A.’

I

The

Mathem

aticalS

tructureof

Tw

oIslam

icA

strological1

ahlesfoi

Casting

theR

as’hi

II

In1N

i)lJk

1.Introduction

and.su,n

e

Inm

edievalIslam

icash

oPg

tilesun

thm

oonand

achplanet

i’crc

believedto

castseven

ias,

meeting

theccli

ttcin

sesn

astrologicallysignificant

pointsT

hes

nI

Sal

infelt

)ln

eai

pe

lipticallongitude

oftheir

point1

itsr

ctioi.vi

thcli

lieleft

sextileray,

theleft

quartileai

.the

lefttrine

ratb..

rasto

thedianietricallv

oppositepoint

theright

time

iiii.

thericht

quartileray

andthe

rightsextik.

rayhe

‘dev

illvi

sii

isot

Cnerally

believedthat

thesextik

.uar

crid

Ii0

iid

ni

nedby

thesides

ofa

regularhesagon

asquare

andan

equilateraltriangle

respectively.T

herew

erehow

evertw

odii

letent

metnods

for

po

si

tiom

ng

these

regula

rp

oI

usA

ccordingto

thesim

pis

th. th

.t

nsIi

c0

esenhed

inthe

eclipticin

sucha

isav

thatthe

planetis

at

one

angularpo

int

(Figure

1).T

herays

arecast

alo

ng

thesides

ofthe

polygonsT

husif

thecelestial

longitude)f

Ip

ii1

isA,

tsex

Icira

tI

dtrifle

raysare

castto

thepoint

wO

nlo

ngilu

ck

ill.

±)

t12O

.A

l-Sufi

(903—983)

andA

lB

irãni(972-

1i)4)pri.scrrhe

ibit

un

important

modifications

he

made

incase

theplanet

Isisnon—

zerolatitu

de.

2In

thesecond

theoryi

assamed

thaut.

jvs

ttO

.s1

rits

arerelated

tothe

apparentdaily

otationof

theuniv

rse,so

thatthe

*M

ath

ern

atlc

ai

institu

te,

Stairt

hiB

0s

((

1.chi

the

Netherlands.

t)S9

oI

32pp

1t

2n2

1 72Jan

1’F!ogendzjk

I(

Aoka

aInt’l

173

regularpolygons

haveto

bepositioned

onthe

celestialequator.

Inthe

Masudic

Cation,

Al-B

irUni

givesa

rathervague

descriptionof

thistheory,

which

canbe

summ

arizedas

follo

ws.

3In

Figure

2,N

ES

Wis

thelocal

ho

rizo

nand

N,

E,

Sand

Ware

thecardinal

points.E

MW

isthe

celestialequator,

theplanet

isat

pointF,

andsem

icircleN

PS

intersectsthe

celestialequator

atPt

The

regularpolygons

haveto

heinscribed

inthe

equatorin

sucha

way

thatJ)t

isat

anangular

point.T

henthe

seniicirclesthrough

N,

Sand

theother

angularpoints

(suchas

Qfor

theleft

sextileray

inF

igure2;

arcp*Q

*=

600)determ

inethe

positonsof

therays:

thusP

Qin

Figure

2is

theleft

sextileray.

etc.C

ompare

alsoF

igure7

below.

The

exactcom

putationof

therays

accordingto

thissecond

method

iscom

plicated.A

l-BirU

nidescribes

anapproxim

atecom

putation,w

hichhas

beentreated

byK

ennedyand

Krik

orian

,4

andw

hichw

ew

illdiscuss

below(S

ection2)

inconnection

with

some

geometrical

preli

minaries.

Al-B

irUni

saysthat

theastrologers

alsoused

instruments

andtables

forthe

com

pu

tation

.5

No

instruments

forthis

purposeare

known

tobe

extant,but

two

setsof

tableshave

come

down

tous.

Inthis

paperw

ew

illdiscuss

thesetables,

andgive

some

newinsight

intheir

mathem

aticalstructure.

The

firstset

oftables

was

computed

byM

ubamm

adibn

MusA

al-Khw

irizrni,

1’w

hosem

odernfam

eis

mainly

basedon

histreatises

on

/%

RT

>N

NJ

/

arithmetic

andalgebra

(aithugh

hisistro

iom

rcalssork

sh

im

oresubstantial).

These

tabks

sli

1i.

illc

Ithe

Kv

iibles,

arepresented

inthe

&ok

alc

Iioogv

thatte

dirill

es

sup

erfluous

(aI—m

ughnifiaik

arnal-nujuni)

ofthe

(hr

stianastrologer

IbnH

ibintã(early

lOcenrur\

),w

hoattributes

themto

A!-K

hw

inzm

i,t

andthey

alsooccur

(inI

ains

ipto

tii

a((

liis( 1

2e

cen{u

ry).

The

Khw

ariim

sIlo.serc

puhlhod

yK

oneand

Krikorian

in1972.1

Th

einvolve

afunction

oftw

oxarrahles

hch

hasbeen

explainedarithm

eticallrby

1oomcr

andeo

mctn

eilIby

Kennedy

andK

rikoriana

Ithrs

funeths

ofsin

vsable

which

havenot

beenunder

dh

modern

hb

ra

sr

Ia

sW

edescribe

theK

hwãrizm

itahlcs

anddiscuss

thesignificance

otthe

two

functionsin

Section

3below

.T

heK

hwãrizm

itables

cr

originallycontained

tth

famous

Zij

(astronomical

handbooks

1ah

ks)

ofA

Kh

sarvs

khas

onlycom

edow

nto

usin

agarbled

Latin

translation(

ae

siorw

hichhad

beenrevised

bythe

Andalusian

astro

nom

er

Masla

ma

ibnA

hmad

al-Majrifi,’2

who

diedaround

1007In

thispro

‘assthe

originalK

hwã—

rizmi

tablesfor

castingthc

‘w

replace

aa

sf

air

thesam

epurpose.

computsd

Ka

geographicaatit

ia

q3

3fl3

probablyfor

thecity

of(

ordobaH

encethis

setof

tablesw

as

P

///I\N

////\NN

13/

Y\

II

I\

Igale

I

Figure2

1 74ian

PIlogcndijk

1oo1

175

ccitainlynot

byA

l—K

hwarizm

i,and

we

will

thereforecall

itthe

Majrifi

tables,even

thoughhis

authorshipis

notcom

pletelycertain.

rhe

Majrii

tablesare

ofenorm

ouslength,

theyextend

over24

pageso

fS

uter’s1914

editionof

theL

atinversion

ofA

l-Khw

hrizmi’s

Zij.0

The

Maii

titables

havebeen

studiedby

Suter,

Neugebauer

andK

ennedyand

Krikonan,’°

butthe

underlyingm

athematics

hasrem

aineda

mys—

teiuntil

thepresent.

InS

ection4

we

explainthe

mathem

aticalstructure

ofthe

Majriti

tables.It

turnsout,

notsurprisingly,

thatthese

tablescan

heconsidered

asan

improvem

entof

theK

hwhrizm

itables.

As

we

will

see,A

l-Majriti

solveda

quitedifficult

trigonometrical

problem.

andhe

thenw

entthrough

acom

plicatedand

laboriouscom

putation.‘l’hus

theM

ajriitables

areof

considerableinterest

forthe

historyof

appliedm

athematics

inthe

middle

ages.

2.!‘re

liint,m

arie

.c

Right

riiiiioblique

flSC

CIIS

iOflA

Figures

3and

4display

thecelestial

spherefor

observerson

theterrestrial

equator(3)

andon

theN

orthernhem

isphere,south

ofthe

Arctic

(‘ircle(4).

ES

DW

NfI

isthe

localhorizon

andthe

cardinaldirections

areE,

S,W

,N

,E

VM

Wis

thecelestial

equator,V

isthe

vernalpoint.

DK

VH

ispart

ofthe

ecliptic,S

KM

ZC

Nis

them

eridian,(

isthe

celestialnorth

poleand

Zis

thezenith.

The

intersectionH

ofthe

eclip

ticw

iththe

Eastern

horizonis

calledthe

horoscopusor

ascendant.S

upposeH

hascelestial

longitudeA,

andw

ritee

forthe

obliquityof

theecliptic

andq

forthe

geographicallatitude

(=

0°in

Figure

3).then

A=

arcV

H,

‘—

LE

VI!,

q=

arcC

N.

Definitions:

For

q=

0°(F

igure3),

arcV

Eis

calledthe

rightascension

ofarc

VH

.N

otation:V

E=

For

00<

q<

90°—(F

igure4),

arcV

Eis

calledthe

obliqueascen

sionof

arcV

H.

Notation:

VE

=A

q(A).

The

obliqueascension

Ag(A

)can

hedefined

inan

analogousw

ayfor

southerngeographical

lat

itudes,that

is,for

—(90°

E)

<q

<0°.

The

functionsA

(A)

andA

,(A)

aredefined

forall

values0°

A<

360°and

theyassum

evalues

between

0°and

360°.

The

declination6

ofthe

horoscpus

i’its

distane..t

theequator,

reckonedpositive

ifI/is

nthe

athem

hallof

IIsphere,

andnegative

if//isin

Isoutheir

all,not.

thatIn

I’gure

3,6

—E

li.For

iid

b.orni

Then

theright

ascensionI

Im

dF

51

(‘1)

sinA

((m

6(a

(22)

ifw

ekeep

inm

indthat

AA

)is

inthe

same

quadranta

,tF

urtherA

(A)

=A0(A

),

ssith

sin?

tanà

tançI

(2.3)

and—

90°<

i<

90°;com

parefig

ure

4.in

which

1fF6

FE

=

Note

thatZ

VE

N=

Oth

erw

aysof

computing

theoblique

ascensionare

possible.hut

them

ethodpresented

hereis

essentiallythat

usedby

many

medieval

astronomers.

Tables

ofright

andoblique

ascensions(for

variousla

titudes)

occurin

theA

lmagr’cr

ofP

tolemy

nin

mails

Islamic

istrono—

met

0

Seq

/\ecl

I_

___/_.

-

-.--—

---—

Nc

FH

or

Figure3.

celestialelestial

6’—ro

n

176Jan

PH

ogendijkI

1)5

51)

a177

sphereover

18(1’arou

IN

xs

throngI

Au)

—A

,,(2)(m

oI

6))

iscalk

darc

XY

forgeographical

latitudeq’

losettle

thequestion

foran

arh

itrar

semieH

ele\I1

ssohP

inthe

Eastern

halfof

thecelestial

sphere(F

iguiu5)

se

di assaic

(Pp

erpen

dicularto

semicircle

NP

S.

[hen

(liepo

e4

ofeiicL

\PS

ison

thegreat

circleto

which

areC

Rbelongs.

Lu

‘PS

inietseet

thecelestial

equatorat

P.

Because

Cis

thepole

ofthe

celestialeq

ta

circe

NP

S,

theintersee

if

theequ

NcA

.T

hereforearc

*J

9his

mcai

1can

betreated

as“hot

zoi)Y

eog

rahi

pointR

andeast

poinI

tor

CR

NP

SinA1(ji)

—A5Q

.)i

deL360)

timr.

s.F

inally,if

Pis

inth

rnhalf

ofii

Isphe

mc

canshow

inth

esam

ew

ayII

aX

Ypasses

cvet

V!

5’in

A(p)

A1

)tim

e-degrees,hut

inthis

asethe

latitude.

isnegative.

.C

R{.

Som

eA

rabicastrologers

calleda

circlesuch

as,\‘P.S’

inF

igure5

“ufqb

ãd

ith,e

forw

hichw

ew

illuse

Professor

Kennedvs

translation“in

cidenthonizon’.

They

calledarc

CR

the‘latitude’

ofthe

incidenthorizon

NP

S.

IS

AN

-

N

S

Hgur

5

hor.[ig

urc

4.F

mical

handbooks;further

detailscan

befound

inpublications

of

Kennedy,

King

andP

edersen

.i7

Definition:

thetim

e-degreeis

theinterval

oftim

ein

which

the

celestialsphere

revolvesby

onedegree.

Asidereal

dayconsists

of360

time-degrees,

andone

time-degree

is

alittle

lessthat

4m

inutes.F

orthe

observerof

Figure

4,arc

VH

riseson

theE

asternhorizon

in

A5(2

)tim

e-degrees.If

Xand

Yare

two

arbitrarypoints

onthe

celestialequator

with

longitudes..

andp,

thenarcX

Y(i.e.

thearc

extendingfrom

Xin

the

directionof

increasingcelestial

longitudetow

ardsY

)rises

inA

u)

—

A4i)

(modulo

360)tim

e-degrees.

Oblique

descensionsand

incidenthorizons

We

nowask

thequestion:

howlong

doesit

takeX

Yto

passover

an

arbitrarysem

icirclethrough

Nand

S(other

thanthe

Eastern

horizon)

suchas

NP

Sin

Figure

5.If

thesem

icircleis

halfof

them

eridian(above

orbelow

theh

ori

zon),the

passagetakes

placeinA0(u

)—A0().)

(modulo

360°)tim

e-

degrees.If

thesem

icircleis

theW

esternhorizon,

thepassage

takesA

_,,(u)—

Aq

(?’)

(modulo

360°)tim

e-degrees.T

osee

this,ro

tatethe

celestial

iiIhL

5nessio

n

iqiis

C.s

nsio

nof

iand

.1is

thepole

of

iV

PSis

lbspole

ofi

matic

’N

P*S

1i

CR

hnorth

Ieso

ver

178Jan

P.H

ogendi

1kfz

4F

179

The

following

two

formulas

caneasily

heverified

(Figure

4).L

et2,,

and).,

hethe

celestiallongitudes

ofthe

horoseopusH

andthe

in

tersectionK

ofthe

eclipticw

iththe

meridian

abovethe

horizon.

Because

areE

W180°

andarc

EM

—90

,w

ehave

Au.,,

—180°)

=A

t%,,)

180°(m

odulo360°)

(2.4)

A(%

K)

=A

(2,,)

—90°

(modulo

360°).(2.5)

The

identity(2.4)

shows

thatA

canbe

foundfrom

atable

ofA

,.T

hereforeoblique

descensionsw

erenever

tabulatedseparately.

The

sta

ndard

co

tnp

uta

tion

of

thera

ys

Many

Arabic

sou

rcesdescribe

what

Icall

thestandard

computation

ofthe

astrologicalrays.

This

procedurehas

alsobeen

discussedby

Nallino

andby

Kennedy

andK

rikorian

.2°

Itis

basedon

theabove-

mentioned

theorythat

theregular

polygonsw

hichdeterm

inethe

rayshave

tobe

locatedon

thecelestial

equator.I

beginw

ithintroductory

remarks,

referringto

Figure

2.T

hebasic

ideais

(>w

A()

P*Q

*=

A(P

Q)

where

isthe

latitudeof

theincident

horizonN

PS

.21

Form

ula(2.6)

iscorrect

onlyw

henthe

incidenthorizon

NQ

Shas

thesam

elatitude

asN

PS

,or

when

Qis

thevernal

orautum

nalpoint.

Inall

othercases

(2.6)is

atbest

anapproxim

ation.T

hiscan

beseen

in

Figure

2,w

hereA1(P

Q)

=P

*Q

,w

ithQ’

definedsuch

that/P

P*w

=

ZQ

Q’W

=9

O+

.T

heapproxim

ation(2.6)

canhe

usedif

tablesof

obliqueascensions

forall

latitudesare

available(w

hichw

asoften

thecase).

Inthe

standardcom

putationa

linearinterpolation

ism

ade,so

thatthe

entire

computation

canbe

doneby

means

ofa

tableof

rightascensions

anda

tableof

obliqueascensions

forone’s

own

geographicallatitude.

The

standardcom

putationis

asfollow

sin

modern

notation.L

et).

bethe

givencelestial

longitudeof

theplanet

P,and

Lthe

required

longitudeot

thepoint

shm

ethe

rayhits

theeshptiF

(flis

br

=

PO

oneof

thevalues

±(.

+9()

or12(1

Ici

theecliplk

mtersect

theE

asternhorizon

atIi

lie\t

sternh

osm

1)<in

heni

iolianabo\e

thehorizon

atK

IuI

IlLii

(()

assu

mes

thatthe

longi/

hec

sa<

prlousE

determined:

then

Aw

Iid

by

(21

amearl

18(P,—

Several

caseshave

tobe

onsidered

1.If

Pis

onthe

mend

ii.

cA

(2)A0(2

t

2.If

Pis

onthe

Easter

ioiiz

(2A

).thr

iz

oh.1

(A)

=

A(A

)+i.

3.If

Pis

onthe

Western

hn

won

(AA

),then

2,2

with

4(2.)

=A

(2,)+

,.O

necan

(md

fromi

tableI

h4)

4.If

Pis

inthe

Eastern

mi

Ii

re,then

AI

Aw

ih

Ic=(A

11

(2

)A

,(Aj)

(A,)

41(A

K))’

Is

oon

(asin

Figure

2),and

2(.IA

A(

,)(1

,)(A

,,)lif

Pis

underthe

horizon.(If

thuvernal

pointis

between

Pand

k<

A(A

j:in

thiscase

(.1.(2

,).4.

()j)

hasto

h1replaced

by(.4,(%

,,)—

A,,(%5))

ihtI;

sshatis

inten

ded

isi1ss

‘tInIig

ht

ascensionof

arcP

Km

iIn

idificati

mIc

iin

theo

ther

parts

of

the

fon

a‘C

PK

,etc.).

5.If

Pis

inthe

\\ste

in

uadia

nt,

substituteA

toia:

ml

I)for

II.

Inthe

Great

Introdjuijo,,n,

ActloIogv

otA

buM

a<sh<‘\.I)

787—886)

thestandard

metl

Iis

tbuted

IP

my

II

shas

causedsom

econfusion

cais

notta

0h

moc

irsin

Ptolem

y’sastrological

nrk,

tiT

hrabilbc

i1erncntions

thecasting

ofrays

huthe

gie

no

detailsabout

thecom

puta

tion.sa

lhn

o23

explainedthe

attributionbr

thefact

that

Ptolem

ruses

asim

ilarm

ethodin

aproblem

rdited

tothe

astrologaat

themof

rogressions(A

rabic:ta

syIr),

24

Inthis

ihin

ticco

mp

ites

1u

nA

,tlietim

e-nterval

ittakes

thci

si

phit

Qt

uc

Ir

Ie

cirde

NP

Sby

mean

sof

thecia

lyrotatn

iiof

thcurns

‘sc(F

re2).

NaH

ino’s

argument

issupported

hrthe

facttha.

Piolem

sU

sesa

similar

interpolationcoefficient

2.In

Ptolem

r‘s

solu

tion

,I

ap

pears

ina

with

4

(2.6)

df’(d

s)

417

!o,tca1I

18118(1

Jan

PH

ogendxjk

naturalw

ayas

aconsequence

ofthe

assumption

(actuallystated

byP

tolemy)

thatsem

icircleN

PS

isapproxim

atelyan

hourlin

e.

29

3.The

Khiv

dri:,n

itables

We

nowdescribe

theK

hwãrizm

itables

following

Kennedy

andK

rikorianH6

The

eclipticalsigns

will

henum

beredA

ries=

1, Taurus

2.G

emini

=3.

Cancer

=4.

Leo

=5.

Virgo

=6.

Libra

=7,

Scorpio

8.S

agittarius=

9,C

apricornus=

10.A

quarius=

11.P

isces=

12.E

achsign

isdivided

intothree

equal10°-intervals,

calleddecans

orfaces

(Arabic:

wajh,

Latin:

fricies).A

notationsuch

as[3,2]

will

beused

forthe

seconddecan

ofsign

3(that

isthe

intervalof

pointson

theecliptic

with

celestiallongitude

between

70°and

80°).T

heK

hwãrizm

itables

consistof

432values

ofa

functionF

and36

valuesof

threefunctions

5,Q

andT.

Fis

usedin

allcom

putations,w

hileS

isfor

sextilerays,

Qfor

quartilerays

andT

fortrine

rays.T

able1

containsfor

everydecan

dand

signin

anum

berF

(d,m)

tabulatedin

degreesand

minutes

ofarc.

Follow

ingK

ennedyand

Krikorian,

thecolum

nsin

them

anuscripthave

beenrendered

asrow

s,but

br

therest

thearrangem

entof

them

anuscripthas

beenm

ainlam

ed.T

hereforethe

rowfor

d=

[3.2]begins

notw

ithF

([3,2].1)hut

with

F(I3.21.3).F

orsake

ofclarity,

thesign

number

inappears

in‘lithle

Iabove

eachgroup

ofthree

numbers

F([n.I jin

).F

([n,2].rn).I”(‘lahle

2contains

foreach

decand

anum

berT

(d)in

degreesand

minutes

ofarc.

‘l’heK

hwärizm

itables

alsocontain

valuesS

(d)and

Q(d)

butthese

havenot

beenrendered

herebecause

always

S(d):Q

(d):T(d)

=60:90:120

(3.1)

asnoted

byT

oo

mer.

27

By

means

ofthe

Khw

ãrizmi

tablesone

cancom

putethe

astrologicalrays

fromthe

longitudeof

theascendant

‘tHand

thelongitude

ofthe

planet.

Al-K

hwärizm

i’sinstructions

forthe

useof

thetables

havebeen

preservedin

theB

ookon

astrologythat

makes

(allothers)

superfluousof

lbnFlibintA

,and

theyhave

beentranslated

byK

ennedyand

Krik

orian

.2

0I

paraphrasethese

instructionsby

means

ofa

num

erical

example.

InIll

10

[I,1

20,052541

[1.2422413

29

0[42]

2415

21.0211,55

212.01

15,9’2

[I31

222424.23

12,7”1,

7‘0

27,4.,49

‘9

15

441

-6

12,’21.114

30.3271,03

29‘—

‘.9

,52

223)

23(1

[2,2125,116

34,4)31,17

75”

22,19,‘2:il

24015

l.

[2,5’27.18

92.5159,21’

29,542025

22.2)2429

‘2,5.

54

51,

‘88

‘110

[54’29,11)

54,0032.01

22.5521:19

25,4429,44’

54(k

7[3,2’

34:2134,10

51;962’.51

291423,3’,

31,2429

,.[3

7’

33.043.4,37

5.5

2‘7

.0’

231,620.17

33,0811149

43

6‘1

‘8,

[4,1]14.92

34,02‘80,25

7‘

10,5234,02

9‘

14255,1”

31:59111,25

11,4137,54

..

1,

.5[43

59’4111,90

37,253

.019039

“11

50

01

[5,]]11,44

13,00102

43415)

36[3,2]

1603300

3,29

132]3

[5,3]3049

330010

33)4

31(13

3.‘7

61

1[6,4]

35,3933,00

32.03

713393

1,4

[6,23313

33,695142

9543

435

121)

[6,3]35,13

33,4741,4,

35,4135

5

741

1.74,

35.3534,07

3340’,’’,

‘35(8’

35,99311

7,2’35

41

33;4932,39

32,9.,24

3,1)334,25

9,4

‘8,3,341;s’i

33,1131.511

52.I

‘240

31,66’31

,11

32

419

10II

21

2‘1

[9,4]16,44

34:0031:24

25,3,00,

‘513(8]

6’Ii

58.2]

35,4”32.51

30,41125

5490

2’

1243

41412

1’0’

‘0,1]35:15

34:4]311,03

29‘4

’.0,2

531

2’

910

1]‘

,2

54

9,11‘4.52

30,322.4’2’.

)T10,0.’

‘84923

‘I,

19,2]39,00

29.1127,24

250

103240’

331195’

11‘8

‘‘

5,

,

9,3]11

2423,49

22,4112

53

1’4

21124

5‘

1

1011

423

45

[101]2940

25:4123.15

54,19129,49

3131

[40,2]27:38

24,542515

2,

34

59,1

‘33

[10,3]25,36

24,982015

116

4211,45

3

II42

I1,1]23,34

22,41125,

53

05,

1521

345

11,2]22:24

22,4920,113

301421,4

04

[14.1]21:15

22:4024101

4524,00

,..

93

42I

25

0[12,1]

20,0522:40

27,4224

2,34150,0

497

4[42,2

20,0523,27

24:2111

IS2,549

2(5,140‘1

1‘12

,‘.

..11

[42,3]241,05

24,132935)

32.6

133.2,9.54

20,0’82

.,2417’o

72‘

‘“‘A

’S

1141)1171

.‘9J—

khso1114/1541”

labIt’

of11dm

).

42C

un,o

ru,

XX

’7I[

182Jan

I’H

ogendijkIa

IsOm

4sO

abets

a,Ish/

5183

d1(d)

d1(d)

d1(d)

dT

(d)d

T(d)

dT

(d)

[1,1]06,35

[3,1]116,18

[5,)]129:58

1711133:25

19,11123:42

[11,1]110:02

[1,2]07:41

[3,2]116:46

1521131:07

[7,2]132:16

[9,2]121:14

[11,2]08:53

lII106:5

3.3]121.14

[5:91(2

16

[7.3]131:7

19.3]116:46

[11.3]107:44

2,1]II0J)2

[4,!]i23:42

[6,!]133:25

[6.1]129:56

[10,1]116:18

[12.1]106:35

[2.2]II2M

7[4.2]

125.47[6.2]

133:25[6.2]

127:53[1(1,2]

114:13[12,2]

106:35

[2.1]14.13

[3.3]127:53

[6.3]133:25

[6.3]125:47

[10,31112:07

[12.3]106:35

‘Itthle2.

AI-K

hwirizm

i’stable

forT

(d).

Suppose

,=

77°,2

=133°.

We

wish

tocom

putethe

lefttrifle

ray.B

ecause77°

isin

L3,2],w

eread

inT

able2

T([3,2])

=118;46°,

andw

ethen

considerthe

rowfor

13,21in

Table

1.B

ecauseP

isin

sign5

(i.e.L

eo,the

interval1120°,

150°1),w

eread

offF

([3.2].5)=

31;56°.

SinceP

isin

degree13

ofthe

sign.w

ecom

pute(30—

13)130tim

es31;56°

=

18:6’.W

ethen

performa

number

ofsubtractions:

T(13,21) —18;6° —

100;40°,100;40°—

F([3,2j,6)

—72;50°

72;50 —F

(j3,2],7) —47;39°

47:39F

U3,2].8)

=21;40°,

which

isless

thanF([3,21,9)

=

31:24°.F

inallyw

ecom

pute30(21

;40°)131;24°=

20;42°.T

heleft

trineray

iscast

tothe

point20:42°

Sagittarius.

‘I’husif

theplanet

isin

degreek

ofsign

in(i.e.

ifits

eclipticallongitude

is30f,m

—1)

+k°).

we

subtract(30

k)/30tim

esF(d, m

)and

thenF

(d,rn+1),

F(d

,rn+

2),

...F(d,m

+j—

1)from

T(d)

untilw

eobtain

arem

ainderr

with

r<

F(d,m

+j).

We

thencom

putex

=3O

rlF(d,m

+j).

The

lefttrifle

rayfalls

indegree

xof

signrn+

j.(H

ereand

furtheron.

we

reckonsign

numbers

modulo

12:thus

sign7+

8is

thesam

eas

sign3.)

The

computation

ofthe

numbers

inT

ables1

and2

isnot

explainedin

anyknow

nm

edievalsource.

Toom

erand

Kennedy

haveshow

nthat

Table

1w

ascom

putedfor

q=

33°,s

=23;510,29

ina

way

tobe

men

tionedbelow

.N

otethat

medieval

sou

rces

3°give

valuesnear

33°for

thelatitude

ofB

aghdad,w

hereA

l-Khw

ãrizmi

worked,

andthat

Al

Khw

ãrizmi

usesr

=23;51°

elsewh

ere.3’

Tue

following

xplanII

iI

tIc1

si

ircii

lysnd

Krik

oria

n.

32

InF

igurto,

NC

ale

thN

I],ii

Iipoints

of’the

horizon,P

,....1,

arcth

beginniiiesIf

thein

Ii,d

si.ns,

and

thegreat

circlethrough

N.

.Sand

Pin

ters

ccls

thec’. lestia]

equ

ator

atP

,°.F

igure(3

displaysthe

P.and

P,

[‘ori‘-

1 .2...?

(thegtcat

circlesN

PP

*S

arenot

sho

ii).C

onsidera

fixedsign

in,

and

callP

.P

.the

projectionof

signin

(c

ometiica[ly,

cah

F( din)

ii‘laN

eI

isan

approximation

of

theproject

osign

ra

wttm

nd

cciid

isrising

at

theE

asternh

izIf

decan[m

, 1]is

rtsmg,

ii

tt1h

Fin

sth

itits

projcctionis

approximatcil

h[iqua

cnsion.

11115explains

why

inT

ableI

F(Im,

1]in)=

A(in

.30°)

—A

,,((in—1).3

0j.

(3.2)

Ifdecan

[in+

3,1]or

decan[m

91]

isrisin

sienin

inthe

meridian,

sotha

itsp

cttonis

ap

IlX

tfllat

sion.T

husin

Table

1in

11i

hcolum

n

//cc

N

\\&\<

:1F

igut

6

pioxim

atelght

ascen

F(jn

z+3

,1],in

)=

f([r;l’19, I]

A6(m

0)

4((

)3(

)(3

3)

184Jan

1’Ilogendijk

4rl)L

’12,l’/

185

If(lecan

L,n+

6.l]is

rising,sign

mis

settingon

theW

esternhorizon,

sothat

itsprojection

isapproxim

atelyitso

bliq

ue

descension.T

hereforein

irhIc

Iin

theseventh

column

F([nz+

6,11,,n)=

Aq(m

.30°)—

Aq((m

—1).30°).

(3.4)

As

aconsequence

of(3.2),

(3.3).(3.4)

and(2.4)

we

havethe

linearrelationI’ijm

+3,1J,m

)=

(112)(F([m, 1],m

)+

F([m

+6,1],rn)).

(3.5)

For

i=1,2,4,5,

thevalues

F(tn

+i,1

j,m)

inthe

tablew

erecom

putedby

linearinterpolation:

F(lm4

i,1] ,m)

=(1

—i/6)F(fm

,1j,m)

+(i/6

)F(tm

+6, 1] ,m

).(3.6)

Note

that(3.5)

isthe

casei

=3

of(3,6).

The

valuesin

thetable

alsosatisfy

(3.7)fX

lni+6+i,11,m)

=F

([,n+6—

i, l],m).

We

nowconsider

thesecond

andthird

decansof

signin.

Inthe

firstcolum

nin

Tirhie

Ilinear

interpolatonw

asused:

F([m

,2Jjn

)=

(213)F([rn, 1]in

)+

(113)F([in+

1 ,1],rn+1)

(3.8)

F(lin

,3]jn

)=

(1/3)F([rn,11 jn

)+

(213)F([m

+1

,11,m+

1).(3.9)

One

would

expectthe

same

tobe

truefor

theother

columns.

But

asa

matter

offact,

forthe

fourthcolum

nw

ehave

fork

=2,3,

FU

m+

3,kJ,m)

=F

([m+

3,1

],m).

(3.10)

For

k=

2,3

thenum

bersin

theseventh

column

arerelated

tothose

inthe

fourthand

thefirst

column

by

F([m

+3,k

l,m)

=(112)(F

([m,kl,m

)+

F([m

+6,k

],m)

(3.11)

andthe

nuifibersin

therem

ainingcolum

nsare

definedby

F(lm

+6±

1,kin>

!4[iii+6

i,k]in

).(3.12)

The

identities(3.11)

and(3.(2>

areanalogous

to(.

5>m

d(3.7),

but(3. It))

isodd.

Asaco

nseq

uen

eeol

thisthe

numbers

inthe

seventhcolum

nvar

irregu

larl.

[hisconcludes

theexplanation

of‘lable

1.L

ikeK

ennedyand

Krikorian

Iam

unableto

understandsh

thecom

pilerof

thetable

used(3

It))F

orif

heft

jin3

niin

thefourth

column

hadbeen

obtainedby

linearinterpola’

i)n

et

ceni”(ltn

+3,11

.m)

andT

Urn

i4n

11),the

six

rightu

otthe

tabkw

ouldbe

identicalh

ixleft

columns,

soth

Ihalf

of

thetable

couldbe

dis1x

scI

Aith

(compare

aIxi

fthis

papt. r).V

,enow

turn

toTab

cfunc

lionsN

efirst

expltin

theirarithm

eti1

1t

1)(d)—

F(d,i

i)

(.

I

Note

thatthe

number

Ii

nonly

anapproxim

ationhe

seenfrom

thefact

tini

A1

S

P,,2P

,,..’±

P,.

P.

(3.13)

whereas

ingeneral

in

gcneral

nottl

>t.eltonP,,

.I,,Ic

,hut

hiscan

cP(d)3(pC

(3.14)

Itcan

beem

piricall

verifiedthat

thefunctions

N()

andI

inthe

Kh

irizmi

tablessatisfx

S(d):Q

(d):7’(d15

1)60:90,12()

360(

15)

Note

that(3.15)

isa

,cn

The

functionsS,

Qar

ingreason

Figure

7sh

is

B,

aplanet

atB

castingleft

trifleray

toD

Thcequilateral

triangle,an

Si

iOi

0(3.1)

t.p

robab

lyin

troit

cci

lou

Iit

Aw

hichc

sf

ura

yto

(a

&p

ojections.4

Bhr

J)*A5

0llo

wray

tosting

atm

an)n

the

186Jan

PIlogendrjk

/1o

a1

/

Fig

ure

7...

-

otherhand,

onecan

alsocom

puteB

,C

andD

bym

eansof

Tables

1

and2.

The

computation

ofD

fromA

canbe

characterizedthus:

we

continuouslysubtract

numbers

F(d,m

)in

Table

1from

T(d)

+T

(d)+

assuggested

bythe

numerical

example.

untilw

eobtain

remain

derzero.

The

factthat

Dand

Acoincide

means

thatw

eobtain

remainder

zeroafter

subtractingan

entirerow

ofnum

bersF

(d.m).

in

—1

212

inT

able1.

Therefore

(d)

must

heequal

toT

(d)+

T(d)

±T

(d).A

similar

argument

shows(d

)=

4Q(d)

and(d

)6S

(d).

Therefore

Al-K

hwãrizm

iintroduced

hisT,

Qand

Sbecause

he

wanted

anysuccession

ofthree

trinerays

(orfour

quartilerays,

orsix

sextilerays)

beginningin

Ato

returnto

thesam

epoint

A.

Ifd

isthe

firstdecan

ofa

sign,w

ehave

F(d

,m+

i)=

F(d

,m+

i+6)

for

alli

SO

that

F(d

,in+

i)+

F(d

,m±

i+1

)+

...+

F(d

,nz+

i+5

)=

(d)l2

.(3.16)

This

means

thatif

Acasts

aquartile

rayat

Eand

Ecasts

aquartile

ray

atI

thenA

andF

arediam

etricallyopposite

points,w

hichm

akes

astrologicalsense

(compare

Figures

1and

7);a

similar

propertyholds

fora

successionof

threesextile

rays.N

osuch

niceproperties

are

obtainedif

dis

thesecond

orthird

decanof

asign.

This

astrological

flawis

anotherconsequence

olthe

factthat

Al

Kh\iatiznii

(orw

hoeverelse

compiled

ilibir.I

inits

picscnlform

)defined

thenurn—

hersF([tn, 2] ,m

-F3)

and1

(]ni

}n4

i)in

thetot

ithC

(di

inol

IIde

1

by(3. 10)

andnot

byin

iii

i3

fluto

t

4.T

heM

ajriitables

The

purposeof

theM

at

sthe

com

ut

ionas

olo

gi

calrays

fromthe

longitit

ohot

ose

)Uid

as.ud

ofthe

planetA

sm

en

tned

aby

e,the

aNts

scr1

ipult

If

)ra

geographicallatitude

q3

:(0T

he\lal

rut

tab

les

area

ollection

of72

subtahlcs.E

achsubiahle

islot

afixed

,,

andih

erl

isa

.subtahle

forevery

multiple

of

5ii

1N

e3,

rskH

ho

uie

1p

207is

for)

20).

Ihebe

Rnp

Ih

esu

ltN

tiii

thre

e

numers

q(A1)

31A

thesex

illl

iit

tys.in

Table

3w

efind

sf204

95

q(2

0)

4.5

and1

1)9.S

4’.

Each

subtablealso

co

nta

ins

36num

bers/(2

..,

Iin

sixe

riUm

Trs,

Each

column

isfor

,in

thetw

odiam

etricallyop

posite

siens

\ihosenam

esappear

abovethe

column

In(

tir

t5flid

s

Is‘

1)

Ifis

insign

n(that

is.

SOn

—30

<).

r—30’i

).the

hurstcolum

nis

forin

signsn

andn

±6.

these

cond

colu

mn

islot

1iii

signs

ti!1

andn

+7,

andso

on.F

orlatei

usew

eall

tn(A)

them

xim

aliu

mber

inthe

suhtthle

forA

;m

(1

1)

i0(t

+5)

)mn(

71

atording

toT

able3,

butw

ew

illK

II

wthat

tls

nui

Iii

bereor,

Inthe

headingof

cadof

thes

xcolum

nsth

ilunte

lasthe

word

“horoscopus”.T

hisis

atranslation

ofthe

Arabic

maa1i

.anti

shouldtherefore

heinterpreted

as“ascensions”

‘for

typographicalm

asons,the

abbreviationA

sc,h-is

beenused

inla

hk

3T

heaccom

panyingto

i.x

uns

now

tha

esr

Iused.

The

procedureis

asfo

lbS

ii3

ftrnrn

tati

UP

At

sish

tocom

putethe

positio

nA

fthe

oiltrine

ray.1

sg

thI table

ftr

We

compute

a-

/A,,,A

)+

1(A

11

).

(4.2)

188Jan

1’!(ogendijk

14

(4U

fl1

189

semR

irclcthrough

N,

S(i)>

0if

Pis

inthe

Western

hcmispher

‘1!

f(21,3

0n

)

andt e

lastnum

bersin

I

f(2,

1,30(n

+j)°

)—

f(A

,1,3

0(n

+j

1))

A

The

)th

erf(2

11

,A

,.)

wer

We

will

notdiscuss

thP

utm

(211)

30(determ

med

by

c(2):q

(2).

Exam

ple:in

Iah1c3

18for

thelatitude

ofti

Cirn

21;3(

°is

theobliqu

‘iseincident

horizonthrouo

18;51Y’4-21;39

appeaic1.

SOon.

W&

flOW

turn

toth

lure.L

etP

heany

ponp

rojctio

nof

P(on

thgreat

semicircle

NP

Sad

6).If

2,

isin

signn,

di

i

donof

thearc

ofthe

ep

P.H

erew

eassum

ebe

discussedbelow

,T

huprojection

ofthe

signsA

time

when

thehoruscop

1eguinini

5li

i

.is

tunli

1

lOA

f1

8

1blique

oin

thi

111

)fI

signIi

i

Ihr

ci

inin

Ic

lumbe

ni

Cid

ah

II

Ili

1ticA

ka

equato

i)i

t

II

ale

qi

)i

an

aro

xm

os

emthe

xg

111

180Ii

ismIi

k3

402

isij

lislau

ius

orth

ss

is-sris

rsi

illthaI

iitile

IiS4

3)

uby

(44)

ation

Iarc

Aries

Aijes,

(Ifthe

,29and

strucill

theA

thes2

,5Iro

jec

gnand

80w

illof

theat

theu

puta

ideais

(ITSC

X-

Table

bhque

Tw

entydegrees

Aries

Sextile

49;57°Q

uartile74:56°

Trine

99;54°

Aries

Taurus

Gem

iniC

ancerL

eoV

irgo

1.ihraScorpio

Sagitt.C

apricA

quariusPisces

Ase,

Asc.

Asc.

Asc.

Asc.

Asc.

52;58

21:4345:24

75;54108;21

133:19

105:45

25;2849;49

81;24112;54

136;52

159:02

29;1054;54

86;56117;24

140:12

2012:10

32:3559;57

92;32121:39

143:19

2515;22

36:3265:00

98:04125:49

146:22

3018;50

40:2970:34

103:39129:54

149:32

lb

A

Hull

(

)())

3)

aiS

(i(I(

S

f1

,hvr

tlab

ia3

AbM

ajnti’stable

for),,

-20°.

Ifa

>in

(1)

we

pu

ta’

=a—

in(

11

),

ifa

m(,l,1)

we

leta’

a.

We

thenfind

(bym

eansof

linearinterpolation)

anum

ber2T

such

thatf(

2,,’

2r)

=a’

and2

<+

180°.E

xamples

for.

=20°

(Table

3):

1.2

—40°,

thena

=2

5;2

8°

99:54°=

125;22°=

a’,2,

144;28°.

2.2,,

155°,then

a=

133;19°+99;54°

=233;13°,

a’=

83;41°,2-

=

282;4°.

The

sam

em

ethodis

usedfor

tileleft

quartileand

sextilerays.

The

text

saysthat

theright

trine,quartile

andsextile

raysare

diametrically

oppositethe

leftsextile,

quartileand

trinerays

respectively.

The

extantm

anuscriptsdo

notcontain

information

aboutthe

geo

metrical

significanceand

thecom

putationof

thenum

bersin

the

MajrifT

tables.Investigations

bym

eansof

apersonal

computer

ledto

thefollow

inginsights

inthe

structureof

thefu

nctio

nsf

q,s

andt.

For

didacticalreasons

we

will

firststate

theconclusions,

andpresent

the

numerical

evidenceafterw

ards.

38

We

firstdiscuss

thearithm

eticalstructure

off,referring

toF

igure6.

Assum

ethat

thehoroscopus

isin

signn

(i.e.2,,

=30(n—

1)°+5k

for

some

k,k

=1,2,3,4,5

or6,

inF

igure6

n=

3,k

=2,

2,

=70°).

Let

(i)

betile

latitudeof

theincident

horizonN

P,S

(seeS

ection2),

thatis

the

45)

S

tion

(Sectio

n2)

and

thtK

lA

,ii

mtables

Sethat

thesepro

jectio

n.

ri

IprO

xlm

at

‘Ia

‘I

pla

ins

thew

ords“ascei

siS

iith

headng

sI

3.ih

epro

jectio

nof

t,

sis

appxi

I

190ian

I’H

ogendzjk191

ascensionof

thei’

signfor

thelatitude

ofthe

incidenthorizon

through

thebeginning

ofthe

sign.T

hisis

tosay

thatin

thenotation

of

Figure

6,p*p

A(,)(30i)

—A

,>(30(i—

1)°).(4.6)

This

formula

explains(4.3)

and(4.4).

The

factthat

(4.6)is

onlyan

approximation

hasas

aconsequence

thatarc

which

isalw

ays

equalto

180°,is

“approximated”

bythe

number

nz

1),

which

canbe

verydifferent

from180°.

Thus

inT

able3,

theprojection

ofthe

arc

between

0°A

riesand

0°L

ibrais

approximated

byrn(

2H)

=149;32°.

We

nowdiscuss

thereason

why

Al-M

ajritidid

nottabulate

the

f(

11

,A)

beyondrn

(2

11

.).

Apparently

heknew

atleast

oneof

thefo

llow

ingtw

oidentities,

pip

*=

(4.7)

A1,,(30

i°)

—A

,1(3

0(i—

I)°)=A1,1(30(i+

6)°)

—

(4.8)

Form

ula(4.7)

istrue

becauseN

,P,,

P,<,S,

P,

areon

onegreat

circle,and

soare

N,P

,,P

,*,

S,p

p*

Therefore

theapproxim

ationof

canalw

aysbe

obtained

byadding

tothe

approximation

of(that

isrnQ

H))

theap

pro

x

imation

ofP

,*P

fl46k*,

which

isequal

tothe

approximation

ofJ)J)*

(by(4.7)

or(4.8)).

InT

able3

theprojection

ofthe

firstseven

signs(0°

Aries

to30°

Libra)

isequal

torn

QH

)=

149;32°plus

the

projectionof

thesign

Libra,

which

isequal

tothe

projectionof

the

signA

ries(i.e.

18;50°).T

hereforethere

isno

needto

tabulatem

ore

thansix

columns.

The

functionss,

qand

tinthe

Majrifi

tablescan

beexplained

ina

similar

way

asthe

functionsS,

Qand

Tof

Al-K

hwãrizm

i.A

l-Majrifi

wanted

tom

akesure

thata

successionof

threesextile

rays(or

ofa

trifleand

asextile

ray,or

oftw

oquartile

rays)as

inF

igure7

always

arrivesat

thepoint

diametrically

oppositethe

planet.B

ecausein

generalmo.,1)

180°,he

hadto

introducethe

functionss,

qand

I

satisfying(4.5).

The

Majriti

tablesw

erecom

piledin

orderto

replacethe

Khw

ãrizrni

tables.T

hetw

osets

oftables

arebased

onthe

same

astrological

doctrin

e,and

two

basit’ideas

arcthe

same,

nameR

thatthe

pro

jections”

onthe

celestialcc

uato

tiii

appioxinted

Iyt,cci

ins,and

thatth

reefunctions

(Sii

forA

K)

itd

/for

Al-M

ajri(i)havc

tohe

todit

4in

oid

erget

)ht

lvm

eaningful

results,such

as.d

.4casts

atrifle

ratat

B,and

Brasts

atrine

rayat

C,

thenC

must

easta

trineiav

atA

,etc

Ilo

west

r.A

lM

ajriti

improved

onA

l-Khw

3rim

tin

thtec

speets

1.A

l-Khw

ãrizmi

apprcx

ii‘1

cI

thec

liii.i

)ij

A,(3

0(i—

1)°)hut

AI-M

ajrifico

mp

uted

thesequ

tiltitiesexactly.

2.T

heF

(d,n

).F(d

,n+

1)etc.

inthe

Khw

arizmi

tablesare

app

roxim

ations

ofthe

pro

jections

of

theindividual

signsa,

n±

Ietc.,

hutA

l-Majriti

computcc

apr

xi

ation

sof

sits’

(i.eof

signn

pitissign

aas

rotcitt

kit

Mjn

titables

hadto

make

otlv

inciddition

(antpossib

oneitraetion

ofin(

11

)),

whereas

ant

oneusing

thekh

wariin

utables

so

dd

havehad

tosubtract

quan

titiesF( (I.

In)

C\

r’raltim

es3.

‘1 heM

ajrIIisubtab)

eona

iisix

co

lu,

bho.

Ks3rizm

itables

have12

coluri

ito

ast

quc

sn

ath

em

aticaladvantage

iits

trologi‘al

itet1.

Stise

on

ecom

putesa

successionof

two

quaitilerays

(oioh

threesextile

rays.or

ofa

sextileand

atu

ne

rat)

Iroma

po

int

4ott

thi.ecliptic,

asshow

nby

Figure

7.II

incu

osth

eM

ari

iib

les1

lastis

inthe

rossis

castat

pointI

‘iiall

ip

e4

always

thecase

ifthe

Khw

anes

at

use

Before

we

describeour

reeom

putao

on

andco

nip

artthe

recom

puted

valuesw

iththe

text,one

more

pro

blem

ha

tohe

dic

usstd

Neu

ge

bau

erpointed

outth

atn

inhei

sa

the

vte

1ly

andthat

theycontain

nume

isr

rs.In

geit

01

0able

todistinguish

betseen

asu

ihaltad

aco

mpi

tational

t no

r.4o

eer.

almost

allscribal

errorsin

thevalu

ess(.

0),

q(i,

0),

i(2,)

andin

)2)

canbe

removed

thanksto

therelatio

n(4.5).

For

example.

11ib

le3

(takenfrom

Suter)

we

h’e so

)49,S

7’N

)’)74,5

(20°)=

99:54°,m

(20°)=

149.32vi

c)in

no

talI

irti

.°‘20°):q(2

)°):t(20°)=

60:90,,

ihs

Ihicc

yak

cm

arw

ciccl.

andthus

itturns

outthat

149,32m

ustbe

ascrihal

error

for149:52’.

[hescribal

error

iseasy

toexplain,

becausein

theA

rabic:H

ad-n

oiatio

n

192Jan

F’H

ogendijk7,.

1c/m

oIc!T

Iift,31

Ial/,s

193

x),,

60+

iA

,120+

x),,

—180+

x2,,

240+x

2H=

3()()+x

5147:09

163;59199:18

206:50178;O

O151:38

10147;32

167:45201:50

205:08175:21

148:45

15148:36

170;07203:55

204;49173;18

148:25

20149:52

172:43205:51

202:30169:05

146:42

25151:26

174:17206;02

201:05166:53

146;27

30152:30

178:55208;00

200:39163:33

146:05

35152;43

183:33208:26

197:55?159:27

144;58

40154:43

185:41208:37

195;03158:25

145;55

45154:53?

189:39208;57

193:10154;38

145:55

50157:28

193;15?208;30

186:55154;28

145:55

55160;31

195:36?207:21

182:03152:58

146:15

60163:50

197:47?207:40

183:00151:43

146:31

Fable4.

Reconstructed

valuesof

m(?.,,).

32(1dm

-ha’)and

52(ndn-bd’)

areeasily

confused.In

thisw

ayone

can

reconstructalm

ostall

thevalues

ofm

Q1

1)

which

Al-M

ajriico

rn

pu

ted

.4°

These

restoredvalues

havebeen

collectedin

Table

4.Q

ues

ti()nm

arksindicate

valuesm

()

thatcould

notbe

checked,because

thecorresponding

s(2

,1),

q(

211

).

andt(A

,,.)are

missing

inall

man

u

scripts.T

heirregularities

inT

able4

must

bedue

tocom

putational

errorsm

adeby

Al-M

ajriti.I

nowgive

thedetails

ofm

ycom

putationof

theM

ajriitables,

referringto

Figure

8.L

et ,=

30(iz—1)

+5k°

forfixed

k,Ik6,

andlet

Pbe

thepoint

ofthe

eclipticsuch

that)L

,=

30(n+j—

1)°fo

r1

fixedj,

1 j6.

Points

N,

Eand

Sare

theN

orth

,E

astand

South.

pointsof

thehorizon,

Vis

thevernal

point,H

isthe

horoscopus,C

is

thecelestial

North

pole,and

arcC

Ris

perpendicularto

thein

ciden

t

horizonN

PS

.R

ecallthat

q=

areC

Nis

thegeographical

latitude,e

=

<F

IVE

isthe

obliquityof

theecliptic.

The

risingam

plitudew

ofthe

horoseopusand

theangle

between

eclipticand

horizoncan

becom

putedfrom

andA

H)

bythe

sine

rulein

triangleV

ilE.

CN

\\‘

licuF

e.5.

-

90°w

.T

herefore°=

I’NJI

inlx

Itm

intc

C(

nfcn

t

rule:cos

U)

cot(A%

),)c

sitn

1i’

(410)

The

latitudeof

theincident

honzon.

I’St.an

beto

md

hthe

sinerule

tnA

CR

A:

5if

lC

OS

0S

inq

(4.11)

recallthat

=±

(R

,the

hs

‘nhung

difl

thastern

hemisphere,

andthe

mir

tgr

fR

isin

ti

Wi

sm

nero.T

Ie

obliqueascensions

a4

)and

(44

can

ob

tned

bym

eansof

formulas

(2.1)—(2

ifw

ereplace

qhs

There

areof

coursern

anequivalent

ways

tocom

pute.

andit

would

beof

interestto

knoww

hatm

ethodA

l-Majriti

usedbecause

theproblem

was

ratherdifficult

forhis

time,

The

crucIalstep

inour

argument

isthe

computation

of0

intrianele

hA

Ph

im

eansof

(4.10).T

hisproblem

ism

athernatiii

yi.quivalent

Ithe

dnatio

ofthe

qibla,that

isthe

directiof

Mcca

fiomlx

mu

Ma

thelatitude

ofthe

locality,an

Ith

dfference

inm

gic

‘t

Mecca

andthe

locality(to

seetI

i1

cntifvII

‘th

thh

pokof

thesin

co/sinE=

sinA

q(A

q1)/sin

ip=sin)

11

/cosq.

(4.9)

Intriangle

PU

Nw

eknow

PH

=<

PH

N=

andH

N=

194Jan

P.U

ogendijk

,I

I12

1j

4jS

(‘6

25

45

8.

(1(frolog

Iil’h

1

58:24

(.1)

40,46(-I)

70,39(-4)

(0309(

II)20:03

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(+1)

1019,26(4

40:41(-6)

70,26(-I)

103,11(-II)

(20,30(-65)

447;45(-(6)

1519,1

0(5

)40,40)-I)

70;16).13)103:17541)

12903(*47)

140,35(.1

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‘2’.),

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Table

5R

ecom

puted

valu

esf(261,

30(01+1—1)’),

andthe

differences(values

inthe

Latin

text

minus

recomputed

values).l’he

signnum

bern+I—

1appears

aboveeach

groupof

6values,

earth.P

with

Mecca,

andN

with

thelocality).

Thus

thecom

putation

of0

belongedto

aclass

ofproblem

sthat

were

studiedin

medieval,

Islamic

science(but

thereis

ofcourse

noevidence

thatA

l-Majriti

recognizedany

relationshipw

iththe

determination

ofthe

qibla,also1

notreatm

entof

theqibla

problemby

himis

known

tous).

Iknow

of

onlytw

om

edievaltexts

where

computation

ofis

described.A

l-

BirU

ni’sM

asudicC

anoncontains

acom

putationof

bym

eansof

the

‘l’able6

Recom

putedvalu

cs

f(32(

)511

(6ffs’I*.

IO

HS

01114(cx)

07

i00

5leC

*)I4

l)71.I)ed

values),16

S)1

)0

11

ill1

ap6

rs.860%

6013114)

values.

azimuth

andaltitude

ofP

.3

-A

l-Majriti

probablydid

notuse

thism

ethod,because

theprelim

in’y

deterntrlatlO

fl-

thean

11thand

altitude

of

Pinvolvos

tflSlch

T.eo

essars)rk

“cource

isthe

anonymous

14

ccIarnil

Zi

eIc

ofA

lB

iruni

isre

peate

d.

4W

hatcvcA

lM

aIritm

elci

sas

sthave

involv

eda

lotof

nUm

ericalso

rkT

herecom

putedvalues

Ji

54,

‘O(n

+1,)‘I

aredisp

1a

edin

‘[ahles5

196Jun

1’H

ogenthjka

i/ate

.4tro

di,’icai

Ia)t

I 97

and6.

Each

rowin

Tables

5and

6contains

fromleft

toright

an

argulnent)L,,

30n+5k°

(k=

1.2,3.4,5.6)and

therecom

putedvalues

J,

130(n

+j—

1)°

)for

j1,2,3,4,5,6

indegrees

andm

inutesof

arc.

The

numbers

inparentheses

arethe

differencesin

minutes.

with

the

understandingthat

therecom

putedvalue

plusthe

differenceis

the

valuein

theL

atintext,

For

thesixth

column,

therecom

putedvalue

piusthe

differenceis

therestored

entryin

Table

4.T

ables5

and6

are

basedon

parameters

ql=

38;30°,=

23;51°.R

ecomputed

valuesfor

=

38;3

0°,

23;35°differ

fromthe

textin

asystem

aticalw

ay,hence

Ai-M

ajripm

usthave

usedthe

Ptolem

aicvalue

r=

23;51°,just

as

AiK

hwárizm

idid.

Because

thedifferences

inT

ables5

and6

change

signfrequently,

thesedifferences

canin

allprobability

beattributed

to

randomcom

putationalerrors.

The

lengthof

thecom

putationis

one

likelycause

ofthese

errors,but

asecond

causeis

surelya

certain

carelessnesson

thepart

ofA

1-M

ajriti.45

Note,

however,

thatthe

averagedifference

ofthe

recomputed

valueof

rn(,)

(sixthcolum

n)

minus

thecorresponding

valuein

Table

4is

only—

2m

inutes.

l3’m

eansof

Tables

5and

6one

canidentify

some

furtherscribai

andcom

putationalerrors

inthe

Latin

text.T

heenorm

ousdifference

inthe

firstentry

for2,

75

0and

theabsence

ofsuch

differencesin

the

subsequentcolum

nsshow

thatthe

value25;05

inthe

textm

usthe

a

scrihalerror

for29;05.

For

22300

thedifferences

—82,

—82

and

—83

inthe

second,third

andfourth

column

suggestthat

Al-M

ajrili

addedtw

ocorrect

numbers

toan

incorrectnum

berin

thesecond

column.

Inspite

ofthe

computational

errors,the

MajriP

tablesarc

an

impressive

pieceof

work,

andan

example

ofthe

positiveinfluence

of

astrologyon

mathem

aticsin

theIslam

icm

iddleages.

A’know

ledgemenl

This

paperhas

beenm

uchim

provedthanks

tocom

ments

made

by

Professors

E.

S.K

ennedy(P

rinceton),D

.A

.K

ing(F

rankfurt)and

H.

J.M

.B

os(U

trecht)on

anearlier

version.T

heyare

notinvolved

in

anyrem

aininginadequacies.

ftI

NIs

.\hUSI

ish;tr.

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the

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rti

‘‘

slarntiS

ttenet,’,,

Srk

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asicii

iV

sIt.41

1

and41,2)

I’S

Kennedy,

I11

‘Sstirs

ciat

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tote

tint

0.11all,

I2,7.7,5

((‘ii

(‘Iit’

it‘7

(it,I’

‘it.itinlitctil

Sotitt)

,S

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ctict4o,

pp12

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I.S

Kettnedy,

M.

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I 1)5’(‘ti

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titt

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Band

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SK

ennedy,I-I.

Krik

ort,tn

.

I)72Flie

astrologicaldoctrine

ttfp

tpeeiiitg

therays.

.41.41,,ath

25pp

11).

pitto

edin

SS

Kennedi

,colleagues

andfor

it de

I.5

Jasn

11

at

an

lem

rut

1983,

pp172

31/3S

lAM

S,

seeV

.I’.

Malvievskaya

BA

.R

isetilt,ld

(I.I’

Matvievskaya,

B.

AR

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uiiim

askog

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datael,ti

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Alt

eta

ItI

15

VII

iii’3(M

attieniaticiant,and

astron

oi

ieisf

theIslam

icm

idde

agesan

lb.

iat

Is)t it

Russian).

Mosc

ow

1983,3

vttisA

N:illintt,

i-Ilattainsits’

.Athatenji

it215<

alto),tttift’jfli,

Vol.,

51lan’s

(Ri.

.t(

‘tat

torI

19(5).

()N

ettgebaucr,1962

flu’astronom

icaltablesofA

lK

hwanzm

i,C

openhagen(lltsto

risk.filo

sofisk

cskrifter

ud

gi

vetaf

delK

ongeligeD

anskcV

tdenskabernesS

elskab4

(no.2))

()Peder’,en

1974A

surveyo

fthe

Aim

agest,O

dense.Iholeny1989

leuabiblos,edited

andtranslated

byF.E

.R

obbins,C

ambridge

Mass.

(Loch

Classical

1braty

475)

ISeigin

19741979-

(;ecehuhtedes

arabischenSehriftru,ns,

Band

V:

Mathem

atik,L

eiden1974,

Band

VI,

Astronom

ic,L

eiden197$,

Band

VII:

Astrologie

Meteorologic

undV

erwandtes,

Leiden

1979.11.

Suter,

1914D

o’actronom

tschenlafs’ln

desM

uhamm

ad/hoM

usaal-K

hwdrizini in

derB

earbeitungdes

Mat/am

athu

Aiiried

alM

ad/ripund

derlate/n,

Ubersetzung

desA

thelardvon

Bath

aufG

runddci

Vo,a,heiu’n

vonA

.B

jornhgund

R.

Besthorn

herausgegehenund

komm

entiert,K

open

hagen(I).

Kgl.

I)anskeV

idensk.Selskab

Skrifter

7.R

iekke,H

istoriskog

filosofiskA

id.Ill

I).R

eptintedin

Ii,S

uter,R

ebel/gezur

Geschichte

deeM

athemattk

undA

stronomic

imIslam

.N

ueltd,ucl,se/ncr

Schriftenaus

denJahren

18921922,

Frankfurt

1986,2

vols.(V

eroltentltchungcn

desInstitutes

furG

eschtchteder

arabischislam

ischenW

issenschaftenR

etheIt,

Aht

Mathem

atik,B

and1,1

und1,2).

GI

loomet

1968-A

surseyof

the‘loledan

tables,O

siris15,

5174.

I.V

etnet,M

A.

(‘staB,

1965-1

asobras

mathem

8tieasde

Maslam

ade

Madiid,

AlA

ndalus30,

pp.15—

45,reprinted

inJ.

Vetnet,

1studtos

sobrehistorta

deía

(iette

iam

edieval,B

at celona1979,

pp.241

271.Y

ahtathu

Abi

Mansu,,

lQiin

Ilu

ter/fadaitrontim

icalia/ties

forthe

(‘ahphal

Ma

‘,nun(in

Arabic),

Frankfurt

(FSe/gin

ed,P

iiltlicitionsof

theInstitute

forthe

History

ofA

rabic-Islamic

Sciences,Series

C,

Facsimi

Iceditions,

Vol.

28)

NO

TE

S

Io

nthe

astrologicaldoctrine

of‘casting

therays”

seeB

ouchdL

eclereq1899,

pp.247

251(antiquity);

andN

allino1903,

pp.307—

313,S

uter1912,

pp.98—

102,K

ennedyand

Krikorian

1972(Islam

icm

iddleages).

2.See

Al-I3iruni

1956vol.

3,pp.

13851392,

Russian

translationin

Al-B

trüni1976

pp.470—

474,m

odernsum

mary

inK

ennedyand

Krikorian

1972,pp.

6—7.

3.See

Al-B

iruni1956

vol.3,

pp.1378

1379line

7,R

ussiantranslation

inA

l-Biruni

1976,pp.

465466.

4.K

ennedyand

Krikorian

1972,p.5.

5See

Al

Birum

1956vol.3,

p.1385,

Russian

translationin

Al-B

iruni1976,

p.

470.

6.O

ttA

lK

hwarizm

iseeG

AS

V,pp.

22

24

1,G

AS

VI,

pp.140—

143, MA

MS

II,pp.

40—45

anttie

articleby

G.J.

Toom

erin

DSB

VII,

pp.358—

365.

7.U

tIhn

llibtntaste(1A

bS

Hhints

1987.V

ol2

P6

8.N

otedby

Nallino

1907p

9.1

tetables

wete

desertti

Kr

nnedyand

Ktikot ian

I

10K

ennedyand

Knkortat

II.Isom

er1968,

PP148

51,

12.()t

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Abm

adIs

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194195

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194,

pp.sit

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13.It

transcriptionsof

sexagtsii

m‘sits

ofa

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m,il,

thus.23;51

,20.

ThyI,

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p2)6

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c’wtt

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1914,P

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1987,PP

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herefore

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Suppose

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arrivesafter

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ittakes

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I’S,so

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(A,).Sem

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I’Sis

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herefore

a-

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x(I’))- h

(A),

or

ri(A0(40)

il,,(A,,))

(h(A

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))-

(A,,(A

,,)’—A

,(A,)).

(2.8j

Ptolem

yrestates

(2.8)in

thefollosring

forni(1980

pp

.30

0305):

i-I-

(10(4<,),l(A

j)1.

-{A

,,(20)

A,(A

,.)—

(A,,(40)

.4,,(A)))

will,

k—

71,,(Ap)

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pp.300-303.

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