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Joseph Ray.

Ray's algebra, part first : on the analytic and inductive methods of instruction : with numerous practical exercises designed for common schools and academics online

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IN MEMORIAM
FLORIAN CAJORI




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ECLECTIC EDUCATIONAL SLlIES.



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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 COLLEO£.



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 fold— 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



^♦^►ri« 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 — 64.

The greater part of Chapter 11.

Supplement to Equations of the First Degree, Articles 164 — 177.

Properties of the Roots of an Equation of the Second Degree, Articles
215—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 I— FUNDAMENTAL RULES.

Preliminary Definitions and Principles 1 — 15

Definitions of Terms, and Explanation of Siijns 16— 52

Examples to illustrate the use of the Signs

Addition 53 — 55

Subtraction 56 — 59

Observations on Addition and Subtraction 60 — 64

Multiplication— Rule of the Coefficients 65 — 67

Rule of the Exponents 69

General Rule for the Signs 72

General Rule for Multiplication

Division of Monomials— Rule of the Signs 73 — 75

Polynomials — Rule 79

CHAPTER II— THEOREMS, FACTORING, &c.

Algebraic Theorems 80— 86

Factoring 87— 96

Greatest Common Divisor 97 — 106

Least Common Multiple 107—112

CHAPTER III— ALGEBRAIC FRACTIONS.

Definitions and Fundamental Propositions 113 — 127

To reduce a Fraction to its Lowest Terms 123 — 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 — 138

To multiply one Fractional Quantity by another 139 — 140

To divide one Fractional Quantity by another 141 — 142

To reduce a Complex Fraction to a Simple one .... 143
Resolution of Fractions into Series 144

CHAPTER IV— EQUATIONS OF THE FIRST DEGREE.

Definitions and Elementary Principles 145 — 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 . • 156

Equations of the First Degree containing two Unknown Quau-

titiea 157

5



PAGES.


7— 24


25— 26


26— 31


31— 33


33- 39


39— 43


43— 46


47— 48


48— 50


51


53


54— 65


59— 63


63— 68


68—73


74— 80


80— 82


83— 87


87— 89



92-



95



97— 99
100-103
103—107
107—108
108—109



110—112
112-113
113—115

115-119

119—131

132



VI



CONTENTS



ARTICLES. PAGES.

Elimination — by Substitution 158 132

by Comparison 159 133

by Addition and Subtraction 160 134 — 136

Questions producing Equations containing two Unknown Quan-
tities IGl 136—142

Equations containing three or more Unknown Quantities . . 162 143 — 140

Questions producing Equations containing three or more Un-
known Quantities 163 147 — 150

CHAPTER V— SUPPLEMENT TO EQUATIONS OF THE FIRST DEGREE.

Generalization — Formation of Rules — Examples 164 — 170 150 — 158

Negative Solutions 172 159

Discus.sion of Problems 173 161

Problem of the Couriers 163 — 16:>

Cases of Indetermination and Impossible Problems .... 174 — 177 165 — 167

CHAPTER VI— POWERS— ROOTS— 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 183—186 171—176

Extraction of the Square Root 176

Square Root of Numbers 1S7 — 190 176 — 179

Fractions 101 179

Perfect and Imperfect Squares — Theorem 192 180

Approximate Square Roots 193—194 181—183

Square Root of Monomials 195 183 — 184

Polynomials 196 184^187

Radicals of the Second Degree— 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 — 197

CHAPTER VII— EQUATIONS OF THE SECOND DEGREE.

Definitions and Forms 206-208 197—193

Incomplete Equations of the Second Degree 209—210 198—200

Questions producing Incomplete Equations of the Second Degree, 211 200 — 201

Complete Equations of the Second Degree 212 202

General Rule for the Solution of Complete Equations of the Sec-
ond Degree 212 204—207

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

Questions producing Complete Equations of the Second Degree, 214 209 — 212
Properties of the Roots of a Complete Equation of the Second

Degree 215—218 213—217

E(iuations containing two Unknown Quantities 219 217 — 220

Questions protlucing Equations of the Second Degree, routain-

ing two Unknown Quantities ......... 219 220 — 222

CHAPTER VIII— PROGRESSIONS AND PROPORTION.

Arithmetical Progression 220 — 225 222 — 227

Geometrical Progression 22C— 230 228—232

Ratio 231—239 232—234

Proportion 240—255 234—240



RAY'S

ALGEBRA

PART FIRST.



INTELLECTUAL EXERCISES.



LESSON I



Note to Teachers. — 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. — 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. — 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. — 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. — 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-


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Online LibraryJoseph RayRay's algebra, part first : on the analytic and inductive methods of instruction : with numerous practical exercises designed for common schools and academics → online text (page 1 of 20)