to be taken as a factor. Thus, in a', the number 5 indicates that a is to be taken 5 times as a factor; and the expression is equivalent to aaaaa.

A factor repeated to form a product is called a root; the product itself is called a power; and the figure which indicates how many times the root or factor is taken, is called the exponent of the power. Thus, in the indicated product ao, a is the root, a is the power, called the 5th power of a, and 5 is the exponent of this power. When no exponent is written over a quantity, the exponent 1 may always be understood.

NOTE. For the sake of brevity, the exponent of the power may be called the exponent of the letter or quantity over which it is placed. Thus in a3, 5 may be called the exponent of a.

14. The Sign of Evolution, or Radical Sign, is the character 1. It indicates that some root of the quantity after it is to be extracted. The name or index of the required root is the number written above the radical sign. Thus, Va denotes the cube root of a; Va denotes the 4th root of a; and so on. When no index is written over the sign, the index 2 is understood; thus, ✔a denotes the square root of a.

Fractional Exponents are also used as the sign of evolution, the denominator being the index of the required root. Thus, in a3, the denominator, 3, indicates that the cube root of a is required, and the expression is equivalent to Va.

Fractional exponents are used to denote both involution and evolution in the same expression, the numerator indicating the power to which the quantity is to be raised, and the denominator the required root of this power. Thus, the expression signifies the 4th root of the 3d power of a, and is equivalent to √a3.

SYMBOLS OF RELATION.

15. The Sign of Equality is two short horizontal lines, It indicates that the two quantities between which it is placed are equal. Thus, in a=b+c, the sign,, indicates that is equal to b plus c. An expression of equality between two quantities is called an equation.

16. The Sign of Inequality is the angle, >. It indicates that the quantities between which it is written are unequal, the opening being always turned toward the greater. When the opening is toward the left, it is read greater than; wher the point or vertex is toward the left, it is read less than. Thus, ab signifies that a is greater than b; x+y<z signifies that x plus y is less than z.

17. The Signs of Aggregation are the parenthesis, (), brackets, [], brace, {}, vinculum, ——— -, and bar, . They indicate that the quantities included within, or connected by them, are to be taken collectively and subjected to the same operation. Thus,

(a+b−c)x, [a+b—c]x, {a+b—c}x, a+b−c×x,

+a х

and +b

are expressions signifying that the whole quantity,

a+b-c, is to be multiplied by x. Two or more of these signs may be used correlatively in the same expression, in which case the brackets should include the parenthesis or vinculum, and the brace should include the brackets; thus,

{m—a[c—b(m+d)]+*}·

18. The Sign of Continuation is a succession of points, indicating that a series of quantities may be continued indefinitely according to the same law. Thus, in the expression, a+a+a3+a1+

the points indicate that the series has an infinite number of terms, all formed according to the same law.

19. The Sign of Ratio is two points like the colon, :, placed between the quantities compared. Thus, the expression, a : b, signifies the ratio of a to b.

20. The Sign of Proportion is a combination of the sign of ratio and the sign of cquality, := :; or a combination of points only, Thus, abc: d, signifies that the ratio of a to b is equal to the ratio of c to d; and the expression ab::c: d signifies the same, and may be read, a is to b as c is to d.

21. The Sign of Variation is the character ∞. It signifies that the two quantities between which it is placed, whether equal or unequal, increase or diminish together, so as to preserve constantly

the same ratio.

NOTE. The signs of ratio, proportion, and variation, will be more ful. ly explained hereafter.

COMPOSITION OF ALGEBRAIC QUANTITIES.

22. An algebraic quantity may consist of a single letter or element, or a combination of symbols as factors, or several combinations or parts. The parts are called terms; hence,

23. The Terms of an algebraic quantity are the parts or divisions made by the signs and Thus, in the quantity 5a+262-cx, there are three terms, of which 5a is the first, +262 the second, and cx the third.

24. When a quantity consists of a single term, it is said to be simple; when it is composed of two or more terms, it is said to be compound.

25. Positive Terms are those which have the plus sign; as +x, or +2cd. The first term of an algebraic quantity, if written without any sign, is positive, the plus sign being understood.

26. Negative Terms are those which have the minus sign; as -3a, or -2mx2. The sign of a negative quantity is never omitted.

27. A Coefficient is a number or quantity prefixed to another quantity, to denote how many times the latter is taken. Thus, in 3x, the number 3 is the coefficient of x, and indicates that x is taken 3 times; hence, the expression 3.r is equivalent to x+x+x. In 4ax, 4 may be regarded as the coefficient of ax, or 4a as the coefficient of x. In 5(a+x), 5 is the coefficient of a+x. When no coefficient is written, the unit 1 is understood.

28. It should be observed that in a term having the plus sign, the coefficient shows how many times the quantity is taken additively; and in a term having the minus sign, the coefficient shows how many times the quantity is taken subtractively. Thus,

+3a=+a+a+a

-3a--a-a-a

29. Similar Terms are terms containing the same letters, affected with the same exponents; the signs and coefficients may differ, and the terms still be similar. Thus, 3x2 and 7x' are similar terms; also, 2mď2 and —5mď are similar terms.

30. Dissimilar Terms are those which have different letters or exponents. Thus, axy and ayz are dissimilar; also 3x'y and 3x3y2. 31. A Monomial is an algebraic quantity consisting of only one term; as 3x, or -7xy.

32. A Polynomial is an algebraic quantity consisting of more than one term; as x+y, or 4a2—3x+m.

33. A Binomial is a polynomial of two terms; as a+b, or 3x-z.

34. A Residual is a binomial, the two terms of which are connected by the minus sign; as a-b, or 4x-3y.

35. A Trinomial is a polynomial of three terms; as x+y+%, or 7a-362+d.

36. The Degree of a term is the number of its literal factors. Since the exponents show how many times the different letters are taken as factors, the degree of a term is always found by adding the exponents of all the letters. Thus, x and 5y are terms of the first degree; a2 and 4ab are terms of the second degree; x3, 3x2y, 3xy, and 4xyz are terms of the third degree.

37. A Homogeneous Quantity is one whose terms are all of the same degree; as x3-5x3y+3xyz.

38. A Function of a quantity is any expression containing that quantity. Thus ax* is a function of x; 3y2+2y-4 is a func tion of y.

AXIOMS.

39. An Axiom is a self-evident truth. The following axioms underlie the principles of all algebraic operations:

1. If the same quantity or equal quantities be added to equal quantities, the sums will be equal.

2. If the same quantity or equal quantities be subtracted from equal quantities, the remainders will be equal.

3. If equal quantities be multiplied by the same, or equal quantities, the products will be equal.

4. If equal quantities be divided by the same, or equal quantities, the quotients will be equal.

5. If a quantity be both increased and diminished by another, its value will not be changed.

6. If a quantity be both multiplied and divided by another, its value will not be changed.

7. Quantities which are respectively equal to the same quantity, are equal to each other.

8. Like powers of equal quantities are equal.

9. Like roots of equal quantities are equal.

10. The whole of a quantity is greater than any of its parts. 11. The whole of a quantity is equal to the sum of all its parts.

EXERCISES IN ALGEBRAIC NOTATION.

40. In the examples which follow, it is required of the pupil simply to express given relations in algebraic language.

1. Give the algebraic expression for the square of a increased by by 4 times b. Ans. a2+4b.

y,

2. Give the algebraic expression for 7 times the product of x and diminished by 5 times the cube of z.

3. Indicate the quotient of 12 times the square of a minus 5 times the cube of b, divided by the sum of a and c.

4. If d represent a person's daily wages, what will represent his wages for 6 days? Ans. 6d.

5. An army drawn up in rectangular form, has 6 men in rank, and a men in file; of how many men is the army composed?

6. If a man labor m days in a week at c dollars per day, what will his earnings amount to in 7 weeks?

7. The length of a prism is a, the breadth a-c; required the solid contents.

c, and the altitude Ans. ac(a-c).

8. A has 4m dollars, B has m times as many dollars as A, and C has 3 times as many dollars as B wanting d dollars; how many dollars has C?

9. A dealer sells b sheep and c calves, at an average price of m dollars per head; how much does he receive for all?

10. A man has 3 square lots measuring m rods on a side; how inany acres in the 3 lots?

3m2

Ans.

160

11. From a rectangular piece of land whose length was a rods and whose width was b rods, there were sold c acres; how many acres remained unsold?

12. A ship laden with a barrels of flour, valued at m dollars per