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IN MEMORIAM

FLORIAN CAJORI

-IN

1

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-â€¢<'

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r^V Ci

ECLECTIC EDUCATIONAL SLlIES.

(fcUmnttaru |.Igtka.

liAY^S ALGEBRA,

PART FIRST:

ON THE

ANALYTIC AND INDUCTIVE

METHODS OF INSTRUCTION:

WITH

NUMEROUS PRACTICAL EXERCISES

DESIGNED FOn

COMMON SCHOOLS AND ACADEMIES.

BY JOSEPH RAY, M. D.

PROFLSSOR OF MATHPIMATICS IN WOODWARD COLLEOÂ£.

REVISED EDITION.

VAN ANTWERP, BRAGG & CO.,

137 WALNUT STREET, 28 BOND STREET,

CINCINNATT. NEW YORK.

ECLECTIC EDUCATIONAL SERIES.

RAY'S MATHEMATICS.

EMBRACING

A Thorough, Pfogressive, and Complete Course in Arith^netic, Algebra^

and the Higher Mafhemalics.

Ray's Primary Arithmetic. Ray's Higher Arithmetic.

Ray's Intellectual Arithmetic. Key to Ray's Higher.

Ray's Practical Arithmetic. Ray's New Elementary Algebra.

Key to Ray's Arithmetics. Ray's New Higher Algebra.

Ray's Test Examples in Arith. Key to Ray's New Algebras.

Raifs Plane and Solid Geometry.

By Eli T. Tappan, A. M., Fres't Kenyon College. l2mo., clotli,

276 pp.

Raifs Geometry and Trigonometry.

By Eli T. Tappan, A. M. , Preset Kenyon College. 8vo. , sheep, 420 pp.

Ray's Analytic Geometry.

By Geo. H. Howison, A. M.,Prof. in Mass. Institute of Technology.

TreatLse on Analytic Geometry, especially as applied to the prop-

erties of Conies: including the Modern Methods of Abridged

Notation. 8vo., sheep, 574 pp.

Ray's Elements of Astronomy.

By S. H. Peabody, A. M., Prof, in the Chicago High School,

Handsomely and profusely illustrated. 8vo., sheep, 336 pp.

Ray's Surveying and Navigation.

With a Preliminary Treatise on Trigonometry and Mensuration.

By A. Schuyler, Prof, of Applied Mathematics and Logic in Bald-

win University. Svo., sheep, 403 pp.

Ray's differential and Integral Caleulus.

Elements of the Infinitesimal Calculus, with numerous examples

and api)lications to Analysis and Geometry, By J as. G. Clark, A.

M., /^jv/. in William Jewell College. 8vo., slieep, 440 pj).

Entered according taAct of Congress, in the ye.ir 1848, by Winthrop B. Smith,

in the Clerk's Office of the District Court of the United

States for the District of Ohio.

K2.L

PREFACE.

The object of the study of Mathematics, is two foldâ€” the acqui-

sition of useful knowledge, and the cultivation and discipline of

the mental powers. A parent often inquires, "Why should my

son study mathematics? I do not expect him to be a surveyor, an

engineer, or an astronomer." Yet, the parent is very desirous

that his son should be able to reason correctly, and to exercise,

in all his relations in life, the energies of a cultivated and disci-

plined mind. This is, indeed, of more value than the mere attain-

ment of any branch of knowledge.

The science of Algebra, properly taught, stands among the first

of those studies essential to both the great objects of education.

In a course of instruction properly arranged, it naturally follows

Arithmetic, and should be taught immediately after it.

In the following work, the object has been, to furnish an ele-

mentary treatise, commencing with the first principles, and leading

the pupil, by gradual and easy steps, to a knowledge of the ele-

ments of the science. The design has been, to present these in a

brief, clear, and scientific manner, so that the pupil should not be

taught merely to perform a certain routine of exercises mechani-

cally, but to understand the wJit/ and the wherefore of every step.

For this purpose, every rule is demonstrated, and every principle

analyzed, in order that the mind of the pupil may be disciplined

and strengthened so as to prepare him, either for pursuing the

study of Mathematics intelligently, or more successfully attending

to any pursuit in life.

Some teachers may object, that this work is too simple, and too

easily understood. A leading object has been, to make the pupil

feel, that he is not operating on unmeaning symbols, by means of

arbitrary rules ; that Algebra is both a rational and a practical

subject, and that he can rely upon his reasoning, and the results

3

^â™¦^â–ºriÂ« ikWO^-y

IV PREFACE.

of his operations, with the same confidence as in arithmetic. For

this purpose, he is furnished, at almost every step, with the means

of testing the accuracy of the principles on which the rules are

founded, and of the results which they produce.

Throughout the Avork, the aim has been, to combine the clear,

explanatory methods of the French mathematicians, with the prac-

tical exercises of the English and German, so that the pupil should

acquire both a practical and theoretical knowledge of the subject.

While every page is the result of the author's own reflection,

and the experience of many years in the school-room, it is also

proper to state, that a large number of the best treatises on the

same subject, both English and French, have been carefully con-

sulted, so that the present work might embrace the modern and

most approved methods of treating the various subjects presented.

With these remarks, the work is sul)mitted to the judgment of

fellow laborers in the field of education.

Woodward College, August, 1848.

SUGGESTIONS TO TEACHERS.

It is intended that the pupil shall recite the Intellectual Exercises with

the book open before him, as in mental Arithmetic. Advanced pupils may

omit these exercises.

The following subjects may be omitted by the younger pupils, and passed

over by those more advanced, until the book is revicAved.

Observations on Addition and Subtraction, Articles 60 â€” 64.

The greater part of Chapter 11.

Supplement to Equations of the First Degree, Articles 164 â€” 177.

Properties of the Roots of an Equation of the Second Degree, Articles

215â€”217.

In reviewing the book, the pupil should demonstrate the rules on the

blackboard.

The work will bo found to contain a large number of examples for prac-

tice. Should any instructor deem these too nixmerous, a portion of them

may bo omitted.

To teach the subject successfully, the principles must be first clearly

explained, and then the pupil exercised in the solution of appropriate

examples, until they are rendered perfectly familiar.

CONTENTS.

ARTICLES.

Intellectual Exercises, XIV Lessons,

CHAPTER Iâ€” FUNDAMENTAL RULES.

Preliminary Definitions and Principles 1 â€” 15

Definitions of Terms, and Explanation of Siijns 16â€” 52

Examples to illustrate the use of the Signs

Addition 53 â€” 55

Subtraction 56 â€” 59

Observations on Addition and Subtraction 60 â€” 64

Multiplicationâ€” Rule of the Coefficients 65 â€” 67

Rule of the Exponents 69

General Rule for the Signs 72

General Rule for Multiplication

Division of Monomialsâ€” Rule of the Signs 73 â€” 75

Polynomials â€” Rule 79

CHAPTER IIâ€” THEOREMS, FACTORING, &c.

Algebraic Theorems 80â€” 86

Factoring 87â€” 96

Greatest Common Divisor 97 â€” 106

Least Common Multiple 107â€”112

CHAPTER IIIâ€” ALGEBRAIC FRACTIONS.

Definitions and Fundamental Propositions 113 â€” 127

To reduce a Fraction to its Lowest Terms 123 â€” 129

a Fraction to an Entire or Mixed Quantity . . . 130

a Mixed Quantity to a Fraction ....... 131

Signs of Fractions 132

To reduce Fractions to a Common Denominator 133

the Least Common Denominator . . 134

To reduce a Quantity to a Fraction with a given Denominator 135

To convert a Fraction "^"^ another with a given Denominator . 136

Addition and Subtraction of Fractions 137 â€” 138

To multiply one Fractional Quantity by another 139 â€” 140

To divide one Fractional Quantity by another 141 â€” 142

To reduce a Complex Fraction to a Simple one .... 143

Resolution of Fractions into Series 144

CHAPTER IVâ€” EQUATIONS OF THE FIRST DEGREE.

Definitions and Elementary Principles 145 â€” 152

Transposition 153

To clear an Equation of Fractions 154

Equations of the First Degree, containing one Unknown Quan-

tity 155

Questions pro'lucing Equations of the First Degree, containing

one Unknown Quantity . â€¢ 156

Equations of the First Degree containing two Unknown Quau-

titiea 157

5

PAGES.

7â€” 24

25â€” 26

26â€” 31

31â€” 33

33- 39

39â€” 43

43â€” 46

47â€” 48

48â€” 50

51

53

54â€” 65

59â€” 63

63â€” 68

68â€”73

74â€” 80

80â€” 82

83â€” 87

87â€” 89

92-

95

97â€” 99

100-103

103â€”107

107â€”108

108â€”109

110â€”112

112-113

113â€”115

115-119

119â€”131

132

VI

CONTENTS

ARTICLES. PAGES.

Elimination â€” by Substitution 158 132

by Comparison 159 133

by Addition and Subtraction 160 134 â€” 136

Questions producing Equations containing two Unknown Quan-

tities IGl 136â€”142

Equations containing three or more Unknown Quantities . . 162 143 â€” 140

Questions producing Equations containing three or more Un-

known Quantities 163 147 â€” 150

CHAPTER Vâ€” SUPPLEMENT TO EQUATIONS OF THE FIRST DEGREE.

Generalization â€” Formation of Rules â€” Examples 164 â€” 170 150 â€” 158

Negative Solutions 172 159

Discus.sion of Problems 173 161

Problem of the Couriers 163 â€” 16:>

Cases of Indetermination and Impossible Problems .... 174 â€” 177 165 â€” 167

CHAPTER VIâ€” POWERSâ€” ROOTSâ€” RADICALS.

Involution or Formation of Powers 178 168

To raise a Monomial to any given Power 179 168

Polynomial to any given Power 181 170

Fraction to any Power 182 171

Binomial Theorem 183â€”186 171â€”176

Extraction of the Square Root 176

Square Root of Numbers 1S7 â€” 190 176 â€” 179

Fractions 101 179

Perfect and Imperfect Squares â€” Theorem 192 180

Approximate Square Roots 193â€”194 181â€”183

Square Root of Monomials 195 183 â€” 184

Polynomials 196 184^187

Radicals of the Second Degreeâ€” Definitions 198 187

Reduction 199 188

Addition 200 189

Subtraction 201 190

Multiplication 202 191

Division 203 192

To render Rational, the Denominator of a Fraction containing

Radicals 204 193

Simple Equations containing Radicals of the Second Degree . 205 195 â€” 197

CHAPTER VIIâ€” EQUATIONS OF THE SECOND DEGREE.

Definitions and Forms 206-208 197â€”193

Incomplete Equations of the Second Degree 209â€”210 198â€”200

Questions producing Incomplete Equations of the Second Degree, 211 200 â€” 201

Complete Equations of the Second Degree 212 202

General Rule for the Solution of Complete Equations of the Sec-

ond Degree 212 204â€”207

Hindoo Method of solving Equations of the Second Degree . 213 207

Questions producing Complete Equations of the Second Degree, 214 209 â€” 212

Properties of the Roots of a Complete Equation of the Second

Degree 215â€”218 213â€”217

E(iuations containing two Unknown Quantities 219 217 â€” 220

Questions protlucing Equations of the Second Degree, routain-

ing two Unknown Quantities ......... 219 220 â€” 222

CHAPTER VIIIâ€” PROGRESSIONS AND PROPORTION.

Arithmetical Progression 220 â€” 225 222 â€” 227

Geometrical Progression 22Câ€” 230 228â€”232

Ratio 231â€”239 232â€”234

Proportion 240â€”255 234â€”240

RAY'S

ALGEBRA

PART FIRST.

INTELLECTUAL EXERCISES.

LESSON I

Note to Teachers. â€” All the exercises in the following lessons can

be solved in the same manner as in intellectual arithmetic; yet the instruc-

tor should require the pupils to perform them after the manner here indi-

cated. In every question let the answer be verified.

1. I have 15 cents, which I wish to divide between William

and Daniel, in such a manner, that Daniel shall have twice as

many as William ; what number must I give to each ?

If I give William a certain number, and Daniel twice that num-

ber, both will have 3 times that certain number; but both together

are to have 15 cents ; hence, 3 times a certain number is 15.

Now, if 3 times a certain number is 15, one-third of 15, or 5,

must be the number. Hence, AV^illiam received 5 cents, and Dan-

iel twice 5, or 10 cents.

If, instead of a certain, number, we represent the number of cents

William is to receive, by x, then the number Daniel is to receive

will be represented by 2x, and what both receive will be repre

sented by x added to 2x, or 3x.

If 3a3 is equal to 15,

then la; or X is equal to 5.

The learner will see that the two methods of solving this ques-

tion are the same in principle : but that it is more convenient to

represent the quantity we wish to find, by a single letter, than by

one or more words.

In the same manner, let the learner continue to use the letter a:

to represent the smallest of the required numbers in the following

questions.

7

RAY'S ALGEBRA, PART FIRST.

Note. â€” x is read x, or one x, and is the same as \x. 2x is read two x,

or 2 times x. 3x is read three x, or 3 times x, and so on.

2. What number added to itself will make 12?

Let X represent the number ; then x added to x makes 2x, which

is equal to 12 ; hence if 2x is equal to 12, one ar, which is the half

of 2x, is equal to the half of 12, which is 6.

Verification. â€” 6 added to 6 makes 12.

3. What number added to itself will make 16?

If X represents the number, what will represent the number

added, to itself? What is 2x equal to? If 2x is equal to 16, what

is X equal to ?

4. What number added to itself will make 24 ?

5. Thomas and William each have the same number of apples,

and they both together have 20 ; how many apples has each ?

6. James is as old as John, and the sum of their ages is 22

years ; what is the age of each ?

7. Each of two men is to receive the same sum of money for a

job of work, and they both together receive 30 dollars ; what is

the share of each ?

8. Daniel had 18 cents ; after spending a part of them, he found

he had as many left as he had spent; how many cents had he spent?

9. A pole 30 feet high was broken by a blast of wind ; the part

broken off was equal to the part left standing ; what was the

length of each part ?

Instead of saying x added to x is equal to 30, it is more conven-

ient to say X 2)his X is equal to 30. To avoid writing the word

phis, we use the sign +, which means the same, and is called the

sign of addition. Also, instead of writing the word equal, we use

the sign ^=, which means the same, and is called the sign of

equality.

10. John, James, and Thomas, are each to have equal shares of

12 apples; if x represents John's share, what will represent the

share of James? What will represent the share of Thomas?

What expression will represent x-\-X'tx more briefly. If 3x^=12,

what is the value of x ? AVhy ?

11. The sum of four equal numl)ers is equal to 20 ; if a; repre-

sents one of the numbers, what will represent each of the others?

What will represent x-^x-\-x-rx, more briefly? If4a;=20, what

is X equal to ? Why ?

12. What is x-{-x equal to? Ans. 2x.

13. What is x-\-x^x equal to?

14. What is x-\-x-]rx-\-x equal to?

INTELLECTUAL EXERCISES.

LESSON II.

1. James and John topjether have 18 cents, and John has twice

as many as James ; how many cents has each ?

If a: represents the number of cents James has, what will repre-

sent the number John has ? What will represent the number they

both have? If 3a: is equal to 18, what is x equal to? Why?

Note. â€” If the pupil does not readily perceive how to solve a question,

let the instructor ask questions similar to the preceding.

2. A travels a certain distance one day, and twice as far the

next, in the two days he travels 36 miles ; how far does he travel

each day ?

3. The sum of the ages of Sarah and Jane is 15 years, and the

age of Jane is twice that of Sarah ; what is the age of each ?

4. The sum of two numbers is 16, and the larger is 3 times the

smaller ; w*hat are the numbers ?

5. What number added to 3 times itself will make 20 ?

6. James bought a lemon and an orange for 10 cents, the orange

cost four times as much as the lemon ; what was the price of each?

7. In a store-room containing 20 casks, the number of those

that are full is four times the number of those that are empty;

how many are there of each ?

8. In a flock containing 28 sheep, there is one black sheep for

each six w^hite sheep ; how many are there of each kind ?

9. Two pieces of iron together weigh 28 pounds, and the hea-

vier piece weighs three times as much as the lighter; what is the

weight of each ?

10. William and Thomas bought a foot-ball for 30 cents, and

Thomas paid twice as much as William ; w^hat did each pay?

1 1 . Divide 35 into two parts, such that one shall be four times

the other.

12. The sum of the ages of a father and son is equal to 35

years, and the age of the father is six times that of his son ; w^hat

is the age of each ?

13. There are two numbers, the larger of which is equal to nine

times the smaller, and their sum is 40 ; what are the numbers ?

14. The sum of tw^o numbers is 56, and the larger is equal to

seven times the smaller ; what are the numbers ?

15. What is x-{-2x equal to?

16. What is x-\-Sx equal to?

17. What is x-\-4x equal to?

10 RAY'S ALGEBRA, PART FIRST.

LESSON III.

1. Three boys are to share 24 apples between them ; the second

is to have twice as many as the first, and the third three times as

many as the first. If x represents the share of the first, what will

represent the share of the second? What will represent the share

of the third? What is the sum of x-{-2x-\-'Sx'i If Gx is equal

to 24, what is the value of x? What is the share of the second?

Of the third ?

Verification. â€” The first received 4, the second twice as

many, which is 8, and the third three times the first, or 12 ; and

4 added to 8 and 12, make 24, the whole number to be divided.

2. There are three numbers whose sum is 30, the second is

equal to twice the first, and the third is equal to three times the

first ; what are the numbers ?

3. There are three numbers whose sum is 21, the second is

equal to twice the first, and the third is equal to twice the second.

If X represents the first, what will represent the second ? If 2x

represents the second, what will represent the third ? What is

the sum of x-|-2x+4x? What are the numbers?

4. A man travels 63 miles in 3 days ; he travels twice as far

the second day as the first, and twice as far the third day as the

second ; how many miles does he travel each day ?

5. John had 40 chestnuts, of which he gave to his brother a

certain number, and to his sister twice as many as to his brother ;

after this he had as many left as he had given to his brother; how

many chestnuts did he give to each ?

6. A farmer bought a sheep, a cow, and a horse, for 60 dollars ;

the cow cost three times as much as the sheep, and the horse twice

as much as the cow ; what was the cost of each ?

7. James had 30 cents ; he lost a certain number ; after this

he gave away as many as he had lost, and then found that he had

three times as many remaining as he had given away ; how many

did he lose ?

8. The sum of three numbers is 36 ; the second is equal to

twice the first, and the third is equal to three times the second ;

what are the numbers?

9. John, James, and William together have 50 cents ; John has

twice as many as James, and James has three times as many as

AYilliam ; how many cents has each?

10. What is the sum of x, 2x, and three times 2x?

11. What is the sum of twice 2x, and three times 3a;?

INTELLECTUAL EXERCISES. 11

LESSON IV.

1 . If 1 lemon costs x cents, what will represent the cost of 2

lemons? Of 3 ? Of 4 ? Of 5? Of 6? Of 7?

2. If 1 lemon costs 2x cents, what will represent the cost of 2

lemons ? Of 3 ? Of 4 ? Of 5 ? Of 6 ?

3. James bought a certain number of lemons at 2 cents a piece,

and as many more at 3 cents a piece, all for 25 cents ; if x repre-

sents the number of lemons at 2 cents, what will represent their

cost ? What will represent the cost of the lemons at 3 cents a

piece ? How many lemons at each price did he buy ?

4. Mary bought lemons and oranges, of each an equal number ;

the lemons cost 2, and the oranges 3 cents a piece ; the cost of the

whole was 30 cents ; how many were there of each ?

5. Daniel bought an equal number of apples, lemons, and

oranges for 42 cents ; each apple cost 1 cent, each lemon 2 cents,

and each orange 3 cents ; how many of each did he buy ?

6. Thomas bought a number of oranges for 30 cents, one-half

of them at 2, and the other half at 3 cents each ; how many

oranges did he buy ? Let a;= one-half the number.

7. Two men are 40 miles apart ; if they travel toward each

other at the rate of 4 miles an hour each, in how many hours will

they meet?

8. Two men are 28 miles asunder ; if they travel toward each

other, the first at the rate of 3, and the second at the rate of 4

miles an hour, in how many hours will they meet?

9. Two men travel toward each other, at the same rate per

hour, from two places whose distance apart is 48 miles, and

they meet in six hours ; how many miles per hour does each

travel ?

10. Two men travel toward each other, the first going twice as

fast as the second, and they meet in 2 hours; the places are 18

miles apart; hoAv many miles per hour does each travel?

11. James bought a certain number of lemons, and twice as

many oranges, for 40 cents ; the lemons cost 2, and the oranges

3 cents a piece ; how many were there of each ?

12. Two men travel in opposite directions ; the first travels

three times as many miles per hour as the second ; at the end of

3 hours they are 36 miles apart ; hoAV many miles per hour does

each travel ?

13. A cistern, containing 100 gallons of water, has 2 pipes to

empty it ; the larger discharges four times as many gallons per

1^ KAY'S ALGEBRA, PART FIRST.

hour as the smaller, and they both empty it in 2 hours ; how many

gallons per hour does each discharge ?

14. A grocer sold 1 pound of coffee and 2 pounds of tea for 108

cents, and the price of a pound of tea was four times that of a

pound of coffee : what AA'^as.the price of each ?

If X represents the price of a pound of coffee, what will repre-

sent the price of a pound of tea ? What will represent the cost

of both the tea and coffee ?

15- A grocer sold 1 pound of tea, 2 pounds of coffee, and 3

pounds of sugar, for 65 cents ; the price of a pound of coffee was

twice that of a pound of sugar, and the price of a pound of tea

Avas three times that of a pound of coffee. Required the cost of

each of the articles.

If a: represents the price of a pound of sugar, what will repre-

sent the price of a pound of coffee ? Of a pound of tea ? What

will represent the cost of the whole ?

LESSON V.

1. James bought 2 apples and 3 peaches, for 16 cents; the price

of a peach was twice that of an apple ; what was the cost of each?

If X represents the cost of an apple, what will represent the

cost of a peach ? What will represent the cost of 2 apples ? Of

3 peaches ? Of both apples and peaches ?

2. There are two numbers, the larger of which is equal to twice

the smaller, and the sum of the larger and twice the smaller is

equal to 28 ; what are the numbers ?

3. Thomas bought 5 apples and 3 peaches for 22 cents ; each

peach cost twice as much as an apple ; what was the cost of each?

4. William bought 2 oranges and 5 lemons for 27 cents ; each

orange cost twice as much as a lemon ; what was the cost of

each?

5. James bought an equal number of apples and peaches for 21

cents ; the apples cost 1 cent, and the peaches 2 cents each ; how

many of each did he buy ?

6. Thomas bought an equal number of peaches, lemons, and

oranges, for 45 cents ; the peaches cost 2, the lemons 3, and the

oranges 4 cents a piece ; how many of each did he buy ?

7. Daniel bought twice as many apples as peaches for 24 cents;

each apple cost 2 cents, and each peach 4 cents ; how many of

each did he buy ?

INTELLECTUAL EXERCISES. 13

8. A farmer bought a horse, a cow, and a calf, for 70 dollars ;

the COAV cost three times as much as the calf, and the horse twice

as much as the cow ; what was the cost of each ?

9. Susan bought an apple, a lemon, and an orange, for 16 cents;

the lemon cost three times as much as the apple, and the orange

as much as both the apple and the lemon : what was the cost of

each?

10. Fanny bought an apple, a peach, and an orange, for 18

cents ; the peach cost twice as much as the apple, and the orange

twice as much as both the apple and the peach ; what was the

cost of each ?

LESSON VI.

1. James bought a lemon and an orange ; the orange cost twice

as much as the lemon, and the difference of their prices was 2

cents ; what was the cost of "each ?

If X represent the cost of the lemon, Avhat will represent the

cost of the orange ? What is 2x less x represented by ?

2. What is 3a: less x represented by ? What is 3a; less 2x repre-

^. G^^rl

IN MEMORIAM

FLORIAN CAJORI

-IN

1

:^^

-â€¢<'

:-*v

:|^

r^V Ci

ECLECTIC EDUCATIONAL SLlIES.

(fcUmnttaru |.Igtka.

liAY^S ALGEBRA,

PART FIRST:

ON THE

ANALYTIC AND INDUCTIVE

METHODS OF INSTRUCTION:

WITH

NUMEROUS PRACTICAL EXERCISES

DESIGNED FOn

COMMON SCHOOLS AND ACADEMIES.

BY JOSEPH RAY, M. D.

PROFLSSOR OF MATHPIMATICS IN WOODWARD COLLEOÂ£.

REVISED EDITION.

VAN ANTWERP, BRAGG & CO.,

137 WALNUT STREET, 28 BOND STREET,

CINCINNATT. NEW YORK.

ECLECTIC EDUCATIONAL SERIES.

RAY'S MATHEMATICS.

EMBRACING

A Thorough, Pfogressive, and Complete Course in Arith^netic, Algebra^

and the Higher Mafhemalics.

Ray's Primary Arithmetic. Ray's Higher Arithmetic.

Ray's Intellectual Arithmetic. Key to Ray's Higher.

Ray's Practical Arithmetic. Ray's New Elementary Algebra.

Key to Ray's Arithmetics. Ray's New Higher Algebra.

Ray's Test Examples in Arith. Key to Ray's New Algebras.

Raifs Plane and Solid Geometry.

By Eli T. Tappan, A. M., Fres't Kenyon College. l2mo., clotli,

276 pp.

Raifs Geometry and Trigonometry.

By Eli T. Tappan, A. M. , Preset Kenyon College. 8vo. , sheep, 420 pp.

Ray's Analytic Geometry.

By Geo. H. Howison, A. M.,Prof. in Mass. Institute of Technology.

TreatLse on Analytic Geometry, especially as applied to the prop-

erties of Conies: including the Modern Methods of Abridged

Notation. 8vo., sheep, 574 pp.

Ray's Elements of Astronomy.

By S. H. Peabody, A. M., Prof, in the Chicago High School,

Handsomely and profusely illustrated. 8vo., sheep, 336 pp.

Ray's Surveying and Navigation.

With a Preliminary Treatise on Trigonometry and Mensuration.

By A. Schuyler, Prof, of Applied Mathematics and Logic in Bald-

win University. Svo., sheep, 403 pp.

Ray's differential and Integral Caleulus.

Elements of the Infinitesimal Calculus, with numerous examples

and api)lications to Analysis and Geometry, By J as. G. Clark, A.

M., /^jv/. in William Jewell College. 8vo., slieep, 440 pj).

Entered according taAct of Congress, in the ye.ir 1848, by Winthrop B. Smith,

in the Clerk's Office of the District Court of the United

States for the District of Ohio.

K2.L

PREFACE.

The object of the study of Mathematics, is two foldâ€” the acqui-

sition of useful knowledge, and the cultivation and discipline of

the mental powers. A parent often inquires, "Why should my

son study mathematics? I do not expect him to be a surveyor, an

engineer, or an astronomer." Yet, the parent is very desirous

that his son should be able to reason correctly, and to exercise,

in all his relations in life, the energies of a cultivated and disci-

plined mind. This is, indeed, of more value than the mere attain-

ment of any branch of knowledge.

The science of Algebra, properly taught, stands among the first

of those studies essential to both the great objects of education.

In a course of instruction properly arranged, it naturally follows

Arithmetic, and should be taught immediately after it.

In the following work, the object has been, to furnish an ele-

mentary treatise, commencing with the first principles, and leading

the pupil, by gradual and easy steps, to a knowledge of the ele-

ments of the science. The design has been, to present these in a

brief, clear, and scientific manner, so that the pupil should not be

taught merely to perform a certain routine of exercises mechani-

cally, but to understand the wJit/ and the wherefore of every step.

For this purpose, every rule is demonstrated, and every principle

analyzed, in order that the mind of the pupil may be disciplined

and strengthened so as to prepare him, either for pursuing the

study of Mathematics intelligently, or more successfully attending

to any pursuit in life.

Some teachers may object, that this work is too simple, and too

easily understood. A leading object has been, to make the pupil

feel, that he is not operating on unmeaning symbols, by means of

arbitrary rules ; that Algebra is both a rational and a practical

subject, and that he can rely upon his reasoning, and the results

3

^â™¦^â–ºriÂ« ikWO^-y

IV PREFACE.

of his operations, with the same confidence as in arithmetic. For

this purpose, he is furnished, at almost every step, with the means

of testing the accuracy of the principles on which the rules are

founded, and of the results which they produce.

Throughout the Avork, the aim has been, to combine the clear,

explanatory methods of the French mathematicians, with the prac-

tical exercises of the English and German, so that the pupil should

acquire both a practical and theoretical knowledge of the subject.

While every page is the result of the author's own reflection,

and the experience of many years in the school-room, it is also

proper to state, that a large number of the best treatises on the

same subject, both English and French, have been carefully con-

sulted, so that the present work might embrace the modern and

most approved methods of treating the various subjects presented.

With these remarks, the work is sul)mitted to the judgment of

fellow laborers in the field of education.

Woodward College, August, 1848.

SUGGESTIONS TO TEACHERS.

It is intended that the pupil shall recite the Intellectual Exercises with

the book open before him, as in mental Arithmetic. Advanced pupils may

omit these exercises.

The following subjects may be omitted by the younger pupils, and passed

over by those more advanced, until the book is revicAved.

Observations on Addition and Subtraction, Articles 60 â€” 64.

The greater part of Chapter 11.

Supplement to Equations of the First Degree, Articles 164 â€” 177.

Properties of the Roots of an Equation of the Second Degree, Articles

215â€”217.

In reviewing the book, the pupil should demonstrate the rules on the

blackboard.

The work will bo found to contain a large number of examples for prac-

tice. Should any instructor deem these too nixmerous, a portion of them

may bo omitted.

To teach the subject successfully, the principles must be first clearly

explained, and then the pupil exercised in the solution of appropriate

examples, until they are rendered perfectly familiar.

CONTENTS.

ARTICLES.

Intellectual Exercises, XIV Lessons,

CHAPTER Iâ€” FUNDAMENTAL RULES.

Preliminary Definitions and Principles 1 â€” 15

Definitions of Terms, and Explanation of Siijns 16â€” 52

Examples to illustrate the use of the Signs

Addition 53 â€” 55

Subtraction 56 â€” 59

Observations on Addition and Subtraction 60 â€” 64

Multiplicationâ€” Rule of the Coefficients 65 â€” 67

Rule of the Exponents 69

General Rule for the Signs 72

General Rule for Multiplication

Division of Monomialsâ€” Rule of the Signs 73 â€” 75

Polynomials â€” Rule 79

CHAPTER IIâ€” THEOREMS, FACTORING, &c.

Algebraic Theorems 80â€” 86

Factoring 87â€” 96

Greatest Common Divisor 97 â€” 106

Least Common Multiple 107â€”112

CHAPTER IIIâ€” ALGEBRAIC FRACTIONS.

Definitions and Fundamental Propositions 113 â€” 127

To reduce a Fraction to its Lowest Terms 123 â€” 129

a Fraction to an Entire or Mixed Quantity . . . 130

a Mixed Quantity to a Fraction ....... 131

Signs of Fractions 132

To reduce Fractions to a Common Denominator 133

the Least Common Denominator . . 134

To reduce a Quantity to a Fraction with a given Denominator 135

To convert a Fraction "^"^ another with a given Denominator . 136

Addition and Subtraction of Fractions 137 â€” 138

To multiply one Fractional Quantity by another 139 â€” 140

To divide one Fractional Quantity by another 141 â€” 142

To reduce a Complex Fraction to a Simple one .... 143

Resolution of Fractions into Series 144

CHAPTER IVâ€” EQUATIONS OF THE FIRST DEGREE.

Definitions and Elementary Principles 145 â€” 152

Transposition 153

To clear an Equation of Fractions 154

Equations of the First Degree, containing one Unknown Quan-

tity 155

Questions pro'lucing Equations of the First Degree, containing

one Unknown Quantity . â€¢ 156

Equations of the First Degree containing two Unknown Quau-

titiea 157

5

PAGES.

7â€” 24

25â€” 26

26â€” 31

31â€” 33

33- 39

39â€” 43

43â€” 46

47â€” 48

48â€” 50

51

53

54â€” 65

59â€” 63

63â€” 68

68â€”73

74â€” 80

80â€” 82

83â€” 87

87â€” 89

92-

95

97â€” 99

100-103

103â€”107

107â€”108

108â€”109

110â€”112

112-113

113â€”115

115-119

119â€”131

132

VI

CONTENTS

ARTICLES. PAGES.

Elimination â€” by Substitution 158 132

by Comparison 159 133

by Addition and Subtraction 160 134 â€” 136

Questions producing Equations containing two Unknown Quan-

tities IGl 136â€”142

Equations containing three or more Unknown Quantities . . 162 143 â€” 140

Questions producing Equations containing three or more Un-

known Quantities 163 147 â€” 150

CHAPTER Vâ€” SUPPLEMENT TO EQUATIONS OF THE FIRST DEGREE.

Generalization â€” Formation of Rules â€” Examples 164 â€” 170 150 â€” 158

Negative Solutions 172 159

Discus.sion of Problems 173 161

Problem of the Couriers 163 â€” 16:>

Cases of Indetermination and Impossible Problems .... 174 â€” 177 165 â€” 167

CHAPTER VIâ€” POWERSâ€” ROOTSâ€” RADICALS.

Involution or Formation of Powers 178 168

To raise a Monomial to any given Power 179 168

Polynomial to any given Power 181 170

Fraction to any Power 182 171

Binomial Theorem 183â€”186 171â€”176

Extraction of the Square Root 176

Square Root of Numbers 1S7 â€” 190 176 â€” 179

Fractions 101 179

Perfect and Imperfect Squares â€” Theorem 192 180

Approximate Square Roots 193â€”194 181â€”183

Square Root of Monomials 195 183 â€” 184

Polynomials 196 184^187

Radicals of the Second Degreeâ€” Definitions 198 187

Reduction 199 188

Addition 200 189

Subtraction 201 190

Multiplication 202 191

Division 203 192

To render Rational, the Denominator of a Fraction containing

Radicals 204 193

Simple Equations containing Radicals of the Second Degree . 205 195 â€” 197

CHAPTER VIIâ€” EQUATIONS OF THE SECOND DEGREE.

Definitions and Forms 206-208 197â€”193

Incomplete Equations of the Second Degree 209â€”210 198â€”200

Questions producing Incomplete Equations of the Second Degree, 211 200 â€” 201

Complete Equations of the Second Degree 212 202

General Rule for the Solution of Complete Equations of the Sec-

ond Degree 212 204â€”207

Hindoo Method of solving Equations of the Second Degree . 213 207

Questions producing Complete Equations of the Second Degree, 214 209 â€” 212

Properties of the Roots of a Complete Equation of the Second

Degree 215â€”218 213â€”217

E(iuations containing two Unknown Quantities 219 217 â€” 220

Questions protlucing Equations of the Second Degree, routain-

ing two Unknown Quantities ......... 219 220 â€” 222

CHAPTER VIIIâ€” PROGRESSIONS AND PROPORTION.

Arithmetical Progression 220 â€” 225 222 â€” 227

Geometrical Progression 22Câ€” 230 228â€”232

Ratio 231â€”239 232â€”234

Proportion 240â€”255 234â€”240

RAY'S

ALGEBRA

PART FIRST.

INTELLECTUAL EXERCISES.

LESSON I

Note to Teachers. â€” All the exercises in the following lessons can

be solved in the same manner as in intellectual arithmetic; yet the instruc-

tor should require the pupils to perform them after the manner here indi-

cated. In every question let the answer be verified.

1. I have 15 cents, which I wish to divide between William

and Daniel, in such a manner, that Daniel shall have twice as

many as William ; what number must I give to each ?

If I give William a certain number, and Daniel twice that num-

ber, both will have 3 times that certain number; but both together

are to have 15 cents ; hence, 3 times a certain number is 15.

Now, if 3 times a certain number is 15, one-third of 15, or 5,

must be the number. Hence, AV^illiam received 5 cents, and Dan-

iel twice 5, or 10 cents.

If, instead of a certain, number, we represent the number of cents

William is to receive, by x, then the number Daniel is to receive

will be represented by 2x, and what both receive will be repre

sented by x added to 2x, or 3x.

If 3a3 is equal to 15,

then la; or X is equal to 5.

The learner will see that the two methods of solving this ques-

tion are the same in principle : but that it is more convenient to

represent the quantity we wish to find, by a single letter, than by

one or more words.

In the same manner, let the learner continue to use the letter a:

to represent the smallest of the required numbers in the following

questions.

7

RAY'S ALGEBRA, PART FIRST.

Note. â€” x is read x, or one x, and is the same as \x. 2x is read two x,

or 2 times x. 3x is read three x, or 3 times x, and so on.

2. What number added to itself will make 12?

Let X represent the number ; then x added to x makes 2x, which

is equal to 12 ; hence if 2x is equal to 12, one ar, which is the half

of 2x, is equal to the half of 12, which is 6.

Verification. â€” 6 added to 6 makes 12.

3. What number added to itself will make 16?

If X represents the number, what will represent the number

added, to itself? What is 2x equal to? If 2x is equal to 16, what

is X equal to ?

4. What number added to itself will make 24 ?

5. Thomas and William each have the same number of apples,

and they both together have 20 ; how many apples has each ?

6. James is as old as John, and the sum of their ages is 22

years ; what is the age of each ?

7. Each of two men is to receive the same sum of money for a

job of work, and they both together receive 30 dollars ; what is

the share of each ?

8. Daniel had 18 cents ; after spending a part of them, he found

he had as many left as he had spent; how many cents had he spent?

9. A pole 30 feet high was broken by a blast of wind ; the part

broken off was equal to the part left standing ; what was the

length of each part ?

Instead of saying x added to x is equal to 30, it is more conven-

ient to say X 2)his X is equal to 30. To avoid writing the word

phis, we use the sign +, which means the same, and is called the

sign of addition. Also, instead of writing the word equal, we use

the sign ^=, which means the same, and is called the sign of

equality.

10. John, James, and Thomas, are each to have equal shares of

12 apples; if x represents John's share, what will represent the

share of James? What will represent the share of Thomas?

What expression will represent x-\-X'tx more briefly. If 3x^=12,

what is the value of x ? AVhy ?

11. The sum of four equal numl)ers is equal to 20 ; if a; repre-

sents one of the numbers, what will represent each of the others?

What will represent x-^x-\-x-rx, more briefly? If4a;=20, what

is X equal to ? Why ?

12. What is x-{-x equal to? Ans. 2x.

13. What is x-\-x^x equal to?

14. What is x-\-x-]rx-\-x equal to?

INTELLECTUAL EXERCISES.

LESSON II.

1. James and John topjether have 18 cents, and John has twice

as many as James ; how many cents has each ?

If a: represents the number of cents James has, what will repre-

sent the number John has ? What will represent the number they

both have? If 3a: is equal to 18, what is x equal to? Why?

Note. â€” If the pupil does not readily perceive how to solve a question,

let the instructor ask questions similar to the preceding.

2. A travels a certain distance one day, and twice as far the

next, in the two days he travels 36 miles ; how far does he travel

each day ?

3. The sum of the ages of Sarah and Jane is 15 years, and the

age of Jane is twice that of Sarah ; what is the age of each ?

4. The sum of two numbers is 16, and the larger is 3 times the

smaller ; w*hat are the numbers ?

5. What number added to 3 times itself will make 20 ?

6. James bought a lemon and an orange for 10 cents, the orange

cost four times as much as the lemon ; what was the price of each?

7. In a store-room containing 20 casks, the number of those

that are full is four times the number of those that are empty;

how many are there of each ?

8. In a flock containing 28 sheep, there is one black sheep for

each six w^hite sheep ; how many are there of each kind ?

9. Two pieces of iron together weigh 28 pounds, and the hea-

vier piece weighs three times as much as the lighter; what is the

weight of each ?

10. William and Thomas bought a foot-ball for 30 cents, and

Thomas paid twice as much as William ; w^hat did each pay?

1 1 . Divide 35 into two parts, such that one shall be four times

the other.

12. The sum of the ages of a father and son is equal to 35

years, and the age of the father is six times that of his son ; w^hat

is the age of each ?

13. There are two numbers, the larger of which is equal to nine

times the smaller, and their sum is 40 ; what are the numbers ?

14. The sum of tw^o numbers is 56, and the larger is equal to

seven times the smaller ; what are the numbers ?

15. What is x-{-2x equal to?

16. What is x-\-Sx equal to?

17. What is x-\-4x equal to?

10 RAY'S ALGEBRA, PART FIRST.

LESSON III.

1. Three boys are to share 24 apples between them ; the second

is to have twice as many as the first, and the third three times as

many as the first. If x represents the share of the first, what will

represent the share of the second? What will represent the share

of the third? What is the sum of x-{-2x-\-'Sx'i If Gx is equal

to 24, what is the value of x? What is the share of the second?

Of the third ?

Verification. â€” The first received 4, the second twice as

many, which is 8, and the third three times the first, or 12 ; and

4 added to 8 and 12, make 24, the whole number to be divided.

2. There are three numbers whose sum is 30, the second is

equal to twice the first, and the third is equal to three times the

first ; what are the numbers ?

3. There are three numbers whose sum is 21, the second is

equal to twice the first, and the third is equal to twice the second.

If X represents the first, what will represent the second ? If 2x

represents the second, what will represent the third ? What is

the sum of x-|-2x+4x? What are the numbers?

4. A man travels 63 miles in 3 days ; he travels twice as far

the second day as the first, and twice as far the third day as the

second ; how many miles does he travel each day ?

5. John had 40 chestnuts, of which he gave to his brother a

certain number, and to his sister twice as many as to his brother ;

after this he had as many left as he had given to his brother; how

many chestnuts did he give to each ?

6. A farmer bought a sheep, a cow, and a horse, for 60 dollars ;

the cow cost three times as much as the sheep, and the horse twice

as much as the cow ; what was the cost of each ?

7. James had 30 cents ; he lost a certain number ; after this

he gave away as many as he had lost, and then found that he had

three times as many remaining as he had given away ; how many

did he lose ?

8. The sum of three numbers is 36 ; the second is equal to

twice the first, and the third is equal to three times the second ;

what are the numbers?

9. John, James, and William together have 50 cents ; John has

twice as many as James, and James has three times as many as

AYilliam ; how many cents has each?

10. What is the sum of x, 2x, and three times 2x?

11. What is the sum of twice 2x, and three times 3a;?

INTELLECTUAL EXERCISES. 11

LESSON IV.

1 . If 1 lemon costs x cents, what will represent the cost of 2

lemons? Of 3 ? Of 4 ? Of 5? Of 6? Of 7?

2. If 1 lemon costs 2x cents, what will represent the cost of 2

lemons ? Of 3 ? Of 4 ? Of 5 ? Of 6 ?

3. James bought a certain number of lemons at 2 cents a piece,

and as many more at 3 cents a piece, all for 25 cents ; if x repre-

sents the number of lemons at 2 cents, what will represent their

cost ? What will represent the cost of the lemons at 3 cents a

piece ? How many lemons at each price did he buy ?

4. Mary bought lemons and oranges, of each an equal number ;

the lemons cost 2, and the oranges 3 cents a piece ; the cost of the

whole was 30 cents ; how many were there of each ?

5. Daniel bought an equal number of apples, lemons, and

oranges for 42 cents ; each apple cost 1 cent, each lemon 2 cents,

and each orange 3 cents ; how many of each did he buy ?

6. Thomas bought a number of oranges for 30 cents, one-half

of them at 2, and the other half at 3 cents each ; how many

oranges did he buy ? Let a;= one-half the number.

7. Two men are 40 miles apart ; if they travel toward each

other at the rate of 4 miles an hour each, in how many hours will

they meet?

8. Two men are 28 miles asunder ; if they travel toward each

other, the first at the rate of 3, and the second at the rate of 4

miles an hour, in how many hours will they meet?

9. Two men travel toward each other, at the same rate per

hour, from two places whose distance apart is 48 miles, and

they meet in six hours ; how many miles per hour does each

travel ?

10. Two men travel toward each other, the first going twice as

fast as the second, and they meet in 2 hours; the places are 18

miles apart; hoAv many miles per hour does each travel?

11. James bought a certain number of lemons, and twice as

many oranges, for 40 cents ; the lemons cost 2, and the oranges

3 cents a piece ; how many were there of each ?

12. Two men travel in opposite directions ; the first travels

three times as many miles per hour as the second ; at the end of

3 hours they are 36 miles apart ; hoAV many miles per hour does

each travel ?

13. A cistern, containing 100 gallons of water, has 2 pipes to

empty it ; the larger discharges four times as many gallons per

1^ KAY'S ALGEBRA, PART FIRST.

hour as the smaller, and they both empty it in 2 hours ; how many

gallons per hour does each discharge ?

14. A grocer sold 1 pound of coffee and 2 pounds of tea for 108

cents, and the price of a pound of tea was four times that of a

pound of coffee : what AA'^as.the price of each ?

If X represents the price of a pound of coffee, what will repre-

sent the price of a pound of tea ? What will represent the cost

of both the tea and coffee ?

15- A grocer sold 1 pound of tea, 2 pounds of coffee, and 3

pounds of sugar, for 65 cents ; the price of a pound of coffee was

twice that of a pound of sugar, and the price of a pound of tea

Avas three times that of a pound of coffee. Required the cost of

each of the articles.

If a: represents the price of a pound of sugar, what will repre-

sent the price of a pound of coffee ? Of a pound of tea ? What

will represent the cost of the whole ?

LESSON V.

1. James bought 2 apples and 3 peaches, for 16 cents; the price

of a peach was twice that of an apple ; what was the cost of each?

If X represents the cost of an apple, what will represent the

cost of a peach ? What will represent the cost of 2 apples ? Of

3 peaches ? Of both apples and peaches ?

2. There are two numbers, the larger of which is equal to twice

the smaller, and the sum of the larger and twice the smaller is

equal to 28 ; what are the numbers ?

3. Thomas bought 5 apples and 3 peaches for 22 cents ; each

peach cost twice as much as an apple ; what was the cost of each?

4. William bought 2 oranges and 5 lemons for 27 cents ; each

orange cost twice as much as a lemon ; what was the cost of

each?

5. James bought an equal number of apples and peaches for 21

cents ; the apples cost 1 cent, and the peaches 2 cents each ; how

many of each did he buy ?

6. Thomas bought an equal number of peaches, lemons, and

oranges, for 45 cents ; the peaches cost 2, the lemons 3, and the

oranges 4 cents a piece ; how many of each did he buy ?

7. Daniel bought twice as many apples as peaches for 24 cents;

each apple cost 2 cents, and each peach 4 cents ; how many of

each did he buy ?

INTELLECTUAL EXERCISES. 13

8. A farmer bought a horse, a cow, and a calf, for 70 dollars ;

the COAV cost three times as much as the calf, and the horse twice

as much as the cow ; what was the cost of each ?

9. Susan bought an apple, a lemon, and an orange, for 16 cents;

the lemon cost three times as much as the apple, and the orange

as much as both the apple and the lemon : what was the cost of

each?

10. Fanny bought an apple, a peach, and an orange, for 18

cents ; the peach cost twice as much as the apple, and the orange

twice as much as both the apple and the peach ; what was the

cost of each ?

LESSON VI.

1. James bought a lemon and an orange ; the orange cost twice

as much as the lemon, and the difference of their prices was 2

cents ; what was the cost of "each ?

If X represent the cost of the lemon, Avhat will represent the

cost of the orange ? What is 2x less x represented by ?

2. What is 3a: less x represented by ? What is 3a; less 2x repre-