Arquimedes, Notas de estudo de Matemática

Baixe Arquimedes e outras Notas de estudo em PDF para Matemática, somente na Docsity!

CO DO

OU 166404

5m

^

CQ

co

THE WOli KB

AKCHIMEDES.

PREFACE.

S book is intended to form a

companion

volume to

my

edition of the treatise of

Apollonius

on Conic Sections

lately published.

If it was worth while to

attempt

to make the

work of "the

great geometer"

accessible to the

mathematician

of

to-day

who

might

not be

able,

in

consequence

of its

length

and of its

form,

either to read it in the

original

Greek or in a

Latin

translation, or,

having

read

it,

to master it and

grasp

the

whole scheme of the

treatise,

I feel that I owe even less of an

apology

for

offering

to the

public

a

reproduction,

on the same

lines,

of the extant works of

perhaps

the

greatest

mathematical

genius

that the world has ever seen.

Michel Chasles

has drawn an

instructive distinction between

the

predominant

features

of

the

geometry

of Archimedes and

of the

geometry

which we find so

highly developed

in

Apollo-

nius. Their works

may

be

regarded, says

Chasles,

as the

origin

and basis of two

great inquiries

which seem to share between

them the domain of

geometry. Apollonius

is concerned with

the

Geometry of

Forms and

Situations,

while

in

Archimedes

we find the

Geometry of

Measurements

dealing

with the

quad-

rature of curvilinear

plane figures

and with the

quadrature

and cubature

of

curved

surfaces,

investigations

which

"gave

birth to the calculus of the infinite

conceived and

brought

to

perfection

successively by Kepler,

Cavalieri, Fermat, Leibniz,

and

Newton." But whether Archimedes is viewed as the

man

who,

with the limited means at his

disposal,

nevertheless

succeeded

in

performing

what are

really integrations

for the

purpose

of

finding

the area of a

parabolic

segment

and

a

VI PREFACE.

spiral,

the

surface and volume of a

sphere

and a

segnrent

of a

sphere,

and the volume of

any segments

of the solids

of revolution of the second

degree,

whether he is seen

finding

the centre of

gravity

of a

parabolic segment, calculating

arithmetical

approximations

to the value of

TT, inventing

a

system

for

expressing

in words

any

number

up

to that which

we should write down with 1 followed

by

billion

ciphers,

or

inventing

the whole science of

hydrostatics

and

at

the same time

carrying

it so far as to

give

a most

complete

investigation

of the

positions

of rest and

stability

of a

right

segment

of a

paraboloid

of revolution

floating

in a

fluid,

the

intelligent

reader cannot fail to be struck

by

the remarkable

range

of

subjects

and the

mastery

of treatment.

And if these

are such as to create

genuine

enthusiasm in the student of

Archimedes,

the

style

and method are no less

irresistibly

attractive. One feature which will

probably

most

impress

the

mathematician accustomed to the

rapidity

and directness secured

by

the

generality

of modern methods is

the deliberation with

which Archimedes

approaches

the solution of

any

one of his

main

problems.

Yet this

very

characteristic,

with its incidental

effects,

is calculated to excite the more admiration because the

method

suggests

the

tactics of some

great

strategist

who

foresees

everything,

eliminates

everything

not

immediately

conducive to the execution of his

plan,

masters

every position

in its order,

and then

suddenly

(when

the

very

elaboration of

the scheme has almost

obscured,

in the

mind

of

the

spectator,

its ultimate

object)

strikes the final blow. Thus we read in

Archimedes

proposition

after

proposition

the

bearing

of which is

not

immediately

obvious but which we find

infallibly

used later

on

;

and we are led on

by

such

easy

stages

that the

difficulty

of

the

original problem,

as

presented

at the

outset,

is

scarcely

appreciated.

As Plutarch

says,

"it

is not

possible

to find in

geometry

more difficult and troublesome

questions,

or

more

simple

and lucid

explanations."

But it is

decidedly

a

rhetorical

exaggeration

when Plutarch

goes

on to

say

that we are deceived

viii PREFACE.

even

tolerably readable), though

I have tried to

secure as

mu/sh

uniformity

as

was

fairly possible. My

main

object

has been to

present

a

perfectly

faithful

reproduction

of the treatises as

they

have come down to

us,

neither

adding anything

nor

leaving

out

anything

essential or

important.

The notes are for the most

f

part

intended to throw

light

on

particular points

in the text or

to

supply

proofs

of

propositions

assumed

by

Archimedes as

known;

sometimes I have

thought

it

right

to

insert within

square

brackets after certain

propositions,

and in

the

same

type,

notes

designed

to

bring

out the exact

significance

of those

propositions,

in cases where to

place

such notes in the Intro-

duction or at the bottom of the

page might

lead to their

being

overlooked.

Much of the Introduction is,

as will be

seen,

historical

;

the

rest is devoted

partly

to

giving

a more

general

view of certain

methods

employed by

Archimedes and of their mathematical

significance

than would be

possible

in notes to

separate propo-

sitions,

and

partly

to the discussion of certain

questions arising

out of the

subject

matter

upon

which we have no

positive

historical

data to

guide

us. In

these latter cases,

where

it is

necessary

to

put

forward

hypotheses

for

the

purpose

of

explaining

obscure

points,

I have been careful to call attention to their

speculative

character,

though

I have

given

the historical evidence

where such can be

quoted

in

support

of a

particular hypothesis,

my object being

to

place

side

by

side the authentic information

which we

possess

and the inferences which have been or

may

be

drawn from

it,

in order that the

reader

may

be in a

position

to

judge

for himself how far he can

accept

the latter as

probable.

Perhaps

I

may

be

thought

to owe an

apology

for the

length

of

one

chapter

on the so-called

vevcreis,

or

inclinationes,

which

goes

somewhat

beyond

what is

necessary

for the elucidation of

Archimedes;

but the

subject

is

interesting,

and I

thought

it

well to make my

account

of it as

complete

as

possible

in

order

to round

off,

as it

were, my

studies in

Apollonius

and

Archimedes.

PREFACE. IX

jl

have had

one

disappointment

in

preparing

this book for

the

press.

I was

particularly

anxious

to

place

on

or

opposite

the

title-page

a

portrait

of

Archimedes,

and I was

encouraged

in this idea

by

the fact that the

title-page

of Torelli's edition

bears

a

representation

in medallion

form on which are endorsed

the words Archimedis

effigies

marmorea in veteri

anaglypho*

Romae asservato. Caution

was however

suggested

when I

found two more

portraits wholly

unlike

this

but still

claiming

to

represent

Archimedes,

one of them

appearing

at the

beginning

of

Peyrard's

French translation

of

and the other in

Gronovius' Thesaurus Graecarum

Antiquitatum ;

and I

thought

it well to

inquire

further into the matter. I am now informed

by

Dr A. S.

Murray

of the British Museum that there does

not

appear

to be

any authority

for

any

one of the

three,

and

that writers on

iconography apparently

do not

recognise

an

Archimedes

among existing portraits.

I

was, therefore,

re-

luctantly obliged

to

give up my

idea.

The

proof

sheets have,

as

on the former

occasion,

been read

over

by my

brother,

Dr R. S.

Heath,

Principal

of Mason

College,

Birmingham

;

and I desire to take this

opportunity

of

thanking

him for

undertaking

what

might

well have

seemed,

to

any

one

less

genuinely

interested in Greek

geometry,

a thankless task.

T. L. HEATH.

March,

LIST OF THE PRINCIPAL WORKS CONSULTED.

JOSEPH

TORELLI,

Archimedis

yuae supersunt

omnia cum Eutocii A$ca-

lonitae commentariis. (Oxford, 1792.)

ERNST

NIZZE,

Archimedes van

Syrakus

vorhandene Werke aus dem

griechischen

iibersetzt und mil erldutemden und kriti&chen Anmerk-

ungen begleitet. (Stralsund,

1824.)

J. L.

HEIBERG,

Archimedis

opera

omnia cum commenlariia Eutocii.

(Leipzig, 1880-1.)

J. L. HEIBERG, Quaestiones

Archimedean. (Copenhagen, 1879.)

F.

HULTSCH,

Article Archimedes in

Pauly-Wissowa's Real-Encycloptidie

der

classischen

Altertumswmeiwhaften. (Edition

of

1895,

n.

1, pp.

507-539.)

C. A.

BRETSCHNEIDKR,

Die Geometric uiid die Geometer vor Euklide*.

(Leipzig,

1870.)

M.

CANTOR,

Vorlesungen

fiber Ges^hichte der

Mathematik,

Band

I,

zweite

Auflage. (Leipzig, 1894.)

G.

FRIEDLEIN,

Procti Diadochi in

primum

Euclidis elcmentorum

libmm

commentarii.

(Leipzig, 1873.)

JAMES

(row,

A short

history of

Greek Mathematics.

(Cambridge, 1884.)

SIEGMUND

OUNTHER,

Abriis der Getchichte der Mathematik und der

Naturwissenschaften

ini Altertum in I wan von Mailer's Handbuch der

klassischen

Altertumswi&senschaft,

. 1.

HERMANN

HANKEL,

Zur Geschichte der Mathematik in Alterthum und

Mittelalter.

(Leipzig, 1874.)

J. L.

HEIBERG, Litterarge&chichtlichc

Studien iiber Euklid.

(Leipzig,

1882.)

J. L. HEIBKRG,

Euclidis elemerta.

(Leipzig,

1883-8.)

(

F.

HULTSCH,

Article Arithmetica in Pauly-Wissowa^s

Real- Encyclopedic,

II.

1, pp.

Xll LIST OF PRINCIPAL

WORKS

CONSULTED.

F.

HULTSCH,

fferonis Alexandrini

geometricorum

et stereometricorum

reliquiae. (Berlin,

1864.)

(

F.

HULTSCH, Pappi

Alexandrini collectionis

quae supersunt. (Berlin,

1876-8.)

QINO LORI

A,

II

periodo

aureo della

geometria greca.

(Modena, 1895.)

MAXIMILIEN

MARIE,

Histoire des sciences

mathe'matiques

et

physiques,

<-

Tome I.

(Paris,

1883.)

J. H.

T.

MULLER, Beitriige

zur

Terminologie

der

griechischen

Mathematiker.

(Leipzig,

1860.)

Q. H. F.

NESSELMANN,

Die

Algebra

der Griechen. (Berlin, 1842.)

F.

SUSEMIHL,

Geschichte der

griechischen

Litteratur in der

Alejcandrinerzeit,

Band I.

(Leipzig, 1891.)

P.

TANNERY,

La

Geome'trie

grecque,

Premi6re

partio,

Histoire

ge'ne'rale

de la

Geometric tltmentaire.

(Paris, 1887.)

H. G.

ZEUTHEN,

Die Lehre von den

Kegelschnitten

im Altertum.

(Copen-

hagen, 1886.)

H. G.

ZEUTHEN,

Geschichte der Mathematik im Altertum und Mittelalter.

(Copenhagen,

1896.)

xiv CONTENTS.

PAGE

CHAPTER V^ UN THE PROBLEMS KNOWN AS NEY2EI.

.

^

Nevcrctff

referred to

by

Archimedes ... c

2. Mechanical constructions : the conchoid

of Nico-

medes cv

Pappus'

solution of the vcva-ts referred to in

Props. 8,

On

Spirals

cvii

4. The

problem

of the two mean

proportionals

. ex

5. The trisection of an

angle

.... cxi

6. On certain

plane

vcvo-ets cxiii

CHAPTER VI. CUBIC

EQUATIONS

cxxiii

CHAPTER VII. ANTICIPATIONS BY ARCHIMEDES OF THE INTE-

GRAL CALCULUS cxlii

CHAPTER

VIII. THE TERMINOLOGY OF ARCHIMEDES ... civ

THE WORKS OF ARCHIMEDES.

ON THE SPHERE AND

CYLINDER,

BOOK

1 1

BOOK

II. ... r>

MEASUREMENT OF A CIRCLE 91

ON CONOIDS AND SPHEROIDS 99

ON SPIRALS 151

ON THE

EQUILIBRIUM

OF

PLANES,

BOOK I. ... 189

BOOK II. ... 203

THE SAND-RECKONER 221

QUADRATURE

OF THE PARABOLA 233

ON FLOATING

BODIES,

BOOK 1 253

BOOK II 263

BOOK OF LEMMAS. 301

THE CATTLE-PROBLEM 319

INTRODUCTION.

CHAPTER I.

ARCHIMEDES.

A LIFE of Archimedes was written

by

one

Heracleides*,

but

this biography

has not survived,

and such

particulars

as are known

have to be collected from

many

various sources f.

According

to

TzetzesJ

he died at

the

age

of

75, and,

as he

perished

in the

sack

of

Syracuse (B.C. 212),

it follows that he was

probably

born about

B.C. He

was the son

of Pheidias the

astronomer,

and was

on intimate terms

with,

if not related to.

king

Hieron and his

Eutocius mentions this work in his commentary

on Archimedes' Measure-

ment of

the circle, ws QrjGiv 'HpaK\eiSijs

tv

r$ 'ApxiM^Sou* pUp.

He alludes to it

again

in his

commentary

on

Apollonius'

Conies

(ed. Heiberg,

Vol. n.

p. 168),

where, however,

the name is

wrorn^v given

as

'Hpd/tXetos.

This Heracleides

is

perhaps

the same as

the Heraciei^s mentioned by Archimedes

himself

in the

preface

to

his book On Spiral*.

t

An exhaustive collection of the materials is

given

in

Heiberg's Quaestiones

Archimedcac (1879).

The preface

to Torelli's edition also gives

the main points,

and the same work (pp.

370. quotes

at

length

most of the

original

references to the mechanical inventions of Archimedes. Further,

the article

Archimedes

(by Hultsch)

in

Pauly-Wissowa's Real-JKncyclopfitlie

der cfassischen

Altertunuwi*enchaftcH

gives

an

entirely

admirable

summary

of all the available

information. See also Susemihl's Geschichte der

gricchitchen

Litteratur in der

Alexandrinerzcit,

i.

pp.

723

t Tzetzes, Chiliad.,

n. 35,

Pheidian

is mentioned in the Sand-reckoner of Archimedes,

rwv

Trpartpuv

dffrpoXbywv Ev$6$ov

..ct5la 8t TOV d/uou -rrarpos (the

last words

being

the correction

of Blass for rov

'

AKOVTTO.TPOS,

the

reading

of the

text).

Of. Schol. Clark, in

(^regor.

Nazianz. Or. 34, p.

355 a

Morel. 4>et5ias

rb /ueV 7^0? yv

Zi'

6

ARCHIMEDES. XVJi

catapults

so

ingeniously

constructed as to be

equally

serviceable

at

^ong

or short

ranges,

machines for

discharging

showers of

missiles through

holes made in the

walls,

and others

consisting

of

long^

moveable

poles projecting beyond

the walls

which either

dropped heavy weights upon

the

enemy's ships,

or

grappled

the

prows by

means of an iron hand or

a

beak like that of a

crane,

then lifted them into the air and let them fall

again*.

Marcellus

is said to have derided his own

engineers

and artificers with the

words,

Shall we not make an end of

fighting against

this

geo-

metrical Briareus

who,

sitting

at ease

by

the

sea, plays pitch

and

toss with our

ships

to our

confusion,

and

by

the multitude of

missiles that he hurls at us outdoes the hundred-handed

giants

of

mythology?t";

but the exhortation had no

effect,

the Romans

being

in such

abject

terror that

if

they

did but see a

piece

of

rope

or wood

projecting

above the

wall, they

would

cry

l

there it is

again,' declaring

that Archimedes was

setting

some

engine

in motion

against them,

and would turn their backs and run

away,

insomuch

that Marcellus

desisted from

all conflicts and

assaults, putting

all

his

hope

in a

long siege J."

If we are

rightly

informed,

Archimedes

died,

as he had

lived,

absorbed in mathematical

contemplation.

The accounts of the

exact circumstances of his death differ in some details. Thus

Livy says simply

that,

amid the scenes of confusion that followed

the capture

of

Syracuse,

he was found intent on some

figures

which

he had drawn in the

dust,

arid was killed

by

a soldier who did

not know who

he was. Plutarch

gives

more than one version in

the following passage.

Marcellus was most of all afflicted at

the death of Archimedes

; for,

as fate would have

it,

he was intent

on

working

out some

problem

with a

diagram and, having

fixed

his mind and his

eyes

alike on his

investigation,

he never noticed

the incursion of the Romans nor the

capture

of the

city.

And

when a soldier came

up

to him

suddenly

and bade him follow to

Polybius,

Hi*t. vin.

78; Livy

xxiv. 34;

Plutarch, Marcellus,

t Plutarch, Marcellus, 17.

ibid.

Livy

xxv.

Cum

multa irae,

multa auaritiae

foeda

exempla ederentur,

Archimedem memoriae proditum

est in tanto tumultu, quantum

pauor captae

urbia in discursu diripientium

militum ciere poterat,

intentura formis, quas

in

puluere descripaerat,

ab

ignaro

milite quis

easet interfectum ; aegre

id Marcellum

tulisse sepulturaeque

curam habitam,

et propinquis

etiam iuquisitis

honori

praesidioque

nomen ao memoriain eius fuisse.

H. A. b

XV111 INTRODUCTION.

Marcellus,

he refused to do so until he had worked out his

problem

to a demonstration

;

whereat the soldier was so

enraged

that he-

drew his sword and slew him. Others

say

that the Roman

ran

up

to him with a drawn sword

offering

to kill him

; and,

when

Archimedes saw

him,

he

begged

him

earnestly

to wait a short

time

in order that he

might

not

leave his

problem incomplete

and

unsolved,

but the other took no notice and killed him.

Again

there

is a third account to the effect

that,

as he was

carrying

to

Marcellus some of

his mathematical

instruments, sundials, spheres,

and

angles adjusted

to the

apparent

size of the sun to the

sight,

some

soldiers met him

and, being

under the

impression

that lie carried

gold

in the vessel,

slew him*." The most

picturesque

version of the

story

is

perhaps

that which

represents

him as

saying

to a Roman

soldier

who came too

close,

Stand

away,

fellow,

from

my diagram,"

whereat

the man was so

enraged

that he killed

him

t-

The addition

made to this

story by

Zonaras, representing

him as

saying Trapu

Ka\dv

KOL

fjirj Trapd ypafjifjidv,

while it no doubt recalls the second

version

given by

Plutarch,

is

perhaps

the most far-fetched of the

touches

put

to the

picture by

later hands.

Archimedes

is said to

have

requested

his

friends and relatives

to

place upon

his tomb a

representation

of a

cylinder circumscribing

a sphere

within it, together

with an

inscription giving

the ratio

which the

cylinder

bears to

the

sphere

J ;

from which we

may

infer that he himself regarded

the

discovery

of this ratio

[fM

the

Sphere

and

Cylinder,

I.

33, 34]

as his

greatest

achievement. Cicero,

when

quaestor

in

Sicily,

found the tomb in a

neglected

state and

restored it.

Beyond

the above

particulars

of the life of

Archimedes,

we

have

nothing

left

except

a number of

stories, which,

though perhaps

not

literally accurate, yet help

us to a

conception

of the

personality

of the most

original

mathematician of

antiquity

which we would

not

willingly

have altered.

Thus,

in illustration of his entire

preoccupation

by

his abstract studies,

we are told that he would

forget

all about

his food and such

necessities of

life,

and would

be

drawing geometrical figures

in the ashes

of the

fire, or,

when

Plutarch, Marcellus,

t Tzetzes,

Chil.

n.

35,

135 ;

Zonaras

ix. 5.

J Plutarch,

Marcellus, 17 ad fin.

Cicero,

Tutc. v. 64 sq.