32 2n2 - jphogendijk.nl
Transcript of 32 2n2 - jphogendijk.nl
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
(-14)(47,09
(+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
)
20iR
,16)*14)4*2:42)-Il)
70,11(.23)
103:261,13)(29,44
11(3)149:31
(.21)
25643
(-3)48,47
(-32)70:09
(+31)(0319
(.26)(5
02
6(.30)
(50:34+52)
3018,52
(2140,53
(.10)7
0I2
(*18)103,56(,19)
13116
(-.19)151:45
(.33)
23
45
67
3522,04
(-4)51,15
(-60105:131
10)4131091
16)434,03
(20)
453:06)23)
402207
(*13)51,13(416)
01,7
5(5
)1(3,59
(-4)135,19
5-.))(54:371.6)
4522,))
(-6)51;14
(-6,85,39)-R
I)114,53
(-82)(36:4-4
(-73)156:17
(-94)
5022,10547)
51:18(.15)
85.54(-20)
1132(2
9)
130,16(-23)
158:091-41)
5522,26
(-I)51.23
(.18)86:14
(*11)(16,55
(*31)439,57
(.44)
160:11(.20)
6022:371,19)
54:36(.24)
86:36(*29)
118,02(.58)
(41:40(109)(62.25
(+8))
34
56
78
6520,59
(.4)
64.08(-10)
96.22(-9)
420-5)(.3
)1
42
08
)I)
105.51(-52)
7029,01
(-I)64,191-9)
97,4954-3)(221391*8)
144:195443)467,27
(*10)
7529,041
239)64;20
1-7)98:13
(-7)124:29
(*6)(6
0,6
0)6
)170,13
(-6)
8029,0
9(.II(
64.30)-4
899:101.25)
426:21(.32)
149:49(-41)
(73,09(-26)
8529,10
)+12(64:47
(-65)105,0)
(-41)120,13
(-53)(51,56
(-66)176,11
(-114)
9029,22
(*1)64,55(.9)
49849(,45)
430:03)17)
454:33)26)179:18)
23)
45
67
09
953532
(+8)459(,2
I(402:18(430)
127:41(032)
152.561+52)
182:271.66)
6035:29
(-4)72.241+
1)403:46
).7)
430295(-8)
455,541.4)182:351+6)
0’55:24)4-)
72:41(*18)
404401+29)
(32,22(.4
7)
(58:48(+56)
(88.401+59)
III35:10
))7250
)((8
6:1
31
)134:29
))(61:33
)‘)191:371.98)
1153
5:0
9(9
72
51
(9407:08(9
436,24)9(6
4:0
6(9
494:261+701
12*35.101()
72:4525)407:49)9
130:05))
106:31(9
(97.02.47)
56
78
910
12537:42)66)
73.271*36)104:42)-IS
)(33:50)4)
(64:36(*44)
499:26(-8)
13037.341*16)
73,52(*18)
486,04(+6)
(35
561.6
)166;56
1(66)
201:341.16)
13537,21
(*19)74)10,(*35)
107:40(47)137:45
(.40)(69,02
(*33)203,26
(*29)
4133:03)*
10(74336
(.27)100:04
(.2,2)(39:191*24)
170:5)(+37)
205:03(*48)
44’36:42
(+51)73.56
(.571406:361.7)
(40,37(-.5)
472:24(-4)
206,22(-20)
1543
6)9
(.2)
73.35)+20)
106,561*14)(41:381+22)
473:4)1’S”)
207:231.3
5)
67
89
1011
(5537.14
(.11)
73091*5)
186,32).20)138:90)*21)
172,20(.6
)200.)
11*15)
6637,04
(.1)
73:2
9)9
)(07:301*10)
140.04(*8)
173:19(-7)
200:4114)
(653
6:4
91
4)
73.36)20)
100:14(-(0)
145:0
4)))
474,84(-2)
208,55(.2
)
17))36.27)
4)73.31
)-5)(08:44
(-3)141,49
(-8)434:35)
16)2011:53
(-23)
17536701
23)73.13
(-17)109.60)-22)
442:205-44)
174,51(-65)
208:5))74)
(803
5:3
1)1
)72,407-3)
109:021-2)(42,57
(.27)474:53
(.18)200,02
(.22)
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n73
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‘2’.),
.)‘-
-
‘2
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.
I 985.lit
gi
atIntl tidu<
no,to
‘a
Aii,
og
t(
ihiS<
gin.
pt.hlin
ifthe
Institutt.loi
thlisiti
iof
•\r:tbiolsitm
iaS
ien
<’.
I ,tcstmie
edit‘‘it.,
StI
21)
Al
Fir
irn,
I!Q
inunu
1. I/asit
(iiiSi ‘.I’t
i/s<B
‘thd
t()sm,ttit
Iiw
itt,eP
tiltsit
tollu
ron
t),3
Al
Bit
nt.178
Attn
Raik
ha
iil3et urn
(93
1481
hian
iti’rot
et
isiii
Vp
K,
iiiM
at.
’uda
knii.
VX
itrttnsl
13A
Re
enft.
1A
AS
ned
is(in
ftussiitnlB
tL.a
AB
ttuehcI
eclereq,
189<.
I‘A
.ttroiogiegisvqut
P,t
J.D
ie.’ki’r.
I92
131’T
hs’ori<dIr
Sc,ins’nuften
Io
p<p
1I/and
II
wferun
1a
Isott
/1,,s
‘irita
nn
itIdan
IcI.)
Die
Gest hi
Itt,
Inito
anI
dtit
3SF
,
1)72‘A
lt:(‘
V.
(Allispt
tedt.
13C
lint:
ofS
seniifts’I/iot’i
<1185,N
aS
rhI
tol
plus
sup
Iplus
trdtees
(1S
tt.ce
FS
e, gin.
I SnII
btiiiit1987.
litt otn
p1<It
hoc
kto
air
t1<1
tiii
ftalt
itlit
07
5it
dS
esgin
ineoll.thtration
alb
Ni.
Ant,
it.S
Jtil.ttt.liaI
Net
haiti.I
ian!. I
ati
sl’ltcstt’tnsof
the
insiiiuiefor
theh
istot
01A
rti
‘‘
slarntiS
ttenet,’,,
Srk
sI
asicii
iV
sIt.41
1
and41,2)
I’S
Kennedy,
I11
‘Sstirs
ciat
islantie
tote
tint
0.11all,
I2,7.7,5
((‘ii
(‘Iit’
it‘7
(it,I’
‘it.itinlitctil
Sotitt)
,S
ew.S
ctict4o,
pp12
I‘7
I.S
Kettnedy,
M.
Hkennede
I 1)5’(‘ti
eraphitsi/coordinates
of10,
dii‘t
Irtni
humt
iite
eI
titt
It(F
Seeginc,d
\‘ei,itientliehungendes
inslttih
’sflit.
Iohi
lilt.It
Sr
It‘a
1’IsI
tnt‘lu’
V,
no
h,tflen,
Rethe
A,
Band
2)I’
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
osencit
1t85.M
att’ntug,/ij
,4sjro
no,n
iM
uiiim
askog
Sr
datael,ti
vut
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
Mslam
sthu
Abm
adIs
IIPP
194195
(alM
’ijtitt
st‘
Sutet
194,
pp.sit
S
13.It
transcriptionsof
sexagtsii
m‘sits
ofa
semicolon
‘in]
m,il,
thus.23;51
,20.
ThyI,
steS
utet1914.
p2)6
P
t,ittle
forasttolahe
c’wtt
Pris,
3ihliothbqueN
itionaliI,
4C
nnpite
SuIts
1914,P
l(
atd
Kennedy
1987,PP
ISS
ite1914.
PP206
22)t
hilots
-
a’A
sB
itout
1lietiass
iges
9.hr
listset.
Ktrtncdy
an)
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tI
ritttd
ittY
ah1
iln
Issiinilr
hasttarrant.ed
IeF
uttdn
Ahu
Ma
shat1)8
2(1N
illims
190),p.
.111(haset
At
Bttiint
1956vol
3,PP
1K
rttnecl
andK
rikotian19
21.
AB
trutituses
thelet
tstnt
rt
P1384
lineI
22.A
toM
a’shat1985,
p(1
Ii23.
Nilliits
1903p.311.
24.Pt
slcmy
1981).PP
290-30.
251t
tlettt1980,
p299
‘ttlesI)
eeLr
1925,P
l12
20ti
thetout
linesarc
greateirtl
Because
lttoletttys
textit
expl,iirthits
etntpsttato
nit
198Jan
PU
ogendijk
16St
te1914,
plO
ll,N
i gUt
17S
‘K
tnn
td1956,
P140
pi‘s/ia
ofIslam
Ied
et(B
18F
tcte
tuts
foundft0
‘xim
Ashe
2528,I
9ah
e23
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et
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theE
asternhalf
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ilarform
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forother
positions
at
I’and
Q.A
ssume
thaIc’r.
A,
Aand
A(or
A,,,
cf.(2
5))
areknow
nand
thatone
hasto
conip
ute
thetim
e-intervalu
ittakes
Qto
reachthe
greatcircle
NI’S
inF
igure2
bym
eansof
thedaily
ostat ionIn
generala
I”Q
N
F’roler,isfirst
com
putcsa(P
).that
isthe
distancefrom
Pto
them
eridianin
seasonalhours
x(I’)(/1
0(A,.)
—
isherehU
e)is
thelenittli
ofone
seasonalhour
when
thesun
haslongitude
A.
One
seasonal
houris
onesixth
ofthe
periodhctaeen
sunriseand
noon.T
herefore
(116)’(90°4A
,,(A.)
A,(A
)).(2.7)
Siniilarlvfoi
11inl(1
(A,,(20)
Suppose
that(1
arrivesafter
arotation
ofa
time-degrees
atpoint
Pon
semicircle
NPS.
Then
ittakes
Qx(Q
)a(P
’)seasonal
hoursto
reachN
I’S,so
thatu
—(x(Q
)—x(P
’))-4
(A,).Sem
ii.ircleN
I’Sis
alniostan
hourline,
thatis
tosay
that.r(!’)
a(I”).T
herefore
a-
(x(Q)
x(I’))- h
(A),
or
ri(A0(40)
il,,(A,,))
(h(A
)Ih(A
))-
(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
,.)—
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will,
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.4,(A
)}I{A
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4R)}.
(2.9)
I?is
thep
oin
tof
intersectionof
theecliptic
andthe
meridian
ifw
eturn
thecelestial
sphere
suchthat
I’is
onthe
Eastern
horizon.(‘oinpare
Ptolem
y1980.
pp.300-303.
26.K
ennedyand
krikorian1972,
pp.7
12,for
theA
rabictext
seelbn
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1987V
ol.2.pp
66438
27.‘loom
er196$.
p.147.
28.K
ennedyand
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1972,pp.
13—14.
29.‘Ioom
er1968,
pp.148
149,K
ennedyand
Krikorian
1972,pp.
9—10. T
hecorrect
valuesof
the
obliqueascensions
forthe
firstsix
signsare
asfollow
sfor
r=
23;51°,q
=330;
20:08°,23:33
-
29:38’,34;56”,
36;l5°,35;32°.
Table
1is
basedon
thevalues
20:05°,23;34°,
29;40°,34;52’.
36;14’35;35°,
seethe
firstcolum
n.
30.O
nthe
latitudeof
Baghdad
inm
edievalIslam
icsources
seeK
ennedyand
Kennedy
1987,
p.85.31.
Suter
1914,pp.
19,58;
Neugehauer
1962,p.
47.
32.K
ennedyand
Krikorian
1972,p.
Ii.
33.K
ennedyand
Krikorian
1972,p.
10.
34.V
ariationsof
afew
minutes
ofarc
occuras
aconsequence
ofrounding
errorsand
dueto
201
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