction must consist of Ternaries of places; as three, six, nine, &c. EXAMPLES. 1 What is the Cube Root of 2? Answer 352 2 What is the Cube Root of 1941? Ans. 4608 3 What is the Cube Root of $48? Ans. § SURDS. 3000 4 What is the Cube Root of 4? Ans. .763+ 5 What is the Cube Root of ? Ans. 949+ 6 What is the Cube Root of? Ans. 693++ OF THE CUBE ROOT OF A MIXED NUMBER. Q. How do you extract the Cube Root of a mixt number? A. 1 Reduce the fractional part to its lowest terms. 2 Reduce the mixt number to an improper fraction. 3 Extract the Cube Roots of the Numerator and Denominator, for a new Numerator and Denominator. 4 If the mixt number given be a Surd, reduce the fractional part to a Decimal, and annex it to the whole number, and extract the Cube Root from the whole. EXAMPLES. 1 What is the Cube Root of 57812? Ans. 8 Q. SURDS. 4 What is the Cube Root of 87 ? Ans 2.013+ 5 What is the Cube Root of 73? Ans. 1.966+ OF THE BIQUADRATE ROOT. WHAT is the Biquadrate Number? A. Any Number involved four times produces a Biquadrate. Q. How is the Biquadrate Root extracted? A. First extract the Square Root of the given Resolvend, and then extract the Square Root of that Square Root, for the Biquadrate Root required. 1 EXAMPLES. 1 What is the Biquadrate of 48 ? Ans. 5308416. 2 What is the Biquadrate of 96 ? Ans.84934656. 3 What is the Biquadrate Root of 5308416? Ans. 48.. 4 What is the Biquadrate Root of 84934656? Ans. 96. 5 What is the Biquadrate Root of 21743271936 ? OF THE SURSOLID ROOT. o.WHAT is a Sursolid? Ans. 384. A. Any Number involved 5 Times produces a Sirsolid. Q. How is the Sursolid Root, or the Root of any other higher Power extracted? A. By the following general Rules. If any even power be given, let the Square Root of it be extracted, which reduces it to half of the given Power, then the Square Root of that Power reduces it to half of the same power, and so on till you come to a square of a Cube. For example: Suppose a 24th Power be given the Square Root of that reduces it to a 12th Power, the Square Root of the 12th Power reduces it to a 6th Power; and the Square Root of the 6th Power to a Cube. 2 If any odd Power be given, as the 17th, &c. observe [1] From the Unity Place, both ways, point off at every such Number of figures as is the Index of the power for a Resolvend. [2] Seek in the Table of Powers, for such a power (being the save Power with the Index) as comes nearest the first Period, whether greater or less, calling its Root accordingly more than just, or less than just. [3] Annex so many Cyphers to the Root, as there are |