24. James and John together have 18 cents, and John has four fifths as many as James; how many has each?

25. Two places, A and C, are 40 miles apart; between them is a village which is two thirds as far from C as it is from A; what is its distance from each of the places?

26. The sum of two numbers is 21, and the smaller number 's three fourths of the larger; what are the numbers?

27. Thomas and Charles have 35 cents, and Charles has half as many more cents as Thomas; how many cents has each?

28. The double of a certain number, increased by one third of itself, is equal to 21; what is the number?

29. William, James, and Robert, together, have 33 cents; James has twice as many as William, and Robert has one third as many as James; how many cents has each?

30. What number is that, which being increased by its half and its fourth, equals 21?

31. What number is that, which being increased by its half, its fourth, and 4 more, equals 25?.

32. A boy, being asked how much money he had, replied, that if one half and one third of his money, and 9 cents more, were added to it, the sum would be 20 cents; how much money had he?

33. There are three numbers, whose sum is 44; the second is equal to one third of the first, and the third is equal to the second and twice the first; what are the numbers?

34. There are four towns in the order of the letters, A, B, C, and D; the distance from B to C is one fifth of the distance from A to B, and the distance from C to D is equal to twice the distance from A to C; the whole distance from A to D is 72 miles. Required the distance from A to B, from B to C, and from C to D. 35. What number is that, to which if its half, its fourth, and 26 more be added, the sum will be equal to 5 times the number?

36. There is a fish whose head is 6 inches long, and the tail is as long as the head and half the body, and the body is as long as the head and tail; what is the length of the whole fish?

37. A gentleman being asked his age, replied, "If to my age you add its half, its third, and 28 years, the sum will be equal to three times my age." Required his age.

The preceding exercises will serve to give the learner some idea of the nature of Algebra, and of the manner in which it may be applied to the solution of problems. We shall now proceed to consider the subject in a regular and scientific manner.

ELEMENTS OF ALGEBRA.

CHAPTER I.

PRELIMINARY DEFINITIONS AND PRINCIPLES.

NOTE TO TEACHERS.-In general, the Introduction, embracing ArLicles 1 to 1, need not be thoroughly studied until the pupil reviews the book.

ARTICLE 1. In Algebra, numbers and quantities are represented by symbols. These symbols are the letters of the alphabet. ART. 2. Quantity is anything that is capable of increase or decrease; such as numbers, lines, space, time, motion, &c.

ART. 3. Quantity is called magnitude, when presented or ccasidered in an undivided form, such as a quantity of water.

ART. 4. Quantity is called multitude, when it is made up of individual and distinct parts, such as three cents, which is a quantity composed of three single cents.

ART. 5. One of the single parts of which a quantity of multitude is composed, is called the unit of quantity, or measuring unit; thus, one cent is the measuring unit of the quantity three cents. The value or measure of every quantity, is the number of times it contains its measuring unit.

ART. 6. In quantities of magnitude, where there is no natural unit, it is necessary to fix upon an artificial unit, as a standard of measure; and then to find the value of the quantity, we must ascertain how often it contains its unit of measure. Thus, to measure the length of a line, we take a certain assumed distance called a foot, and applying it a certain number of times, say five, we ascertain that the line is five feet long; in this case, one foot is the unit of measure.

ART. 7. The numerical value of any quantity, is the number that expresses how many times it contains its unit of measure. Thus, in the preceding example, the line being 5 feet long, its numerical

REVIEW.-1. How are numbers and quantities represented in Algebra? What are symbols? 2. What is a quantity? 3. When is quantity called magnitude? 4. When is quantity called multitude? 5. What is the unit of quantity? 6. How is the value of a quantity ascertained, when there is no natural unit? 7. What is the numerical value of any quantity?

value is 5. The same quantity may have different numerical values, according to the unit of measure that is assumed.

ART. 8. A unit is a single or whole thing of an order or kind. ART. 9. Number is an expression denoting a unit, or a collection of units. Numbers are either abstract or concrete.

ART. 10. An abstract number denotes how many times a unit is to be taken. A concrete, or applicate number, denotes the units that are taken.

Thus, 4 feet is a concrete number; while 4 is an abstract number, which merely shows the number of units that are taken. A concrete number may be defined to be the product of the unit of measure by the corresponding abstract number. Thus, 6 dollars are equal to 1 dollar multiplied by 6, or 1 dollar taken 6 times. ART. 11. In Algebra, quantities are represented by numbers and letters; the letters used, stand for numbers.

ART. 12. There are two kinds of questions in Algebra, theorems and problems.

ART. 13. In a theorem, it is required to demonstrate some relation or property of numbers, or abstract quantities.

ART. 14. In a problem, it is required to find the value of some unknown number or quantity, by means of certain given relations existing between it and others, which are known.

ART, 15. Algebra is a general method of solving problems and demonstrating theorems, by means of figures, letters, and signs. The letters and signs are sometimes called symbols.

DEFINITION OF TERMS, AND EXPLANATION OF SIGNS. ART. 16. Known quantities are those whose numerical values are given, or supposed to be known: unknown quantities are those whose numerical values are not known.

ART. 17. Known quantities are generally represented by the first letters of the alphabet, as a, b, c, &c.; and unknown quantities by the last letters, as x, y, z.

ART. 18. The following are the principal signs used in Algebra: = +, −, X, ÷, ( ), >, √. Each of these signs is the representative of certain words; REVIEW.-8. What is a unit? 9. What is number? 10. What does an abstract number denote? What does a concrete number denote? 11. What do the letters used in Algebra represent? 12. How many kinds of questions are there in Algebra? What are they? 13. What is a theorem? 14. What is a problem? 15. What is Algebra? 16. What are known quantities? What are unknown quantities? 17. By what are known quantitics represented? By what are unknown quantities represented? 18. Write on a slate, or a blackboard, the principal signs used in Algebra. What do the signs represent? For what purpose are they used?

they are used for the purpose of expressing the various operations, in the most clear and brief manner.

ART. 19. The sign of equality, is read equal to. It denotes that the quantities between which it is placed are equal to each other. Thus, a=3, denotes that the quantity represented by a is equal to 3. ART. 20. The sign of addition, +, is read plus. It denotes that the quantity to which it is prefixed, is to be added to some other quantity.

If a 2 and

, is read minus. It denotes

Thus, a+b denotes that b is to be added to a. b=3, then a+b=2+3, which are =5. ART. 21. The sign of subtraction, that the quantity to which it is prefixed is to be subtracted. Thus, a-b denotes that b is to be subtracted from a. If a 5 and b=3, then 5-3-2.

ART. 23. Every quantity is supposed to be preceded by one or the other of these signs. Quantities having the positive sign are called positive; and those having the negative sign are called negative. When a quantity has no sign prefixed to it, it is considered positive. ART. 24. Quantities having the same sign are said to have like signs; those having different signs are said to have unlike signs. Thus, a and b, or a and b have like signs; while +c and -d have unlike signs.

ART. 25. The sign of multiplication, X, is read into, or multiplied by. It denotes that the quantities between which it is placed, are to be multiplied together.

A dot or point is sometimes used instead of the sign X. Thus, ab and a.b, both mean that b is to be multiplied by a. The dot is not used to denote the multiplication of figures, because it is used to separate whole numbers and decimals.

The product of two or more letters is generally denoted by writing them in close succession. Thus, ab denotes the same as aXb, or a.b; and abc means the same as aXbXc, or a.b.c.

REVIEW.-19. How is the sign of equality,, read? What does it denote? 20. How is the sign read? What does it denote? 21. How is the sign. -read? What does it denote? 22. What are the signs plus and minus called, by way of distinction? Which is positive, and which negative? 23. When quantities are preceded by the sign plus, what are they said to be? By the sign minus? When a quantity has no sign prefixed, what sign is understood? 24. When do quantities have like signs? When unlike signs? 25. How is the sign X read, and what does it denote? What other methods are there of representing multiplication, besides the sign X?

ART. 26. Quantities that are to be multiplied together, are called factors. The continued product of several factors, means that the product of the first and second is to be multiplied by the third, this product by the fourth, and so on. Thus, the continued product of a, b, and c, is expressed by aXbXc, or abc.

If a=2, b=3, and c=5, then abc=2×3×5=6X5=30.

ART. 27. The sign of division, ÷, is read divided by. It denotes that the quantity preceding it is to be divided by that following it. The division of two quantities is more frequently represented, by placing the dividend as the numerator, and the divisor as the denominator of a fraction. Thus, a÷b, or, means, that a is to be divided by b. If a=12 and b=3, then a÷÷b=12÷3=4; or

a

Division is also represented thus, alb, where a denotes the dividend, and b the divisor.

ART. 28. The sign >, is called the sign of inequality. It denotes that one of the two quantities between which it is placed, is greater than the other, the opening of the sign being turned towards the greater quantity.

Thus, ab denotes that a is greater than b. It is read, a greater than b. If a 5, and b=3, then 5>3.

Also, c<d denotes that c is less than d. It is read, c less than d. If c=4 and d=7, then 4<7.

ART. 29. The sign ∞, denotes a quantity greater than any that can be assigned; that is, a quantity indefinitely great, or infinity. ART. 30. The numeral coëfficient of a quantity is a number prefixed to it, to show how often the quantity is to be taken. Thus, if the quantity represented by a is to be added to itself several times, as a+a+a+a, we write it but once, and place a number before it, to show how often it is taken.

Thus, a+a+a+a=4a; and ax+ax+ax=3ax.

ART. 31. The literal coëfficient of a quantity, is a quantity by which it is multiplied. Thus, in the quantity ay, a may be considered the coefficient of y, or y may be considered the coefficient of a. The literal coefficient is generally regarded as a known quantity.

REVIEW.-26. What are factors? How many factors in a? In ab? In abc? In 5abc? 27. How is the sign read, and what does it denote? What other methods are there of representing the division of two quantities? 28. What does the sign of inequality, >, denote? Which quantity is placed at the opening? 29. What does the sign ∞ denote? 30. What is the numeral coefficient of a quantity? How often is ax taken in the expression 3ax? In 5ax? In 7ax? 31. What is the literal coöfficient of a quantity?