E have seen that logarithmic tables may be used as a substitute for many lengthened operations in arithmetic. It is evident that the value of all methods of computation lies in their brevity. Algebra must be considered as one of the most important departments of mathematical science, on account of the extreme rapidity and certainty with which it enables us to determine the most involved and intricate questions. The term algebra is of Arabic origin, and has a reference to the resolution and composition of quantities. In the manner in which it is applied, it embodies a method of performing calculations by means of various signs and abbreviations, which are used instead of words and phrases, so that it may be called the system of symbols. Although it is a science of calculation, yet its operations must not be confounded with those of arithmetic. All calculations in arithmetic refer to some particular individual question, whereas those of algebra refer to a whole class of questions. One great advantage in algebra is, that all the steps of any particular course of reasoning are, by means of symbols, placed at once before the eye, so that the mind, being unimpeded in its operations, proceeds uninterruptedly from one step of reasoning to another, until the solution of the question is attained. Symbols are used to represent not only the known, but also the unknown quantities. The present custom is to represent all known quantities by the first letters of the alphabet, as a, b, c, etc and the unknown quantities by the last letters, x, y, z. The symbols used in arithmetic to denote addition, subtraction, etc., beieng properly to algebra. Thus the sign + plus denotes that one quantity is to be added to another, and is called the positive or additive sign, all numbers to which it is prefixed are called positive. The sign minus denotes that one quantity is to be subtracted from another; it is called the negative or subtractive sign, and all quantities to which it is prefixed are called negative. If neither + nor be prefixed to a quantity, then the sign + plus is understood. The general sign to denote that one number is to be multiplied by another is x; but it often occurs that one letter has to be multiplied by another, and this is represented by placing those letters one after another, generally according to the order in which they stand in the alphabet; thus a multiplied by b is expressed by ab. The multiplication of quantities consisting of more than one term, as, for instance, a + b by c + d, may be represented by any one of the following methods: a+bxc+d, or a+be+d, or (a+b) (c+d). The bar drawn over a+b and c+d, which in the first two examples marks them as distinct quantities, is called a vinculum, but brackets or parentheses for the same purpose, as in the last example, are now in more frequent use. When a letter is multiplied by any given number, it's usual to prefix that number to the letter. Thus, twice, three times b, four times c, six times x, etc., are expressed thus: 2a, 3b, 4c, 6x; and the numbers 2, 3, 4, 6, thus prefixed, are called the coefficients of the letters before which they stand. The sign between two numbers shows, as in arithmetic, that the former of those numbers is to be divided by the latter; thus, ab means that a is to be divided by b. It is, however, more usual to place the number to be divided above that by which it is to be divided, with a small line between, in the form of a fraction; thus denotes that a is divided by b. In arithmetic the powers of quantities are denoted by a small figure, called the exponent or index of the power. Thus axa, or the square of a, is expressed by a2; bxbxb, or the cube of b, is expressed by 63, etc. The cube of 1+b is expressed thus: (a+b)3. The roots of quantities are represented by the sign with the proper index affixed; thus a, or, more simply, /a, expresses the square root of a; a the cube root of a; Va+b represents the 4th or biquadratic root of a+b. Fractional indices are also frequently used to denote the roots of quantities, thus: a is the square root of a. at is the cube root of a. al is the 4th root of a, etc. Again, a is the cube root of a2, or of the square of ́. a is the square root of a3, or of the cube of a. a is the 5th root of a2. When two or more letters or quantities are connected together by signs, the combination is called an algebraic e-pression, and each letter or quantity is called a term. Quantities of one term are called simple quantities; as a 2a, 36, etc. A quantity of two terms, as b+c, is called a binomial. When the binomial expresses the difference between wo quantities, it is called a residual, as a-b. A quantity consisting of 3, 4, or many terms, are called -espectively trinomials, quadrinomials, multinomials. The sign placed between two quantities shows as in arithmetic, the equality of those quantities. = When quantities are connected by this sign, the expression is called an equation: thus, 2+4=6, is an equation, as also a+b=c-f. The symbol> or < is called that of inequality, it being placed between two quantities, of which one is greater than the other; the open part of the symbol is always turned towards the greater quantity: thus, a > b denotes a to be greater than b; and c <d denotes d to be greater than c. The sign of difference, is only used when it is uncertain which of two quantities is the greater; thus e~ f denotes the difference between e and ƒ when it is uncertain which is the greater. The word therefore, or consequently, often occurring in algebraical reasoning, the symbol .. has been chosen to represent it: thus, the sentence Therefore a + b is equal to c + d," is thus expressed in algebra, .'. a + b=c + d. Like quantities are such as consist of the same letter or letters, or power of letters: thus, 6 a and 2 a are like quantities, and also 4 abc and 9 abc. Unlike quantities are such as consist of different letters: as, 4a, 5b, 6ax2, 4cd, which are all unlike quantities. — 6 5 = − 17 6. But act often happens that like quantities which are to be added together have unlike signs, addition has in algebra a far more extended signification than in arithmetic. Thus, to add 7 + 4 a to 8 a 3 a, it is evident that, after 7 a + 4 a + 8 1 have been added according to the usual method, 3 a must be subtracted. Hence the general rule for the addition of le quantities with unlike signs is to add first the coefficients of the positive terms, and then to add those of the negative terms; the less sum must be subtracted from the greater, and to this difference the sign of the greater must be annexed, with the common letter or letters. Thus, let it be required to add 7 2 - 3 a + 4 a + 5 a − 6 a 2a and 9 a; 25 a will be found the sum of the positive terms, and II a that of the negative; Ira, being the less number, must there. fore be subtracted from 25 c, the greater, leaving a remainder of 14 a, which is the required amcant. Unlike quantities can only be added by collecting them in one line, and prefixing the proper sign of each; thus, the sum of 3 a + 2 b + 4 c 2 d can only be rendered 3 a + 2 b + 4 c — 2 d; this will be evident by reflecting that different letters in the same algebraical expression always represent different quantities, which cannot of course be added into one sum unless their precise value be known. Thus, the addition of a and b cannot be represented by 2 a or 2 b, because that would imply that a is equal to b, whieh it is not necessarily; neither could it be represented by ab, because ab denotes the multiplication of the two quantities; the only method then of expressing these sums is thus, à + b. When like and unlike quantities are mixed together, as in the following example, the like quantities must first be collected together according to the method above described, and all unlike quantities must be annexed in order : - b is from subtracting 2 a from 9 a; but if to 3 a we add the other term, namely, 4 a, the sum will be the remainder sought, because 3 a +4a=7a; and if 2 a be subtracted from 9 a, which is just the same question in another form, for 6 a 4 a is 2 a, the remainder is just 7 a as before. So, if a to be subtracted from c, the remainder would be ca + b, and for the same reason. It may therefore be given as a general rule, that all the signs of a quantity which is required to be subtracted from another must be changed: thus, when 4x-3y is subtracted from 7 a + 5 b, the remainder is written thus, 7a+56 4 x + 3y. When like quantities are to be subtracted from each other, it is usual to place them in two rows, the one above the other; the signs of the quantities to be subtracted must, for the reason above adduced, be conceived to be changed; and the several quantities must be added, as shown in the following example : From 5 ax + 7 xy - 2 y In division, all letters common to both quantities must be omitted in the quotient; and when the same letters occur in both with different indices, the index of the letter in the divisor must be subtracted from that in the dividend; thus, abx+ ab, or 6 a 2 a or abx ab 6 a 2 a3 =x: and = 31 3 a2 When the exponent of any letter in the divisor exceeds that of the same letter in the dividend, the latter exponent must be subtracted from the former, and the quotient will be in the form of a fraction; thus, - 12 a3x2 ÷ 8 ax5 = — 122 a3x2 8 axs = 3 a2 23 When the number to be divided is a compound quantity, and the divisor a simple one, then each term of the dividend must be divided separately, and the result will be the answer; thus, 3 + 12 6 + 4 a + 6 c When the divisor and dividend are both compound quantities the rule is the same as that of long division in arithmetic. When there is a remainder, it must be made the numerator of a fraction, under which the divisor must be put as the denominator; this fraction must then be placed in the quotient, as in arithmetic. The compound quantities must, however, be previously arranged in a particular way, namely, according to the descending powers of some letter, as of b in the following example; and this letter is called the leading quantity. The following is an example of the division of compound quan tities: 5 αυ + may be thus reduced to a fraction: 2 x x 6 e 12 ex, 6e and as 5 ab must be added to form the numerator, and the former denominator be retained, the required fraction is the 12 ex + 5 ab following: An operation exactly the reverse of 6e this would of course be requisite, were it proposed to reduce a 12 ex + 5 ab fraction to a mixed quantity. Thus, the fraction- бе may be reduced to a mixed number by dividing the numerator by the denominator; the numerator of the fractional part must be formed by that term which is not divisible without a remainder; the following is therefore the required mixed 5 ab quantity: 2 a + A fraction is reduced to its lowest 6 e terms, in algebra as in arithmetic, by dividing the numerator and denominator by any quantity capable of dividing them both without leaving a remainder. Thus, in the fraction 10 a 20 ab + 5 a2 it is evident that the coefficient of every 35 a2 term can be divided by 5, and as the letter a enters into every term, 5 a may be called the greatest common measure of this fraction, because it can divide both the numerator and the denominator. The numerator, (10 a3 + 20 ab + 5 a2) ÷ 5a= 2 a2 +46 + a; and the denominator, 35 a2 ÷ 5 a 7 a; 2 a2 + 4 b + a hence the fraction, in its lowest terms, is 7 a Sometimes the greatest common measure of two quantities is not so obvious as in the example just adduced, in which case recourse must be had to the following operation :-The quantity, the exponent of whose leading letter in the first term is not less than that in the other, must first be divided by the other; the divisor must then be divided by the remainder; each successive remainder is made the divisor of the last divi sor, until nothing remains, when the divisor last used will be the greatest common measure. Quantities which have no common measure or divisor except 1, are called incommensurable; thus, 7, 5, 3, and II, are incommensurable quantities, and are also said to be prime to each other. When fractions are required either to be added or to be subtracted, they must necessarily be first reduced to a common denominator, which is effected by multiplying each numerator by every denominator but its own, to produce new numerators, and all the denominators together for the common denominator. The new numerators can then be either added or subtracted according as the case may require, and the new denominator must be left unchanged. Multiplication of fractions is performed by multiplying all the numerators together for a new numerator, and their denominators together for a new denominator; it is then usual to reduce the resulting fraction to its lowest terms. Division of fractions is effected by multiplying the dividend by the reci. procal of the divisor. The reciprocal of any quantity is unity, or I, divided by that quantity, or simply that quantity inverted : thus, the reciprocal of a oris and the reciprocal of a b a I I a is b a 8 a2 dend, 5 4 a 40 a2 ; therefore, X ; this last fraction, divi16 a ded by its greatest common measure, 8 a, is the fraction re quired, namely, The raising of a quantity to any required power is called involution, and is performed by multiplying the quantity into itself as often as it is indicated by the given power. When the quantity has no index, it is only necessary to place the given power above it, in order merely to indicate the power: thus, the 4th power of a is a', and the cube or 3d power of a + b is (a + b)3. When the quantity has an index, that index must be multi. plied by the given power; thus, the fourth power of a2 is a3, because 2 x 4 = 8. If the quantity required to be raised be a fraction, both the numerator and the denominator must be a1 multiplied by the given power: thus, the square of is d3 When the sign of the quantity is +, then all the powers to which it can be raised must be +; if, then all the even powers will be +, and all the odd powers - Thus xxx = x2; -a x − a = + a2; -ax-axα=- - a3. A compound quantity, that is, one consisting of more than |