the signs of equality and inequality, together with the letters of the alphabet, are the elements of the algebraic language. The interpretation of the language of Algebra is the first thing to which the attention of a pupil should be directed; and he should be drilled in the meaning and import of the symbols, until their significations and uses are as familiar as the sounds of the letters of the alphabet. All the apprehensions, or elementary ideas, are conveyed to the mind by means of definitions and arbitrary signs; and every judgment is the result of a comparison of such impressions. Hence, the connection between the symbols and the ideas which hey stand for, should be so close and intimate, that the one shall always suggest the other; and thus, the processes of Algebra become chains of thought, in which each link fulfils the double office of a distinct and connecting propos tion. ELEMENTS OF ALGEBRA. CHAPTER I. DEFINITIONS AND PRELIMINARY REMARKS. 1. QUANTITY is anything which can be increased or dimin ished, and measured. 2. MATHEMATICS is the science which treats of the measurement and relations of quantities. 3. ALGEBRA is a branch of mathematics, in which the quantities considered are represented by letters, and the operations to be performed upon them are indicated by signs. signs are called symbols. The letters and 4. In algebra two kinds of quantities are considered: 1st. Known quantities, or those whose values are known or given. These are represented by the leading letters of the alphabet, as, a, b, c, &c. 2d. Unknown quantities, or those whose values are not given. They are denoted by the final letters of the alphabet, as, x, y, z, &c. Letters employed to represent quantities are sometimes written with one or more dashes, as, a', b'', c''', x', y', &c., and are read, a prime, b second, c third, x prime, y second, &c. 5. The sign +, is called plus, and when placed between two quantities, indicates that the one on the right is to be added to the one on the left. Thus, ab is read a plus b, and indicates that the quantity represented by b is to be added to the quantity represented by a. 6. The sign, is called minus, and when placed between two quantities, indicates that the one on the right is to be subtracted from the one on the left. Thus, c d is read c minus d, and indicates that the quantity represented by d is to be subtracted from the quantity represented by c. The sign, is sometimes called the positive sign, and the quantity before which it is placed is said to be positive. The sign, is called the negative sign, and quantities affected by it are said to be negative. 7. The sign ×, is called the sign of multiplication, and when placed between two quantities, indicates that the one on the left is to be multiplied by the one on the right. Thus, a × b, indi cates that a is to be multiplied by b. The multiplication of quantities may also be indicated by placing a simple point between them, as a.b, which is read a multiplied by b. The multiplication of quantities, which are represented by letters, is generally indicated by simply writing the letters one after another, without interposing any sign. Thus, ab is the same as a xb, or a.b; and abc, the same as a b× c, or a.b.c. It is plain that the notation last explained cannot be employed when the quantities are represented by figures. For, if it were required to indicate that 5 was to be multiplied by 6, we could not write 5 6, without confounding the product with the number 56. The result of a multiplication is called the product, and cach of the quantities employed, is called a factor. In the product of several letters, each single letter is called a literal factor. Thus, in the product ab there are two literal factors a and b; in the product bed there are three, b, c and d. 8. The sign, is called the sign of division, and when placed between two quantities, indicates that the one on the left is to be divided by the one on the right. Thus, a - b indicates that a is to be divided by b. The same operation may be indicated by writing b under a, and drawing a line between them, as a b ; or by writing on the right of a, and drawing a line between them, as alb. 9. The sign, is called the sign of equality, and indicates that tne two quantities between which it is placed are equal to each other. Thus, a − b = c +d, indicates that a diminished by b is equal to c increased by d. 10. The sign >, is called the sign of inequality, and is used to indicate that one quantity is greater or less than another. Thus, a > b is read, a greater than b; and a <b is read, a less than b; that is, the opening of the sign is turned toward the greater quantity. 11. The sign ~ is sometimes employed to indicate the difference of two quantities when it is not known which is the greater. Thus, ab, indicates the difference between a and b, without showing which is to be subtracted from the other. 12. The sign ∞, is used to indicate that one quantity varies as 1 to another. Thus a ∞ b' 1 indicates that a varies as b 13. The signs: and ::, are called the signs of proportion; the first is read, is to, and the second is read, as. Thus, a: b :: cd, is read, a is to b, as c is to d. The sign, is read hence, or consequently. 14. If a quantity is taken several times, as a+a+a+a+a, it is generally written but once, and a number is then placed before it, to show how many times it is taken. Thus, a + a + a +a+a may be written 5a. The number 5 is called the co-efficient of a, and denotes that a is taken 5 times. Hence, a co-efficient is a number prefixed to a quantity, denoting the number of times which the quantity is taken. When no co-efficient is written, the co-efficient 1 is always under. stood; thus, a is the same as la. 15. If a quantity is taken several times as a factor, the product may be expressed by writing the quantity once, and placing a number to the right and above it, to show how many times it is taken as a factor. Thus, αχαχαχαχα may be written as. The number 5 is called an exponent, and indicates that a is taken 5 times as a factor. Hence, an exponent is a number written to the right and above a quantity, to show how many times it is taken as a factor. If no exponent is written, the exponent 1 is understood. Thus, a is the same as a1. 16. If a quantity be taken any number of times as a factor, the resulting product is called a power of that quantity: the exponent denotes the degree of the power. For example, a2 = a × a is the second power, or square of a, a2 = a × a × a xa is the fourth power of a, a5 = axaxaxaxa is the fifth power of a, in which the exponents of the powers are, 1, 2, 3, 4 and 5; and the powers themselves, are the results of the multiplications. It should be observed that the exponent of a power is always greater by one than the number of multiplications. The exponent of a power of a quantity is sometimes, for the sake of brevity, called the exponent of the quantity. 17. As an example of the use of the exponent in algebra, let it be required to express that a number a is to be multiplied three times by itself; that this product is then to be multiplied three times by b, and this new product twice by c; we should write a xaxax a x b x b x bx c x c = a4b3c2. If it were further required to take this result a certain number of times, say seven, we should simply write 7a4b3p* |