greek mathematics ii loeb

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THE LOEB CLASSICAL LIBRARYFOUNDED BY JAMES LOEB,EDITED BYfT. E. PAGE,O.H., LiTT.D.

LL.D.

tE. CAPPS,L. A.

PH.D., LL.D.

tW. H.

D.

ROUSE,

litt.d.

POST,

M.A.

E. H.

WARMINGTON,

m.a., f.b.hist.soo.

GKEEK MATHEMATICSII

SELECTIONSILLUSTRATING THE HISTORY OF

GREEK MATHEMATICSWITH AN ENGLISH TRANSLATION BY

IVOR

THOMAS

FORMERLY SCHOLAR OF ST, JOHN'S AND SENIOR DEMY or MAGDALEN COLLF.fiE, OXFORD

IN

TWO VOLUMESII

FROM ARISTARCHUS TO PAPPUS

LONDON

WILLIAM HEINEMANN LTDCAMBRIDGE, MASSACHUSKITS

HARVARD UNIVERSITY PRESS

First printed 1941 Reprinted 1951, 1957

mPrinted in Oreat Britain

.

CONTENTS OF VOLUMEXVI. Aristarchus of Samos(a)(6)(c)

II

. .

General

.

.

.,

Distances of the sun and

moon

^ 414

Continued fractions

(?)

#

XVII. Archimedes(a)(b)

General

Surface and volume of the cylinder and sphere (c) Solution of a cubic equation . , (d) Conoids and spheroids(i)(ii)(iii)

......

18

40126

Preface

.

..

Two lemmas

,164 .164,

Volume of a segment of aparaboloid of revolution170

(e)

The(i)(ii)

spiral of

Archimedes. .

Definitions

,,

Fundamental property

(iii)

(iv)

A verging Property of the subtangent,,

182 184 186 190194.

(J")

Semi-regular solids

V

CONTENTS

PAOB

(g)(A)

of System numbers

expressing. .

large.

Indeterminate analysis Problem

....:

.198202

the Cattlegravity

(i)

Mechanics(i)(ii)

:

centres.

of.

(iii)

Postulates . Principle of the lever . Centre of gravity of a parallelo.

.206208

gram(j)(^)

Mechanical methodHydrostatics(i)(ii)

(iii)

(iv)

Postulates Surface of fluid at rest Solid immersed in a fluid Stability of a paraboloid . revolution ..

.... ....in

216

geometry

.

220 242 244 248252

..

of.

XVIII. Eratosthenes(a)(b)(c)

.

General

On means

(d)

The Plaionicus Measurement of the earth

..... ..... .

.

.

260 262 264 266

XIX. Apollonius of Peroa(a)

...

The(i)(ii)

conic sections

Relation to previous works

Scope of the workDefinitions

(iii)

.....

(iv)

Construction of the sections

(v)(vi)

Fundamental propertiesTransition to

. .

new diameter

276 280 284 288 304 328

CONTENTS()

CONTENTSXXI. Trigonometry1.

..

2.

Hipparchus and Menelaus Ptolemy(a)(6)

406408

General Table of sines(i)(ii)(iii)

.....

Introductionsin 18

,. ,.

(iv)

and sin 36 =1 sin2 ^+cos2 Ptolemy's theorem

.412 .414 .420

(v) sin

(-) = 5cossin sinsin

cos,

-

.

42242 4281 AH HA, for HA>HN. Therefore AM MA exceeds > AH HA that is,bysimilartriangles,: : : ::

,

;

:

the ratio of the polygon to the surface of the pyramid.

88

GREEK MATHEMATICS

7

ive^av

,8.

' dpae^etrj

iv

Be

6

6

iv

A

iv

icrrlv

iv

iLava

['\}

. '.ovhk

, iviv

il

iSe

ilv

8,

iavea

,

jLterafu

,iiK

" , , ,84

ihv

i8o.

8

, iB

,

ARCHIMEDESscribed in the circle A has to the polygon inscribed in the circle a ratio greater than that which the same polygon [inscribed in the circle A] has to the surface of the pyramid therefore the surface of the pyramid is greater than the polygon inscribed in B. Now;

the polygon circumscribed about the circle has to the inscribed polygon a ratio less than that which has to the surface of the cone the circle by much more therefore the polygon circumscribed about the has to the surface of the pyramid inscribed circle in the cone a ratio less than that which the circle which is impossible." has to the surface of the cone Therefore the circle is not greater than the surface of the cone. And it was proved not to be less ; therefore it is equal.; ;

Prop. 16

If an

isosceles cone be cut by

a plane parallel

to the

base, the portion

of

the surface

of

the cone between the

parallel planes

is equal to a circle whose radius is a mean proportional between the portion of the side of the cone between the parallel planes and a straight line equal to

sum of the radii of the circles in the parallel planes. Let there be a cone, in which the triangle through and let it be cut by a plane the axis is equal to parallel to the base, and let [the cutting plane] make and let BH be the axis of the cone, the sectionthe

,TO

,

For the circumscribed polygon is greater than the but the surface of the inscribed pyramid is less than the explanation to this the surface of the cone [Prop. 12] effect in the text is attributed by Heiberg to an interpolator.circle B,;

*

^

...

TotJ

om. Heiberg.

VOL.

D

85

rov

GREEK MATHEMATICSrrjs

Kevrpov

,

,

re

6

,6

^ ,6

)"

Se

.,

8

,IS!Z,

^

,

,. ,, " , , ,rfj Tjj

, he

ttj

,,,

6 86

, ,

?.

ARCHIMEDESandlet there

be set out a

circle

mean proportional between HA, and let be the circleis

whose radius and the sum of

is

,

a

;

I

say that the

circle

equal to the portion of the surface of the cone

betweenFor

let

square of the radius ofcontained by

,. ,,.

the circles A,

be set out, and let the be equal to the rectanglelet

andis

the square of the radius

of A be equal to the rectangle containedtherefore the circle

by BA,

AH

cone

,

A

equal to the surface of theis

while the circle[Prop. 14].

equal to the surfacesince

of the cone

BAbecause

=is

.

+(...

And

+AH)

parallel to AH,'* while the square of

the radius ofradius of

A isisis

equal to

AB AH, the square of the

equal to equal to

the radius of

, (.

and the square of + AH), therefore the

square on the radius of the circle

A

is

equal to the

sum

of the squares on the radii of the circles K,

;

so that the circle

A

is

equal to the

sum

of the circles

The proof is

*

,

GREEK MATHEMATICS

ju,ia

, , , ? . " , ^ ^^, ?els

.

8 ,iarl

^

rfj

^evdeiai

*

-

,,,, , ,, ,. , , , U , [ ]^ , ,, , . , ? ,

, 'etvatjuia

6

iv

'^,

^

otl

^,

'

00,

*

iva

.

,

.

om. Heiberg.

88

ARCHIMEDES

is equal to the surface of the cone But BE ; while is equal to the surface of the cone therefore the remainder, the portion of the surface is of the cone between the parallel planes equal to the circle

,.

,

.

,,

Prop. 21

If a

regular polygon

tvith

an even number of sides be

inscribed in a circle,the angles "

and

straight lines be

drawn joiningto be

of

the polygon, in such

a manner as

parallel to any one whatsoever

of the lines subtended by two sides of the polygon, the sum of these connecting lines bears to the diameter of the circle the same ratio as thehalf the sidesless

straight line subtended bythe side

one bears

to

of the polygon. Let be a circle, and in it let the polygon be inscribed, and let EK, HN, be joined then it is clear that they are parallel to a straight line subtended by two sides of the polygon * I say therefore that the sum of the aforementioned straight lines bears to the diameter of the circle, the same ratio as bears to EA. For let ZK, AB, be joined then ZK is

,

,

;

;

parallel to

A,

BA;

and

to

to ZK, also therefore

,:

,

;

to

BA,

to

But

ES:HA=KH:HO. KH HO =Zn 0,:

[Eucl.

vi.

4

" Sides " according to the Archimedes probably wrote*

text,

= lKZA For, because the arcs EZ are equal, [Eucl. iii. 27] ; therefore EK is parallel to and so on. ; " For, as the arcs AK, EZ are equal, ^AEK = ^EKZ, and therefore AE is parallel to ZK ; and so on.89

,

but Heiberg thinks where we have nXevpas.

ju,ev

7/30

, ^^ ? ,, ,? , , ,," ^

GREEK MATHEMATICS

,?cos Setj

Be

tJ

-

^

/cat

en,

?, ?, ?,Tf

NT

he -

NT

ert,

MX