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

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Online LibraryLeonhard EulerAn 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)
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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-
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


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Online LibraryLeonhard EulerAn 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)