a divisor, and the other three for a dividend; when inverse, multiply the third and fourth terms together for a divisor, and the other two for a dividend.

INVOLUTION AND EVOLUTION. Involution is the multiplying any number into itself, and that product by the former multiplier; and so on, and the several products which arise are called powers. The number denoting the hight of the power, is called the index, or exponent of that power. Evolution is dividing a number into equal factors. A root of any number, is that number which multiplied one or more times into itself, will produce that number. A root takes its name or number, from the number of times it is contained as a factor in the corresponding power. Rule for extracting the square root. 1. Having pointed off the given number into periods of two figures each, subtract from the highest period the greatest square contained in it, place the root in the quotient, and to the remainder bring down the next period for a dividend. 2. Double the root already found, (understanding a cypher at the right,) for a divisor, and divide the dividend by it, for the next figure of the root. 3. Annex this figure to the divisor, which, so increased, multiply by the same figure for a subtrahend. 4. Subtract the subtrahend from the dividend, to the remainder bring down the next period for a new dividend, and so proceed. The proof is by involution. Rule for the extraction of the cube root. 1. Having pointed off the given number into periods of three figures each, subtract from the highest period the greatest cube contained in it, place the root in the quotient, and, to the remainder bring down the next period for a divisor. 2. Square the root already found, (understanding a cypher at the right,) and multiply it by three for a divisor. Divide the dividend by the divisor for the next figure of the root. Multiply the divisor by the quotient, multiply three times the square of the quotient by the part of the root previously found, finally, cube the quotient, and add these three results together for a subtrahend. 4. Subtract the subtrahend from the dividend, to the remainder bring down the next period for a new dividend, and so proceed.

3.

Arithmetical and Geometrical Progression. Any rank of numbers more than two, increasing by a common excess, or decreasing by a common difference, is said to be in arithmetical progression. The sum of all the terms may be found

by multiplying the sum of the extremes by the number of terms, and dividing the product by 2. The common differ. ence of the terms may be found, by dividing the difference of the extremes by the number of terms less one. If the difference of the extremes be divided by the common difference, the quotient increased by one will be the number of terms. Geometrical progression is when any rank or series of numbers is increased by a common multiplier or decreased by a common divisor. When the extremes and ratio are given, the sum of the terms is found by multiplying the greatest term by the ratio, from the product subtracting the least term, and dividing the remainder by the ratio less one. When the first term and number of terms are given, the last term is found by multiplying the first term by that power of the ratio whose index is one less than the number of terms.

Alligation is a general name given to the mixing of simple things of different qualities, so as to form a compound of a medium or mean quality. An annuity is a sum of money payable every year, or for a certain number of years, or forever. Permutation of quantities is the showing how many different ways any given number of things may be changed.

ALGEBRA.

ALGEBRA is a general method of investigating the rela tions of quantities by letters and other symbols. Among the advantages of algebra over common arithmetic, are 1. that calculations may be greatly abridged by the use of letters, 2. the quantities brought into calculation may be preserved distinct from each other, and 3. by means of letters unknown quantities may be treated as if they were known, &c. A simple quantity is either a single letter, or a number of letters not connected by the signs and A compound quantity consists of a number of simple quantities connected by the sign + or -. If there are two terms in a compound quantity, it is called a binominal. A reciprocal of a quantity is the quotient which arises from dividing a unit by that quantity. One quantity is a multiplier of another, when the former contains the latter a certain number of

times without a remainder. One quantity is the measure of another when the former is contained in the latter, any number of times without a remainder. A negative quantity is one which is required to be subtracted. An axiom is a selfevident proposition. Addition is the connecting of several quantities, with their signs, in one algebraic expression. Quantities are added, by writing them one after another, without altering their signs. Several terms are reduced to one, when the quantities are alike and the signs alike, by adding the co-efficients, annexing the common letter or letters, and prefixing the common sign. To reduce several terms to one, when the quantities are alike, but the signs unlike, take the less co-efficient from the greater; to the dif ference annex the common letter or letters, and prefix the sign of the greater co-efficient. Subtraction is finding the difference of two quantities or sets of quantities. Subtraction is performed in algebra by changing the signs of all the quantities to be subtracted, and then proceeding the same as in addition. Multiplying by a whole number is taking the multiplicand as many times as there are units in the multiplier. Multiplying by a fraction is taking a certain portion of the multiplicand, as many times as there are like portions of a unit in the multiplier. Multiplication by a nega. tive quantity has the same relation to multiplication by a positive quantity, which subtraction has to addition. When the letters have numerical co-efficients, these must be multiplied together, and prefixed to the product of the letters. If the signs of the factors are alike, the sign of the product will be affirmative; but if the signs of the factors are unlike, the sign of the product will be negative. General rule for multiplication in algebra. Multiply the letters and co-efficients of each term in the multiplicand, into the letters and co-efficients of each term in the multiplier; and then prefix to each term of the product, the sign required by the principle, that like signs produce and different signs Division is finding a quotient, which multiplied into the divisor will produce the dividend. When the divisor is found as a factor, the division is performed by canceling this factor. In division the same rule is to be observed respecting the signs as in multiplication. If the letters of the divisor are not found in the dividend, the division is expressed by writing the divisor under the dividend, in the form of a vulgar fraction.

FRACTIONS. If the denominator remains the same, multiplying the numerator by any quantity, is multiplying the value by that quantity; and dividing the numerator is dividing the value. If the numerator remains the same, multiplying the denominator by any quantity, is dividing the value by that quantity, and dividing the denominator is multiplying the value. If the numerator and denominator be both multiplied or both divided, by the same quantity, the value of the fraction will not be altered. If the sign prefixed to a fraction, or all the signs of the numerator, or all the signs of the denominator, be changed, the value of the fraction will be changed from positive to negative, or from negative to positive. If all the signs both of the numerator and denom. inator, or the signs of one of these with the sign prefixed to the whole fraction, be changed at the same time, the value of the fraction will not be altered. A fraction may be reduced to lower terms, by dividing both the numerator and denominator, by any quantity which will divide them without a remainder. Fractions of different denominations may be reduced to a common denominator, by multiplying each numerator into all the denominators except its own, for a new numerator; and all the denominators together, for a common denominator. To reduce an improper fraction to a mixed quantity, divide the numerator by the denominator. Fractions may be added by reducing them to a common de. nominator, making the signs before them all positive, and then adding their numerators. For the subtraction of fractions, change the fractions to be subtracted from positive to negative, or the contrary, and then proceed as in addition. To multiply fractions, multiply the numerators together for a new numerator, and the denominators together for a new denominator. A fraction is multiplied into a quantity equal to its denominator, by canceling the denominator. Reducing a compound fraction to a simple one, is the same as multiplying fractions into each other. To divide one frac tion by another, invert the divisor, and then proceed as in multiplication. The reciprocal of a fraction is the fraction inverted.

SIMPLE EQUATIONS. An equation is a proposition, expressing in algebraic characters, the equality between one. quantity or set of quantities and another, or between different expressions for the same quantity. The reduction of

an equation, consists in bringing the unknown quantity by it. self, on one side, and all the known quantities on the other side, without destroying the equation. When known quan. tities are connected with the unknown quantity by the sign + or the equation is reduced by transposing the known quantities to the other side, and prefixing the contrary sign. When the unknown quantity is divided by a known quantity, the equation is reduced by multiplying each side by this known quantity. An equation is cleared of fractions by multiplying each side into all the denominators. When the unknown quantity is multiplied into any known quantity, the equation is reduced by dividing both sides by this known quantity. A proportion is converted into an equation, by making the product of the extremes one side of the equation; and the product of the means, the other side. On the other hand an equation may be converted into a proportion, by resolving one side of the equation into two factors, for the middle terms of the proportion; and the other side into two factors for the extremes.

INVOLUTION AND POWERS. When a quantity is multiplied into itself, the product is called a power. A co-efficient shows how often a quantity is taken as a part of the whole. An exponent shows how often a quantity is taken as a factor in a product. Rule for involving a quantity. Multiply the quantity into itself, till it is taken as a factor, as many times as there are units in the index of the power to which the quantity is to be raised. The power of the product of seve ral factors, is equal to the product of their powers. The square of a binominal, the terms of which are both positive, is equal to the square of the first term, twice the product of the two terms, the square of the last term. When the root is positive all its powers are positive also; but when the root is negative, the odd powers are negative while the even powers are positive. A quantity which is already a power, is involved by multiplying its index into the index of the power to which it is to be raised. A fraction is involved by involving both numerator and denominator. Powers may be added by writing them one after another, with their signs. Subtraction of powers is to be performed in the same manner as addition, except that the signs of the subtrahend are to be changed. Powers may be multiplied like other quantities, by writing the factors one after another, either with or