Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's Muʿādalāt

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Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's Muʿādalāt

Sharaf al-Dīn al-Ṭūsī, oeuvres mathématiques: Algèbre et géométrie au XIIe siècle by RoshdiRashedReview by: J. L. Berggren and Sharaf Al-Dīn Al-TūsīJournal of the American Oriental Society, Vol. 110, No. 2 (Apr. - Jun., 1990), pp. 304-309Published by: American Oriental SocietyStable URL: http://www.jstor.org/stable/604533 .

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INNOVATION

AND

TRADITION

IN SHARAF

AL-DIN

AL-TUSI'S

AL-MUcADALAT*

J. L.

BERGGREN

SIMON

FRASER UNIVERSITY

The

largest treatise

in Roshdi

Rashed's

edition of

the mathematical

works of Sharaf

al-Din

al-Tiisi

is Al-mu'ddaldt,

a work

devoted

to the solution

of

cubic

equations. This

treatise

shows

that Islamic

authors

went

considerably beyond

the achievements

of 'Umar al-Khayyami

in

three

areas: (1)

finding

conditions

for the existence

of solutions

to

cubic

equations, (2)

discovering

algorithms

for calculating

these solutions,

and (3) justifying

these algorithms.

Although

so

much

is

clear, it is

still no easy

task to understand

Sharaf

al-Din's

thought processes

and so

achieve

an

understanding

of the

historical

filiations

of the

document.

Rashed has argued

that Sharaf

al-Din

discovered

the derivative

of cubic polynomials

and realized

its

significance

for investigating

conditions

under

which cubic equations

were solvable;

however, other scholars

have

suggested

quite

different

explanations

of Sharaf

al-Din's thinking,

which connect

it with mathematics

found

in Euclid

or Archimedes.

Our

purpose

in the

present

essay review

is to decide

which of

these

three

readings

seems best

to

fit

the

text.

HISTORIANS

OF

SCIENCE,

NO LESS than

scientists

themselves,

are accustomed

to

seeing

their

discipline

change

slowly.

Most

scholars spend

their

lifetimes

making

small

changes

or

additions,

and

their

satisfac-

tion

comes

from knowing

that their

work,

if done

well,

will

be something

that

the next generation

can

build

upon.

Rarely

does

a historian

find

a document

that

forces

major

changes

of fundamental

ideas

about

a

given

field,

and

for

that reason there was both excite-

ment

and

caution

when

Prof.

Rashed

announced

in

1974

his

discovery

of

a

document

that

would

compel

us to

revise

substantially

our history

of

Muslim

achievements

in the

theory

of the solution

of

algebraic

equations.

The

history

of the

theory

of equations

has

already

provided

several

instances

of discoveries

of

source

material

resulting

in major

changes

in our

understand-

ing

of

the

history

of

mathematics,

an

outstanding

instance

being

the

decipherment

of the

mathematical

cuneiform

texts.

These

showed

that

the

Babylonians

of Hammurabi's time knew both the exact solution to

the general

equation

of

degree

2 and good

numerical

methods

to solve

the equation

x2

=

m.

*

This

is a review

article

of: Sharaf

al-DTn

al-

Tus,

oeuvres

mat/Umatiques:

Algebre

et

geometrie

au Xiie

siecle.

Edited

and

translated

by

ROSHDI

RASHED.

vols.

Collection

Sci-

ences

et Philosophie

Arabes:

Textes

et

Etudes.

Paris:

SOCIfTi

D'EDITION

LESBELLES

ETTRES,"

986.

Pp.

cxci

+

129

+

xxvi

(Arabic

summary),

and

cxliii +

166.

FF

640.

Another

instance

is

Fr.

Woepcke's

discovery

in

1851 of

Omar Khayyam's

theory

of cubic

equations

based

on the

theory

of

the conic sections.

Woepcke's

discovery,

together

with

the extracts

that

he presented

from the

other

works

of the

10th-1 1th

centuries,

made

it clear

that

not only

had

Omar Khayyam

found

geometric

solutions

to

all

possible

cubic

equations,

but also

that

his work

was part

of

a major current

of

Islamic work on problems whose origins lay in such

classical

problems

as

duplicating

the

cube

and

trisect-

ing

an

arbitrary

angle.

Rashed

neatly

characterizes

this

work

as

involving

a

"double-translation,"

first

of a geometric

problem

(such

as

the ones

referred

to above)

into

the

algebraic

problem

of solving

a cubic

equation

and

then

of

that

algebraic

problem

into

the

geometric

problem

of

intersecting

conic

sections.

Thus,

after

the

second

translation

one

is

back

to a geometric

problem,

but

one

of

a

standard

type

and

very

different

from

the

initial

problem.

A third

major

discovery

in this

field was

that

of

P. Luckey,

about

a century

after

Woepcke's

publica-

tion of

Khayyam's

Algebra,

who

showed

that by

the

time

of al-Kashli,

Muslim mathematicians

had

not

only

discovered

an

efficient numerical

method

for

finding

roots

of equations,

known today

as the

Ruffini-

Horner

method,

but they

had

also developed

the full

theory

of decimal

fractions.

There

were,

however,

some

gaps

in the story

as

it

had evolved

thus

far. For

example,

Omar

Khayyam's

304

This content downloaded from 188.72.126.55 on Fri, 13 Jun 2014 05:51:03 AMAll use subject to JSTOR Terms and Conditions






BERGGREN: Sharaf al-Din al- TasT's

Al-mu'adalt

305

discussion

of the existence of solutions

for

certain

types of cubics was

incomplete.

In

certain

cases,

in

fact,

he

could

only argue

from

a

geometrical

diagram,

and, judged

in

terms

of the

logical

standards

of

his

day, Khayyam's

theory

needed

further

work.

In addition, certain other developments in Euro-

pean mathematics

that were relevant

to

the

theory

of

equations were

altogether

absent

from medieval

Mus-

lim

mathematics.

First of

all, Omar

Khayyam

said

he

had tried and

failed

to

find algebraic

solutions for the

cubic

and quartic

equations,

and

it was left

to

mathe-

maticians of

16th-century

Italy

to

discover

the

formu-

lae for the

roots.

Also absent

was

what Rashed calls

"local analysis"

of the

way

that values

of a

polynomial

f(x) vary in the

vicinity

of

a real number

x..

Such

studies

led

to theoretical

and

practical

advances

in

the

theory

of

equations

in

the

hands

of

17th-century

mathematicians

like

Fermat

and

Newton,

but

there

was no

trace

of

them

in

medieval Islam.

Finally

there

was

the

question

of

proving

the

validity

of the nu-

merical

procedures

for

finding

roots

of

algebraic

equations.

Although

Omar

Khayyam

claims to

have

given

proofs

of

methods

for

solving xm

=

r,

and there

are

reports

of

a

longer

treatise

by

al-KdshT

hat

could

conceivably

justify

one of his

numerical

methods,

it

was still

the case

that

the first known

proofs

of

the

validity

of

numerical methods were

European.

This

was the situation

when, in 1974,

the noted

historian of

mathematics

in

medieval

Islam,

Roshdi

Rashed, gave the

scholarly world

its first view

of

the

contents of a treatise on cubic equations by the 12th-

century

mathematician Sharaf

al-Din

al-TIsT,

who

had been known

up

to

that

time

principally

as

the

inventor of

a linear

form of the

astrolabe. The

ac-

count

Rashed

provided made

it

appear

that

Sharaf

al-Din

al-TusT's treatise

would alter

fundamentally

our

opinions

on the

last two of the

three

matters

mentioned

above, and

so

historians

of

mathematics

have

been

waiting

with

considerable interest

for the

publication

of an edition

of the complete

treatise

promised in

Rashed's paper of

1974.

In

1986

this edition

appeared, and what

may have

been

some

scholars'

impatience at what

appeared to

be an

excessive

delay must,

on

inspection

of the

final

product, turn to admiration

that Rashed

was able to

finish such a

difficult project during a time

when he

was

also

publishing the

newly discovered

four books

of

Diophantus

in

Arabic as

well as (with A.

Djebbar)

editions

of

Omar

Khayyam's algebraic works.

Simply as a

piece

of

mathematical writing Sharaf

al-D-n's Arabic text

is full of

ambiguities.

Had Rashed

done

nothing beyond

producing an

exemplary edition

of

the

Arabic text

and a French

translation which

manages

fidelity

to that text

while,

at the same

time,

generally

clarifying

its

ambiguities,

he would

deserve

our

gratitude.

However,

in

addition

to

the

ambiguities

of the

text,

which may be laid at

the

doorstep

of Sharaf

al-D-n

al-Tusl, there is the complicating fact that the scribe

omitted the tableaux

illustrating

the

many

numerical

algorithms

that the

text

describes.

Such

algorithms

are one of the

chief

reasons the text is so

interesting,

but they are difficult to follow

without

the

tableaux,

and this reviewer can

only

salute

Rashed's

success

in

restoring them.

Unfortunately,

the

restorations

as

presented

are

not

easy to

follow,

and Rashed

would

have

done better

to

present a series of tableaux for each

case,

with each

tableau

showing

what would have been on

the

board

(takht) at a

given

time,

rather than

trying

to

show

all

of the

steps

in one

diagram.

Moreover,

Rashed's

claim that Sharaf al-D-n's

takht refers

only

to a table

rather than, as it earlier

did,

to a

board covered with

some fine

powder,

is

unconvincing.

Such boards are

extremely

convenient

for

doing

the

Islamic arithmetic

algorithms,

and Sharaf al-Din's statements create the

definite

impression

that

he

was

working

with three or

four

rows

of

numbers and

that,

rather

than

accumu-

lating rows,

he

replaced

one row

by

another as the

algorithm progressed.

Although, on

the whole,

Rashed has done

a finejob

in

editing a

difficult text, there

are occasions

on which

he has

intervened

unnecessarily. One

such, in this

reviewer's opinion, occurs on p. 34 of vol. I (and at

similar

points in

the

mathematical

exposition in the

following

pages, e.g., lines 16

and 19 of p.

26) where

Rashed has

inserted

the

phrase wa

'l-a'll after the

word

al-'asfal.

In

fact

there

is

no

reason

to

move the

"3"

in the

upper

row to

the

right. We can

just leave it

where

it

is, provided

that it

does

not

bother us to

see

"321" written

"3

2

1."

And

it

is interesting to

note that

in

a

case where the row

of the

answer also serves

to

record

part

of

the

coefficients of

powers of x, as in

the

case on

pp. 78-79, so

it

must

be

shifted, Sharaf al.-D-n

does

mention the

shifting.

Some

other

problems with

the

editing are listed in

the paper

"Sharaf al-D-n

al-

TUsTon

the

number

of

roots of

cubic equations"

by

J. P.

Hogendijk, to

appear in

Historia

Mathematica.

This

also has a list of

misprints,

which we

supplement

with

the

following.

In

each

entry the

numerals,

n,m(*)

refer to

line m

(from the

bottom) of page n.

Following

this

comes the error

and then the

correction.

MISPRINTS: Vol

I.

xx,

12*, al/2, b1/2; XXII,

5*, c,

cO;

XXVII, 1*,

Pk,

Pk;

LVI, 10, je,

(j+l)e; LVI, 12,

ne

(second

time), je; LXI,

8, <, <;

LXII,

7*, <,

<;

LXV,

12,







306 Journal of the American Oriental Society 110.2

(1990)

Sn,

sn; LXIX,

6*,

A,

a; LXXVII,

9,

SI, S.;

LXXIX, first

ine

after tableau,

f, F;

35 (Arabic),

note

8, 455765,

45576;

15, 1*, carr6, carre;

34, 6, le

zero,

les

zeros; 79,

a "4"

under the

first "9"

on line

16 was omitted;

119,

10*,

Tk, 3Tk;

120,

16,

Tk2

and

Tk,

Tk

and

Tk2.

Vol.

II.

xviii, 2*, x2+, x3+; 3, 12, DC et, DC a BC et; 35, 8,

"le nombre

en question"

does

not

make it clear

the

Arabic refers

to

the number

in the

original

cubic

equation;

51,

13, AB en

G,

AB

en deux

moities

a

G;

37, 8,

JK,

IK.

MATHEMATICAL:

The statement

(vol. I, p.

LIX) that

if

c

>

0

f may have

a single,

simple

positive

root is

incorrect.

If there

were

a positive

root in

this case

it

would have

to be

a double

root. Also,

on p.

LXII the

statement

in

the last paragraph

that a cubic

equation

has a unique,

positive

simple

root

when it

can be

written

in the form

x3

+

bx

=

ax2

+ N (with

a, b and

N positive) is false, as can be seen from the example

x3

lIx

=

6x2

+

6, which

has the three

distinct

posi-

tive

roots 1,

2 and 3.

With such

matters

out

of the way

we may

turn

to

the

question

of historical

understanding,

for

Rashed

makes claims

for

the contents

of the

treatise that,

if

correct,

would

force

us

to

revise radically

our estimate

of what was possible

for medieval mathematicians.

In

particular

Rashed finds

in

this

text evidence

for

the

following concepts

and approaches,

which up

to the

present,

have

been

seen as

characteristic

of 17th-

century

mathematics:

1.

The notion

of local

variation

of

a

function.

2.

The

application

of the derivative

to locate

max-

imum values

of functions

over

a

given

domain

of

numbers,

and

this

being

not

by

chance

but as

a result

of

(1).

3. The use

of

graphs

to

study

polynomial

equations.

Although

Rashed

makes a strong

case

for

the

first

two,

it

seems

that there

are

good

reasons for

believing

that

Sharaf

al-Din

is an

heir of

the

Hellenistic

mathe-

maticians,

not

a forerunner

of

Fermat;

but,

to see

why

this is so we

must

summarize

the

contents

of this

remarkable work. (To avoid constant repetitions we

state

once

and for

all that

a "solution"

to an

equation

refers

to

a

"positive

solution"

to

the

equation.)

Sharaf

al-Din

opens

with some

lemmas on

the

construction

of conic

sections

that his

work

uses,

and

he then

proceeds

to

enumerate

the 25

types

of

equa-

tions

of

degree

less

than

four.

(There

are

so

many

possibilities

because

medieval

mathematicians

did

not

accept negative

numbers

as coefficients.

Thus,

e.g.,

they

would

write

x3

+ 4X2-7x-3

=

0

as

x3

+

4x2

=

7x

+

3 and

would

regard this

as of a

type

different

from

x3 + 7x

=

4X2

+

3.)

Immediately

differences

with

the work of

Omar

Khayyam

show themselves.

As Rashed

points

out,

Omar

Khayyam orders

the

25 cases

according

to their

apparent complexity whereas for Sharaf al-D-n the

overarching

principle

for equations

of the

third

degree

is whether

or not

a given

equation

has a

solution.

Accordingly,

Sharaf

al-DTn

first discusses

the 20

cases

where there

is a solution

and,

at the

end, the five

cases

where the equation

has a solution

only

when

the

coefficients

satisfy

certain conditions.

Sharaf al-Din

follows a uniform

method

in all

of

the first

20

cases that

are not

either

trivial (e.g.,

X3

=

a) or do

not reduce

immediately

to a case

of

lower

degree

(e.g., x3

+ ax2

=

bx). He

first gives

a

geometric

construction

of a root,

stating

if necessary

(in

the case

of quadratics)

conditions

that

must

be

satisfied

if

there is

to be a root;

he then shows

how to

use a

tableau

(formed initially

from

the coefficients

of

the

equation)

to calculate

the successive

digits of the

root,

which is

always

321. Since

in many cases

the

arrangement

of

the numbers

on the tableau

varies

according

to the

relative sizes of

the coefficients

there

are

frequently

several

cases

for each type

of equation.

Occasionally

Sharaf al-Din

makes a slip (e.g., p.

27

and p.

35), and

it is typical

of

the care

with

which Rashed has

studied the

MSS that

he

points

these

places out to

the reader,

although

it

might

have been

more useful to a

wider class

of readers

had he also

pointed them out in French notes as well as in the

Arabic.

At the end

of the discussion

of the

tableaux

for

each type

of

equation

Sharaf

al-Din gives

a

justifica-

tion for

the

actual

positioning

of

the

various

coeffli-

cients

in the tableau

as

well as

for

the

algorithm

followed.

We

now have,

for the

first

time,

an

attempt

at

systematic

justification

of the

validity

of

a

wide

class

of

numerical

procedures,

something

that

opens

up a

new dimension

to

medieval

Muslim

mathematics.

The

fundamental

idea

of Sharaf

al-Din's

procedure

is a

simple

one,

namely

that

if

f(x)

is a

polynomial

and

f(x)

=

c

has the

root a

+

b

then

f(a

+

x)

=

c

has

the root b; however, devising the tableaux to record

the

changes

in the coefficients

when

a

e

x is sub-

stituted

for x is

by

no

means

a

simple

task and

Sharaf

al-Din

often shows

admirable ingenuity

in

using

his

tableaux

efficiently.

For

example

in

the case

of

x3

+

12x2

+

102x

=

34,345,395

his

tableau

has 34345395

on

one

line,

the successive digits

of

the

root

(3,2,1)

will

be filled

in

on the

line

above

it

and,

in the

two

lines below,

he keeps

track

of the coefficients

of the

x

and the

x2

terms

respectively.

At

least

it seems

so







BERGGREN:

Sharaf

al-Din

al-

TUsT's

Al-mu'adalat

307

when

he

begins,

but after the

first iteration

it

appears

that,

although

the coefficient

of

the x term

has

been

adjusted

correctly,

he

has not

changed

the

coefficient

of

the x2

term at all.

Only

after

some

thought

does

one

see that in

fact Sharaf

al-Din

has realized

that

the

rows above and below 34345395 together may be used

to

keep track

of

the

coefficient

of

the x2

term

and that

he

is

using

very

elegantly

the

entries on the

table for

a

variety

of

purposes.

Sharaf al-Din

is also alert

to

possibilities

of

making

his

work

easier.

Thus,

once

he

has

discussed the

quadratic

equation

x2

+

ax

=

b,

he

notes that

in

the

case

of the

equation

ax +

b

=

x2

setting

X

+

a

=

x

implies the relation

X2

+

aX

=

b,

which

can

be

solved

by the

previous

method,

and then

all one

has

to

do is

add a to

its root to

find the root

of the

original

equation

ax

+

b

=

x2. Of such

a

possibility

Omar

Khayyam, by

comparison, says

nothing.

Perhaps

he

knew

it,

but since he

was not

interested

in

numerical

methods he

probably

saw no

point

in

going

into

it,

which is

a

pity because

it

does show

a connection

between

two

seemingly

separate

cases,

and

so is

of

some

theoretical interest.

Rashed

has elected to

discuss

these

algorithms as

they

would

apply

to

finding

arbitrary

roots of

poly-

nomials of

arbitrary

degree.

On

the one

hand this

does make

clear what

Sharaf

al-Din

never

discusses,

how to

extend the

algorithms to

find

fractional

parts

of the

roots. On the

other

hand,

Rashed's

account of

the

procedures is

phrased so

generally, with

such an

abundant use of algebraic notation, that the reader

might

find

it

easier to

construct the

general

method

for

himself on

the

basis of the

text

and the

restored

tableaux,

and then return

to

Rashed's

general

form

of

the

algorithms, once

the

concrete

details

are

well in

hand, to

get

an

overview

of

the

procedure.

Rashed

has

quite

reasonably

chosen

to

write

ch. I

("The

Numerical

Resolution

of

Equations and

the

Ruffini-

Horner

Method")

for

a

reader

who

is

assumed

to be

familiar

with the

notation

and some

of the

results of

the

differential

calculus;

however,

he

does

not

make

the

task

of

such

readers

any

easier when

he

displays

what he

calls

SCH(N;A,

AO,

. ..,

An)

on

p.

lxxix

of

vol. I without explaining how to interpret the entries.

In

particular, the

reader

is

never

told

that an

entry

of

the

form

AX

is

not to

be

interpreted

as a

fraction

but

y

rather

as

a

shorthand for

Y

=

Z

+

AX,

where Z is

the

entry

in

the

row

above AX.

Rashed's

fondness

for

abstract

exposition

stems

from

his

evident

desire to

impress

upon the

reader

the

generality

of

Sharaf

al-Din's

methods,

but it

is

hard

to

understand

why

he uses

such

generality

of

exposi-

tion

when the

methods

are not

general.

For

example,

Sharaf

al-Din's idea

of

finding

the first

digit

of

the

root by

what

Rashed

calls "the

dominant"

polynomial

works (in

most

cases ) only because he

is

dealing

with

cubic

polynomials,

as

Rashed

admits,

but he

still

gives

a general exposition of the approach.

With the

beginning

of

the

discussion

of

the 21st

type of

equation

the format of

the

discussion

changes,

for

these are

cases where

a

solution

is

not

guaranteed.

Consequently

any

thorough discussion of

the

equa-

tions

must

consider the

conditions under which

a

solution

exists, and Sharaf

al-Din's

success

in

stating

these

conditions is a

good measure

of the

progress

made

in

Islamic

mathematics

since the time

of

Omar

Khayyam.

(The

importance that

Rashed attaches to

Sharaf

al-Din's

achievement

is

shown

by, e.g.,

his

discussion of

type

25,

where 22

pages

of

Arabic

text

call forth

36 pages of

transcription and another

22

pages of

commentary.)

Since

the cases

(21)-(25)

are

crucial

to

Rashed's

mathematical

analysis of the

treatise we

shall

sum-

marize

Sharaf al-Din's

discussion of

case

21,

since it

provides

a

good context

in

which

to

discuss one

of the

principal

problems

this

reviewer sees with

Rashed's

historical

interpretation.

Sharaf al-Din

begins

his

discussion

of the

equation

x3

+

c

=

ax2

by

showing

first

that a

>

x

and

then

that

x2(a

-

x)

=

c.

(He

thinks

of

the

segment

a

divided

into

two

parts, x

and a

-

x, so

that the

square

of the

first

times the

second is

equal

to

the

number

c.)

He

next shows that if x

=

2 a then x2(a

-

x) is

as

large as

4

33

possible,

namely

4 a3, so

this

gives him

the

necessary

27

condition

for the

existence of

a

solution to

x3

+

c

=

ax2, namely

that c

<

4 a3. It

also

tells him

immedi-

'

~~~

7

ately that

when c

=

4

a3 then x=

2

a

is the

unique

27

3

root

of x3 +

c

=

ax2

and

that

when c

<

4

a3

there

are

27

two

roots,

one in

the

interval

(0,

-

a)

and

the

other

23

in

the

interval (

a, a).

Sharaf

al-Din's

proof that

the

maximum

value

is

attained

when x

=

2

a is

a

straightforward (if

some-

3

what

long)

exercise in

the

manipulations

of

volumes

and

areas in

the

manner

of

Euclid's

Elements, Book

II,

and

its

proof

would

not

have

offered

serious

problems

to

any

good

mathematician

from

the 3rd

century

B.C.

onward.

Its

discovery is,

however,

another

matter, for

it is

one

thing

to

prove a

known

result

correct

and it is







308

Journal of the

American

Oriental

Society 110.2

(1990)

quite

another

to

discover

which

of the infinitely

many

possibilities

is

the correct

one. The question

then

is

how

Sharaf

al-Din

discovered

the right conditions

on

the coefficients

of the equation

in

this and

the

four

other cases.

Rashed has no doubt that it is by "a method

rediscovered

and

developed

by Fermat,

five

centuries

later"

(p.

xxvii).

According

to

Rashed,

at

some

stage

in his

analysis

of

the problem

Sharaf

al-Din

let x0

be

the point

where

the

function

f(x) attains

its

maximum

value

and

X a number

so that

xO

+ X is

less

than

the

upper

limit

for

the

roots. Sharaf

al-Din

then

com-

puted

f(xo)

-

f(x.

+ X)

=

2xO

(xO

+ a)X

- (b

-

x.2)X

+

terms positive

for all

relevant

values

of

X and

so

it

would

have been

evident

that

for

f(xo)

to

be

larger

than

f(xo

+ X) it

is sufficient

that

2xo(x0

+

a)

>

(b

-

x02).

A similar

analysis

would

have

shown

him

that

the

condition

2xo(xo

+

a)

<

(b

-

x02)

is

sufficient

to

guarantee

that

f(xo

-

X)

<

f(xO).

If both

conditions

were

to hold

simultaneously,

then f would

have

a

local

maximum

at

xo,

and

therefore

the equation

2xo(x0

+

a)

=

(b

-

x02)

will guarantee

that

xO

is a

local maximum.

This is,

however,

a

quadratic

equa-

tion for

xO,

so

its roots

are

easy

to find,

and

these

roots

are

precisely

the

roots

of

the derivative.

This is, as

Rashed

admits,

a conjectural

explanation

of Sharaf

al-Din's

thought

process,

based

on specula-

tion about

what

role

the above

expansions

of

f(xo)

-

f(xo

+

X),

and

f(xo)

-

f(x0

-

X)

may

have

played

in

Sharaf

al-Din's thought

processes.

In favor of

Ra-

shed's analysis it must be said that the expansions of

f(xo)

-

f(xo

+

X),

and

f(x.)

-

f(xo

-

X)

given

above

do

occur

in

Sharaf

al-Din's explanation

of

how

the

solutions

of the

cubic

in

question

may

be

derived

from

solutions

of

cubics

that

have

previously

been

solved.

Moreover,

the idea

of

analysis

using

two

unknowns,

xO

and

X,

is one that

goes

back at least

to

Diophantus

and

there

is

nothing

inherently

impossible

in

the notion

that Sharaf

al-Din thought

in this

way.

If

he

did

then

Rashed's

remarks

about

the

use

of

"local analysis"

and Sharaf

al-Din's studying

the

variation

of

a function

in

the

neighborhood

of a

local

extremum

would

be

much

to the

point;

however,

the

reviewer

would

still

have

to

dissent

from Rashed's

conjecture

that Sharaf

al-Din "reasoned

on the

basis

of

the

graph

defined by

x

>

0,

f(x)

>

0"

(p.

xxviii),

for

the

only

evidence

offered for

the conjecture

is

Rashed's

impression,

based

on "repeated

reading

of

the

Treatise."

However,

other

possible

interpretations

of the

text

have been

suggested,

among

them

one

by

Hogendijk

in

the

article

referred

to

above

and

another

in

Studies

in the

Exact

Sciences

in Medieval Islam,

by

al-Daffa

and Stroyls.

Since the suggestion

by Hogendijk

seems

to involve

fewer problems

we will present

it

first.

According

to

this account,

the

probable

course

of

Sharaf

al-Din's

analysis

for a

case like (23)

would

have

been something

like

the following:

If x is

the

point where f(x) =-x3

-

ax2 + bx assumes its max-

imum,

and

y any

point so

that

x

<

y

<

\fi,

then

the

difference

f(x)

-

f(y)

may

be written

(y2 - X2)(X

+

a)

-

(b

-

y2)(y

-X)

(essentially

this expression

can

be

found

in the

text),

and

this

may be

written

f(x)

-

f(y)

=

(y

-

x)[(y

+

x)(x

+

a)

-

(b

-

y2)].

One

may

now

show

that

since

y

>

x

a sufficient

condition

for

f(x)

-

f(y)

>

0

is that

2x(x

+ a)

>

b

-

x2, and

in

a

similar way,

one

can

show that

for

0 < y < x

a

suffici-

ent

condition

that

f(x)

-

f(y)

>

0

is that

2x(x

+ a)

<

b

-

x2.

Hence

a sufficient

condition

that

f(x)

>

f(y)

for both y

>

x and

y

<

x

is that

2x(x

+ a)

=

b

-x,

an

equality

that also

appears

in

the

text.

But this

is

equivalent

to the

condition

that

f'(x)

=

0,

and accord-

ing

to Hogendijk

it

was by

considerations

like

this

that, in each

of

the five

cases,

Sharaf

al-Din

came

to

the defining

condition

for x.

(It

must be

emphasized

that Sharaf

al-Din

stated

these

identities

in

geometri-

cal

language.

Thus

not only

were x3

and

ax2 treated

as

volumes

but even

b

was

represented

as

a

square

of

side

\fi,

so

that

bx

could

be

seen

as a

solid of

height

x.

The identities

were

then

proved

by geometrical

methods,

of

which

specimens

may be

found

in Book

II

of

Euclid's

Elements.)

On

the

other hand,

according

to

al-Daffa and

Stroyls, Sharaf al-Din discovered all the right condi-

tions

as a

consequence

of

his

idea

of reducing

equa-

tions (21)-(25)

to (21),

which

was

a case

that

Archi-

medes

had

solved.

Thus

Sharaf

al-Din

realized

that,

in cases

(22)-(25)

we may

write

f(m)-c

=-f'(m)(x-m)

+

[-I

f"(M)](XM)2-

(X-M)3

....(1

2

and

if we

choose

m

<

x so that

f'(m)

=

0

and

-

I

f"(m)

>

0 then,

if we set X

= x-m

(1)

will

have

2

the form f(m)

-

c = AX2

-

X3, A > 0. Moreover, if

f(m)

-c

>

0

then

this

has

the form

C = AX2

-X3,

A, C

>

0,

which

is an

equation

whose

root

was

found

by

Archimedes

in

Sphere

and

Cylinder,

Book II.

Thus

it is

important

that

f(m)

be a maximum

if

the

reduced

form

of

the

original

equation

is to be

Archi-

medean.

Moreover,

we

have

already required

f'(m)

=

0

in order

to get

the expression

on the

right

of

(1)

to

the

form

AX2

-

X3

so it

only

remains

to test whether

or

not the

root

of

f'(m)

=

0

affords

a

maximum

of

f(m).







BERGGREN: Sharaf al-DTn

al- sT's Al-mu'adalat

309

Fortunately, when there

is

only one root

it

does afford

a

maximum,

and

when there are two roots the

addi-

tional requirement that f"(m)

<

0 (so

- -

f"(m)

>

0)

is

sufficient

to

guarantee

that

f(m)

is a maximum.

In assessing these two alternate explanations, this

reviewer

must

say,

first of

all,

that

Hogendijk's

ac-

count of

how Sharaf

al-Din stumbled

upon

the

deri-

vative

by seeking

an

inequality

he

could

solve seems

to

us to be exactly

the

sort of

good

idea

that a

clever

mathematician,

trained

on texts

like

Euclid and

Apol-

lonius, could

well have

had.

The

expressions

that

appear

in the

analysis

do

in

fact occur in the

syn-

thesis, and Hogendijk's representation

of

the

style

of

Sharaf al-Din's thinking rings

true.

On

the other hand there is

nothing

inherently

implausible about

the

suggestion of

al-Daffa and

Stroyls that Sharaf al-Din came upon the condition

f'(x)

=

0 by his technique of using a

transformation

x

=

m + X

to

reduce

the

problem

in

each case to one

that Archimedes had solved. Certainly,

reducing one

problem

to another was a

common mathematical

technique from the 5th century B.C.

onward. The

principal difficulty with the suggestion is that,

accord-

ing to the theory, the equation f(m)

-

c

=

AX2

-X3

is not

always solvable, and the condition that it be

solvable may be stated f(m)

-

c

<

4

A3,

where A is

27

a

function of m and a. Since f(m) is itself a

polynomial

of degree 3 in m, and m has the form

r+V/t,

it would

not be easy to state sufficient conditions for the

inequality f(m)

<

c

+

4

A3

to

be

satisfied,

nor

are

27

there the

kind of

traces of this

technique

left in

the

text

that one finds of the other

technique.

Hence,

it

seems more

plausible

that

Sharaf

al-Din came

upon

the

derivative by replacing

two

inequalities

he

couldn't

solve

by

two that

he

could solve

than

that he dis-

covered

it

by

what would amount to a

finite

Taylor

expansion.

However,

the

suggestion

of al-Daffa

and

Stroyls

is a serious

one,

and it is

disappointing

that

Rashed gives

no serious

discussion

of

possible

his-

torical

relations between the work of

Archimedes and

that of Sharaf al-DIn.

Rashed's edition of the

mathematical

works of

Sharaf al-Din

has shown us that there

were important

advances in

the theory of

equations in the

Muslim

world after the

time of Omar

Khayyam, especially in

synthesizing the

numerical and

geometric traditions,

in formulation of a whole body of numerical al-

gorithms, and in the

justification of these

algorithms.

Rashed has,

moreover, been able to

show, in a highly

interesting note

in

vol. I,

that the work

initiated by

Sharaf

al-Din

continued

to

stimulate

serious mathe-

matical

investigations in the Muslim

world through at

least the 17th

century.

These are some of

Rashed's

major

achievements in

this publication. In

light of

this, if we

say that the Muslim

work was not

rooted in

the same line

of

thought as that of

Fermat, but rather

was a late

blooming

of

ideas and

techniques that go

back

to the

Hellenistic

world,

we

only

suggest

that

the

past is often a

larger part of

the present than

some-

times even

historians

realize.





Documents

Innovation and Tradition in Sharaf al-Dīn al-Ṭūsī's Muʿādalāt