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2/13

nimadversions

t h

r i g i n s

o

Western cience

By Martin Bernalt

SPENT

THE FIRST FIFTY YEARS

of

my

life trying to escape

from

the

shadow

of

my father,

John

Desmond

Bernal,

and

hence, among

other things,

from science

and the

history

of science.

Therefore,

the

trepidation

hat

is

proper

for

anyone

who is neithera scientist nor

a

historian

of science

writing

for Isis

is

multipliedmanyfold n my case. Nevertheless, I am grateful or the invitationto

put

forward my

views on the

origins

of

Western science.

Any approach

to

this

question immediately

stumbles over

the

definition

of

"science."

As no ancient

society possessed

the modern

concept

of

"science" or

a

word for

it, its application

to

Mesopotamia,

Egypt,

China,

India,

or Greece is

bound

to be

an

arbitrary

mposition.

This lack

of

clarity

is exacerbatedby the

clash

between

historians,

ike David

Pingree,

who are concerned

with

"sciences"

as

"functioningsystems

of

thought"

within

a

particular

society

and

those

who

apply

transhistorical

tandards

and see "science"

as "the

orderly

and

systematic

comprehension,

description

and/or

explanation

of natural

phenomena

. . .

[and]

the tools necessary for the undertaking ncluding, especially, mathematics and

logic."'

I

should

add the words

"real or

imagined"

after "natural

phenomena."

Pingree

denounces

the

claims

of what he calls

"Hellenophilia"

hat "science"

is

an

exclusively

Greek

invention

owing

little or

nothing

to earlier civilizations

and

that it was

passed

on

without

interference

o the

Western

European

makers

of the "scientific

revolution."

Puzzlingly,

the work

of Otto

Neugebauer-and

his

school,

including Pingree

himself-on

the extent

and

sophistication

of

Mesopo-

tamian

astronomy

and

mathematics

and

Greek

indebtedness to

it,

as well as

M. L. West's

demonstration

of the Near

Eastern influences on the

Presocratic

cosmologies, appearsto have left this kind of thinkingunscathed.2

t Departments

of

Government

and Near

Eastern

Studies,

Cornell

University,Ithaca,

New York

14853.

I

could

not

have

begun,

let alone completed,

this

paper

without

many years

of

patienthelp and

encouragement

rom JamilRagep,

who,

it

shouldbe pointedout,

is

far

from accepting

all

my conclu-

sions.

1

For Pingree

see

"Hellenophilia

ersus the

History

of

Science,"

a lectureoriginallypresentedat

the Department

f History

of

Science,

HarvardUniversity,

14

November 1990,

and now publishedn

this special

section. For the passage

cited see G.

E.

R. Lloyd, Early

GreekScience: Thales to

Aris-

totle (New

York/London:Norton, 1970),p. 1, quoting

Marshall

Clagett,

GreekScience in Antiquity:

How Human Reason and Ingenuity First Ordered and Mastered the Experience of Natural Phenom-

ena,

new corrected

ed.

(New

York:

Collier,

Macmillan,1962),p.

15.

2

See

Otto

Neugebauer,

The Exact

Sciences in

Antiquity

(New

York: Dover, 1969);and M. L.

West, Early

Greek

Philosophy

and

the Orient

(Oxford:

Clarendon, 1971)-pace

John Vallance,

"On

Marshall

Clagett's Greek Science

in Antiquity," Isis, 1990,

81:713-721, on p. 715.

See also G. S. Kirk,

"Popper

on

Science

and the

Presocratics,"

Mind, 1960,

69:318-339,esp. pp. 327-328;

and Kirk,

ISIS, 1992,

83 . 596-607

596


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3/13

CULTURES

OF ANCIENT

SCIENCE

597

There are

still defenders

of

the

claim

that

"Thales [seen as

a Greek]was the

first

philosopher

scientist"-the

word "scientist"

beingused here

in the positivist

sense.

According

to G. E. R. Lloyd,

the Greeks

were the

first

to "discoverna-

ture,"

"practice

debate," and

introduce

such specifics

as the study of

irrational

numbers(notably

V2)

and geometrical

modelingfor astronomy.

Lloyd sees the

discovery of

nature as

"the

appreciation

of the

distinctionbetween 'natural'and

'supernatural,'

hat is the recognition

that natural

phenomena

are not

the prod-

ucts

of random

or arbitrarynfluences

but regular

and governed

by determinable

sequences

of cause

and

effect.

3 However,

it is clear that

at least by

the second

millenniumB.C.

Mesopotamian

astronomy and

Egyptian

medicine,

to take two

examples,

were concerned

with

regular

and,

if

possible,

predictable

phenomena

with

relativelylittle supernatural

nvolvement.4

It is true that Egyptian

medicine

contained

some

religion

and magic.

At one

point even the "scientific"Edwin Smith Papyruson surgeryturns to magical

charms.

However,

E.

R.

Dodds

and

others have

shown

how

isolated

the natural

philosophers'

criticism

was

against

the

widespread

Greek belief

in the

efficacy

of

magic.S

Even

Hippocratic

medicine,

which is

generally

regarded

as

highly

ratio-

nal,

was institutionally

centered

on

the

religious

cult

of Asclepius and

his ser-

pents,

which

laid

great

emphasis

on the

religious

practice

of incubation.

Both

the

cult

and

the

practice,

incidentally,

had clear

Egyptian

roots.6

On

the question

of the

alleged uniqueness

of

Greek "scientific"debate, as

we

can

see

from those

in

Gilgamesh,

"debates"

are

at least as old as literature.

Some, such as the "Disputebetween a man and his Ba," which dates back to

Middle

KingdomEgypt,

contain

quite

profound

philosophy.

It is also clear

that

different

Mesopotamian, Syrian,

and Egyptian

cities

had not

merely

different

gods

but

distinct

cosmogonies,

most

of which involved abstract elements

or

forces

without

cults,

of

which

the

priesthoods

of the others

were aware.

There

were also attempted

and

actual

syncretizations,

suggesting

that

there had

been

debates.7

This situation

resembles

that

plausibly

reconstructed

or the

cosmolog-

ical disputes

of the Presocratics.

Later Greek

philosophical

and scientific

debates

clearly

owed a

great

deal

to

the

Sophists,

who came

from the

Greek tradition

of

"persuasion,"

with its close

"Common-Sense

n

the Development

of Greek Philosophy,"Journal

of Hellenic Studies, 1961,

81:

105-117,

pp. 105-106.

3 Lloyd, Early Greek

Science

(cit. n. 1), p.

8.

4

Neugebauer,

Exact Sciences

in Antiquity cit.

n.

2), pp.

29-52;Paul Ghalioungui,

The House

of

Life:

Per Ankh Magic

and

Medical Science

in Ancient Egypt,

2nd ed.

(Amsterdam:

B. M. Israel,

1973);

and Ghalioungui,The Physicians

of

Ancient Egypt (Cairo:

Al-Ahram

Centerfor Scientific

Translation,

1983).

S

E. R. Dodds, The Greeks and

the Irrational

(Berkeley/Los

Angeles:

Univ. California

Press,

1951);

G. E. R. Lloyd,

Magic, Reason,

and

Experience

Cambridge:

Cambridge

Univ.

Press,

1979),

pp. 10-58, 263-264;

and Garth

Fowden,

The Egyptian Hermes:

A Historical

Approach

to the Late

Pagan Mind(Cambridge:CambridgeUniv. Press, 1986),pp. 81-82.

6

J. B. de

C.

M.

Saunders,

The

Transition from Ancient

Egyptian

to Greek Medicine

(Lawrence:

Univ.

KansasPress, 1963),p.

12.

7

See E. A. Wallis

Budge, The

Gods of the Egyptians:

Studies

in

Egyptian Mythology,

2

vols.

(London:Methuen,

1904),

Vol.

1,

pp.

282-287;and

Marshall

Clagett,

Ancient

Egyptian

Science, Vol.

1: Knowledge

and Order,2 pts.

(Philadelphia:American

PhilosophicalSociety,

1989),pp. 263-372.

For

an annotated

translation

of the

"Dispute

between

a man

and his

Ba"

see Miriam

Lichtheim,

Ancient Egyptian

Literature, 3

vols., Vol.

1: The Old and

Middle Kingdoms

(Berkeley/Los

Angeles:

Univ.

California

Press,

1975),pp.

163-169.




4/13

598

BERNAL:

ANIMADVERSIONS

association

with

legal

disputes.Oratory,persuasion,

and

ustice arehighly valued

in nearly

all cultures, but, interestingly,

they received particular

emphasis in

Egypt. The central

scene

in

Egyptian conography

s the

judicialweighing of the

soul of

the dead

person,

and

the

legal

battle between

Horus and Seth is a central

episode in its mythology.Oneof the mostpopularEgyptian exts was that of The

EloquentPeasant,

which its most recent translator

nto English, MiriamLicht-

heim,

describes as "both

a serious disquisitionon the

need for

justice and a

parableon

the utility of fine speech."8

I

have

written

elsewhere

on the

centrality

of

the

image

of

Egyptian

ustice

to

both Mycenaean

and Iron

Age

Greece,

and

there

is

no doubt that

Greeks

of

the

Classical

and Hellenistic

periods

saw

Egyptian

aw

as the ultimatebasis of

their

own.

As

Aristotle wrote at

the end of the Politics: "The

history

of

Egypt

attests

the antiquity

of all political

institutions.

The

Egyptiansare generally

accounted

the oldest people on earth;andthey have alwayshad a body of law anda system

of

politics.

We

ought

to take

over

and use what has

already

been

adequately

expressed

before us and

confine

ourselves

to

attempting

to discover what has

already

been omitted."9

While

the first attestation

of

written law

in

Egypt

comes from the tomb

of

Rekhmire

n

the fifteenth

centuryB.C.,

there

is no reason to doubt that it existed

much earlier.10

n

any

event,

the

Egyptian

New

Kingdom

is

sufficiently

old by

Greek standards.

It is

clear

that what

Aristotle

was

recommending

had not

hith-

erto been

carriedout. Nevertheless,

it would

seem

likely

that Aristotle was

con-

ventional in his belief that, even thoughEgyptianand Greek law were very dif-

ferent

in

his own

day,

the true foundationof Greek

law and

justice

lay

in

Egypt.

The

emphasis

on

law

is

important

both because

of its

promotion

of

argument

and

dialectic and

because

of the

projections

of social law into nature and the

establishment

of

regularities.'1

There

is no doubt

that the

Egyptian

M3t

(Maat:

"truth,""accuracy,"

"justice")

was

central to

both

social

and natural

spheres

in

the same way as

the Greek Moira,

which

derivedfrom

it.

Similarly,

t is

clear

that

the Egyptians applied

the

"justice"

of

scales

to social and

legal

life at least as

early

as

the Middle

Kingdom.12

To

return to some

of the

specific

claims made

for the

originality

of Greek

science, there is now no doubt that Babylonianscholars were concerned with

2

and

Pythagorean riples

as

well

as

having

a

good approximation

of 'a. The

Egyptian

estimate

for wr

was

even

more accurate.

The

standarduse

in land mea-

8

Lichtheim, Ancient

Egyptian

Literature, Vol. 1, p.

169.

9 Aristotle, Politics, 7.10,

trans. Ernest

Barker

n

The

Politics of Aristotle (Oxford:

Oxford

Univ.

Press, 1958), p. 304.

See Martin

Bernal, "Phoenician

Politics and Egyptian

Justice in Ancient

Greece,"

in

Der Anfang

des politischen Denkens

bei den Griechen, ed.

Kurt Raaflaub

(Munich:

Historisches

Kolleg, forthcoming).

1o

A. Theodorides,

"The Concept

of Law

in

Ancient Egypt,"

in The

Legacy

of Egypt, ed.

J. R.

Harris Oxford:ClarendonPress, 1971),pp. 291-313;andAnne Burton,Diodorus SiculusBookI: A

Commentary Leiden:

Brill, 1972), pp.

219-225.

"

See JosephNeedham,

Science and Civilizationn China, Vol.

2

(Cambridge:Cambridge

Univ.

Press,

1956), pp. 518-583.

12

Bernal,

"PhoenicianPolitics

and EgyptianJustice'

(cit.

n. 9). Clagett subtitles

Volume

1

of

Ancient

EgyptianScience "Knowledge

and Order" cit.

n. 7), pp.

xi-xii. Thesewords are

translations

of the Egyptianrht

and

MY1t,

which he sees as the Egyptian

"rudimentary"cience.

See the constant

referencesto balances

and plumblines

as symbolsof justice

in TheEloquent

Peasant, trans.

in Licht-

heim,

Ancient Egyptian

Literature (cit. n. 7), Vol. 1,

pp. 170-182.




5/13

CULTURES OF ANCIENT

SCIENCE

599

surementof the

diagonal

of

a

square

of one

cubit

was

the

so-called double

remen,

that is to say

\/2

times

the

cubit.

3

Thus the irrational

number

par

excellence

was

employed

in

Egyptfromthe beginning

of

the

second millenniumB.C. at the

latest;

whether or not its irrationalitywas proved

in

Euclidean

fashion, its use provides

circumstantial vidence that Egyptianscribeswere aware of the incommensura-

bility of the side and

the

diagonal.

Modern scholars

have

poured scorn on

the

widespread

ancient

tradition

hold-

ing that Egyptians had known of

the

"Pythagorean" riangle. However, the very

cautious

Gay

Robins

and

Charles

Shute have

accepted

Beatrice

Lumpkin'sargu-

ment that knowledge of

it

is shown

by

the

use

in

Late

Old

Kingdompyramidsof

a

seked of 54 palms,

which

imposed

"a

half-base width to

height

of 3:4 and

so

could

have been modelled on

a

3:4:5

right angled triangle."14

I

shall discuss

the

strong

possibility

that

geometry, thought

to be

typically

Greek, came fromEgypt. However, at this point it would seem difficult o argue

that before the second half of

the

fourth

centuryB.C. any aspect

of

Greek "sci-

ence"

-with the

possible exception

of

axiomatic mathematics-was

more ad-

vanced

than

that of

Mesopotamia

or

Egypt.

WAS

NEUGEBAUER

RIGHT

TO

DISMISS ANCIENT

TRADITIONS OF

EGYPTIAN

SCIENCE?

In

this section

I

should

like

to take

it

as

given

that R. 0.

Steuer, J.

B. de

C. M.

Saunders,

and Paul

Ghalioungui

have established not

merely

that

Egyptianmed-

icine contained considerable "scientific"elements long before

the

emergence of

Greekmedicine,but thatEgyptianmedicineplayeda centralrole in the develop-

ment of Greek medicine.

15

Similarly,

he work of

Neugebauer

and

his school has

made

it

impossible to deny that some Mesopotamianmathematicians nd astron-

omers

were "scientific"

n

the

positivist

sense and that

Mesopotamian

"science"

in

these areas was

crucial to the

creation

of Greek

mathematics

and

astronomy.

However,

I

should

like to

challenge

these scholars' dismissal of claims that there

was an

Egyptian

mathematics that could

have had

a

significant influence on

Greek thinkers.

13

For the

triples

see the

bibliography

t

the end of Olaf

Schmidt,

"On

Plimton322:

Pythagorean

Numbers n

BabylonianMathematics,"Centaurus,1980,24:4-13,

on

p.

13. On the

Egyptian

estimate

for

IT

see Richard

Gillings,

Mathematics in the

Time

of

the Pharaohs

(New

York:

Dover, 1972), pp.

142-143;

and Gay Robins

and

Charles Shute, The Rhind

Mathematical Papyrus:

An

Ancient Egyptian

Text(London:

British Museum

Publications, 1987), pp.

44-46. On

the double remen

see Gillings,

Mathematics

in the Time

of

the

Pharaohs, p.

208.

14

Gay

Robins and Charles Shute,

"MathematicalBases

of

Ancient

Egyptian Architectureand

GraphicArt,"

Historia

Mathematica,

1985, 12:107-122,

on

p. 112;

and Beatrice

Lumpkin,

"The

Egyptian

and

PythagoreanTriples," ibid.,

1980,

7:186-187. See also

Gillings,

Mathematics

in the

Time of the Pharaohs, app. 5.

15

See

Ghalioungui,

House

of

Life (cit.

n.

4); Ghalioungui, Physicians of Ancient

Egypt (cit. n. 4);

R. 0. Steuer and J. B. de C. M.

Saunders,

Ancient

Egyptian

and Cnidian Medicine: The

Relationship

of

Their

Aetiological Concepts of

Disease

(Berkeley/Los Angeles:

Univ. California

Press, 1959);

and

Saunders, Transitionfrom Ancient Egyptian to Greek Medicine (cit. n. 6). Pace J. A. Wilson, "Med-

icine

in Ancient

Egypt," Bulletin of Historical

Medicine, 1962,

36(2):114-123;

Wilson, "Ancient

Egyptian

Medicine," editorial

in

Journal of the International College of

Physicians (Sect. 1), June

1964, 4J(6):665-673; and G.

E. R.

Lloyd, introduction to The

Hippocratic

Writings, ed. G. E. R.

Lloyd

(London: Penguin, 1983),p. 13n.

Even

the skepticalHeinrichvon

Staden,

who is

very reluc-

tant to concede Egyptian nfluenceson

Hellenisticmedicine, admits hat the

study of pulses and their

timingby waterclocks, for whichhis

subject Herophilus

of

Alexandriawas famous, probablycame

from the

Egyptian

tradition: von

Staden, Herophilus:

The Art

of

Medicine in

Early

Alexandria

(Cam-

bridge:

Cambridge

Univ.

Press, 1989),p.

10.




6/13

600

BERNAL:

ANIMADVERSIONS

Despite his early passionfor

ancient

Egypt

and

his

considerablework on Egyp-

tian astronomy, throughout

his

long

life

Neugebauer

insisted that

the

Egyptians

had no original or

abstract ideas

and

that

mathematically

and

scientificallythey

were not on the same

level

as

the

Mesopotamians.

He claimed that the

accurate

astronomicalalignments

of the

pyramids

and

temples

in

Egypt

and

the

use of

ir

and

4

could

all be

explained

as

the

results of

practical

knacks

rather

than

of

profound thought.

An

example

of

this

approach

is

the

following:

"It

has

even

been claimed that

the

area

of

a hemispherewas correctly

found in an

exampleof

the Moscow papyrus, but

the text

admits also

of

a

much more

primitive nterpre-

tation which

is

preferable."

16

In

his Exact Sciences in AntiquityNeugebauer

did not

argue

with the

pyrami-

dological school;

he

simply

denounced

it, recommending

hat

those interested

in

what

he

admitted o be

"the

very complex historical

and

archaeologicalproblems

connected with the pyramids"read the books by I. E. S. Edwards and J. F.

Lauer

on the

subject.

17

While

Edwards does not involve himself with

the

pyramidologists

and their

calculations, the surveyor

and

archaeologist

Lauer

did,

in the

face of

opposition

from Egyptologists, who

were

"astonished

that

we

shouldgive so much impor-

tance to

the

discussion of theories

which have never had

any

credit in the

Egyp-

tologicalworld." Lauer's

work had a certain

contradictoryquality.

He

admitted

that

the

measurementsexhibitedby

the

pyramids

do have

some remarkableprop-

erties;

that one can

find

such relations as

ir, +,

and

Pythagoras's triangle from

them; and that these facts generallybear out the claims Herodotus and other

ancient writers made for

them.

On

the

other

hand,

he

denounced the

"fantasies"

of

pyramidologists

and claimed that the

formulas

according

to which

the

pyra-

mids

were

aligned

and the

extraordinarydegree

of

sidereal

accuracy they

exhib-

ited were

purely

the

result

of

"intuitiveand utilitarian

mpiricism."18

A

conflict between

the

acceptance

of the

extraordinary

mathematical

precision

of the

Great

Pyramid

and

a

"certainty"

that

the Greeks

were

the

first "true"

mathematicians unsthroughout

Lauer's

many writings

on the

subject.The strain

is

made

still

harder o bear

by Lauer's awareness

that

some

Greeks

had been

told

about many

of this

pyramid's extraordinary

eatures and that

they

believed the

Egyptians

to

have been the

first mathematicians

and

astronomers. Moreover,

there

was

the

problem

that

so many

of the Greek

mathematicians

and

astrono-

mers

had studied

in

Egypt.

Lauer's

honest

attempt

to deal with

these difficulties

was

the

following:

Even

though up

to

now,

no

esoteric

Egyptian

mathematicaldocument has been dis-

covered,

we

know,

if we can

believe

the

Greeks,

that the

Egyptianpriests

were

very

jealous

of the secrets of their science and

that

they occupied themselves,

Aristotle

tells

us,

in

mathematics.

It

seems then

reasonably probable

that

they

had

been

in

possession of an esoteric science erected, littleby little, in the secrecy of the temples

during

he

long

centuries that

separate

the

constructionof the

pyramids,

owards the

16

Neugebauer,Exact Sciences in Antiquity

cit.

n.

2), p.

78

(italics added).

I shall

challenge

this

interpretation elow.

17

Ibid., p.

96.

18

J.

F.

Lauer, Observations sur les

pyramides (Cairo: Institut

Frangais

d'Arch6ologie

Orientale,

1960), pp. 11,

10, 4-24 (here andelsewhere,translations re

my

own unless

otherwise

ndicated).




7/13

CULTURESOF ANCIENTSCIENCE

601

year 2800 [I shouldput

it two hundred

years earlier],

to the eve of

Greek mathemat-

ical

thought

n

the sixth

century

B.C.

As far as

geometry

s

concerned,

the

analysis of

buildings

as famous as the Great

Pyramid

would take

a notable

place

in

the

researches

of these priests; and

it is

perfectly

conceivable that

they could

have

succeeded in

discovering

n

it, perhaps ong

after their

erection,

chance

qualities hat hadremained

totally unsuspected

to the constructors.19

The

question

of

when

Egyptians developed

this

sophisticated mathematical

knowledge is not directly

relevant to the

topic

of this article.

However, apart

from

the

precision

and

intricacy

of

many

of

the architectural

constructions

of

the

Old

Kingdom,

there is another

argument

for

the

existence of

relatively

"ad-

vanced" mathematics

in

the first half of

the third millennium B.C.

This is that

although the

two

great

mathematical

texts

that have

survived,

the

Moscow and

the Rhind papyri, come from

the

Middle Kingdom

in the

twentieth and nineteenth

centuries

B.C.,

some of the problems set in them use measures that belong to the

Old

Kingdom,

which had

been

discarded

by

this time.20

Lauer's solution still

allowed some

later

Egyptians

to

have been

capable of

relatively advanced thought. He continued:

For the whole

length

of

the three

thousand

years

of

her

history, Egypt thus,

little

by

little, prepared

he

way

for the Greek scholars

who

like

Thales, Pythagoras,

and

Plato

came

to

study,

then even

to

teach,

like Euclid at the school

in

Alexandria.But it

was

in

theirphilosophicspirit,which

knew how to draw from the treasureamassed

by the

technicalpositivism

of the

Egyptians,

that

geometry

came

to

the

stage

of a

genuine

21

science.

Even this

degree

of

recognition

was too

much for

Neugebauer.

As

he

put

it

at

one

point:

"Ancient science was

the

product

of

a

very

few men and

these

few

happened

not to be

Egyptians."

In

1981 he

published

his

note "On the

Orienta-

tion of Pyramids,"

in

which

he

showed how

accurate

alignments

could be

made

without

sophisticated astronomy, simply by measuring

and

turning

the shadow

of

a

model

pyramid

or

the

capstone

over

a

period

of some weeks. There

is no

evidence,

one

way

or

the

other,

whether

this

was the method

used,

but it

would

seem

plausible,

if

only

because

pyramids appear

to

have

had

solar

rather than

stellar

cultic

associations.

Nevertheless,

the

requirement

of what

Neugebauer

concedes

to be "remarkable

accuracy

of

.

..

orientation of

the

Great

Pyramid,"

a

structure

of

extraordinary sophistication,

indicates

very

serious

religious

and

theoretical

concerns.22

Thus, Neugebauer's

choice of the word

"primitive"

to

describe

the

alignment

seems

inappropriate;

the word

is-as

we

shall see-indic-

ative of his

general opinion

of

the ancient

Egyptians.

There

is

little doubt that this modern

view of the

Egyptians'

lack

of

mathemat-

ics and science has

been influenced

by

a

distaste

for the

theology

and

metaphys-

ics

in

which

much

of

Egyptian-and Platonic-knowledge

was embedded

and

by

progressivist views that no one who lived so early could have been so sophisti-

19

Ibid., pp. 1-3. For a more skeptical view of this see Robins and Shute, "Mathematical ases"

(cit. n. 14), p. 109.

20

Robins and Shute,

Rhind

Mathematical Papyrus (cit.

n.

13), p.

58.

21

Lauer, Observations sur les pyramides (cit. n. 18), p. 10.

22

Neugebauer,ExactSciences inAntiquity cit. n. 2), p. 91;andNeugebauer,"Onthe Orientation

of Pyramids," Centaurus, 1980, 24:1-3, on p. 2.




8/13

602

BERNAL:

ANIMADVERSIONS

cated.

It

may also

have

been reinforced

by assumptions,

almost universal

in

the

nineteenth and early

twentieth

centuries,

that

no Africans

of any sort could have

been capable of such great

intellectual achievements.

An indication

that

such

attitudes

may

have had an

impact

even

on such a

magnificentchampion of

liberalism and foe to racism

as Neugebauer comes in

one

of

his bibliographicalnotes,

where

the

first

book

he

recommended

"for

a

deeper understanding

of the

background

hat

determined

he

characterof

Egyp-

tian arithmetic" was Lucien

Levy-Bruhl's Fonctions mentales dans les societes

inferieures.Levy-Bruhl

was far

from the

worst

of his

generation.Nevertheless

he

belonged

to

it,

and it

was appropriate

hat his work was translated nto English as

How Natives

Think.23

Having said this,

there is no doubt

that

Neugebauer

had

some substantialar-

guments

to back his case.

The

strongest

of these

were

his claims

that

none of the

survivingmathematicalpapyrifrompharaonicEgyptcontained what he believed

to be sophisticated

calculations

and that

the

Egyptians'systems

of

numbersand

fractions were too crude

for profound

mathematicaland

astronomical hought of

the kind that had been

attributed

to

them. There are

seven

major arguments

against this position.

1. The strong possibility

that-pace Neugebauer-the survivingEgyptianpapyri do

contain "advanced"

mathematics.

2. Parallels rom Mesopotamia

and PtolemaicEgyptshowing that one cannot rely on

the papyrological

record

to

gauge

the full

range

of

pharaonic

Egyptian

"science."

3. The general agreement hat Egyptiangeometrywas equal to or better than that of

Mesopotamia, in conjunction

with the conventional wisdom that one

of

the chief

contributions

of the Greeks

to

Mesopotamian

"arithmetic"was

geometric modeling,

which suggests

that the

geometrical nput may

well

have come from Egypt.

4. The coordinationof

sophisticated geometry

and

computation

n

Egypt

with

ex-

traordinarypractical

achievements.

5.

The Greek insistence that

they

learned

mathematics-and medicine-not

from

Mesopotamia

but

from

Egypt.

6. The Greek

adoption

of an

Egyptian

rather han a

Mesopotamian

alendar.

7. The facts that much

of Hellenistic and Roman

science

took

place

in

Egypt,

not

Greece,

and that

although hey

wrote in Greek some of

its

practitioners, ncluding

he

astronomerPtolemy, were Egyptian.

The

first

argument

s buttressed

by

the

fact

that,

as

we have

seen, Neugebauer

preferred

"more

primitive

interpretations"

nd therefore could have

overlooked

evidence of

more

sophisticated

work. Thus

we

must

allow

for the

possibility

that

the

surviving

texts

contain or

refer to

elements

that are

more

sophisticated

than

he and

some

other twentieth-century

historians

of science have

supposed.

There

is little doubt about

the

employment

of

irrational

numbers,

mentioned

above,

and

the use of

arithmeticaland

geometricalprogressions

n the

Rhind

Papyrus prob-

lems 40

and

79.24

The Soviet scholar

V. V.

Struve,

who

was the

first to

study

the

Moscow Math-

ematical

Papyrus,

was much more

respectful

than

Neugebauer.

He

wrote,

for

instance,

that "we

must admit that

in

mechanics

the

Egyptians

had

more knowl-

23

Neugebauer, Exact Sciences

in Antiquity, p. 92; and Lucien Levy-Bruhl, How Natives

Think

(New York: Knopf, 1926).

24

Robins and Shute,

Rhind Mathematical

Papyrus (cit.

n.

13), pp. 42-43,

56.




9/13

CULTURES

F ANCIENT CIENCE

603

edge than we wanted to

believe."

He

was convinced that

this

papyrus and

the

Rhind Mathematical Papyrus

demonstrated

a theoretical

knowledge

of the

vol-

ume of a truncated pyramid,

and he

has

been followed

in

this

interpretation by

later scholars. Given the many pyramids successfully constructed during the Old

and Middle Kingdoms,

this

would not

in

itself

seem

unlikely. Archimedes,

how-

ever, maintained

in the third

century

B.C. that the volumes of

pyramids

were

first

measured by Eudoxos of Knidos

a hundred

years

earlier.25

Here, as

in

some

other

instances,

Archimedes

was

knowingly

or

unknowingly

mistaken.

Even

so,

it

is possible

that Eudoxos

was the first

to transmit the

for-

mulas

to Greece. Eudoxos

spent many years

in

Egypt

and

was

reported

to

have

learned Egyptian

and

to

have made

translations,

some

of which may

well have

come from

the

Book of the Dead,

into Greek. As

Giorgio

de

Santillana

pointed

out,

it is

unlikely

that Eudoxos

translated

these texts

merely

for their entertain-

ment value; it is much more probable that he believed that they contained eso-

teric astronomical information.26 This raises

the

important suggestion that Egyp-

tian

religious

and

mystical writings

and

drawings may

well

contain

esoteric

mathematical

and

astronomical

wisdom.

To

return

to

earth with

the

particular

case

of the measurement of

the

surface

area of either

a

semicylinder

or

a

hemisphere

in

the

Moscow

Papyrus:

Richard

Gillings,

who

believes the measurement

refers

to

the

latter, describes

the

Egyp-

tian

operations

and writes:

If this interpretation.. is the correctone, then the scribe who derived the formula

anticipated

Archimedes

by 1,500 years

Let

us, however,

be

perfectly

clear

[that]

in

neither case has

any proof

that either

Acylinder

=

1-rTdhr

Ahemisphere

=

27Tr2been

2

established

by

the

Egyptian

scribe

that is at all

comparable

with

the

clarity

of

the

demonstrations

f the Greeks Dinostratosand

Archimedes.

All we can

say

is

that,

in

the

specific

case

in

hand,

the mechanical

operations performed

are

consistent

with

these

operations

which would be made

by

someone

applying

these

formulas even

though

the order and

notation

might

be different.27

In

general,

it

is clear

that the

specifically mathematical papyri give consider-

able indications of sophisticated operations. As Struve put it in the conclusion of

his study

of the Moscow

Papyrus:

These

new facts

through

which the Edwin Smith and Moscow

papyri

have

enriched

our

knowledge, oblige

us to make

a radical

revision of

the

evaluationmade

up

to now

of

Egyptian

"science"

[Wissenschaft].

Problems such as the research

into

the

func-

tions

of

the brainor the surfacearea of a

sphere

do

not

belong

to the

range

of

practical

"scientific"

questions

of a

primitive

culture.

They

are

purely

theoretical

problems.

25

V.

V. Struve, "MathematischerPapyrus

des

staatlischen

Museums

der schonen kuinstein

Moskau," Quellen und Studien zur Geschichte der Mathematik (Pt. A), 1930, 1:184; and Paul Ver

Eecke, Les oeuvres completes d'ArchimedeParis:Blanchard,1960),p.

xxxi.

Among

later

scholars

who have supportedStruve's view

(on pp. 174-176)

see

Gillings, Mathematics

n

the Time of the

Pharaohs (cit.

n.

13), pp. 187-194;

and Robins and

Shute,

Rhind

Mathematical

Papyrus,

p. 48.

26

Giorgio

de

Santillana,

"On

Forgotten

Sources in the

History

of

Science,"

in

Scientific Change:

Historical Studies in the Intellectual, Social, and Technical Conditions for Scientific Discovery and

Technical Invention, from

Antiquity to the Present, ed.

A.

C. Crombie (London: Heinemann,

1962),

pp.

813-828,

on

p.

814.

27

Gillings, Mathematics in

the

Time

of

the

Pharaohs (cit.

n.

13), p.

200.




10/13

604

BERNAL: ANIMADVERSIONS

Or

earlier:

The Moscow Papyrus ... confirms

n

a strikingway

the mathematicalknowledge of

the Egyptian scholars and

we no

longer

have

any reason

to reject the claims of the

Greek writers that the Egyptianswere the teachersof the Greeksin geometry.28

Objectionsby Neugebauer

and others

to

Struve's

specificinterpretation f

the

surface

area of

a

hemisphere

have now been answered.29

Similarly,as mentioned

above, claimsfor the use of "Pythagorean"riangles

and the

sophisticationof the

measurementof

the

volume of

the

truncated

pyramid

have both survived earlier

skepticism.

If these bases of

Struve's

general

case still

stand,

should one

accept

Neugebauer's

dismissal of

it?

Even if one were to concede Neugebauer's argumentthat the mathematics

containedin these papyriis merely practicaland primitive,there is the second

argument:

he

strong

likelihood

that more

sophisticated

work

was recorded on

others that

have

not

been

preserved.

Lauer raised

the

point

that

all

reports

indi-

cate

that the

Egyptianpriests

were

secretive about

their

writings;thereforethere

would have

been few

copies

and

the

chances of

their

survival would have been

slim.

It should

be

emphasized

that

relatively

few

papyri

of any

kind

have sur-

vived. This is very

different rom

Mesopotamia,

where the

baked clay tablets are

remarkablydurable and hundreds of thousands

of them have been

discovered.

The problem

with

Mesopotamian

exts is not

a

lack

of them but the

difficultyof

finding enough Assyriologists

to read and

publish

them.

Even

here, however,

there are gaps in what exists. Neugebauerpoints out that the "greatmajority"of

mathematical

tablets come from one of

two

periods,

the

Old

Babylonian pe-

riod-of

two hundred

years-in

the

first

half

of

the second millennium

B.C. and

the Seleucid period. Continuities

between

the two

sets

of texts

make

it

clear that

sophisticated

mathematicswas

carried out

in the

twelve or more centuries that

intervened. However,

there

is

no record of this.30

The

situation

is far

worse

in

Egypt,

and there

is

no

doubt

that

most of the

papyri

written and all

of those

that have survived

were texts

used for

teaching

scribes

techniques

that

were useful for

practical

accounting

ratherthan "state

of

the art" advanced

mathematics.31

An instructiveparallelcan be seen in the Ptole-

maic

period. Many

more

mathematical

papyri

have been found

from

these

few

centuries

than from the whole

pharaonic period, yet

none of these

go beyond

book

1

of Euclid

or

give any

indicationsof

the

extraordinary ophistication

of the

work

we know from textual transmission

o have been

takingplace

in

Hellenistic

Egypt.

Thus

the

argument

from

silence,

which should

always

be

applied spar-

ingly,

should

be

used

with

particular

aution

in

evaluating

he

absence of textual

proof

of advanced

Egyptian

mathematics.

Against this,

it is

argued

that the few texts that do exist show

a

consistency

of

techniquesand notationthat makes it impossiblefor the Egyptiansto have pro-

duced

sophisticated

mathematics. This

brings

us to

the third

argumentagainst

skepticism:Egyptian

numericalnotation

may

not have been as flexible and

help-

28

Struve, "Mathematischer apyrus" cit.

n. 25), pp. 183, 185.

29

Gillings,Mathematics

in the Time

of

the Pharaohs (cit.

n.

13),pp. 194-201.

30

Neugebauer,Exact Sciences in Antiquity

(cit. n. 2), p. 29.

31

Robins and Shute,

Rhind Mathematical

Papyrus (cit. n. 13), p. 58.




11/13

CULTURES

OF

ANCIENT SCIENCE

605

ful

as

that

of

the

Mesopotamians,

but it

was,

if

anything,

better than that in

which

the

Greeks wrote their

sophisticated formulas. There is no doubt

that Egyptian

mathematicswas

based on

very simple

principles;

on the other

hand,

the

existing

papyri show

that

extraordinarily

laborate mathematical

tructureswere

erected

upon them.

Neugebauer

admits that while the

Egyptians

were

not as

good

in

their

arithme-

tic as the Babylonians,

their

geometry

was

equally good;

and if we

are to

believe

other scholars' interpretationsof the Moscow

Papyrus, Egyptians were able

to

carry

out

geometricaloperations

that were

beyond

those

of the

Mesopotamians.

The

notion

that the

Egyptians

were

the

better

geometers

fits both with

their

unparalleledarchitectural achievements

and with

their

reputation

among

the

Greeks

as

the founders of

geometry

and their

teachers

in

it.32

Given

this

concern

with

geometry,

it

is

not

surprising

that there

are

many

direct and indirectproofs that Egyptiansrelied on plans for their architectural

constructions.

Struve

may

have been

exaggerating

when he

wrote,

"The

Egyp-

tian

plans are as correct as those

of

modern

engineers."33Nevertheless, there is

no

reason to suppose

that

they were inferior o those of

the

Greeks and

Romans.

According

to

the

Egyptians,

the

tradition

of

makingplans

went

back to

Imho-

tep, at the beginning of

the

third

dynasty, circa 3000 B.C., but

most modern

scholars

have

understood this claim

merely

as

a

mythical

projection

onto the

deified

prototype

of

all

architects.

However,

it

is now

proven

that

architectural

plans were used

during

the Old

Kingdom

and that

Imhotep

did

design

the

Step

Pyramidand the elaboratecomplex of buildingsaroundit. Furthermore,an os-

tracon

found

at

the

Step Pyramid

does contain measurements or a vault.34

This

coordination

of

geometry

and

computation

with

architecture

constitutes

the fourth

argumentagainst

modern

denials that

the

Egyptianspossessed a supe-

rior mathematics.

While

the textual evidence for

such

knowledge

can be con-

strued

as

ambiguous,

the

case

for it is

greatly strengthenedby

the

architectural

evidence.

In

addition to the

pyramids

there were

temples,

granaries,

and

irriga-

tion networks

on

huge scales

that

required

extraordinary lanning

and the

ability

to visualize

these

structures

n

advance on

writing

or

drawing

surfaces.

The

fifth

reason

for

supposing

that the

Egyptians

had

sophisticated

mathemat-

ics is

that the Greeks

said so.

Writerson the

subject

were

unanimous

hat

Egyp-

tian mathematics

and

astronomy

were

superior

to their own and

that while

only

two

Greek mathematicianswere

supposed

to have

studied

in

Mesopotamia,

the

majority

of Greek

scientists, astronomers,

and

mathematicianshad

studied or

spent

time

in

Egypt.

These

reports

are treatedwith

skepticism

by

modernhistoriansof

science,

who

know

that there was

no

Egyptian

science or mathematicsworth

studying.

How-

32

See Herodotos,

2.109; DiodorosSikeliotes, 1.69.5,

81.3, 94.3; Aristotle,Metaphysics1.1.

(981b);

Hero, Geometria2; Strabo, 16.2, 24 and 17.1, 3; and Clementof Alexandria,Stromateis 1.74.2. See

also Cheikh Anta Diop, Civilization or

Barbarism:

An

Authentic Anthropology, trans.

Yaa-Lengi

Meema Ngemi (New York:Lawrence

Hill, 1991),pp. 257-258.

33

Struve,"Mathematischer

apyrus"

cit.

n.

25), pp. 163-165,

on

p. 165.

34

SergioDonadoni, "Plan," n Lexikon

der Agyptologie,ed.

WolfgangHelck

and

EberhardOtto,

5

vols. (Wiesbaden:

Harrassowitz,1977-1984),

Vol.

4, cols. 1058-1060. For

my dating see

Martin

Bernal, Black Athena:

The Afroasiatic Roots of Classical

Civilization, Vol. 2: The Archaeological

and Documentary

Evidence(London:

Free

Association

Books;

New

Brunswick,

N.J.: RutgersUniv.

Press, 1991),pp.

206-216.




12/13

606

BERNAL: ANIMADVERSIONS

ever, as

de Santillanawrote about

Eudoxos,

who

undoubtedlystudied in Egypt:

"We

are asked

to

admit, then,

that the

greatest

mathematician f

Greece learned

Egyptian and tried to work on astronomy

n

Egypt without realizingthat he was

wasting his time."35

There is little doubt that after the Persianconquests the mathematicsand as-

tronomy of Egypt

drew from both

Egyptian

and

Mesopotamiansources. How-

ever, the Greek belief that it was an Egyptiantraditionstrengthens he case that

the native component was significant.

The

sixth argumentagainst

the

skeptics

is the fact that the

Greeksadoptedan

Egyptian rather than a Mesopotamiancalendar.

Apart

from the

greater conve-

nience of the Egyptiancalendar,this adoption s indicativeof what seems to have

been

a

wider Greek tendency to drawfromnearbyEgyptrather hanmore distant

Mesopotamia.

The final argument is that in Hellenistic times, while Athens remainedthe

center of

Greek

philosophicalstudies, nearly

all "Greek"

science took place in

Egypt. This was partly the result of Ptolemaic

patronage,

but if

we are to believe

Greek

and Roman

sources, they

also drew and built on

Egyptian

wisdom. It is

striking

that

Euclid worked

in

Egypt

at the

very beginning

of the

Ptolemaic

period,

that

is

to

say

a mere

fifty years

after Eudoxos had felt

the need to learn

Egyptian

in

order to study mathematics and

astronomy. Thus,

it

would seem

more

accurate to

view

Euclid's work as

a

synthesis

of

Greek and

Egyptian ge-

ometry

than as an

imposition

of the Greek rational mind

on muddled oriental

thinking.

While it is true that Babylonianmathematicsand astronomyflourishedunder

the

Seleucids,

as

already noted,

most

of

the

great

"Greek"

scientists

wrote

in

Greek

but

lived in

Egypt,

and some indeed

may

have been

Egyptian.

For exam-

ple,

the astronomer

Ptolemy

was known

in

early

Arabic

writings

as DAc

acid,

"the

Upper Egyptian.

36

It

seems to

be

generally accepted

that the

great

Greek

contribution

o mathe-

matics

and

astronomy

was the introductionof

geometric modeling,

in

particular

the

transposition

of

Mesopotamian

arithmetical

astronomical

cycles

into

rotating

spheres.37However,

the Greeks themselves believed that

geometry developed

in

Egypt, a view supportedby Egyptianarchitectural ophisticationand the math-

ematical

papyri. Furthermore,

hose most

responsible

for the Greek view

of the

heavens

as

spinning spheres,

Plato and

Eudoxos,

were

reported

to have

spent

time in

Egypt

and were known for

their

deep

admiration

of

Egyptian

wisdom.38

We

have seen how

particularly

close

Eudoxos's

association

was with

Egyptian

priests,

and

it

was

precisely

Eudoxos who

established the new

astronomy

of

complex

concentric

spheres.

I

believe that these seven

arguments

present

a

very strong

case indeed that

there

were

rich

mathematical-particularlygeometrical-and

astronomical radi-

tions in Egypt by the time Greek scholars came in contact with Egyptian earned

priests.

After

the

first Persian

conquest

of

Egypt,

in

the

sixth

centuryB.C., Egyp-

3

De Santillana,"ForgottenSources" (cit. n. 26), p. 814.

36

J. F. Weidler,Historia astronomiae (Wittenberg:Gottlieb, 1741),p.

177.

37

Pingree, "Hellenophilia ersus the History of Science" (cit. n. 1).

38

See the bibliography

n

WhitneyDavis, "Plato on EgyptianArt,"Journal of Egyptian Archae-

ology, 1979, 66:121-127,on p. 122,

n. 3.




13/13

CULTURES

OF

ANCIENT

SCIENCE

607

tian mathematics

and

astronomy

were

substantially nfluencedby

Mesopotamian

"scientific"

hought,

a

process

which

continued

n

Ptolemaicand Roman

Egypt.39

The Egyptianmedical tradition

appears

to have been less

affected

by

Mesopota-

mia.

In

general,

the

"scientific"

riumphs

of

Hellenistic

Egypt

would

seem to be

the result of

propitioussocial,

economic,

and

politicalconditions

and

the

meeting

of three "scientific"

raditions,

those

of

Egypt,

Mesopotamia,

and

Greece. How-

ever, the two former

were

much older than

the

third,

reaching

back to the

third

millennium

or

beyond,

and more

substantial.

It

should also be noted

that the

point

at

which

the

Greeks

"plugged

nto" Near Eastern "science"

was

Egypt;

this

was the reason

that the

Greeks

always emphasized

he

depth

and extent

of

Egyp-

tian wisdom.

The arbitrarinessof the

applicationof the word "science" to ancient

civiliza-

tions was noted

at

the

beginningof this essay.

I

suppose,

like

Humpty-Dumpty,

we can use wordsmore or less as we please. However, the only way to claimthat

the

Greeks

were

the first Western scientists is to define

"science" as

"Greek

science."

If

less circular definitions are

used,

it is

impossible

to

exclude the

practice

and

theory

of

some

much earlier

Mesopotamians

and

Egyptians.

3 A clear

example

of

this can be seen in the

fragment

discussed

by Neugebauer

n

his

exquisite

swan song:Neugebauer,

"A

Babylonian

Lunar

Ephemeris

rom Roman

Egypt,"

in A

ScientificHu-

manist: Studies in

Memory of

Abraham

Sachs,

ed. E.

Leichty

et al.

(Occasional

Publications of

the

Samuel Noah

Kramer

Fund, 9)

(Philadelphia:University Museum,

Univ.

Pennsylvania, 1988), pp.

301-304.

Documents

Bernal, Martin_Animadversions on the Origins of Western Science_Isis, 83, 4_1992_596-607