Note. Any of the convenient examples in the Rule of Three or Rule of Five in Integers, or Vulgar Fractions, may be taken as proper examples to the same rules in Decimals. The following example, which is the first in Vulgar Fractions, is wrought here to show the method. If of a yard of velvet cost al, what will yd. cost? LUODECIMALS, or CROSS MULTIPLICATION, is a rule made use of by workmen and artificers, in computing the contents of their works. Dimensions are usually taken in feet, inches, and quarters; any parts smaller than these being neglected as of no consequence. And the same in multiplying them together, or casting up the contents. RULE.-Set down the two dimensions, to be multiplied together, one under the other, so that feet stand under feet, inches under inches, &c. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand; omitting however what is below parts of inches, only carrying to these the proper number of units from the lowest denomination. Or, instead of multiplying by the inches, take such parts of the multiplicand as these are of a foot. Then add the two lines together, after the manner of Compound Addition, carrying 1 to the feet for 12 inches, when these come to so many. INVOLUTION is the raising of Powers from any given number, as a root. Root, a certain number of times continually by itself. Thus, 22 is the root, or-first power of 2. 2×2= 4 is the 2d power, or square of 2. 2×2×2= 8 is the 3d power, or cube of 2. 2×2×2× 2 = 16 is the 4th power of 2, &c. And in this manner may be calculated the following Table of the first nine powers of the first nine numbers. 981 729 6561 59049 531441 4782969 43046721 387420489 The Index or Exponent of a Power, is the number denoting the height or degree of that power; and it is 1 more than the number of multiplications used in producing the same. So 1 is the index or exponent of the 1st power or root, 2 of the 2d power or square, 3 of the 3d power or cube, 4 of the 4th power, and so on. Powers, that are to be raised, are usually denoted by placing the index above the root or first power. When two or more powers are multiplied together, their product will be that power whose index is the sum of the exponents of the factors or powers multi 1 Note. Any of the convenient examples in the Rule of Three or Rule of Five in Integers, or Vulgar Fractions, may be taken as proper examples to the same rules in Decimals. The following example, which is the first in Vulgar Fractions, is wrought here to show the method. If of a yard of velvet cost al, what will yd. cost? LUODECIMALS, or CROSS MULTIPLICATION, is a rule made use of by workmen and artificers, in computing the contents of their works. Dimensions are usually taken in feet, inches, and quarters; any parts smaller than these being neglected as of no consequence. And the same in multiplying them together, or casting up the contents. RULE.-Set down the two dimensions, to be multiplied together, one under the other, so that feet stand under feet, inches under inches, &c. Multiply each term in the multiplicand, beginning at the lowest, by the feet in the multiplier, and set the result of each straight under its corresponding term, observing to carry 1 for every 12, from the inches to the feet. In like manner, multiply all the multiplicand by the inches and parts of the multiplier, and set the result of each term one place removed to the right hand of those in the multiplicand; omitting however what is below parts of inches, only carrying to these the proper number of units from the lowest denontination. Or, instead of multiplying by the inches, take such parts of the multiplicand as these are of a foot. Then add the two lines together, after the manner of Compound Addition, carrying 1 to the feet for 12 inches, when these come to so many. INVOLUTION is the raising of Powers from any given number, as a root. Root, a certain number of times continually by itself. Thus, 2= 2 is the root, or first power of 2. 2×2= 4 is the 2d power, or square of 2. 2×2×2= 8 is the 3d power, or cube of 2. 2×2×2× 2 = 16 is the 4th power of 2, &c. And in this manner may be calculated the following Table of the first nine powers of the first nine numbers. 8 64 512 4096 32768 262144 2097152 16777216 134217728 9 81 729 6561 59049 531441 4782969 43046721 387420489 The Index or Exponent of a Power, is the number denoting the height or degree of that power; and it is 1 more than the number of multiplications used in producing the same. So 1 is the index or exponent of the 1st power or root, 2 of the 2d power or square, 3 of the 3d power or cube, 4 of the 4th power, and 50 on. Powers, that are to be raised, are usually denoted by placing the index above the root or first power. When two or more powers are multiplied together, their product will be that power whose index is the sum of the exponents of the factors or powers multiplied. Or the multiplication of the powers, answers to the addition of the indices. Thus, in the following powers of 2. 1st 2d 3d 4th 5th 6th 7th 8th 9th 10th EVOLUTION, or the reverse of Involution, is the extracting or finding the roots of any given powers. The root of any number, or power, is such a number, as being multiplied into itself a certain number of times, will produce that power. Thus, 2 is the square root or 2d root of 4, because 2o = 2 × 2 = 4; and 3 is the cube root or 3d root of 27, because 3o = 3x3x3 = 27. Any power of a given number or root may be found exactly, namely, by multiplying the number continually into itself. But there are many numbers of which a proposed root can never be exactly found. Yet, by means of decimals we may approximate or approach towards the root, to any degree of exactness. Those roots which only approximate, are called Surd roots; but those which can be found quite exact, are called Rational roots. Thus, the square root of 3 is a surd root; but the square root of 4 is a rational root, being equal to 2: also the cube root of 8 is rational, being equal to 2; but the cube root of 9 is surd or irrational. Roots are sometimes denoted by writing the character ✓ before the power, with the index of the root against it. Thus, the third root of 20 is expressed by20; and the square root or 2d root of it is 20, the index 2 being always omitted, when the square root is designed. When the power is expressed by several numbers, with the sign + or between them, a line is drawn from the top of the sign over all the parts of it: thus, the third root of 45 12, or thus,(45 - 12), inclosing the numbers in parentheses. 12 is/45 But all roots are now often designed like powers, with fractional indices: 1. 1 thus, the square root of 8 is 82 the cube root of 25 is 253, and the 4th root of |