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ELEMENTS OF ALGEBRA,

LEONARD EULER,
TRANSLATED FKOM THE FRENCH;
WITH THE

NOTES OF M. BERNOULLI,

&c.

AND THE

ADDITIONS OF M. DE LA GRANGE.
FOURTH EDITION,
CAREFULLY REVISED AND CORRECTED.

BY THE REV. JOHN HEWLETT,
TO WHICH

^

i^Tcmoir of

^z

IS

B.D. F.A.S. &c.

PREFIXED,

%x\z aniy

(J^j^aractcr of iEuler,

BY THE LATE

FRANCIS HORNER,

ESQ., M. P.

LONDON
PRINTED FOR LONGMAN, REES, ORME, AND
PATERNOSTER-ROW.
1828.

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CO.


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M;

(i/jai

Sc. ,ices

1%^S

THIRD EDITION.

Having prefixed my name
of Euler's Algebra,

it

to the present edition

may be

account of the Translation

proper to give some

which I

;

the greater pleasure, because

vorable opportunity of associating

with those of

my

tical

first

my own

discovered

placed under

my

uncommon powers
thirst for

docility of temper,

which rendered

Horner, M. P.

tuition, at the cri-

of

intellect,

and the

knowledge, united with a

and a sweetness of

disposition,

instruction, indeed, a " delightful

His diligence and attention were such, as

to require the frequent interposition of
tional

labors,

and interesting age of seventeen, he soon

most ardent

task."

a fa-

distinguished pupil, and most

excellent friend, the late Francis

When

do with

shall

furnishes

it

amusement,

in

some

ra-

order to prevent the in-

tenseness of his apphcation from injuring a constitution,

which, though not delicate, had never

been robust.
Perceiving that

mind

the natural tendency of his

led to the exercise of reason, rather than to

the indulgence of fancy

;

— that he was particularly

interested in discussing the merits of

theory, in exposing fallacies,

and

some specious

in

forming

legi-

timate inductions, from any premises, that were
a 2

459399
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ADVERTISEMENT.

IV

\

supposed to rest on the basis of truth
led to

;

but

findin)

from imitation and habit, he had bee

also, that,

think

highly of those metaphysical

which abound

speculations,

annex no

too

distinct ideas,

in

terms to which

we

and which often require

the admission of principles, that are either unintelincapable of proof; I

ligible, or

his

notice Euler's

recommended

to

Algebra, as affording an ad-

mirable exercise of his reasoning powers, and the
best means of cultivating that talent for analysis,
close investigation,

and

logical inference,

which he

possessed at an early period, and which he after-

At

wards displayed in so eminent a degree.

same time,

the

was of opinion, that to translate a

I

work from the French into
English, when he wished to vary his studies, would
improve his knowledge of both languages, and be
the best introduction for him to the mathematics.
part of that excellent

He

was soon delighted with

this occasional

em-

ployment, which seemed to supply his mind with
food, that was both solid

and nutricious

;

and he

generally produced, two or three times a week, as

much

as I could find

time to revise and correct.

In the course of the

first

twelvemonth, he had

translated so large a portion of the
that

it

two volumes,

was determined to complete the whole, and

to publish

it

for the benefit of English students

but he returned to Scotland before the manuscript

was ready

for the press

of editing

it

I

;

and, therefore, the labor

necessarily devolved on me.

wished to give

this short history

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of the Trans-


M; tiiomatical

1^*"'.

My
THIRD EDITION.

Having

prefixed

of Elder's Algebra,

my name to
it

may be

account of the Translation

tical

first

^^

do with
a fa-

my own

labors,

distinguished pupil, and most

Horner, M.

excellent friend, the late Francis

When

I shall

furnishes

it

vorable opportimity of associating

my

<'

proper to give some

the greater pleasure, because

with those of

r!

the present edition

which

;

/

placed under

my

P.

tuition, at the cri-

and interesting age of seventeen, he soon

discovered

uncommon powers

most ardent

thirst for

of

intellect,

and the

knowledge, united with a

docility of temper, and a sweetness of disposition,
which rendered instruction, indeed, a " delightful

task."

His ddigence and attention were such, as

some

ra-

prevent the

in-

to require the frequent interposition of

;

tional

il

amusement,

in order

to

tenseness of his application from injuring a constitution, which,

though not

delicate,

had never

been robust.
Perceiving that

mind

>

the natural tendency

led to the exercise of reason, rather than to

the indulgence of fancy

;

— that he was particularly

interested in discussing the merits of

^
;^

l

^

of his

theory, in exposing fallacies,

and

in

some specious
forming

legi-

timate inductions, from any premises, that were
a 2

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supposed to rest on

from imitation and

also, that,

led to

of

tlie basis

think

speculations,

annex no

too

higlily

liabit,

of tliose

which abound

distinct ideas,

Iriith

in

;

but findin

he had bee
metaphysical

we

terms to which

and which often require

the admission of principles, that arc either unintelincapable of proof; I

ligible, or

his

notice Euler's

recommended

to

Algebra, as affording an ad-

mirable exercise of his reasoning powers, and the
best means of cultivating that talent for analysis,
close investigation, and logical inference,

possessed at an early period,

which he

and which he

At

wards displayed in so eminent a degree.

same time,

I

after-

the

was of opinion, that to translate a

work from the French into
English, when he wished to vary his studies, would
improve his knowledge of both languages, and be
the best introduction for him to the mathematics.
He was soon delighted with this occasional employment, which seemed to supply his mind with
and he
food, thai was both solid and nutricious
part of that excellent

;

generally produced, two or three times a week, as

niuch as I could find time to revise and correct.
In the course of the

first

twelvemonth, he had

two volumes,
complete the whole, and

translated so large a portion of the
that

it

was determined to

to publish

it

for the benefit of English students

but he returned to Scotland before the manuscript

was ready

for the press

of editing

it

X

;

and, therefore, the labor

necessarily devolved on me.

wished to give

this short history

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of the Trans-


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V

ADVERTISEMENT.
lation at

racter

first,

and

without any eulogium on his cha-

talents,

while living, of course

modestly, though, at the same time,

opposed

it,

;

but he

resolutely

saying that whatever merit or emolu-

ment might be attached to the work, it belonged
to me.
The same proposal was made to him,
on publishing the second edition *

;

but he

still

persisted in his former determination.

From

the pleasure and instruction which he re-

ceived from Euler's Algebra,

him

wish to

to

it

was natural

know something more

of the

for
life

and character of that profound mathematician.

Having therefore

some measure

in

satisfied

and collected the necessary

curiosity,

his

materials,

by consulting the ordinary sources of information,

way of literary exercise, to draw
up a biographical Memoir on the subject. He
and this may
readily complied with my wishes
I

advised him, by

;

be considered as one of his earliest productions.
Its merits

writer

would do

credit, in

and therefore

;

reader will not

my

opinion, to any

appreciating them, the

in

deem any apology necessary on

account of the author's youth.
I

have been led into

stances,

first,

because

this short detail of
1

disdain the contemptible

vanity of shining in what

may be thought

rowed plumes, and because
pleasure
*

The

in

speaking of

my

I feel a

bor-

melancholy

highly valued, and

care of correcting the press for this edition was en-

trusted to Mr. P. Barlow, being
in

circum-

the laborious

engaged myself,

employment of editing the

Bible.

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at that lime,


MEMOIR
OF THE

AND CHARACTER OF EULER,

LIFE

BY THE LATE

FRANCIS HORNER,

ESQ., M. P.

Leonard Euler was the son of a clergyman

in

the neighbourhood of Basil, and was born on the

His natural turn for mathe-

15th of April, I707.

matics soon appeared, from the eagerness and
cility

fa-

with which he became master of the elements

under the instructions of his

father,

was sent to the university of Basil

There, his

abilities

and

his

by

whom

he

at an early age.

application were so

distinguished, that he attracted the particular notice

of John

Bernoulli.

That excellent mathe-

matician seemed to look forward to the youth's
future achievements

in

science,

while his

own

kind care strengthened the powers by which they

were to be accomplished.
his studies,

which

far

In order to superintend

outstripped the usual routine

of the public lecture, he gave him a private lesson
regularly once a

week

;

when they conversed

to-

gether on the acquisitions, which the pupil had

been making since
whatever

their last interview, considered

difficulties

might have occurred

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in his


EULER.
progress,
for the

IX

and arranged the reading and exercises

ensuing week.

Under such eminent advantages, the capacity
of Euler did not fail to make rapid improvements ;
and

degree of Master

in his seventeenth year, the

On

of Arts was conferred on him.

he received high applause
discourse, the subject of

this occasion,

his probationary

for

which was a comparison

between the Cartesian and Newtonian systemg.
His

father,

having

along intended him for

all

his successor, enjoined

him new

to relinquish his

mathematical studies, and to prepare himself by
those of theology, and general erudition, for the

After some time, however,

ministerial functions.

had been consumed,

this plan

man of
own views

was given up.

The

father, himself a

learning and liberality,

^abandoned

for those, to

inclination

his

and

were of them-

talents of his son

selves so powerfully directed

;

which the

persuaded, that in

thwarting the propensities of genius, there
sort

of impiety against nature,

would be

real injustice to

is

a

and that there

mankind

in

smothering

those abilities, which were evidently destined to

Leonard was

extend the boundaries of science.

permitted, therefore, to resume his favorite pursuits;, and, at the age of nineteen, transmitting

two

dissertations to the

Paris,

Academy

one on the masting of

of Sciences at

ships,

and the other

on the philosophy of sound, he commenced that
splendid career, which continued, for so long a
period, the admiration

About the same

and the glory of Europe.

time, he stood candidate for a

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X

EULER.

vacant professorship in the university of Basil

but having

lost the election,

he resolved,

in con-

sequence of

this disappointment, to leave his na-

tive country

;

and

in

1727 he

set

out for Peters-

burg, where his friends, the young Bernoullis, had

about two years before, and where he

settled

flattered himself with prospects of literary success

under the patronage of Catherine

spects, however,

nor was

it

were not immediately realised

after

till

Those pro-

I.

he had been frequently and

long disappointed, that he obtained any prefer-

His

ment.
to

the

first

chair

appointment appears to have been

of natural philosophy

;

and when

Daniel Bernoulli removed from Petersburg, Euler

succeeded him as professor of mathematics.
In this situation he remained for several years,

engaged

in the

most laborious researches, enrich-

ing the academical collections of the continent

with papers of the highest value, and producing
almost daily improvements in the various branches

of physical,
science.

and,

more

particularly,

analytical

In 1741, he complied with a very press-

ing invitation from Frederic the Great, and re-

Throughout this period, he continued the same literary labors, directed by the same wonderful sagacity and comsided at Berlin

prehension of

own

till

I766.

intellect.

discoveries

As he advanced with

and inventions, the

field

of know-

ledge seemed to widen before his view, and
subjects
tion.

still

The

his

new

multiplied on him for further specula-

toils

of intense study, with him, seemed

only to invigorate his future exertions.

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Nor

did


EULER.

the energies of Euler's

XI

mind give way, even when

body were overpowered

the organs of the

:

for in

the year 1735, having completed, in three days,

which the

certain astronomical calculations,

demy

called for in haste

aca-

but which several ma-

;

thematicians of eminence had declared could not

be performed within a shorter period than some

months, the intense application threw him into a
fever, in

which he

of one eye.

lost the sight

Shortly after his return to Petersburg, in I766,

he became totally blind.
however, suffered

His passion for science,

no decline

;

the powers of his

mind were not impaired, and he continued

Though

defatigable as ever.

likewise were

had passed

latter period

pain,
his

and

now crowding

loss

of his

the distresses of age

fast

his sixtieth year

;

upon him,

yet

under

life,

as in-

it

for

was in

he

this

infirmity, bodily

of sight, that he produced some of

most valuable works

;

such as

command our

astonishment, independently of the situation of

the author, from the labor and originality which

they display.

In

fact,

his

habits

of study and

composition, his inventions and discoveries, closed

only with his
died, he

life.

The very day on which he

had been engaged

in calculating the orbit

of Herschel's planet, and the motions of aerostatic

machines.

His death happened suddenly in Sep-

tember 1783, from a

fit

of apoplexy,

when he was

in the seventy-sixth year of his age.

Such

The

is

the short history of this illustrious man.

incidents of his

life,

like that

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of most other

EULER.

Xll

laborious students, afford very scanty materials for

biography

little

;

more than a journal of

and a catalogue of publications
find

ample compensation

ter of his

mind.

An

studies

but curiosity

:

surveying the charac-

in

object of such magnitude,

so far elevated above the ordinary range of
intellect,

may

human

cannot be approached without reverence,

nor nearly inspected, perhaps, without some de-

Should an apology be ne-

gree of presumption.

attempting the following

therefore, for

cessary,

estimate of Euler's character, let
that

we

it

be considered,

can neither feel that admiration, nor offer

that homage, which

is

worthy of genius, unless,

aiming at something more than the dazzled sensations of

mere wonder, we subject

amination, and compare

it

human nature in general.
Whoever is acquainted
those great men, to

whom

it

to actual ex-

with the standards of

with the memoirs of
the

human

race

is

in-

debted for the progress of knowledge, must have
perceived, that, while mathematical genius

excellence,
versity.

it

The

so extensive

it is

di-

from the other departments of intellectual

stinct

times,

is

likewise admits in itself of

and so various,

and present

so

many

natural for a person,

cast, to

especially in

di-

become
modern

interesting aspects, that

whose

talents are of this

devote his principal curiosity and attention

to particular views of the science.

pens, the faculties of the
facility

much

subjects of its speculation are

When

mind acquire

this hap-

a superior

of operation, with respect to the objects

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EULER.

XIU

towards which they are most frequently directed,

and the invention becomes habitually most active
and most acute

The

in that channel of inquiry.

truth of these observations

by the character of Euler.

illustrated

and discoveries lay not among the

strikingly

is

His studies

lines

and

figures

of geometry, those characters, to use an expression of Galileo in

which the great book of the

universe

;

written

is

had a turn

nor does he appear to have

by experiment, and

for philosophising

advancing to discovery through the rules of
ductive investigation.

The

dehghted

was that of pure

He

to speculate,

surveyed

region,

the properties

and

in

in-

which he
intellect.

affections

quantity under their most abstracted forms.

of

With

the same rapidity of perception, as a geometrician
ascertains the relative position of portions of extension,

Euler ranges among those of abstract quan-

tity,

unfolding their most involved combinations,

and tracing

their

most

intricate proportions.

That

admirable system of mathematical logic and language, which at once teaches the rules of just

and furnishes an instrument

inference,

for prose-

cuting deductions,

free

from the defects which

obscure and often

falsify

our reasonings on other

subjects

;

the different species of quantity, whether

formed in the understanding by
tions, or derived

its

own

abstrac-

from modifications of the repre-

sentative system of signs

;

the investigation of the

various properties of these, their laws of genesis,

the

limits

of comparison

among

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the

different


EULER.

XIV

and the method of applying

species,

solution of physical problems

searches on which the

;

all this

to the

these were the re-

mind of Euler delighted

to

which he never engaged without

dwell,

and

in

finding

new

objects of curiosity, detecting sources

of inquiry, which had passed unobserved, and exploring fields of speculation and discovery, which

before were unknown.

The

subjects,

merated, form,

which we have here

when taken

Modern Analysis

the

:

together,

slightly enu-

what

is

called

a science eminent for the

profound discoveries which

it

has revealed

;

for

the refined artifices that have been devised, in

order to bring the most abstruse parts of mathematics within the compass of our reasoning powers,

and

for applying

phfenomena,

them

to the solution of actual

as well as for the

remarkable degree

of systematic simplicity, with which the various

methods of investigation are employed and combined, so as to confirm and throw light on one

The

another.

materials,

lecting for years,

indeed,

had been

col-

from about the middle of the

seventeenth century

;

the foundations had been

by Newton, Leibnitz, the elder BernouUis,
and a few others but Euler raised the superstruc-

laid

;

was reserved for him to work upon the
materials, and to arrange this noble monument of
ture

:

it

luiman industry and genius in
metry.

labors, the ultimate

he

its

present sym-

Through the whole course of his

set his

scientific

and the constant aim on which

mind, was the perfection of Calculus

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XV

EULER.

Whatever physical inquiry he be-

and Analysis.
gan with,

always came in view, and very

this

fre-

quently received more of his attention than that

which was professedly the main

subject.

His

ideas ran so naturally in this train, that even in

met with images
that would recall the associations of his more familiar studies, and lead him back, from the fairy

the perusal of Virgil's poetry, he

scenes of fiction, to mathematical abstraction, as
to the element,

most congenial to

his nature.

That the sources of analysis might be ascertained
in their full extent, as well as the various modifica-

tions of

form and restrictions of rule that become

necessary in applying

nature

it

to

different

he appears to have nearly gone through a

;

complete course of philosophy.
rational mechanics,

The

theory of

the whole range of physical

astronomy, the vibrations of elastic
the movements

as

views of

fluids, as

of those which

pressible, naval architecture

and

trine of chances, probabilities,

are

tactics,

and

well

incom-

the doc-

political arith-

metic, were successively subjected to the analytical

method

;

and

all

these sciences received from

fresh confirmation
It cannot

tention

*

A

is

him

and further improvement*.

be denied

that,

in general,

more occupied with the

complete edition of

his works,,

his at-

analysis itself,

comprising the numerous

papers which he sent to the academies of St. Petersburg,
Berlin, Paris,

and other public

on Curves, the Analysis of
Calculus, &c.

societies, his separate Treatises

Infinites, the differential

would occupy,

and integral

at least, forty quarto volumes.

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XVI

EULER.

than with the subject to which he

applying

is

and that he seems more taken up with

it;

his instru-

ments, than with the work, which they are to assist

him in executing. But this can hardly be made a
ground of censure, or regret, since it is the very
circumstance to which we owe the present perfection of those instruments

a perfection to which

;

he could never have brought them, but by the unremitted

and enthusiastic preference

attention

which he gave

If he

to his favorite object.

now

and then exercised his ingenuity on a physical, or
perhaps metaphysical, hypothesis, he must have

been aware, as well

as

any one, that

his conclusions

would of course perish with that from which they
were derived. What he regarded, was the proper
means of arriving

at those conclusions

;

the

new

views of analysis, which the investigation might

open
it

;

and the new expedients of calculus, to which

might eventually give

form pursuit

;

all

This was his uni-

birth.

other inquiries were prosecuted

with reference to

it

;

and

in

this

consisted the

peculiar character of his mathematical genius.

The

faculties that are subservient to invention

he possessed

in a

memory was

at

very remarkable degree.

His

once so retentive and so ready,

that he had perfectly at command

all

those nu-

merous and complex formulae, which enunciate
the rules and more important theorems of analysis.

As
the

is

reported of Leibnitz, he could also repeat

^neid from beginning

to

end

trust his recollection for the first

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;

and

and could
last lines in


XVll

EULER.

every page of the edition, which he had been ac-

These are instances of a kind
of memory, more frequently to be found where
customed to

use.

the capacity

is

accompanying

than

But

inferior to the ordinary standard,

seem to have been not so much
of natural constitution, as of his most

in Euler, they

the result

wonderful attention

dence,
is

is

we

con-

sufficient

evi-

a faculty, which, if

;

testimony of

sider the

It

genius.

scientific

original,

Newton *

the great constituent of inventive power.

that complete retirement of the

itself,

during which the senses are

that intense meditation, on

idea can intrude

;

mind within
locked up

which no extraneous

that firm, straight-forward pro-

no

gress of thought, deviating into

irregular sally,

which can alone place mathematical objects in a

them

light sufficiently strong to illuminate

fully,

and preserve the perceptions of " the mind's eye"

in the

same order that

Two

it

moves along.

M.

of Euler's pupils (we are told by
himself)

a pupil

had

calculated

term

series as far as the seventeenth

on comparing the written

results,

this

difference

to

;

but found,

that they dif-

fered one unit at the fiftieth figure

municated

Fuss,

a converging

:

they com-

their master,

who

went over the whole calculation by head, and his
For the
decision was found to be the true one.

—

purpose of exercising his

extraction of roots, he has been
* This

opinion

grandson in the

little

known

of Sir Isaac Newton

is

to

recorded by Dr.

Pemberton.

b

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form to