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FIRST LESSONS

ALGEBRA,

EMBRACING THE ELEMENTS

SCIENCE.

BY CHARLES DAVIES,

tUTHOR OF MENTAL AND PRACTICAL ARITHMETIC, ELEMENTS OF SURVEYING,

ELEMENTS OF DESCRIPTIVE AND ANALYTICAL GEOMETRY, ELEMENTS OF

DIFFERENTIAL AND INTEGRAL CALCULUS, AND A TREATISE ON

A. S. BARNES & Co, Hartford.— WILEY & PUTNAM; COLLINS,

KEESE & Co, New-York.— PERKINS & MARVIN, Boston.—

CUSHING & SONS, Baltimore.

1839.

X>3f

DAVIES' COURSE OF MATHEMATICS.

DA VIES' MENTAL and PRACTICAL ARITHMETIC— Designed
for the use of Academies and Schools. It is the purpose of this
work to explain, in a brief and clear manner, the properties of numbers,
and the best rules in their various applications.

DAVIES' KEY— To Mental and Practical Arithmetic.

DAVIES' FIRST LESSONS in ALGEBRA— Being an introduction
to the Science.

DAVIES' BOURDON'S ALGEBRA— Being an abridgment of the
work of M. Bourdon, with the addition of practical examples.

DAVIES' LEGENDRE'S GEOMETRY and TRIGONOMETRY
— Being an abridgment of the work of M. Legendre with the addition
of a treatise on Mensuration of Planes and Solids, and a table of
Logarithms and Logarithmic sines.

DAVIES' SURVEYING— With a description and plates of, the Theo-
dolite, Compass, Plane-Table and Level, — also, Maps of the Topo-
graphical Signs adopted by the Engineer Department, and an explana-
tion of the method of surveying the Public Lands.

DAVIES' ANALYTICAL GEOMETRY— Embracing the Equations
of the Point and Straight Line — of the Conic Sections — of the Line
and Plane in Space — also, the discussion of the General Equation
of the second degree, and of surfaces of the second order.

DAVIES' DESCRIPTIVE GEOMETRY— With its applications to
Spherical Projections.

DAVIES' DIFFERENTIAL AND INTEGRAL CALCULUS—

With numerous applications.

Entered according to the Act of Congress, in the year one thousand
eight hundred and thirty-eight, by Charles Davies, in the Clerk's
Office of the District Court of the United States, for the Southern
District of New York.

Stereotyped by Henry W. Rees,
32 Ann Street, New York.

PREFACE.

Although Algebra naturally follows Arithmetic in a
course of scientific studies, yet the change from num-
bers to a system of reasoning entirely conducted by
letters and signs is rather abrupt and not unfrequently
discourages and disgusts the pupil.

In the First Lessons it has been the intention to
form a connecting link between Arithmetic and Algebra,
to unite and blend, as far as possible, the reasoning on
numbers with the more abstruse method of analysis.

The Algebra of M. Bourdon has been closely fol-
lowed. Indeed, it has been a part of the plan, to furnish
an introduction to that admirable treatise, which is justly
considered, both in Europe and this country, as the best
work on the subject of which it treats, that has yet
appeared.

111869

4 PREFACE.

This work, however, even in its abridged form, is too
voluminous for schools, and the reasoning is too elaborate
and metaphysical for beginners.

It has been thought that a work which should so far
modify the system of Bourdon as to bring it within the
scope of our common schools, by giving to it a more
practical and tangible form, could not fail to be useful.
Such is the object of the First Lessons. It is hoped
they may advance the cause of education, and prove
a useful introduction to a full course of mathematical
studies.

Hartford, September , 1838.

CONTENTS.

CHAPTER I.

PRELIMINARY DEFINITIONS AND REMARKS.

ARTICLES!

Algebra — Definitions — Explanation of the Algebraic Signs, - 1 — 23

Similar Terms — Reduction of Similar Terms, - 23 — 26

Addition — Rule, - - -..- 26 — 28

Subtraction — Rule — Remark, - - - - - - 28 — 33

Multiplication — Rule for Monomials, - - - 33 — 36

Rule for Polynomials and Signs, - - - 36 — 38

Remarks — Properties Proved, - - - 38 — 42

Division of Monomials — Rule, - - - 42 — 45

Signification of the Symbol a% - ., - 45 — 46

Of the Signs in Divison, - - - - 46 — 47

Division of Polynomials, - - - 47 — 49

CHAPTER II.

ALGEBRAIC FRACTIONS.

Definitions — Entire Quantity — Mixed Quantity, • - 49—52

To Reduce a Fraction to its Simplest Terms - - . 52

To Reduce a Mixed Quantity to a Fraction, - - - 53

To Reduce a Fraction to an Entire or Mixed Quantity, - 54

To Reduce Fractions to a Common Denominator, - 55

To Add Fractions, - - 56

6

CONTENTS.

To Subtract Fractions,
To Multiply Fractions,
To Divide Fractions, -

ARTICLES.

57

58
59

CHAPTER III

EQUATIONS OP THE FIRST DEGREE.

Definition of an Equation — Properties of Equations,
Transformation of Equations — First and Second, -
Resolution of Equations of the First Degree — Rule,
Questions involving Equations of the First Degree,
Equations of the First Degree involving Two Unknown

Quantities, - - - - - - - • -

Elimination — By Addition — By Subtraction — By Comparison, -
Resolution of Questions involving Two or more Unknown

Quantities, - - - - -

60-

-66

66-

-70

70

71-

-72

72

73-

-76

76—79

CHAPTER IV.

OP POWERS.

Definition of Powers, - - - - 79

To raise Monomials to any Power, - - - 80

To raise Polynomials to any Power, - - - 81

To raise a Fraction to any Power, - - - 82 — 83

Binomial Theorem, - - - - 84 — 90

CHAPTER V.

Definition of Squares — Of Square Roots — And Perfect

Squares, - - - 90—96

Rule for Extracting the Square Root of Numbers, - - 96 — 100

Square Roots of Fractions, - - - 100 — 103

Square Roots of Monomials, 103 — 107

Calculus of Radicals of the Second Degree, - 107—109

CONTENTS. 7

ARTICLES.

Subtraction of Radicals, - - - - HO

Multiplication of Radicals, - - - - - - 111

Division of Radicals, - - - - - - - 112

Extraction of the Square Root of Polynomials, - 113 — 116

CHAPTER VI.

Equations of the Second Degree, - - - » - 116

Definition and Form of Equations, - 116 — 118

Incomplete Equations, - - - - H8 — 122

Complete Equations, - - - - 122

Four Forms, - 123—127

Resolution of Equations of the Second Degree, - - - 127 — 128

Properties of the Roots, - - 128—134

CHAPTER VII.

Of Progressions, - - - .- 135

Progressions by Differences, - - - 136 — 138

Last Term, 138—140

Sum of the Extremes — Sum of the Series, - - 140 — 141

The Five Numbers — To Find any Number of Means, - 141 — 144

Geometrical Proportion and Progression, - - - - 144

Various Kinds of Proportion, - - - 144 — 166

Geometrical Progression, - - - - 166

Last Term— Sum of the Series, 167— -171

Progressions having an Infinite Number of Terms, * - 171 — 172

The Five Numbers— To Find One Mean - - - - 172—173

FIRST LESSONS

IN

ALGEBRA

CHAPTER I.
Preliminary Definitions and Remarks.

1. Quantity is a general term embracing every thing
which can be increased or diminished. ,

2. Mathematics is the science of quantity.

3. Algebra is that branch of mathematics in which the
quantities considered are represented by letters, and the ope-
rations to be performed upon them are indicated by signs.
These letters and signs are called symbols.

4. The sign +, is called plus ; and indicates the addition
of two or more quantities. Thus, 9 + 5, is read, 9 plus 5,
or 9 augmented by 5.

If we represent the number nine, by the letter a, and
the number 5 by the letter b, we shall have a-\- b, which is
read, a plus b ; and denotes that the number represented by
a is to be added to the number represented by b.

5. The sign—, is called minus; and indicates that one

Quest. — 1. What is quantity 1 2. What is Mathematics 1 3. What
is Algebra 1 What are these letters and signs called 1 4. What does the
sign plus indicate 1 5. What does the sign minus indicate \

10 FIRST LESSONS IN ALGEBRA.

quantity is to be subtracted from another. Thus, 9— 5 is
read, 9 minus 5, or 9 diminished by 5.

In like manner, a—b, is read, a minus b, or a diminished
by b.

6. The sign x , is called the sign of multiplication; and
when placed between two quantities, it denotes that they
are to be multiplied together. The multiplication of two
quantities is also frequently indicated by simply placing a
point between them. Thus, 36 x 25, or 36.25, is read, 36
multiplied by 25, or the product 36 by 25.

7. The multiplication of quantities, which are represented
by letters, is indicated by simply writing them one after the
other, without interposing any sign.

Thus, ab signifies the same thing as a X b, or as a.b ;
and abc the same as axbxc, or as a.b.c. Thus, if we
suppose a = 36, and Z>z=25, we have

ab — 36x25 = 900.

Again, if we suppose c=2, b=3 and c=4, we have

abc=z2x3x4=24.

It is most convenient to arrange the letters of a product
in alphabetical order.

8. In the product of several letters, as abc, the single let-
ters, a, b, and c, are called factors of the product. Thus,
in the product ab, there are two factors, a and b ; in the
product abc, there are three, a, b, and c.

Quest. — 6. What is the sign of multiplication 1 What does the sign
of multiplication indicate 1 In how many ways may multiplication be
expressed 1 7. If letters only are used, how may their multiplication be
expressed] 8. In the product of several letters, what is each letter
called 1 How many factors in ab 1 — In abc ? — In abed 1 — In obcd.fi

DEFINITION OF TERMS. 11

9. There are three signs used to denote division. Thus,

a-^-b denotes that a is to be divided by b.
— denotes that a is to be divided by b.
a\b denotes that a is to be divided by b.

10. The sign =, is called the sign of equality, and is
read, is equal to. When placed between two quantities, it
denotes that they are equal to each other. Thus, 9 — 5=4 :
that is, 9 minus 5 is equal to 4 : Also, a-\-b — c, denotes that
the sum of the quantities a and b is equal to c.

If we suppose « = 10, and b=z5, we have

a-\-b = Cj and 10-f5:=c = 15.

1 1. The sign >, is called the sign of inequality, and is
used to express that one quantity is greater or less than
another.

Thus, a > b is read, a greater than b ; and a < b is
read, a less than b ; that is, the opening of the sign is turned
towards the greater quantity. Thus, if a=9, and b = 4, we
write, 9>4.

12. If a quantity is added to itself several times as
a+a+a+a+a+a, we generally write it but once, and
then place a number before it to show how many times it
is taken. Thus,

a-\-a-j-a-\-a-{-a=i5a.

Quest. — 9. How many signs are used in division'? What are they?
10. What is the sign equality"? When placed between two quantities,
what does it indicate'? 11. For what is the sign of inequality used]
Which quantity is placed on the side of the opening 1 12. What is a co-
efficient 1 How many times is ab taken in the expression ab 1 In dab 1
In 4:ab 1 In 5ab 1 In Gab 1 If no co-efficient is written, what co-efficient
is understood 1

12 FIRST LESSONS IN ALGEBRA.

The number 5 is called the co-efficient of a, and denotes
that a is taken 5 times.

When the co-efficient is 1 it is generally omitted. Thus,
a and \a are the same, each being equal to «, or to one a.

13. If a quantity be multiplied continually by itself, as
axaxaxaxa, we generally express the product by writing
the letter once, and placing a number to the right of, and a
little above it : thus,

aXaXaXaX a=a 5 .

The number 5 is called the exponent of a, and denotes
the number of times which a enters into the product as a
factor. For example, if we have a 3 , and suppose a = 3,
we write,

a 3 = axaxa=3 3 =:3x3x3 = 27.
If a=4, a 3 = 4 3 = 4x4x4 = 64,

and for a — 5, a 3 =5 3 = 5x5 x5 = 125.

If the exponent is 1 it is generally omitted. Thus, a 1 is
the same as a, each expressing a to the first power.

1 4. The power of a quantity is the product which results
from multiplying the quantity by itself. Thus, in the example

<z 3 : _4 3 = 4x4x4 : _64,

64 is the third power of 4, and the exponent 3 shows the
degree of the power.

15. The sign V , is called the radical sign, and when

Quest. — 13. What does the exponent of a letter show'? How many
times is a a factor in a2 ] In a* 1 In a* 1 In as 1 If no exponent is
written, what exponent is understood 1 14. What is the power of a
quantity 1 What is the third power of 2 1 Express the 4th power of a.
15. Express the square root of a quantity. Also the cube root. Also
the 4th root.

DEFINITION OP TERMS. 13

prefixed to a quantity, indicates that its root is to be ex-
tracted. Thus,

tyoTox simply -/tTdenotes the square root of a.

-y/o~denotes the cube root of a.

tfa denotes the fourth root of a.

The number placed over the radical sign is called the in-
dex of the root. Thus, 2 is the index of the square root, 3
of the cube root, 4 of the fourth root, &c.

If we suppose a = 64, we have

V64 = 8, ^6T=4.

16. Every quantity written in algebraic language, that
is, with the aid of letters and signs, is called an algebraic
quantity, or the algebraic expression of a quantity. Thus,

is the algebraic expression of three times
the number a ;
^ 2 i is the algebraic expression of five times
( the square of a ;

r is the algebraic expression of seven times

7a 3 b 2 ■? the product of the cube of a by the square

( of b;

o _ci ( is the algebraic expression of the difference

( between three times a and five times b ;

is the algebraic expression of twice the

2 hiAL2 square of «, diminished by three times

I the product of a by J, augmented by four

times the square of b.

1. Write three times the square of a multiplied by the
cube of b. Ans. 3a 2 b 3 .

Quest. — 16. What is an algebraic quantity 1 Is bah an algebraic
quantity 1 Is 9a \ Is 4y ! Is Zb — x l

2

3 i

la <

14 FIRST LESSONS IN ALGEBRA.

2. Write nine times the cube of a multiplied by b, dimin-
ished by the square of c multiplied by d. Ans. 9a 3 b—c 2 d.

3. If a=2, b = 3, and c = 5, what will be the value of
3a 2 multiplied by b 2 diminished by a multiplied by b multi-
plied by c. We have

3a*b 2 —abc=z3 X 2 2 X 3 2 -2 X 3 X 5=78.

4. If a=4, b = 6, c — 7, d=zS, what is the value of
9a 2 + 6c— adt Ans. 154.

5. If a=z7, b = 3, c=7, d=l, what is the value of
Qad+3b 2 c — 4d 2 1 Ans. 227.

6. If a — 5, b = 6, c=z6, d=:5, what is the value of

7. Write ten times the square of a into the cube of b into
c square into d 3 .

17. When an algebraic quantity is not connected with
any other, by the sign of addition or subtraction, it is called
a monomial, or a quantity composed of a single term, or sim-
ply, a term. Thus,

3a, 5a 2 , 7a 3 b 2 ,

are monomials, or single terms.

18. An algebraic expression composed of two or more
parts, separated by the sign + or — , is called a polynomial,
or quantity involving two or more terms. For example,

3a— 5b and 2a 2 —3cb+4b 2

are polynomials.

19. A polynomial composed of two terms, is called a bi-
nomial ; and a polynomial of three terms is called a trinomial.

Quest. — 17. What is a monomial 1 Is Sab a monomial 1 18. What
is a polynomial 1 Is 3a — b a polynomial] 19. What is a binomial?
What is a trinomial 1

DEFINITION OF TERMS. 15

20. Each of the literal factors which compose a term is
called a dimension of this term : and the degree of a term is
the number of these factors or dimensions. Thus,

* is a term of one dimension, or of the first

■x

is

second degree.

degree.

j is a term of two dimensions, or of the

3 , 2 _ 7 , ( is of six dimensions, or of the sixth
( degree.

2 1 . A polynomial is said to be homogeneous, when all
its terms are of the same degree. The polynomial

3a—2b+c is of the first degree and homogeneous.

— Aab+b 2 is of the second degree and homogeneous.

5a 2 c—4c 3 +2c 2 d is of the third degree and homogeneous.

8a 3 +4ab+c is not homogeneous.

22. A vinculum or bar , or a parenthesis ( ),

is used to express that all the terms of a polynomial are to
be considered together. Thus,

a+b-fixb, or (a+b+c)xb,

denotes that the trinomial a+b+c is to be multiplied by b ;

also, a+b+cxc+d+f, or (a+b+c) X (c+d+f),

denotes that the trinomial a+b+c is to be multiplied by
the trinomial c+d+f.

When the parenthesis is used, the sign of multiplication
is usually omitted. Thus,

(a+b+c) xb is the same as (a+b + c)b.

Quest. — 20. What is the dimension of a terml What is the degree
of a term 1 How many factors in 3abc 1 Which are they 1 What
is its degree 1 21. When is a polynomial homogeneous 1 Is the polyno-
mial 2a36+3a2&2 homogeneous'? Is 2a4& — fcl 22. For what is the
vinculum or bar used 1 Can you express the same with the parenthesis \

16 FIRST LESSONS IN ALGEBRA.

23. The terms of a polynomial which are composed of
the same letters, the same letters in each being affected
with like exponents, are called similar terms.

Thus, in the polynomial

lab + 3ab — 4 a 3 b 2 + 5a 3 b 2 ,

the terms lab, and 3ab, are similar : and so also are the
terms — 4a 3 b 2 and 5a 3 b 2 , the letters and exponents in both
being the same. But in the binomial 8a 2 b-\-7ab 2 , the
terms are not similar ; for, although they are composed of
the same letters, yet the same letters are not affected with
like exponents,

24. When an algebraic expression contains similar
terms, it may be reduced to a simpler form.

1. Take the expression 3ab + 2ab, which is evidently
equal to bob.

2. Reduce the expression 3ac+9ac+2ac to its simplest
form. Ans. \4ac.

3. Reduce the expression abc-\-4abc-{-5abc to its sim-
plest form.

In adding similar terms together we abc

take the sum of the coefficients and 4abc

annex the literal part. The first term, babe

abc, has a coefficient 1 understood, \0abc
(Art. 12).

25. Of the different terms which compose a polynomial,
some are preceded by the sign +» and the others by the
sign — , The first are called additive terms, the others,
subtractive terms.

Quest. — 23. What are similar terms of a polynomial? Are 3a2ja
and 6a2&2 similar 1 Are 2a2j2 and 2a.3& 2 ] 24. If the terms are positive
and similar, may they be reduced to a simpler form t In what way 1

DEFINITION OF TERMS. 17

The first term of a polynomial is commonly not preceded
by any sign, but then it is understood to be affected with the
sign +.

1. John has 20 apples and gives 5 to William: how-
many has he left ?

Now, let us represent the number of apples which John
has by a, and the number given away by b : the number he
would have left would then be represented by a— b.

2. A merchant goes into trade with, a certain sum of
money, say a dollars ; at the end of a certain time he has
gained b dollars : how much will he then have ?

Ans. a-\-b dollars.

would he have had 1 Ans. a — b dollars.

Now, if the losses exceed the amount with which he
began business, that is, if b were greater than a, we must
prefix the minus sign to the remainder to show that the
quantity to be subtracted was the greatest.

Thus, if he commenced business with \$2000, and lost
\$3000, the true difference would be —1000 : that is, the
subtractive quantity exceeds the additive by \$f000.

3. Let a merchant call the debts due him additive, and
the debts he owes subtractive. Now, if he has due him
\$600 from one man, \$800 from another, \$300 from another,
and owes \$500 to one, \$200 to a second, and \$50 to a
third, how will the account stand? Ans. \$950 due him.

4; Reduce to its simplest form the expression
3a 2 b + 5a 2 b — 3a 2 b + 4a 2 b —6a 2 b — a 2 b,

Quest. — 25. What are the terms called which are preceded by the
sign -f- ? What are the terms called which are preceded by the sign — .
If no sign is prefixed to a term, what sign is understood ? If some of the
terms are additive and some subtractive, may they be reduced if similar ?
Give the rule for reducing, them, . Does the reduction affect the expo-
nents, or only the coefficients ?

2*

18 FIRST LESSONS IN ALGEBRA.

+ 3a 2 b — 3a 2 b

+ 5a 2 b — 6a 2 b

+ 4a 2 5 — a 2 b

Sum + I2a 2 b Sum — lO^F.

But, 12a 2 b-l0a 2 b=2a 2 b.

Hence, for the reduction of the similar terms of a polyno-
mial we have the following

RULE.

I. Form a single additive term of all the terms preceded by
the sign plus ; this is done by adding together the coefficients
of those terms, and annexing to their sum the literal part.

II. Form, in the same manner, a single subtractive term.

III. Subtract the less sum from the greater, and prefix to
the result the sign of the greater.

Remark. — It should be observed that the reduction affects
only coefficients, and not the exponents.

• EXAMPLES.

1. Reduce to its simplest form the polynomial

-\-2a 3 bc 2 — 4a 3 bc 2 +6a 3 bc 2 — 8a 3 bc 2 + 11 a 3 bc 2 .

Find the sum of the additive and subtractive terms sepa-
rately, and take their difference : thus,

+ 2a 3 bc 2 — 4a 3 bc 2

+ 6a 3 bc 2 ' — Sa 3 bc 2

+ Ua 3 bc 2 Sum — I2a 3 bc 2
Sum + I9a 3 bc 2

Hence, the given polynomial reduces to
1 9a 3 bc 2 — 1 2a 3 bc 2 = 7a 3 bc 2 .

2. Reduce the polynomial 4a 2 b — 8a 2 b — 9a 2 b-\-lla 2 b to
its simplest form. Arts. —2a 2 b.

3. Reduce the polynomial 7abc 2 —abc 2 — 7abc 2 -\-8abc 2
+ 6abc 2 to its simplest form. Ans. I3abc 2 .

4. Reduce the polynomial 9cb 3 — 8ac 2 +l5cb 3 -\-8ca-\- 9ac 2
— 24cb 3 to its simplest form. Ans. ac 2 -\-8ca.

The reduction of similar terms is an operation peculiar to
algebra. Such reductions are constantly made in Algebraic

26. Addition in Algebra, consists in finding the simplest
equivalent expression for several algebraic quantities, con-
nected together by the sign plus or minus. Such equivalent
expression is called their sum.

1. What is the sum of

3ax-\-2ab and —2ax + ab. — £^^

3ax-{~2ab

We reduce the terms as in Art. 25, — 2ax-\- ab

and find for the sum ax-\-3ab

2. Let it be required to add together
the expressions :

3a
5b
2c

The result is 3a+5b+2c

an expression which cannot be reduced to a more simple
form.

Quest. — 26. What is addition in Algebra ? What is such simplest
and equivalent expression called 7

£0 FIRST LESSONS IN ALGEBRA.

s 4a 2 P

Again, add together the monomials ? 2a 2 b 3

( 7a 2 P

The result after reducing (Art. 25), is . . I3a 2 b 3

c 2a 2 —4ab
3. Let it be required to find the sum \ q2_qt > _la2

of the expressions

2ab — 5b 2

Their sum, after reducing (Art. 25) is . 5a 2 — Sab— 4b 2

27. As a course of reasoning similar to the above
would apply to all polynomials, we deduce for the addition
of algebraic quantities the following general

RULE.

I. Write down the quantities to be added so that the similar
terms shall fall under each other, and give to each term its
proper sign.

II. Reduce the similar terms, and annex to the results the
terms which cannot be reduced, giving to each term its respec-
tive sign.

EXAMPLES.

1. What is the sum of 3ax, 5ax, — 2ax and I3ax. ?

Ajis. \§ax.

2. What is the sum of 4ab-\-8ac and 2ab— 7ac+d. 1

Ans. &ab-\-ac-\-d.

3a 2 — 2b 2 —4ab, 5a 2 — b 2 -\-2ab, and 3ab— 3c 2 — 2b 2 .
The term 3a 2 being similar to
5a 2 , we write 8a 2 for the result
of the reduction of these two terms, <
at the same time slightly crossing
them, as in the first term.

Quest. — 27. Give the rule for the addition of Algebraic quantities.

3^2_4^__ 2 & 2
5# 2 -f-2^- b 2

+ 3\$b—2b 2 —3c 2
8a 2 -f- ab — bb 2 —3c 2

(?)

3ab

(8)
3ac

5ab
Sab

Sac
11 at

(11)
12a— 6c

— 3a— 9c