Leonhard Euler. # An introduction to the elements of algebra, designed for the use of those who are acquainted only with the first principles of arithmetic. Selected from the algebra of Euler online

. **(page 1 of 16)**

Online Library → Leonhard Euler → An introduction to the elements of algebra, designed for the use of those who are acquainted only with the first principles of arithmetic. Selected from the algebra of Euler → online text (page 1 of 16)

');
}
else
{
document.write('');
}
//-->

Font size

/;.

^

€1^

/^■/^; j^-^^^l^-4-

/ /^ I

V^*^

b

''■i ,

Ui'^iku^

INTRODUCTION

ELEMENTS OF ALGEBRA.

^=^-^S^

AN

mtRODUCTION

ELEMENTS OF ALGEBRA,

DESIGNED FOB THE USE OF THOSE

I WHO ARE ACQUAINTED ONLY WITH THE FIRST PRINCIPLES

ARITHMETIC.

Selected from the Algebra of Euler.

CAMBRIDGE, N. ENG.

PRINTED BY HILLIARD AND METCAIF,

At the Unirersity Press.

SOI.D BT W.HILLIARD, CAMBRIDGE, AND BT CUMHIBTGS AND BILLIARD,

NO. 1 CORNHILL, BOSTON.

1818.

DISTRICT OF MASSACHUSETTS, TO WIT :

District Clerk's Office.

BE it remembered, that on the ninth day of February A. D. 1818, and in the forty second year

of the Independence of the United States of America, JOHN FARRAR of the said District has de-

lo^»^ng, to wit :

• An Introduction to the Elements of Algebra, designed for the use of those who are acquainted

only with the first pi-incipies of Arithmetic. Selected from the Algebra of Euler.'

In conformity to the act of the Congress of the United States, entitled, " An Act for the en-

couragement of learning, dy securing the copies of maps, charts and books, to the authors and

proprietors of such copies, during the times therein mentioned :" and also to an Act, entitled

" An Act supplementary to an Act, entitled An Act for the encouragement of learning, by

securing the copies of maps, charts and books, to the authors and proprietors of such copies

during the times therein mentioned : and extending the benefits thereof to the arts of designmg,

engraving and etching historicai and other prints."

Txin iir T\^\Trc S ^^^^ of the District

mo. W. DAVIS, ^ oj-j^asiachusetts.

A

^x.

ADVERTISEMENT.

None but those who are just entering upon

the study of Mathematics need to be informed

of the high character of Euler's Algebra. It

has been allowed to hold the very first place

among elementary works upon this subject.

The autlsor was a man of genius. He did not, like

most writers, compile from others. He wrote

from his own reflections. He simplified and im-

proved what was known, and added much that

was new. He is particularly distinguished for tlie

clearness and comprehensiveness of his views.

He seems to have the subject of which he treats

present to his mind in all its relations and

bearings before he begins to write. The parts

of it are arranged in the most admirable order.

Each step is introduced by the preceding, and

leads to that which follows, and the whole taken

together constitutes an entire and connected

piece, like a highly wrought story.

This author is remarkable also for his illus-

trations. He teaches by instances. He presents

one example after another, each evident by

vi Advertisement.

itself, and each throwing some new light npon

the subject, till the reader begins to anticipate

for himself the truth to be inculcated.

Some opinion may be formed of the adapta-

tion of this treatise to learners, from the cir-

cumstances under which it was composed. It

was undertaken after the author became blind,

and was dictated to a young man entirely with-

out education, who by this means became an

expert algebraist, and was able to render the

author important services as an amanuensis.

It was written originally in German. It has

since been translated into Russian, French, and

English, with notes and additions.

The entire work, consists of two volumes

octavo, and contains many things intended for

the professed matlicmatician, rather than the

general student. It was thought that a selec-

tion of such parts as would form an easy intro-

duction to the science would be well received,

and tend to promote a taste for analysis among

the higher class of students, and to raise the

character of mathematical learning.

Notwithstanding the high estimation in which

this work has been held, it is scarcely to be met

with in the country, and is very little known in

England. On the continent of Europe this

author is the constant theme of eulogy. His

writings have the character of classics. They

are regarded at the same time as the most

Mvertisement. vU

profound and the most perspicuous, and as

affording the finest models of analysis. They

furnish the germs of the most approved ele-

mentary works on the different branches of this

science. The constant reply of one of the first

mathematicians* of France to those who con-

sulted him upon the best method of studying

mathematics was, " study Euler.^^ " It is need-

less," said he, ' to accumulate books ; true

lovers of mathematics will always read Euler ;

because in his writings every thing is clear,

distinct, and correct ; because they swarm with

excellent examples ; and because it is always

necessary to have recourse to the fountain

head."

The selections here offered are from the first

English edition. A few errors have been cor-

rected and a few alterations made in the

phraseology. In the original no questions were

left to be performed by the learner. A collec-

tion was made by the English translator and

subjoined at the end with references to the

sections to whicli they relate. These have been

mostly retained, and some new ones have been

added.

Although this work is intended particularly

for the algebraical student, it will be found to

contain a clear and full explanation of the fun-

damental principles of arithmetic ; vulgar frac-

• Lagrange.

\iu Mveriisement

tions, the doctrine of roots and powers, of the

different kinds of proportion and progression,

are treated in a manner, that can hardly fail to

interest the learner and make him acquainted

T\ ith the reason of those rules which he has so

frequent occasion to apply.

A more extended work on Algebra formed

after the same model is now in the press and will

soon be published, This will be followed by

other treatises upon the diflPerent branches of

pure mathematics.

JOHN FARRAR,

Professor of :Mathematics and Natural Philosophy in the

§ UniTersity at Cambridge.

Cambridge^ February, 1818.

CONTENTS,

SECTION I.

OP THE DIFFERENT METHODS OF CALCULATING SIMPLE QUANTITIES.

CHAPTER I.

€f Mathematics in general, 1

CHAPTER II.

Explanation of the Sign>s + Plus and — Minus. -3

CHAPTER III.

Of the Multiplication of Simple Quantities, 6

CHAPTER IV.

Of the nature of wJiole J^umhers or Integers^ with respect to

their Factors, 9

CHAPTER V.

Of the Division of Simple ^lantities, 1 1

CHAPTER VI.

Of the properties of Integers with respect to their Divisors, 14

CHAPTER VII.

Of Fractions in general, \7

CHAPTER VIII.

Of the properties of Fractions, 22

CHAPTER IX.

Of the Addition and Subtraction of Fractions, 24

CHAPTER X.

Of the Multiplication and Division of Fractions, 27

CHAPTER XI.

Of Square Mmhers, 31

CHAPTER XII.

Of Square Roots, and of Irrational JVambers residting from

them* 33

X (jontents.

CHAPTER XIII.

Of Impossible o?- Imaginary Quantities) whicJi arise from the

same source. 58

CHAPTER XIV.

Of Cubic lumbers. 41

CHAPTER XV.

Of Cube Roots, and of Irrational J\'^umbers resulting from them, 42

CHxVPTER XVI.

Of Powers in general, 4 5

CHAPTER XVII.

Of tJie calculation of Powers, 49

CHAPTER XVIII.

Of Roots with rdation to Powers in general, 51

CHAPTER XIX.

Of the Method nf representing Irrational JS*umbers by Fractional

Exponents, 5S

CHAPTER XX.

Cjf the different Methods of CalculatioUf and of their mutual

Connexion, 56

SECTION SECOND.

OF THE DIFFERENT METHODS OF CALCULATING COMPOUNp QUANTITIES.

CHAPTER I.

Of the Addition of Compound ^naniities, 59

CHAPTER il.

Of the Subtraction of Compound Quantities, 61

CHAPTER III.

Of the Multiplication of Compound Quantities, 62

CHAPTER IV.

Of the Division of Compound Quantities, 68

CHAPTER V.

Cff the Resolution of Fractions into hifinite Series, 72

CHAPTER VI.

Of the Squares of Compound Quantities, 81

CHAPTER VII.

Of the Extraction of Roots applied to Compound (Quantities. 84

Contents, xi

CHAPTER VIII.

(jftlie caleidation of Irrational Quantities, 88

CHAPTER IX.

Of Cubes, and of the Extraction of Cube Roots, 92

CHAPTER X.

Of the higher Powers of Compoimd Quantities, 94

CHAPTER XL

Of the Transposition of the Letters, an whicJf, the demonstra'

tration of the preceding Rule is founded. 10#

CHAPTER XII.

Of the expression of Irrational Powers by Infinite Series, 104

CHAPTER XIII.

Of tJie resolution of Mgative Powers, 107

SECTION THIRD.

OF RATIOS AND PROPORTIONS.

CHAPTER I.

Of Arithmetical Ratio, or of the difference between two JV^wm-

bers. 111

CHAPTER II.

Of Arithmetical Proportion, 1 is

CHAPTER III.

Of Arithmetical Progressions, 1 16

CHAPTER IV.

Of the Summation of Arithmetical Progressions* 120

CHAPTER V.

Of Geometrical Ratio, 124

CHAPTER VI.

Of the greatest Common Divisor of two given numbers, 126

CHAPTER VII.

Of Geometrical Proportions, 13©

CHAPTER VIII.

Observations on the Rules of Proportion and their utilittj, 133

CHAPTER IX.

Of Compound Relations, ' 13S

xii (Contents.

CHAPTER X.

Of Geometrical Progressions. 144

CHAPTER XI.

Of Infinite Decimal Fractions. 150

SFXTIOl^f FOURTH.

OF ALGEBRAIC EQUATIONS, AND OF THE RESOLUTION OF THOSE EQUATIONS.

CHAPTER I.

Of the Solution of Problems in general. 15S

CHAPTER II.

Of the Resolution of Simple Mquatiatis, or Equations of the

first degree, 159

CHAPTER III,

Of the Solution of Questions relating to the preceding chapter. 163

CHAPTER IV.

Of the Resolution of two or more Equations of the First Degree, 1 73

CHAPTER V.

Of the Resolutim (f Pure Quadratic Equations. 182

CHAPTER VI.

Of the Resolution ofMixt Equations of the Second Degree. 18S

CHAPTER VII.

Of the JVature of Equations of the Second Degree. 196

Questions for Practice^ 202

A'^otes. -214

INTRODUCTION

ELEMENTS OF ALGEBRA.

SECTION I.

OF THE DIFFERENT METHODS OF CALCULATING SIMPLE QUANTITIES.

CHAPTER I.

OJ Mathematics in generaU

ARTICIE I.

Whatever is capable of increase, or diminution, is called

magnitude or quantity.

A sum of money therefore is a quantity, since we may in-

crease it and diminish it. It is the same with a weight, and

other things of this nature.

2. From this definition, it is evident, that the different kinds

of magnitude must be so many as to render it difficult even to

enumerate them : and this is tlie origin of tlie different branches

of the Mathematics, each being employed on a particular kind

of magnitude. Mathematics, in general, is the science of quan-

tity ; or the science whicli investigates the means of measuring

quantity.

3. Now we cannot measure or determine any quantity,

except by considering some other quantity of the same kind

as known, and pointing out their mutual relation. If it were

proposed, for example, to determine the quantity of a sum of

money, we should take some known piece of money (as a louis,

a crown, a ducat, or some other coin,) and shew how many of

I

g Algebra. Sect. 1.

these pieces are contained in the given sum. In the same man-

ner, if it were proposed to determine the quantity of a weight,

w^e should take a certain known weight ; for example, a pound,

an ounce, &c. and then shew how many times one of these

weights is contained in that which we are endeavouring to

ascertain. If we wished to measure any length or extension,

•we should make use of some known length, such as a foot.

4. So that the determination, or the measure of magnitude of

all kinds, is reduced to this : fix at pleasure upon any one known

magnitude of the same species with that which is to be deter-

mined, and consider it as the measure or unit ; then, determine

the proportion of the proposed magnitude to this known mea-

sure. This proportion is always expressed by numbers; so

that a number is nothing but the proportion of one magnitude

to another arbitrarily assumed as the unit.

5. From this it appears, that all magnitudes may be expressed

by numbers ; and that the foundation of all the mathematical

sciences must be laid in a complete treatise on the science of

numbers, and in an accurate examination of the different pos-

sible methods of calculation.

This fundamental part of mathematics is called Analysis, or

Algebra, [l.p

6. In Algebra then we consider only numbers which repre-

sent quantities, without regarding the different kinds of quantity.

These are the subjects of other branches of the mathematics.

7. Arithmetic treats of numbers in particular, and is the

science of numbers properly so called ; but this science extends

only to certain methods of calculation which occur in common

practice ; Algebra, on the contrary, comprehends in general

all the cases which can exist in the doctrine, and calculation of

numbers.

* The numbers thus included in crotchets refer to notes at the

end of this introduction.

Chap. 2. Of Simple Quantities. ^

CHAPTER II.

Explanation of the Signs + Plus and — Minus.

8. When we have to add one given number to another, this

is indicated by the sign + which is placed before the second

number, and is read plus, Tiius 5 -f 3 signifies that we must

add 3 to the number 5, and every one knows that the result is

8 ; in the same manner 12 + 7 make 19 ; 25 -f 16 make 41 ; the

sum of 25 +41 is 66, &c.

9. We also make use of the same sign + plus, to connect

several numbers together ; for example, r + 5 + 9 signifies that

to the number 7 we must add 5 and also 9, which make 21.

The reader will therefore understand what is meant by

8 + 5 + 13 + 11+1 +3 + 10;

vi». the sum of all these numbers, which is 51.

10. All this is evident ; and we have only to mention, that,

in Algebra, in order to generalize numbers, we represent them

by letters, as a, &, c, d, &c. Thus, the expression a + 6, signifies

the sum of two numbers, which we express by a and 6. and

these numbers may be eitlier very great or very small. In the

same manner, / + w + & + a?, signifies the sum of the numbers

represented by these four letters.

If we know therefore the numbers that are represented by

letters, we shall at all times be able to find by arithmetic, the

sum or value of similar expressions.

11. When it is required, on the contrary, to subtract one

given number from another, this operation is denoted by the

sign — , wliich signifies miiius, and is placed before the number

to be subtracted : thus 8 — 5 signifies that the number 5 is to be

taken from the number 8 ; which being done, there remains 3.

In like manner 12 — 7 is the same as 5 ; and 20 — 14 is the same

as 6, &c.

12. Sometimes also we may have several numbers to subtract

from a single one ; as, for instance, 50 — 1 — 3 — 5—7—9. This

signifies, first, take 1 from 50, there remains 49 ; take 3 from

that remainder, there will remain 46 ; take away 5, 41 remains ;

take away 7, 34 remains ; lastly, from that take 9, and there

4 Mgebra. Sect. I.

remains 25 ; this last remainder is the value of the expression.

But as the numbers 1, 3, 5, 7, 9, are all to be subtracted, it is

the same thing if we subtract their sum, which is 25, at once

from 50, and the remainder will be 25 as before.

IS. It is also very easy to determine the value of similar

expressions, in which both the signs + phis and — minus are

found : for example ;

12 — 3 — 5 -f- 2 — 1 is the same as 5.

We have only to collect separately the sum of the numbers that

have the sign -\- before them, and subtract from it the sum of

those that have the sign — . The sum of 12 and 2 is 14 ,• that

of 3, 5 and 1, is 9 ; now 9 being taken from 14, there remains 5.

14. It will be perceived from these examples that the order

in which we write the numhers is quite indifferent and arbitrary^

provided the proper sign of each be preserved, AYe might with

equal propriety have arranged the expression in the preceding

article thus ; 12-f2 — 5 — 3 — l,or2 — 1 — 3 — 5-fl2, or2 +

12 — 3 — 1 — 5, or in still different orders. It must be observed,

that in the expression proposed, the sign -|- is supposed to be

placed before the number 12.

15. It will not be attended with any more difficulty, if, in

order to generalize these operations, we make use of letters

instead of real numbers. It is evident, for example, that

(J — 6 — c+d — e,

signifies that we have numbers expressed by a and <Z, and that

from these numbers, or from their sum, we must subtract the

numbers expressed by the letters &, c, e, which have before them

the sign — .

16. Hence it is absolutely necessary to consider what sign is

prefixed to each number : for in algebra, simple quantities are

numbers considered with regard to the signs which precede, or

affect them. Further, we call those positive quantities, before

which the sign -f is found ; and those are called negative quan-

titiesy which are affected with the sign — .

17. The manner in which we generally calculate a person's

property, is a good illustration of what bas just been said. We

denote what a man really possesses by positive numbers, using,

or understanding the sign -f 5 whereas his debts are represent-

Chap. 2. Of Simple Quantities. 5

ed by negative numbers, or by using the sign — . Thus, when

it is said of any one that he has 100 crowns, but owes 50, this

means that his property really amounts to 100 — 50 ; or, which

is the same thing, -f 100 — 50, that is to say 50.

18. As negative numbers may be considered as debts, because

positive numbers represent real possessions, we may say that

negative numbers are less than nothing. Thus, when a man

has nothing in the world, and even owes 50 crowns, it is certain

that he has 50 crowns less than nothing ; for if any one were to

make him a present of 50 crowns to pay his debts, he would

still be only at the point nothing, though i^ally richer than

before.

19. In the same manner therefore as positive numbers are

incontestably greater than nothing, negative numbers are less

than nothing. Now we obtain positive numbers by adding 1 to

0, that is to say, to nothing ; and by continuing always to

increase thus from unity. This is the origin of the series of

numbers called natural numbers ; the following are the leading

terms of this series :

0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10,

and so on to infinity.

But if instead of continuing this series by successive additions,

we continued it in the opposite direction, by perpetually sub-

tracting unity, we should have the series of negative numbers :

0, — 1, — 2, — 3, ^ 4, — 5, — 6, — 7, — 8, -^ 9, — 10,

and so on to infinity.

20. All these numbers, whether positive or negative, have the

known appellation of whole numbers, or integers, which conse-

quently are eitlier greater or less than nothing. We call them

integers, to distinguish them from fractions, and from several

other kinds of numbers of which we shall hereafter speak. For

instance, 50 being greater by an entire unit than 49, it is easy

to comprehend that there may be between 49 and 50 an infinity

of intermediate numbers, all greater than 49, and yet all less

than 50. We need only imagine two lines, one 50 feet, tlie

other 49 feet long, and it is evident that there may be drawn an

infinite number of lines all longer than 49 feet, and yet shorter

than 50.

& Mgehra. Sect. 1,

21. It is of the utmost importance throu8;h the whole of

Algebra, that a precise idea be formed of those negative quanti-

ties about whicli we liave been speaking. I shall content my-

self with remarking here that all such expressions, as

+ 1 — 1, 4-2 — 2, +3 —3, +4,-4, &c.

are equal to or nothing. And that

+ 2 — 5 is equal to — 3 .

For if a person has 2 crowns, and owes 5, he has not only

nothing, but still owes S crowns : in the same manner

7 — 12 is equal to — 5, and 25 — 40 is equal to — 15.

22. The same observations hold true, when, to make the

expression more general, letters are used instead of numbers :

0, or nothing will always be the value of -f a — a. If we wish to

know the value -fa — h two cases are to be considered.

The first is when a is greater than h; h must then be sub-

tracted from a, and the remainder, (before winch is placed or

understood to be placed the sign -{-,) shews the value sought.

Tlie second case is that in which a is less than h : here a is

to be subtracted from 6, and the remainder being made negative,

by placing before it the sign — , will be the value sought.

CHAPTER III.

Of the Multiplication of Simple Quantities,

23. When there are two or more equal numbers to be added

together, the expression of their sum may be abridged ; for

example,

a + a is the same with 2 x a>

a-fa-f-a 3Xflj

a-fa-f-a-ffl Ax a, and so on ; where x is the sign

of multiplication. In this manner we may form an idea of mul-

tiplication ; and it is to be observed that,

2 X a signifies 2 times, or twice a

3 X ft 3 times, or thrice a

4 X a 4 times a, &c.

Chap. 3. Of Simple ^tantities. f

£4. If therefore a number expressed ly a letter is to he multiplied

by any other number, we simply put that riumber before the letter ;

thus,

a multipled by 20 is expressed by 20a, and

b multiplied by 30 gives 30&, &c.

It is evident also that c taken once, or Ic, is just c.

25. Further, it is extremely easy to multiply such products

again by other numbers ; for example :

2 times, or twice 3a makes 6a,

3 times, or thrice 4b makes 12&,

5 times 7x makes SScc,

and these products may be still multiplied by pther numbers at

pleasure.

26. When the number, by which we are to multiply, is also re-

presented by a letter, we place it immediately before the other letter ^

thus, in multiplying b by a, the product is written ab ; and pq^

will b6 the product of the multiplication of the number q by p>

If we multiply this pq again by a, we shall obtain apq,

27. It may be remarked here, that the order in which the letters

are joined together is indifferent ; that ab is the same thing as ba;

for b multiplied by a produces as much as a multiplied by &.

To understand this, we have only to substitute for a and 6

known numbers, as 3 and 4 ; and the truth will be self-evident i

for 3 times 4 is the same as 4 times 3.

28. It will not be difficult to perceive, that when you have to

put numbers, in the place of letters joined together, as we have

described, they cannot be written in the same manner by put-

ting them one after the other. For if we were to write 34 for

3 times 4, we should have 34 and not 12. When therefore it is

required to multiply common numbers, we must separate them

by the sign x, or points : thus, 3x4, or 3*4, signifies 3 times 4,

that is 12. So, 1 X 2 is equal to 2 ; and 1x2x3 makes 6. In

like manner Ix2x3x4x 56 makes 1344 ; and 1x2x3

X4x5x6x7x8x9xl0is equal to 362^800, &c.

29. In tlie same manner, we may discover the value of an

expression of this, form, 5*7*8' abed. It shews that 5 must be

multiplied by 7, and that this product is to be again multiplied

by 8 ; that we are then to multiply this product of the three

Jlhehra. Sect. 1*

numbers by a, next by 6, and then by c, and lastly by d. It may

be observed also, that instead of 5 x 7 X 8 we may write its value,

280; for we obtain this number when we multiply the product

of 5 by 7, or 35 by 8.

SO. The results which arise from the multiplication of two or

more numbers are called products ; and the numbers, or indivi-

dual letters, are called factors,

31. Hitherto we have considered only positive numbers, and

there can be no doubt, but that the products which we have

seen arise are positive also : viz. -}- a by + 6 must necessarily

give + ab. But we must separately examine what the multi-

plication of + a by . — ft, and of — a by — 6, will produce.

32. Let us begin by multiplying — a by 3 or -f- 3 ; now since

— a may be considered as a debt, it is evident that if we take

that debt three times, it must thus become three times greater,

and consequently the required produ'^t is — Sa. So if we multi-

ply — a by -f &, we shall obtain — ha^ or, which is the same things

— ah. Hence we conclude, that if a positive quantity be multi-

plied by a negative quantity, the product will be negative;

and lay it down as a rule, that 4- by + makes +, or pliis^ and

that on the contrary + by — , or — by + gives — , or minus,

33. It remains to resolve the case in which — is multiplied by

— ; or, for exami>le, — a by — h. It is evident^ at first sight,

with regard to the letters, that the product will be ah; but it is

^

€1^

/^■/^; j^-^^^l^-4-

/ /^ I

V^*^

b

''■i ,

Ui'^iku^

INTRODUCTION

ELEMENTS OF ALGEBRA.

^=^-^S^

AN

mtRODUCTION

ELEMENTS OF ALGEBRA,

DESIGNED FOB THE USE OF THOSE

I WHO ARE ACQUAINTED ONLY WITH THE FIRST PRINCIPLES

ARITHMETIC.

Selected from the Algebra of Euler.

CAMBRIDGE, N. ENG.

PRINTED BY HILLIARD AND METCAIF,

At the Unirersity Press.

SOI.D BT W.HILLIARD, CAMBRIDGE, AND BT CUMHIBTGS AND BILLIARD,

NO. 1 CORNHILL, BOSTON.

1818.

DISTRICT OF MASSACHUSETTS, TO WIT :

District Clerk's Office.

BE it remembered, that on the ninth day of February A. D. 1818, and in the forty second year

of the Independence of the United States of America, JOHN FARRAR of the said District has de-

');
}
else if (getClientWidth() > 430)
{
document.write('');
}
else
{
document.write('');
}
//-->

posited in tnis office the title of a book, the right whereof he claims as proprietor, in the words fol- lo^»^ng, to wit :

• An Introduction to the Elements of Algebra, designed for the use of those who are acquainted

only with the first pi-incipies of Arithmetic. Selected from the Algebra of Euler.'

In conformity to the act of the Congress of the United States, entitled, " An Act for the en-

couragement of learning, dy securing the copies of maps, charts and books, to the authors and

proprietors of such copies, during the times therein mentioned :" and also to an Act, entitled

" An Act supplementary to an Act, entitled An Act for the encouragement of learning, by

securing the copies of maps, charts and books, to the authors and proprietors of such copies

during the times therein mentioned : and extending the benefits thereof to the arts of designmg,

engraving and etching historicai and other prints."

Txin iir T\^\Trc S ^^^^ of the District

mo. W. DAVIS, ^ oj-j^asiachusetts.

A

^x.

ADVERTISEMENT.

None but those who are just entering upon

the study of Mathematics need to be informed

of the high character of Euler's Algebra. It

has been allowed to hold the very first place

among elementary works upon this subject.

The autlsor was a man of genius. He did not, like

most writers, compile from others. He wrote

from his own reflections. He simplified and im-

proved what was known, and added much that

was new. He is particularly distinguished for tlie

clearness and comprehensiveness of his views.

He seems to have the subject of which he treats

present to his mind in all its relations and

bearings before he begins to write. The parts

of it are arranged in the most admirable order.

Each step is introduced by the preceding, and

leads to that which follows, and the whole taken

together constitutes an entire and connected

piece, like a highly wrought story.

This author is remarkable also for his illus-

trations. He teaches by instances. He presents

one example after another, each evident by

vi Advertisement.

itself, and each throwing some new light npon

the subject, till the reader begins to anticipate

for himself the truth to be inculcated.

Some opinion may be formed of the adapta-

tion of this treatise to learners, from the cir-

cumstances under which it was composed. It

was undertaken after the author became blind,

and was dictated to a young man entirely with-

out education, who by this means became an

expert algebraist, and was able to render the

author important services as an amanuensis.

It was written originally in German. It has

since been translated into Russian, French, and

English, with notes and additions.

The entire work, consists of two volumes

octavo, and contains many things intended for

the professed matlicmatician, rather than the

general student. It was thought that a selec-

tion of such parts as would form an easy intro-

duction to the science would be well received,

and tend to promote a taste for analysis among

the higher class of students, and to raise the

character of mathematical learning.

Notwithstanding the high estimation in which

this work has been held, it is scarcely to be met

with in the country, and is very little known in

England. On the continent of Europe this

author is the constant theme of eulogy. His

writings have the character of classics. They

are regarded at the same time as the most

Mvertisement. vU

profound and the most perspicuous, and as

affording the finest models of analysis. They

furnish the germs of the most approved ele-

mentary works on the different branches of this

science. The constant reply of one of the first

mathematicians* of France to those who con-

sulted him upon the best method of studying

mathematics was, " study Euler.^^ " It is need-

less," said he, ' to accumulate books ; true

lovers of mathematics will always read Euler ;

because in his writings every thing is clear,

distinct, and correct ; because they swarm with

excellent examples ; and because it is always

necessary to have recourse to the fountain

head."

The selections here offered are from the first

English edition. A few errors have been cor-

rected and a few alterations made in the

phraseology. In the original no questions were

left to be performed by the learner. A collec-

tion was made by the English translator and

subjoined at the end with references to the

sections to whicli they relate. These have been

mostly retained, and some new ones have been

added.

Although this work is intended particularly

for the algebraical student, it will be found to

contain a clear and full explanation of the fun-

damental principles of arithmetic ; vulgar frac-

• Lagrange.

\iu Mveriisement

tions, the doctrine of roots and powers, of the

different kinds of proportion and progression,

are treated in a manner, that can hardly fail to

interest the learner and make him acquainted

T\ ith the reason of those rules which he has so

frequent occasion to apply.

A more extended work on Algebra formed

after the same model is now in the press and will

soon be published, This will be followed by

other treatises upon the diflPerent branches of

pure mathematics.

JOHN FARRAR,

Professor of :Mathematics and Natural Philosophy in the

§ UniTersity at Cambridge.

Cambridge^ February, 1818.

CONTENTS,

SECTION I.

OP THE DIFFERENT METHODS OF CALCULATING SIMPLE QUANTITIES.

CHAPTER I.

€f Mathematics in general, 1

CHAPTER II.

Explanation of the Sign>s + Plus and — Minus. -3

CHAPTER III.

Of the Multiplication of Simple Quantities, 6

CHAPTER IV.

Of the nature of wJiole J^umhers or Integers^ with respect to

their Factors, 9

CHAPTER V.

Of the Division of Simple ^lantities, 1 1

CHAPTER VI.

Of the properties of Integers with respect to their Divisors, 14

CHAPTER VII.

Of Fractions in general, \7

CHAPTER VIII.

Of the properties of Fractions, 22

CHAPTER IX.

Of the Addition and Subtraction of Fractions, 24

CHAPTER X.

Of the Multiplication and Division of Fractions, 27

CHAPTER XI.

Of Square Mmhers, 31

CHAPTER XII.

Of Square Roots, and of Irrational JVambers residting from

them* 33

X (jontents.

CHAPTER XIII.

Of Impossible o?- Imaginary Quantities) whicJi arise from the

same source. 58

CHAPTER XIV.

Of Cubic lumbers. 41

CHAPTER XV.

Of Cube Roots, and of Irrational J\'^umbers resulting from them, 42

CHxVPTER XVI.

Of Powers in general, 4 5

CHAPTER XVII.

Of tJie calculation of Powers, 49

CHAPTER XVIII.

Of Roots with rdation to Powers in general, 51

CHAPTER XIX.

Of the Method nf representing Irrational JS*umbers by Fractional

Exponents, 5S

CHAPTER XX.

Cjf the different Methods of CalculatioUf and of their mutual

Connexion, 56

SECTION SECOND.

OF THE DIFFERENT METHODS OF CALCULATING COMPOUNp QUANTITIES.

CHAPTER I.

Of the Addition of Compound ^naniities, 59

CHAPTER il.

Of the Subtraction of Compound Quantities, 61

CHAPTER III.

Of the Multiplication of Compound Quantities, 62

CHAPTER IV.

Of the Division of Compound Quantities, 68

CHAPTER V.

Cff the Resolution of Fractions into hifinite Series, 72

CHAPTER VI.

Of the Squares of Compound Quantities, 81

CHAPTER VII.

Of the Extraction of Roots applied to Compound (Quantities. 84

Contents, xi

CHAPTER VIII.

(jftlie caleidation of Irrational Quantities, 88

CHAPTER IX.

Of Cubes, and of the Extraction of Cube Roots, 92

CHAPTER X.

Of the higher Powers of Compoimd Quantities, 94

CHAPTER XL

Of the Transposition of the Letters, an whicJf, the demonstra'

tration of the preceding Rule is founded. 10#

CHAPTER XII.

Of the expression of Irrational Powers by Infinite Series, 104

CHAPTER XIII.

Of tJie resolution of Mgative Powers, 107

SECTION THIRD.

OF RATIOS AND PROPORTIONS.

CHAPTER I.

Of Arithmetical Ratio, or of the difference between two JV^wm-

bers. 111

CHAPTER II.

Of Arithmetical Proportion, 1 is

CHAPTER III.

Of Arithmetical Progressions, 1 16

CHAPTER IV.

Of the Summation of Arithmetical Progressions* 120

CHAPTER V.

Of Geometrical Ratio, 124

CHAPTER VI.

Of the greatest Common Divisor of two given numbers, 126

CHAPTER VII.

Of Geometrical Proportions, 13©

CHAPTER VIII.

Observations on the Rules of Proportion and their utilittj, 133

CHAPTER IX.

Of Compound Relations, ' 13S

xii (Contents.

CHAPTER X.

Of Geometrical Progressions. 144

CHAPTER XI.

Of Infinite Decimal Fractions. 150

SFXTIOl^f FOURTH.

OF ALGEBRAIC EQUATIONS, AND OF THE RESOLUTION OF THOSE EQUATIONS.

CHAPTER I.

Of the Solution of Problems in general. 15S

CHAPTER II.

Of the Resolution of Simple Mquatiatis, or Equations of the

first degree, 159

CHAPTER III,

Of the Solution of Questions relating to the preceding chapter. 163

CHAPTER IV.

Of the Resolution of two or more Equations of the First Degree, 1 73

CHAPTER V.

Of the Resolutim (f Pure Quadratic Equations. 182

CHAPTER VI.

Of the Resolution ofMixt Equations of the Second Degree. 18S

CHAPTER VII.

Of the JVature of Equations of the Second Degree. 196

Questions for Practice^ 202

A'^otes. -214

INTRODUCTION

ELEMENTS OF ALGEBRA.

SECTION I.

OF THE DIFFERENT METHODS OF CALCULATING SIMPLE QUANTITIES.

CHAPTER I.

OJ Mathematics in generaU

ARTICIE I.

Whatever is capable of increase, or diminution, is called

magnitude or quantity.

A sum of money therefore is a quantity, since we may in-

crease it and diminish it. It is the same with a weight, and

other things of this nature.

2. From this definition, it is evident, that the different kinds

of magnitude must be so many as to render it difficult even to

enumerate them : and this is tlie origin of tlie different branches

of the Mathematics, each being employed on a particular kind

of magnitude. Mathematics, in general, is the science of quan-

tity ; or the science whicli investigates the means of measuring

quantity.

3. Now we cannot measure or determine any quantity,

except by considering some other quantity of the same kind

as known, and pointing out their mutual relation. If it were

proposed, for example, to determine the quantity of a sum of

money, we should take some known piece of money (as a louis,

a crown, a ducat, or some other coin,) and shew how many of

I

g Algebra. Sect. 1.

these pieces are contained in the given sum. In the same man-

ner, if it were proposed to determine the quantity of a weight,

w^e should take a certain known weight ; for example, a pound,

an ounce, &c. and then shew how many times one of these

weights is contained in that which we are endeavouring to

ascertain. If we wished to measure any length or extension,

•we should make use of some known length, such as a foot.

4. So that the determination, or the measure of magnitude of

all kinds, is reduced to this : fix at pleasure upon any one known

magnitude of the same species with that which is to be deter-

mined, and consider it as the measure or unit ; then, determine

the proportion of the proposed magnitude to this known mea-

sure. This proportion is always expressed by numbers; so

that a number is nothing but the proportion of one magnitude

to another arbitrarily assumed as the unit.

5. From this it appears, that all magnitudes may be expressed

by numbers ; and that the foundation of all the mathematical

sciences must be laid in a complete treatise on the science of

numbers, and in an accurate examination of the different pos-

sible methods of calculation.

This fundamental part of mathematics is called Analysis, or

Algebra, [l.p

6. In Algebra then we consider only numbers which repre-

sent quantities, without regarding the different kinds of quantity.

These are the subjects of other branches of the mathematics.

7. Arithmetic treats of numbers in particular, and is the

science of numbers properly so called ; but this science extends

only to certain methods of calculation which occur in common

practice ; Algebra, on the contrary, comprehends in general

all the cases which can exist in the doctrine, and calculation of

numbers.

* The numbers thus included in crotchets refer to notes at the

end of this introduction.

Chap. 2. Of Simple Quantities. ^

CHAPTER II.

Explanation of the Signs + Plus and — Minus.

8. When we have to add one given number to another, this

is indicated by the sign + which is placed before the second

number, and is read plus, Tiius 5 -f 3 signifies that we must

add 3 to the number 5, and every one knows that the result is

8 ; in the same manner 12 + 7 make 19 ; 25 -f 16 make 41 ; the

sum of 25 +41 is 66, &c.

9. We also make use of the same sign + plus, to connect

several numbers together ; for example, r + 5 + 9 signifies that

to the number 7 we must add 5 and also 9, which make 21.

The reader will therefore understand what is meant by

8 + 5 + 13 + 11+1 +3 + 10;

vi». the sum of all these numbers, which is 51.

10. All this is evident ; and we have only to mention, that,

in Algebra, in order to generalize numbers, we represent them

by letters, as a, &, c, d, &c. Thus, the expression a + 6, signifies

the sum of two numbers, which we express by a and 6. and

these numbers may be eitlier very great or very small. In the

same manner, / + w + & + a?, signifies the sum of the numbers

represented by these four letters.

If we know therefore the numbers that are represented by

letters, we shall at all times be able to find by arithmetic, the

sum or value of similar expressions.

11. When it is required, on the contrary, to subtract one

given number from another, this operation is denoted by the

sign — , wliich signifies miiius, and is placed before the number

to be subtracted : thus 8 — 5 signifies that the number 5 is to be

taken from the number 8 ; which being done, there remains 3.

In like manner 12 — 7 is the same as 5 ; and 20 — 14 is the same

as 6, &c.

12. Sometimes also we may have several numbers to subtract

from a single one ; as, for instance, 50 — 1 — 3 — 5—7—9. This

signifies, first, take 1 from 50, there remains 49 ; take 3 from

that remainder, there will remain 46 ; take away 5, 41 remains ;

take away 7, 34 remains ; lastly, from that take 9, and there

4 Mgebra. Sect. I.

remains 25 ; this last remainder is the value of the expression.

But as the numbers 1, 3, 5, 7, 9, are all to be subtracted, it is

the same thing if we subtract their sum, which is 25, at once

from 50, and the remainder will be 25 as before.

IS. It is also very easy to determine the value of similar

expressions, in which both the signs + phis and — minus are

found : for example ;

12 — 3 — 5 -f- 2 — 1 is the same as 5.

We have only to collect separately the sum of the numbers that

have the sign -\- before them, and subtract from it the sum of

those that have the sign — . The sum of 12 and 2 is 14 ,• that

of 3, 5 and 1, is 9 ; now 9 being taken from 14, there remains 5.

14. It will be perceived from these examples that the order

in which we write the numhers is quite indifferent and arbitrary^

provided the proper sign of each be preserved, AYe might with

equal propriety have arranged the expression in the preceding

article thus ; 12-f2 — 5 — 3 — l,or2 — 1 — 3 — 5-fl2, or2 +

12 — 3 — 1 — 5, or in still different orders. It must be observed,

that in the expression proposed, the sign -|- is supposed to be

placed before the number 12.

15. It will not be attended with any more difficulty, if, in

order to generalize these operations, we make use of letters

instead of real numbers. It is evident, for example, that

(J — 6 — c+d — e,

signifies that we have numbers expressed by a and <Z, and that

from these numbers, or from their sum, we must subtract the

numbers expressed by the letters &, c, e, which have before them

the sign — .

16. Hence it is absolutely necessary to consider what sign is

prefixed to each number : for in algebra, simple quantities are

numbers considered with regard to the signs which precede, or

affect them. Further, we call those positive quantities, before

which the sign -f is found ; and those are called negative quan-

titiesy which are affected with the sign — .

17. The manner in which we generally calculate a person's

property, is a good illustration of what bas just been said. We

denote what a man really possesses by positive numbers, using,

or understanding the sign -f 5 whereas his debts are represent-

Chap. 2. Of Simple Quantities. 5

ed by negative numbers, or by using the sign — . Thus, when

it is said of any one that he has 100 crowns, but owes 50, this

means that his property really amounts to 100 — 50 ; or, which

is the same thing, -f 100 — 50, that is to say 50.

18. As negative numbers may be considered as debts, because

positive numbers represent real possessions, we may say that

negative numbers are less than nothing. Thus, when a man

has nothing in the world, and even owes 50 crowns, it is certain

that he has 50 crowns less than nothing ; for if any one were to

make him a present of 50 crowns to pay his debts, he would

still be only at the point nothing, though i^ally richer than

before.

19. In the same manner therefore as positive numbers are

incontestably greater than nothing, negative numbers are less

than nothing. Now we obtain positive numbers by adding 1 to

0, that is to say, to nothing ; and by continuing always to

increase thus from unity. This is the origin of the series of

numbers called natural numbers ; the following are the leading

terms of this series :

0, + 1, + 2, + 3, + 4, + 5, + 6, + 7, + 8, + 9, + 10,

and so on to infinity.

But if instead of continuing this series by successive additions,

we continued it in the opposite direction, by perpetually sub-

tracting unity, we should have the series of negative numbers :

0, — 1, — 2, — 3, ^ 4, — 5, — 6, — 7, — 8, -^ 9, — 10,

and so on to infinity.

20. All these numbers, whether positive or negative, have the

known appellation of whole numbers, or integers, which conse-

quently are eitlier greater or less than nothing. We call them

integers, to distinguish them from fractions, and from several

other kinds of numbers of which we shall hereafter speak. For

instance, 50 being greater by an entire unit than 49, it is easy

to comprehend that there may be between 49 and 50 an infinity

of intermediate numbers, all greater than 49, and yet all less

than 50. We need only imagine two lines, one 50 feet, tlie

other 49 feet long, and it is evident that there may be drawn an

infinite number of lines all longer than 49 feet, and yet shorter

than 50.

& Mgehra. Sect. 1,

21. It is of the utmost importance throu8;h the whole of

Algebra, that a precise idea be formed of those negative quanti-

ties about whicli we liave been speaking. I shall content my-

self with remarking here that all such expressions, as

+ 1 — 1, 4-2 — 2, +3 —3, +4,-4, &c.

are equal to or nothing. And that

+ 2 — 5 is equal to — 3 .

For if a person has 2 crowns, and owes 5, he has not only

nothing, but still owes S crowns : in the same manner

7 — 12 is equal to — 5, and 25 — 40 is equal to — 15.

22. The same observations hold true, when, to make the

expression more general, letters are used instead of numbers :

0, or nothing will always be the value of -f a — a. If we wish to

know the value -fa — h two cases are to be considered.

The first is when a is greater than h; h must then be sub-

tracted from a, and the remainder, (before winch is placed or

understood to be placed the sign -{-,) shews the value sought.

Tlie second case is that in which a is less than h : here a is

to be subtracted from 6, and the remainder being made negative,

by placing before it the sign — , will be the value sought.

CHAPTER III.

Of the Multiplication of Simple Quantities,

23. When there are two or more equal numbers to be added

together, the expression of their sum may be abridged ; for

example,

a + a is the same with 2 x a>

a-fa-f-a 3Xflj

a-fa-f-a-ffl Ax a, and so on ; where x is the sign

of multiplication. In this manner we may form an idea of mul-

tiplication ; and it is to be observed that,

2 X a signifies 2 times, or twice a

3 X ft 3 times, or thrice a

4 X a 4 times a, &c.

Chap. 3. Of Simple ^tantities. f

£4. If therefore a number expressed ly a letter is to he multiplied

by any other number, we simply put that riumber before the letter ;

thus,

a multipled by 20 is expressed by 20a, and

b multiplied by 30 gives 30&, &c.

It is evident also that c taken once, or Ic, is just c.

25. Further, it is extremely easy to multiply such products

again by other numbers ; for example :

2 times, or twice 3a makes 6a,

3 times, or thrice 4b makes 12&,

5 times 7x makes SScc,

and these products may be still multiplied by pther numbers at

pleasure.

26. When the number, by which we are to multiply, is also re-

presented by a letter, we place it immediately before the other letter ^

thus, in multiplying b by a, the product is written ab ; and pq^

will b6 the product of the multiplication of the number q by p>

If we multiply this pq again by a, we shall obtain apq,

27. It may be remarked here, that the order in which the letters

are joined together is indifferent ; that ab is the same thing as ba;

for b multiplied by a produces as much as a multiplied by &.

To understand this, we have only to substitute for a and 6

known numbers, as 3 and 4 ; and the truth will be self-evident i

for 3 times 4 is the same as 4 times 3.

28. It will not be difficult to perceive, that when you have to

put numbers, in the place of letters joined together, as we have

described, they cannot be written in the same manner by put-

ting them one after the other. For if we were to write 34 for

3 times 4, we should have 34 and not 12. When therefore it is

required to multiply common numbers, we must separate them

by the sign x, or points : thus, 3x4, or 3*4, signifies 3 times 4,

that is 12. So, 1 X 2 is equal to 2 ; and 1x2x3 makes 6. In

like manner Ix2x3x4x 56 makes 1344 ; and 1x2x3

X4x5x6x7x8x9xl0is equal to 362^800, &c.

29. In tlie same manner, we may discover the value of an

expression of this, form, 5*7*8' abed. It shews that 5 must be

multiplied by 7, and that this product is to be again multiplied

by 8 ; that we are then to multiply this product of the three

Jlhehra. Sect. 1*

numbers by a, next by 6, and then by c, and lastly by d. It may

be observed also, that instead of 5 x 7 X 8 we may write its value,

280; for we obtain this number when we multiply the product

of 5 by 7, or 35 by 8.

SO. The results which arise from the multiplication of two or

more numbers are called products ; and the numbers, or indivi-

dual letters, are called factors,

31. Hitherto we have considered only positive numbers, and

there can be no doubt, but that the products which we have

seen arise are positive also : viz. -}- a by + 6 must necessarily

give + ab. But we must separately examine what the multi-

plication of + a by . — ft, and of — a by — 6, will produce.

32. Let us begin by multiplying — a by 3 or -f- 3 ; now since

— a may be considered as a debt, it is evident that if we take

that debt three times, it must thus become three times greater,

and consequently the required produ'^t is — Sa. So if we multi-

ply — a by -f &, we shall obtain — ha^ or, which is the same things

— ah. Hence we conclude, that if a positive quantity be multi-

plied by a negative quantity, the product will be negative;

and lay it down as a rule, that 4- by + makes +, or pliis^ and

that on the contrary + by — , or — by + gives — , or minus,

33. It remains to resolve the case in which — is multiplied by

— ; or, for exami>le, — a by — h. It is evident^ at first sight,

with regard to the letters, that the product will be ah; but it is