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T

E.S.

Kennedy

On

theC

ontentsand

Significanceof

theK

htqänIZ

ijby

JarnshidG

hiyãthalD

inaL

Kãshi

1998

Institutefor

theH

istoryof

Arabic

1slamic

Scienceat

theJohann

Wolfgang

Goethe

Urnversit

Frankfurt

amM

ainI

iian

a

Kh

au3

nLu

PrefaceT

hisis

adescription

ofa

rnun

iija

rin

iiiliii

teeastronO

ierastroI

,W

ellos

1imdred

tibcxisted

(a)st

ofm

ttenol

extanthe

.ssi.

ate

theexac

ciencei

oc\

amd

tci

Neerth

eless.thi.

pecinir

orpubi

solit

1Its

author.Jam

shidahkashi

sis

hefirst

dit

.r

u.iluc

1S

amarqand.

fora

fewrears,

thes.

entd

itcapital

‘itthe

send

computational

mathem

aticsm

arkedthe

culminatm

onof

‘isle

aiJs

which

was

characteristicot

maih

enia

tic,

nthe

world

o’coneem

poranes,w

hocontinued

tocalculate

mine

abanlar

cs51C

r

of

sexaflesima!

fraclacom

bineds

Iinn

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describesa

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andthe

intihaeach

having

severalvarieties.

These

indicatorsw

eresupposed

toenable

theprediction

ofevents

in

thelives

ofindividuals

andin

thew

orldat

large.T

hetreatises

tendto

havea

standardform

atT

heyhave

anintroduction

andtw

o

chapters.T

heintroduction

isusually

auseful

glossaryof’

technicalterm

sused

inthe

treatiseC

hapter1

isdivided

intonum

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rulesfor

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computations,

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plesof

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results.It

is

convenientand

illuminating

todisplay

suchexpressions

herein

algebraicsym

bolism.

butof

coursethere

isno

traceof

symbolism

inthe

originaltext

Rules

arew

rittenout

verballyC

hapter2

consistsof

proofsalso

usuallyin

numbered

sections,som

etimes

subdividedinto

numbered

rules(qa’idat,

sing.qa

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nfortunatelythe

same

sectior

number

inthe

two

chaptersdoes

notusually

correspondto

thesam

esubject

Forcross

referencepurposes

below,

passagesin

thetext

areindicated

bycom

binationsof

numbers:

thetreatise

inR

oman

numerals,

thenthe

chapter,then

thesection

numbers.

precededby

asem

icolon.N

otethatthe

introductionsto

thetreatises

haveno

numbered

sections.R

eferencesto

thetables

aregiven

asfolio

numbers

of

theIndia

Office

manuscript

Num

bersin

squarebracket

arereferences

tothe

bibliographyat

theend

f

thepaper.

The

Author

ofthe

Zij

Jamshid

Ghiyãth

al-Din

al-Kãshi

(oral-K

ãshãni)w

asa

nativeof

Kashan,

huthe

seems

tohave

spenta

goodpart

ofhis

lifein

otherparts

of

iranFor

atim

ehis

patron

atS

hiraiw

asanother

grandsonof

Tim

ur,Iskandar

b.U

mar

Shaykh,the

rulerof

bars

([381, p.105)K

ashihim

selfm

entionshaving

beenin

Isfahan,(1101, p.176),

butby

the

time

of theeclipses

of1407

hew

asback

inK

ashanlie

was

alsothere

in1416

when

he

completed

thefirst

versionof his

Vu:har

uI-Jladtiiq,a

descriptionof

anequatorium

,an

analogcom

puterhe

inventedfor

determining

planetarypositions.

By

1420he

must

hase

joinedthe

teamof

scientistsU

lughB

eghad

assembled

atS

amarqand

tobuild

and

operatethe

observatoryIn

1424K

ãshicom

pletedafR

zvdlataI-’vluhxnya

[11)his

unprecedentedlyprecise

determination

ofthenum

berit

am

asterpieceof com

putational

techniqueT

hreeyears

laterhe

finishedanother

major

work,

theiM

ftãhal-/u

sa/i10J.

This

coversthe

whole

fieldofarithm

etic,and

containsthe

earliestcom

pletedescription

ofoperations

with

decimal

fractions.O

nthe

morning

of

Wednesday

19R

amaddn,

83211(2’

June1429),

atthe

observatory,K

dshidied,

leavingincom

pletethe

observationsrequired

forU

lughB

egs

zil1-us

successoras

director,Q

ádizadaal-R

umi

alsodied

with

thew

orkunfinished.

Their

much

youngercolleague,

All

Qushehi,

tookover

andsucceeded.

Man

userp

tsof

the/

jIndia

Oflice

U.endon

Ms

-0tl.

the-‘

The

material

presentedhcow

isbased

primpa

lsupou

colO

mof

thism

anuscriptR

eferencesto

itpise

thefjlio

andIn

rr

‘,.i

ASolya

IstarM

SR

cfersnes

tt

ita

itt

CIpt

torepresent

t vovery

os

rathcm

idit-al

cts

Sri

oemI

totbei

irye

frequentlybeen

rephrased,presurnahls

hrthe

authotI here

.:‘hi

e:dicalionfrom

thesim

plifiedtables

oflatitudes

forM

ercursin

reatiseP

the’ar

hethe

earlierversion,

however

thisis

anth

‘butc

irlu

sty

)thcrop

esa

rydera

ar\s

tiS

ne

)aral-K

uoh

Ca

oM

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SC

RIP

TIO

NO

FT

HE

ZIJ

fIr

1itle

PageO

nthe

frontoftheflyleaf preceding

folioI

isa

notereporting

thedale

ofKãshi

s

deathas

givenabove.

f.Iv: I(A

SOv:1)

Invocationand

Dedication

The

bookopens

with

elaboratethanks

andpraise

tothe

Creator

of

theterrestrial

sphereand

thesurrounding

heavens,then

passingon

tothe

ProphetM

uhamm

adhis

family

andcom

panionsT

heauthor

nownam

eshim

selfashaving

longlabored

inthe

fieldsof

scienceuntil

som

uchm

aterialaccum

ulatedthat

hefelt

itincum

bentto

compose

azij.

I hishe

setabout,

much

of

thetim

esecluded

ina

housein

Kashan,

anddiscouraged

But

“thesun

ofgraciousnessburst

shiningfrom

thehorizon

of

happiness”.

andso

onfor

more

thana

pageof

extravagantfigures

ofspeech,

many

ofthem

astronomical,

alllauding

thevirtues

ofthe

Khaqãn

Ulugh

Beg.

Being

acceptedat

his

court,K

äshicom

pletedthe

zijand

presenteditto

theim

periallibrary,

The

actis

likenedto

thatof

anant

ho

(asrelated

intw

opoetic

quotationsone

Arabic

theother

Persian)gave

toSolom

ona

locuss

leg,saying

“thegift

is

proportionedto

thesize

of thegiver”.

9A

tablefor

costing

In‘iate

athe

cOs

0orricti

cc

ine

oth

i

40A

nN

md

‘irc

a4,

Cc

ectiors

rhe

plaiate’.

aits

60D

etermination

ofthe

aseendenttram

tieshadoss

a0’.

,‘cL

ita

ssall61.

Detein

ation

oftheascer’dent

fromthe

obsessane

‘a

,s,

taish

chhas

thesam

eazim

uth69,

Responses

tocriticism

sra

de

bycom

mentators

toi

I“in’

7i

f4r:15A

Sr22)

Ii

sboa

onar

Ma

Motit

of

Three

Iunit

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hed

att

eelre

30Shahrivar

I)ld

Styk07

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isfand775

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400,(O

ppoliei4

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1407(O

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The

method

ofPtolemy

isused,

with

minor

‘ariaare

reported

infull

Am

ongthe

xc’.Is

arcbr

thtunas

Ii

I70

‘475

fhisi

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e

f2v

I(A

S1v.7)

AS

tatement

asto

How

theZII

Was

Com

posedT

heauthor

decidedthat,

exceptfor

thelunar

parameters, he

would

adoptthe

mean

motions

usedin

theZ

ij-i flkhãni,thesehaving

beenobtained

fromobservations

directed

byN

asiral-D

inal-T

fisi(d

1274)at

theM

araghaobservatory

inw

esternIran.

The

equations,how

ever,w

ouldhe

denvedanew

,and

variousim

provements

adoptedso

that,liar

instance,the

true

longitudesof

Mars

would

come

out

sometim

essaith

adifference

of

almost

adegree

asbetw

eenthe

two

zijes

t’34

Therefore

forthis

work

thenam

e(A

S2r.13)

ZK

haqanidaI’aA

mIl

ZijOs

ilkhani(The

Khäqani

Zin

Com

pletionof

theIlk

hni

Zij)

would

headopted.

givenh

I”.ul

ial-I)

rilauhrih

opaul

dm

Forthe

dailsanom

alistictfls,

inm

otio

nf\ash

ihas

133

.M

’oi(i3

7’O

”

This

isserv

Clo

seto

Ptolemy

,a

‘orrccb

o19

(AS

compose

3,’.1,’.6,a.7

S

lui,civ

hiM

m

S

‘liii

aliens

0‘

f.3r:6(A

S2r:18)

AN

umbered

listcom

prisingseventx

specifictopics

demonstrating

thesuperiority

ofthis

zijover

theZ

ij-iIlkhãni

andits

comm

entariesin

particular.and

toall

otherzijes

ingeneral.

Fixamples

are:2

(ilossarmes

of technicalterm

s3

Presentation

ofproofs

oK

a

rva

431

on’.Os

tOcalculatm

orn3

lit.11

mti

f,6v:9(A

Slir:1)

TR

FA

I’ISE

I,O

X(‘,1,F

Xl)%

,l4S

(Dr

Renno

sanOs

len,

Iri’.i

Meyer

Sfrank

Mai

prograta’

mee

AIi

well

ad

inal

(on

‘is‘e

c1i

c‘0”

‘ci’

on,R

ohc1325

(us

Li

‘Oft

I

Iit

addir

tein

4S

E.S

Kim

srnK

snKh5ciSn.

73

oneofthese

isentered,

theprogram

imm

ediatelydisplays

thecorresponding

Julianday

number,

theday

ofthe

week,

andthe

equivalentdate

inall

ofthe

othercalendars)

f,6v:10

Introduction:O

nD

etermining

Years,

Months,

andD

aysB

riefdefinitionsof

fundamental

conceptsare

givensolar

andlunar

years,m

onths.and

days,and

therelations

between

them.

The

notionof

erasdating

fromm

emorable

eventsis

introduced.

17r:I

(AS

6r14)

Chap

ter1.

On

Seleucid

(Rhm

i),llijri,

andY

azdigirdD

ates,and

theE

xtractionof

One

fromthe

Oth

erFor

eachofthe

threecalendars

mentioned

inthe

title,year

lengthsare

giventhe

names

andlengths

of them

onths,and

anypeculiarities

oftheparticular

calendar.T

heinitial

week

day(m

ad/thai)ofthe

eraof

eachis

given,as

well

asthe

number

ofdays,

indecim

alsand

sexagesimals.betw

eenpairs

oferas.

f7v

(AS

7r)T

ablesof

theE

levatedD

aysof

theH

ijri.,S

eleucid,and

Yazdigird

Calen

dars

For

years1,

2,3,

,60,

120180,

...

1860of

thethree

calendarsnam

ed,the

entriesgive

thedays

fromeach

epoch(in

decimals

andpure

sexagesimals)

andthe

madkhal

(initialday

of the

week)

orincrem

entsof

madkhals

f8r

Table

ofthe

Elevated

Days

ofthe

Madkhal

Argum

entsof

theT

hreeC

alendarsW

ithinitial

month

daysof

thesam

ecalendars

asargum

ent,the

entriesare

asdefined

forthe

precedingtables.

0tue

f9(A

SSt

tabe

ofhe

esee

fth

f.9(A

S9r3

Table

ofthe

(‘binesiand

Persian

am

cs

ofilit

Dais

andT

emperam

entsof

theD

uodecimal

(‘ycleof

Choices

f. I iv:2I

(AS

Iir7

Subsection(n

Derm

ining:h

So

aram

arons.

theiru

eL

unarM

onhs.

andIn

‘op\‘h

Shim)

Both

thcins

anI

:i

let:itF

functionho

,rph

smoothly

loinedparahol

tient

ancientB

abylonianrule

equnrng

non..

if12v,13r

AS

l2r,v

Table

ofthe

(hinese

\ear

Madkhak

andthe

Bos

of11w

lunar

Argum

entfor

theY

ears78l8

8l

aidig

irdFor

years781,

782,783

.,882Y

aipird.

I,)

,(Ii

aand

ilit

idualm

onths,entrie

gise,

prei

tensI

(‘hbr

13Ib

tonalcycle

ofday

,he

nc

n‘I

argiments

i’iicrenw

nn

f

if. 13v(A

S13r’) T

ableofthe

Chinese

am

es

ofthe

Solar

‘ear

Dis

usionsand

Related

Param

etersFor

eachC

hinesem

onththe

ame

entrucsare

eren0

is.‘

‘haho\e

f14r(A

S13

)T

ableof

thS

thiri

io

n

The

irgumer

tis

3,

fI4r

AS

Iv

fable

of

Im

arT

heargum

entis

1...

3.entries

infans.

flSv

AS

5)

aolo

M4

aeg

in

kuIn

thistab

k,

frsear

tries1

out

Fth

\l.t

en

s

soil

v.

hein

pci

om

alu

iiciiths

ocat

Ida

ythe

f8v

II(A

S8r:I)

Chap

ter2.

On

the

Chin

ese4ighur

(‘alendar(T

hiscalendar,

widely

usedas

theM

ongolem

pire,is

explaind

in[8

1heearlier

publication[22],

isincom

plete)

Section

1.O

nthe

Determ

inationof

Year

andM

onthB

eginningsG

ivenare

definitionsof

thelen

(aten

thousandthof

aday).

andother

Chinese

units,their

transcribedC

hinese,and

sometim

esT

urkishnam

es,descriptions

ofoperations,m

adkhals,and

parameters,

includingthose

fordeterm

iningtrue

Chinese

netsm

oonsT

hereis

considerablecritic

ismol’the

methods

usedin

theIlkhãni

tij.

f9r

(AS

8r)T

ableof

theC

hãghand

Kib

Arg

um

ents

AchIgh

isa

twelfth

ofa

day,hence

two

hours,each

chaghis

dividednto

igh

t

lit

on

‘48(

0B

aIts0

itau’

snt

Ca

antIii

dav

sL

f.15rT

abular

Worked

Exam

pleo

f\aid

igird

tot

ughur(‘o

ne

suer

Tw

om

ethodsarc

cxt

led0131

rkedout

.ishi

isa

78

rden

6

F.S

Kinnir

i1

HNA

theC

alendarsi

iN’a

dgirdh

-

resand

theirirad

equalonad

idhn,il

the

sisa

firi

freuen

tR

.4B

leg

‘1U

Iin

C.rorno

n’i

medics

a’

The

arguments

areC

hinesem

onthsand

yeardivisions.

Entries

givethe

increasef22v

(AS

20r)T

ableof

aw

Ru7

Nes

Year)

Madkhak

andso

larI rue

longitu

des

mY

azdigirddays

form

eanm

onthbeginnings,

solarand

lunaranom

alisticarg

um

ents

,for

Years

3341390M

aliki(fir

arrestni

loftitude

88in

rnte

T.lCSi

andm

adkhalA

rguments

ailalik

eaI

Ii

inm

l

ofnes

yea1

a

f. 16r:1(A

SI5v:1)

Section2.

On

Extracting

aC

hineseD

atefrom

aH

unD

ate(O

nthe

AS

film,

following

theexposure

which

shows

fl4vand

15r,there

isa

t123r,v(A

S2O

sC

hap

ter3.

On

Determ

iningthe

1’estiak

ofiJ

photographof a

pairof

pagesw

hichseem

notto

belongto

thism

anuscriptbut

which

There

areii

of1

1esi1s

thelalik

llijrSr’le

hasbeen

laidon

itfor

theleft

edgeof

AS

P16r

isvisible

Both

pageshave

holesin

calendarstables

on

rat

id

4k

s

them.

Thefirst

pageone

theright

seems

tohe

aversion

of

Section2

named

above.It

mansions

foM

alim

onas,

inhe

rigs

(‘anti

kae

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osm

alltables.

The

secondpage.

onthe

left,is

blank,hut

seems

tohave

writing

onthe

otherside.)

24r(A

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[RI’

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if)fl

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entsfor

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CT

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fheargum

entis

thebeginnings

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lunarargum

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lunarequation,

thecom

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andthe

fénsof

madkhals

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du

ction

Definition

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erms

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entsare

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years801,

802,803,.

901,1, 2,3,

..

10centune

, andeach

versedsine

Vers

LBor

ani.Fe

iiid

trt.

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FLjri

month.

Entries

areY

azdigird‘,ears,

days,and

fêns.and

elements

ofthe

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xT

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two

aredefined

asshadow

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he

unitsus

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lengths

sexagesimal

ccle

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I?is

usualh(‘0

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fihetr

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distinguish

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our

P19r

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3C

oncerningD

ifficultiesand

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forthe

Uighur

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ihe

siherea

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atn

ri‘id

Kãshi

attempts

tosupply

explanationand

theoreticalunderpinning

forthe

rulesto

introducethe

elements

ofN

ristott’an

cosrnotoi

w11

1‘0

andhis

variantsby

theuse

of

afigure.

How

everthere

isno

evidencethat

ageom

etricm

otions.T

hethree

systems

ofp

heiin

iicoordinates

arehi

model

was

employed

1w

theoriginators

ofthe

systemecliptic

wu

hits

iodiacali

ns,i

aC

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tsrrtr

mln

standardte

ns

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ular

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amp

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theD

etermination

ofU

ighurT

rue13)

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larkin

rc’ii’

aptN

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kh

alsfor

Year

81411introduced,

theyare

thendefined

i)0i$

\C,.

theso

aic

u

invoked,hut

undefined.H

encei

i.Je0ned

here

f2Iv

13(A

Sl9

r.23)(’h

apter3

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theM

aliki(Jalali’)C

alendar

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sphcal

itft

‘Icin

an,lt

I‘0

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tothe

usualm

onthnam

es,and

soon,

theexplanation

includesthe

oppositethen

oyid

pt1

ea

s1

astronomical

determination

of yearand

month

beginnings,placem

entof

epagomenal

Rule

ofFour

statesthat

iam

adt

neh

arigiw

ill11

,C

days.and

theuse

ofthe

tables.fit

Aun

,‘jwn

unu

Aseparate

Ilkhani(or

Khãni)

erais

alsogiven.

Itis

(f22r:15)

thevernal

eouunoxB

ecausethe

functionsanpe’i

i suosthe

theorem,i

w’

ii

of

thesear

of theenthronem

entof

(iháianK

hanI

Farvardn

224M

aliki13

March

functions

1302Julian

day2

196685,

[heange

artti

idot

or

ph

tr

fan

iS

i1

1,

8

FS

Krm

ixriwK

(aaaI/i

where

Ais

thecelestial

longitudeof

ths

givenpoin

t,and

as

iscusto

na

toe

pre

fixarc

beforea

functionm

dcate

iti

si.

.s

For

thesecond

declinationthe

ruleis

areaol Sin

anA

lternativerules

av

pmen

fti

hot‘e

Icci

nna

iris

f27v.7(A

S23v

9ScU

ioi

)c

rnni

psi

ni

Pa

n11,2,3

below.)

Fourrules

arcgiven

PmcalculuS

iiripht

astn

sii

is

aLA)

src( ‘osR

(‘Os

A(‘O

Sô(?

thena

arS

inS

inA

fO(

(osh

QlI

andrS

i[li

6a

finally6

Aw

herethe

exponentd

mdi

aa

nit

SCS

SCC

ICit

ca

i

Rules

aregiven

forco

nva

opascnsiO

flS01

ciiptis

pon’ii’

.;o

rt

quadrantinto

ascensionsin

eachof

theothet

threeqs

adraot,

f.27v16

(AS

23vThI

Sectionn

Deten

iiflifl!the

LquaIlO

rot

Dc

oli

ii.R

isiiii

Am

plituds,ant

OhI

iA

stii

sia

Ra

ap

ixr

iI

ST

hesm

etric

sfthe

eqintion

of davhghtfunction

anh

iasocetso

theequinoctial

andsolstitial

pointsare

rcaarke

IN

Pa

ric

olc

iti

it

ro

hIf

daylight,eq

aq(A)

arenm[

.an6(A

lfan

where

isthe

lattude

of0

sto

atC

\let

thisis

Al

Spi

it

thnsln

p

amplitude

wthe

disance

truirthe

ii’Lom

Ion

lbs

ixn

hisit

pO

int

ol

acelestia

ohwct

ii(

IRD

oÔ

(AI

(‘Os

dd

q(‘

,irut,i

,[I(

A(‘us

6?

qat

Sn(

()S

iA

sfor

obliqueascensions.

ait

it ede

‘hnauunor

ticP

iSix

osoptic

pointis

aotherw

isethe

wo

ouantitissare

addsd‘flit

authorm

tes0

at oldetw

sis

Al ibsput au

‘Set

,:Sir

‘etionsur

givenfo

theis

ofthe

obhqusa

censiontubles

it etidin

glos,

P.us

thesouthern

f26r’13(A

S22v:l0

Chap

ter1.O

nC

arryin

gO

utO

perationsSection

1.L

inearInterpolation

f.26v3(A

S22v:21)

Section2.

On

Determ

iningthe

Sineand

theV

ersedSine

(The

relatedproofs

arein

11,2,2below

)T

heexplanation

isconfined

tothe

useof

asine

tablew

ithentries

forthe

firstquadrantonly,

Sothe

relationsbetw

eenthese

andthe

sineof

arcsor

anglesin

theother

threequadrants

aregiven

There

follows

aw

orkedexam

pleof

two

methods

ofinterpolation

one“easier”,

theoth

er

“more

precise”T

hefirst

isindeed

approximate

yieldingprecise

resultsonly

ifthe

tabulatedfunction

islinear.

The

secondm

ethodis

thelinear

interpolationexplained

inthe

precedingsection.

The

same

tableis

usedto

findvalues

oftheversed

sineusing

thedefinition

ofthe

function,and

thefactthat

thecosine

ofan

angleis

thesine

ofits

complem

ent.T

heuse

ofthe

sineto

findvalues

ofthe

con’espondinginverse

functionsis

alsoexplained

P27r:11(A

S23r20)

Section3.O

nD

etermining

theT

angentand

Cotangent(T

herelated

proofsare

inlI,22

below)

These

functions,called

inthe

textthe

“firstshadow

’and

“secondshadow

’respectively,

aredefined

hereas

Tanx

RS

inx/C

osx

andC

otxR

Cosx

/Sin

x.

ifRis

put equalto

twelve

orseven,the

unitsofthe

shadowlength

arecalled

digitsor

feetrespectively.

Relations

between

thetw

ofunctions

involvingcom

plements

andreciprocals

aregi

en

f27r.2()(A

S23v3)

Section4.

Determ

iningthe

Declinations

ofPoints

onthe

Fcliptic

fromthe

Celestial

Equator

(The

relatedproofs

arein

li,2;3below

.)T

hedeclination,

6,of

anecliptic

pointis

definedas

itsdistance

tothe

celestialequator.

Ifthe

greatcircle

arcm

easuringthe

distanceis

normal

tothe

ecliptic(rather

thanto

theequator)

thedistance

iscalled

thesecond

declination,62.

Sym

metries

ofthedeclination

functionsw

ithrespectto

theequinoxes

andsoistitial

pointsare

slated.T

hem

aximum

declination(e,the

inclinationof

theecliptic)

asdeterm

inedby

theIlkhãni

observations,is

saidto

he23,30”.

To

calculatea

declinationthe

textgives

therule

8arehin(S

inA

Sinc

R),

whence

Afourth

rulis

north,

10

FS

Kiaia

U’

sm

i/

ii

f3lr)0

Ar1

5)

Bst

iC

i(

knowand

valuesrr

roximati

sexageslmal

placesK

a’3llkhãni

sinetable

which

L

ulIi

inI

im

dti

,IL”.

(iito

itth

ie“r

ha

orreLred

rhi

[a‘:n

mm

la”aerlih

iis

aim’

m’(npm

cnicn(Al

icto

l

hemisphere,

andfinding

valuesof

inverseoblique

ascensions.

f28v,5(A

S24r

alongm

argin)Section

7D

etermin

ing

theIerrestria

Longitudes

andL

atitud

esof

Localities

Tw

ozero

meridians

aresaid

tohe

incom

mon

use,they

differingby

tend

egrees,

Inthis

zij

longitudesare

reckonedfrom

theFortunate

Isles,the

Azores.

SeeIV

,I.1lbelow

onthe

same

topic.

f28v.l0

(AS

24r.16)C

hapter2.

Geo

metric

Proofs

ofthe

Op

eration

sin

theP

recedingC

hapterSection

1D

etermination

ttthe

Sineand

Versed

Sine( Y

hesedem

onstritionsare

insupport

of11,1,2

above.)R

ule1.

On

theD

etermination

oftheC

hordFunction

1he

chordsofarcs

of180

120,72,

60,and36°

ared

i3pla

yed

thenthe

sine

ofhalt

ofeach

ofthesearcs

Foreach

individualcom

putation,reference

ism

adeto

thetheorem

of

Euclid’s

[flements

which

justifiesit.

The

sectionends

msith

thevalue

ofSm

18°carried

toeight

fractionalsexagesim

alplaces

f29r:13

(AS

24v’13)

Rule

2.D

etermination

ofthe

Cosine

By

means

ofa

figure,the

PythagoreanT

heoremis

appliedto

obtainthe

relationbetw

eenthe

cosin

eof

anarc

andits

sine.A

worked

example

gises(‘os

18’to

ninesexagesim

alplaces.

29v:4(A

S24v:22)

Rule

3.The

Half

Angle

Rule

forSines

1he

rulenam

edin

thetitle

isderived,

againby

means

ofa

figure,and

isthen

appliedtw

iceto

obtainthe

sinesot

9”and

15’to

sixfractional

placescad

All

theinterm

ediateresults

ofthe

computations

areshow

n

f.30r:4(A

S25r: 18)

Rule

4.R

ulefor

theSine

ofaSum

orD

ifferenceT

heexpressions

forSin(,4

B)

interm

sof

thesines

andcosines

ofAand

Bare

stated,then

provedby

means

ofafigure.

As

usual,references

arem

adeto

propositionsof

theE

lements

ofEuclid

justifyingthe

variousstep

sA

sexam

plesthe

calculationsare

shown

fordeterm

iningSin

33°and

Sin3”

from

Stn(

I 8°i1 5°)A

longgloss

alongthe

lefim

arginof

f3Or

state’;and

proses

with

the

aidof

figure,the

expressionbr

thesine

ofdoublean

areof

known

sine

31r.2(A

S26r:l)

Rule

5.A

nIntroductory

Statem

entA

ccoidingto

Which

theSine

of

One

Degre

(m

nnoiB

enow

nI

Icoft)1

lbI c’s,

ol’‘rn

itic

Iw

i

isan

aIRm

pe

hr1

1c

1

itshorizocita

r dset

etes

it

itsi(

ecas

first,fiv

iqually

spae

sLrs

,n4 hew

‘(si’

Ia

eachar.

starting

from

thein

terscdtio

n‘fthr

horizontaldiam

eterss

am’

.ie3

endingw

ith

oneof the

fivepoints.

The

rmagnitudc

01each

successise

ar,i’t)e’°

‘io

n.

predeceNsor

bya

constant.The

projectionsof

‘hetH

eorig;nal

poiri

tmm

on‘r,.

‘ermicai

diameter

determine

thesines

otthe

(Em’are”

Itis

Dosed

tha’thr

d’

er ‘shissem

nsuccesis

C

projections.hence

thed

I”r

‘i’m’s

hc’tssfl

SlimCm

SI\

(‘‘n

’deereas’rma

sequencec

nceci

1isli

IS

I

“0

tC

r,in,(

tIV

,4

forcak

a‘iti

nSin

adste

in,

iI

toI

significanexagesim

alc

1.3

49

,43

14,44

i1)

i’

Asecond

gloss,b\

someone

elsem

sritien1

Iler

K4

,h‘.

,iea’;.as

tfat

intact

the

paperas

unfinishedw

henhe

liedT

heelegant

iterativalgorism

tieo

iiinatc

dhas

hra’

‘-4hr

\abo

eifl

[I’

buta

caret11

examinalior

th‘S

cnass4

mirrht

0011

ri

ii’

It

ilesc

sracuon

mered1

ti’(

f32r

14(A

S26v. 18)

Secon

)etm’rnm

li’atitmn

o’the

insupprt

of11,1:3

aboveheft

aigure

shotthe

tang‘

nlire

and,lep

ndion

(1i

lx’F

tm

taku

s’i

tsi

f32v

is

JrIS1

,[\‘

,“

‘Ia

(This

suppliesproofs

Ic4

andI

Uhe

tigure

forthi

cooniltastr

tIesthe

si

101ii,

‘,

.a

.m

ci’

OjlS

1fl\O

\LC

12

I’S

Ki

‘SNH

Iheauthor

usesitto

proveô(À

)arcS

m(S

inSin

AR

),by

applicationof

thesine

theorem(referred

tohere

asthe

c/ilk!:m

ughni,substitution

theorem),

then8itA

)arc

Iançan

SinA

Ic;by

what

iscalled

thetangenttheorem

(huIdz:iIIi’)

Alternative

rulesare

provedby

thesine

theoremand o

arcSin[RSin

0(A)

(Os

0(A)l

02

arc(os[R

(‘Os

(‘os8(A)l

hthe

sinetheorem

I orrightascensions,

threerules

aregiven,

cz(A)

arc(os[R

(os

A(‘os

8(A)]

1’ythe

“firstcase’

ofthe

5mw

theorem.

aarcS

in[(o

crSin

A(‘os

O(A

llby

thesine

theorem,

anda

arcSinlR

Tan

8(A)

‘Ianrj

bythe

tangentcase

ofthe

Rule

of[our

34r,6(A

S28r

4)Section

4.D

etermination

ofthe

Iquation

ofD

aylightand

Oblique

Ascensions

Inthis

sectionthere

aretw

ofigures

coveringthe

same

sublectm

atterto

illustratevariantdispositions

onthe

celestialspherethese

axeused

toprove

bythe

tangentcaseofthe

Rule

oft

our,theequation

ofdaylight

eq

(A

,4)

arcSin[fan

8(A)

‘Tan4

/R]

then,by

thesine

theoreman

expressionfor

therising

arnphtudew

(A4)

arcSin{R

Sin8(A

)/Cos4],

alsocq

arcSin[SinSin

a(‘os

8]Erorn

theabove,oblique

ascensionsare

calculatedby

therule

a4,(A)

a(A)

Ihere

followelaborate

rulesgiving

relationsbetw

eenthe

variousfunctions

when

thegiven

eclipticpoint

isin

aquadrant

otherthan

thefirst

At

onestage

KA

shigivesa

referenceto

apassage

inthe

Alm

agest[351to

establishthe

validityof

am

it

if35vJ9

v(A

S29v32r)

‘Table

oftheS

ineF

unctionT

hercarc

entricsfor

tendegiees

oneach

pagct

thchead

uf

eachuf

theten

columns

perpage

isan

entryfor

Sinn

forii

0,1,

2,

89’In

thecolum

ns,underneath

thetop

line, forin

1,2,3,,,6

0m

inutesof

arcentries

givethe

increrneit

tob

eadd

edlo

the

aluat

aretofo

nst

ificat

xc

tt40r4

Ir(

55‘

411ih

eft.ai

‘iii

eachcolum

ris

itC

ry‘I

tablepreced

irt,e

iis

fentry

atthe

topf

Ito

i

significantt’

tat

iiip

ff,4lv

42r

(/S

34ablc

t‘the

irsten

iIi

Tan

60

nto

thrrxa

orcolu

mn

ais

ch

as

‘mi

mm.

of

prt’tmcmnmm

astht

I‘ci

0,1

5,.

05

hit

integerparts

olih

’ii

tangent fueliot

p’

o‘I

minutes

forart urn.

t,I

05

,0,

5,0

3”

n1

5i

thatth

esela

t

ff42v4lx

AS

lhh

ncio

1is

tfrornA

tm

lbr

to0

°and

)0’

Hc

itm

userrnusliernrrnhah

negative, a.V

t‘v

)ik

takesap

og

Li

f44r(A

S36

AIab

kof

S.

rlh

forn

atfoh

[except

thitt

rinc

oIa

Hence

thec

ito

nI

Is

i

onepagei

ic’?r

I(

ff44v12r

(AS

3o

S)

)LT

hess

11

all

“I

unrtii

i

S

C

CIi

abh5CC’

I

I

alt

Ins

u-irC

‘atr

Ig

“

C)iti

LiI

XC

iitt

C

iat

‘14

ES.

KiN

Nin

Ys

KhaqS

n1Th

thisstage

Kishi,

not botheredby

thetim

eexpended

incom

putation,but

perhapsw

orried

bythe

amount

ofpaper

beingused,

compressed

thetables

without

changeof

format

to

thenoint

where

hegot

two

tablesper

page.T

hisis

thesituation

for48

49.50,

6166,30(

90’-c),and

75°In

thetable

for61

°allthe

placesfor

entrieshave

beenleft

blank.In

thelast

two

tables,m

anyplaces

forentries

areem

pty,presum

ablyfor

values

ofthe

argument

forw

hichthe

functionfails

toexist.

Forall

thetables

therange

of theargum

entis

A0,

1,2,360’

andentries

are

toseconds.

fhefirst

tableof

theseries,

since1

0°is

infact

atable

ofright

ascensions

f172v74v

(AS

52v-

54v)

AG

eographicallab

le.A

List

ofC

itiesG

ivingT

heir

Longitudes

fromthe

Fortunate

Islesand

Their

Latitudes

This

givesthe

geographicalcoordinates,

tom

inutes.of

515localities,

mostly

cities,arranged

accordingthe

seven“clim

ates’of classical

antiquity.T

helist

hasbeen

publishedin

[15]w

itha

facsimile

ofthe

textIts

entriesare

alsolisted

in[14],

where

theircoordinates

may

becom

paredw

iththose

of some

seventy-foursim

ilargeographical

tables,m

ostlyof

medieval

Islamic

origin.M

ost of thecities

named

arefrom

theN

earand

Middle

Fast

andC

entralA

sia,but

thereare

many

fromE

urope,especially

theiberian

Peninsula,

India,and

East

and

Central

Africa.

Afew

citiesin

China

areincluded.

Not

surprisingly,the

tableseem

s

most

closelyrelated

tothose

inother

Persianzijes

f.75r;1(A

S55r:i)

TR

EA

TIS

EIII.

ON

TH

ED

ET

ER

MIN

AT

ION

OF

PL

AN

ET

AR

YP

OS

ITIO

NS

Introduction:D

efinitionsof T

echnicalT

erms

Used

inT

hisT

reatise

The

definitionsinclude

trueand

mean

celestiallongitude,

celestiallatitude,

the

pareclipticorb

(Ja1akim

umaihihal),

theinclined

orb,node,

deferent,superior

and

inferiorplanets,

epicycle,deferent

andepicyclic

apogeesand

perigees,equant,

the

severalm

otions,true

andm

ean,civil

day(nychthem

eron),the

equationof

time,

anomalistic

anddeferent equations,

solar, lunar,andplanetary

adjustedcenters,

minutes

ofthe

argument,

nomenclature

of

theplanetary

latitudecom

ponents.true,

mean,

and

apparentconjunctions

andoppositions.

theastrological

aspects.tem

peraments.

cardines,

andhouses,

transfer(iahw

il)apparent

diameters

arealand

absoluteeclipse

digits,the

latitudeof visible

climate,

altitude,azim

uthadlusted

andaltitude

parallaxand

arcsof

visibilits’.

f,77r20

(AS

58v:C

hapterL

cnn

anoa

ifhr

aat

dL4

ngtu

&s

ofthe

Planets,

andO

perations(

onnected‘4

ith‘I hem

Sr

tionI)

teni

ae

iai

iiF

t1

los

The

materia

inthis

sctft

h’s.bee

dcscthe

ind

I-

a)I

Int

courseof his

lengthsexplanation.

Kathi

citesand

makes

useot’the

patamelert

employed

inthe

ilkhani Zij,

bit

nirs

iiI

itliii

itionit

tlx

iid

I etth

passiix

f

time

sincethe

ear

syr

ii

f78r: 10(A

S59r:12)

Section1) ‘tnrm

mat

ottof

.he\ieai

i’os’’’oo’.tho

Planetsat

An

Tim‘Ibis

isan

explanationol

1’w

tol’t

inI

mI

eH

I10

r‘i

toI’tbl

sth

me

a

longitudes,apsidal

longitudes,nodal

positionsaid

atom

altsuram

,’r,mt,

tiltof

thesun

moon,

andplane

sfot

anyen

ste‘id

1ie

I,slet

hu

toi)us

nentto

I e

made

itthe

longiude

oth

Cv

cci

or‘ie

soal

f,78v:9(A

S59v

3)Section

3i)et,,rn’ijnatjon

ofthe

Iin

’e

t”tuo,

of

theSesen

Planets

atA

n‘1

ne

ct,

II1

4d

I

The

standardPtolernaa,

pioedure

(asset

In-tb

orit

stancr

isem

ployedhere

Com

mencing

with

thesim

plerm

odeltom

thw

a.II

finallythe

planetshe

Ie

lailx

tola

irm

hrIs

andepicyclic

com

pone

stc

or-c

alaI

r.‘ig

CII

at1

n

with

them

odifiedequation.

‘(‘hisi’

addedalgebm

aicallsto

th-non’

thetru

elongitude

ofthe

hod1in

mist

on

f.79r:l6(A

SSOs

181Section

4

(Proofsare

in111,2,3.

6.7.t

ivento

ea

um

cin

.A

th

obtainedfrom

thetable

onfo

lios

Calculating

thelatitudes

o

compl

catedPh

ss‘iI

ir

f,79v’14

(AS

60r64

SecTion1)e

te°”

i’iinatio

noft0r

1. S,’s.

Sectorsand

Retrogra

lationsro

oa

aiv

)

The

dr-termnat

onof

whc

‘it’plar

tas

Idi

cio

1ft

mon

(optds

uponits

situationw

ithrespect

toits

retrogradeand

lot ward

staton’

untamed

tromthe

tableo

fsta

tion

sIf

iiis

1,on

Ithe

‘tro‘rade

r-stus

aI

sit

t‘t

ii’,edat

the

‘P(P

l1-2(24

emsdir

moon,

aido

hoc

(emi

ind

oc

am

plode

togm

’a

Dotc

rrmr

nOon

ofe

ii‘iv

is,,

‘natars

1.atmtud

xi

id111

isad

uIt

rids

atm

(aS

Sin138s

and139r

theplanets,

esr

islb

thn,-

Itn

hes

ism

ore

tica

ht’

it

F

167

PS

Kiun

DY

of(

apricorio

thc(

rightasce

o

ii

,s

wherA

uppeill

sr

His

theccc

ndnt

‘rom11

provdcdna

sc

rule

f8lv

AS

(S

nein

cmofth

ey

zgy

Caa

With

anephen

is

nac

occusom

etimhe

andm

ooncc

olum

inaristo

ccs

iti

fromthe

frs0

0

when

sn

ct

rateoft

og

tioni

oe.

hriaa

om

anner1IIQ

iI]

solving‘t

cchan

‘hisseuti

po

anelaborat

p)C

U

andm

oons

Ias

f83

r20

AS

(i

Oppositi

nh

timc

hth

onaitc

otl

ci

tIcasc

nsiond

ftcnv

rseob

qic

aS

forward

stationitis

retrogradeO

therwise

itis

inforw

ardm

otionThis

isrelated

tothe

subjectofsectors

(nitaqat)studied

in[I91.

Background

onthe

topicis

tobe

foundin

[4]

f. 80r.16(A

S60r

18)Section

6.A

Simplified

Method

forDeterm

inationofthe

Planetary

True

Longitudes

andL

atitudes(Proofs

aresupplied

in111,2

8below

.)T

hestandard

Ptolem

aicm

ethodfor

determining

atrue

longitudenvolves

individualtables

forthe

equationsofthe

centerand

theanom

aly,and

acom

binationof

theresults

fromthese

bythe

useof an

interpolationtable

Kashi

substitutesfor

thesea

singletable,

butw

ithtw

oindependentvariables

them

eanand

theanom

alIn

principle.this

isindeed

asim

plification,since

itsubstitutes

oneoperation

forseveral

How

everthe

two

numbers

tobe

enteredw

illusually

hebetw

eenpairs

ofvaluesfor

which

thetable

hasbeen

computed

Then

atw

oway

interpolationis

inevitable,w

hichis

tediousand

difficultT

he‘ongitude

tableshave

heei

recomputed

andthc

resultsreportc.d

r39

1m1

latitudetables

haveyet

tobe

studied.

f80v

SA

Note

onInterpolation

inT

heseT

ablesIn

interpolatinglinearly

acrossa

regionw

hichcontains

alocal

maxim

um,

Kãsh

suggestsreplacing

onetabular

entryby

thesum

ofthe

maxim

umand

thedifference

between

them

aximum

andthe

entrybeing

replacedT

hereis

ananalogous

rulefor

am

inimum

fSO

y: 18(A

S60v4)

Section7

)nPlanetary

Distances

fromthe

Center

oftheI

niverse(P

roofis

inliI,21

Ibelow

.)T

hepurpose

ofthe

determination

isonly

tofind

whether,

atthe

giventim

ethe

planetis

recedingfrom

theearth

orapproaching

itH

encethere

isno

needto

useabsolute

unitsof

distancein

allcases

thedeferent

radiusis

putat

60.

f81

r,4(A

S60v. 8)

Section8

On

Calculating

Sequencesof

Planetary

Urue

Iongitudes

andI

atitudesFor

thepreparation

ofephemerides,

sequencesof planetary

truepositions

atfixed

intervalsare

requiredT

hissection

describesan

interpolationschem

ew

hichpieces

togetherparabolic

segments

[hepassage

istranslated

andthe

procedue

xpresnd

inm

odemnotation

in120]

f8

is:4

(AS

60v,23)S

ection9

Determ

inationofH

alf Daylight

(P0

0s

ren

111,2,12below

)Kashiuses

herea

functionhe

calls“right

ascensionbeginning

fromthe

iistpoint

SF

TO

h

Ici

0

hT

int

I

18

IK

irosii

To

thirule

Kashi

addssuggestions

forim

provingthe

result

f83

v1

0(A

S6)si)

Section12

Determ

inationo

aL

unarL

clipseby

Com

putation(Proofs

arein

111,213

below)

With

thetim

eofthe

oppositionin

hand,aiest of

distancefroiii

theiioik

isapplicd

tosee

ifan

eclipseis

possibleIf

itis

thetim

eof

them

iddleof

theeclipse

sthen

calculated,as

well

asthe

apparentdiam

etersof

them

oonand

theshadow

oneIrom

themthe

magnitude

andduration

aredeterm

ined

f84v20

(AS

62v,14)Section

13D

etermination

ofa

Iunar

Iclipse

bythe

Use

ofIibles

11woperations

required,facilitated

bythe

useof tables

aredescribed

ir111

213,

atfJl6r:8

‘85i10

tAb

o2v’2

)S

eceun4

l)utcrimnatiu

o’a

SolarI

dipseby

(umpw

atiun(Proofs

arein

Ill)14

helowat

1119r

113A

ftercarrying

outa

preliminary

testto

make

surean

eclipseis

possible,th

restof

theprocedure

isso

lengthythat

ithas

beendivided

intothe

folow

ingsix

steps

85r15

(AS

63r: I)Introduction

II )eterm

rnationof

the,atitude

ofV

isibe

Clim

ate(the

anglethe

eclipticm

akesw

iththe

localhoriaon,

forw

hichthree

methods

aregiven)

8S

vII

(AS

63r:6)Introduction

2D

etermination

ofa

Lum

inarys

IrunA

litude

(isdistinguished

fromits

apparentaltitude

which

isaffected

byparallax

86r16

(AS

63r24)

Introduction3

Determ

irirtionof

Adjusted

Iim

arParal

axand

Apparent

/i.nith

l)istanee

f86v

6A

S63v

16)introduction

4D

etermination

oftheA

pparentIunar

Longitude

andatitu

dc

(hcalculating

andallow

ingfbi

thelongitude

andlatitude

comporents

ofparallax)

87s14

(AS

64r18)

Introduction5

Determ

inationof

thelim

eb

Apparent

Conjunction

88r. 13(A

S64s

5)Introduction

6D

etermination

ofthe

krs

Subterded

bythe

Apparent

Rad

iof

Sunand

Muon

at

theinne

uf

theIdpse

(1

bu

lrra

e‘tI’

calculationof

theeclipse

magnitude

andthe

durationsof

imm

ersionand

talit

tables

f89

vIA

Yi

I‘

(Ti

I

theA

pprat

s’n

lIl

eM

usn

lendbeginning

fiv

vtI

1

Ilencit

sfiit

.

aparticulat

wr

lrt

iirrp

ose

ofsunset

salci.

ar

seenotis

iI

IIC

C1S

tiS

C

heor’ic

af

v’riO

ii

ofthei

Ift

I,

Doggett

andSd

ì@

overthe

Uu

t’dS

ihih

liograp

hi

oist

The

aDIt

Ii

andn

atte3t

s

fXr1

5G

ven

Pos

tr

firso

tr

Itstra

isft(a

,,

ou

us

atthefiv

enoi

ti

tti

24

di

slert

it

Iis

theso

att

isIi

1..r

Ihisa

prn

i

comprtat

nI

‘

f91I(\S

6oi

.&

i

P1m

etsU

f.89r8

Secten

Here

igantm

(tth

)I

0

C

itI

f°lr

6(\S

I

FS

Kiecu

mK

tiufij

Apair

ofplanets

issaid

tobe

insuch

andsuch

anastrological

aspectw

hentheir

longitudesdiffer

bya

fixednum

berof

zodiacalsigns

dependingupon

theparticular

aspectcextile

istw

osigns,

quarnieis

three,uppoatzc)Z

issix

(See[21.

p225)

The

method

of

findingthe

time

ofam

valat

anaspect

alsoresem

blesthat

ofSection

10

above

f,9Ir2

3(A

S66r

23)Section

20O

nthe

Equalization

ofthe

Astrological

Houses

(Proofsare

in111,2; 17

below)

The

housesare

twelve

divisionsof

thecc

ipticw

hichunlike

thezodiacal

signs,

varydepending

uponthe

time

andthe

localityfor

which

theFare

calculatedThe

initial

pointsof

fourof

thehouses

arethe

intersectionsof

theecliptic

with

thelocal

horizon

andthe

mendian

l’orthedeterm

inationofthe

remaining

eightcusps,

asth

eare

called,

thereare

anum

berof

differentm

ethodsK

äshigw

esthree

ofthese,w

hichare

inturn

describedin

[281.

f92r3

(AS

66v*l6,15?)Section

21D

etermination

oftheT

emporal

(Unequal)

Hours

An

unequalhour

isa

twelfth

ofthe

lengthof

daylightor

night,hence

itslength

dependson

theseason

andthe

locallatitude.

Kãshi

givesrules

forconversions

between

times

inequal

andunequal

hours

f.92i16(A

S66v22)

Section22

1)etermination

ofthe

Floursof

Sanctuary(7

host)

This

sectiondescribes

anastrological

cyclew

hichcom

mences

fromthe

time

of

aconjunction

andrnns

throughthe

sevenplanets

insuccession

untilthe

next

eon(unctionassigning

twelve

unequalhours

toeach

planet

192v:Q(A

S67r5)

Section23

On

f)etenniningthe

1)urationof

theE

ffectso

Solarand

Lunar

Eclipses

Again

purelyastrological, this

sectionconverts

thetim

eand

durationof

aneclipse

intolonger

spansthrough

which

itseffects

will

last.

f.92v*l9(A

S67r:l0)

Section24

On

thePositions

ofSom

eof

theFixed

Stars

The

authorrem

arksthat

sincein

thecopies

andtranslations

ofthe

Alm

agestthere

aredifferences

inthe

coordinatesofthe

fixedstars

andin

thellkhãni

observationsriot

allthe

starsw

ereinvestigated,

furtherobservations

areto

behoped

fe(IThese

were

indeedcarried

outby

Kashi

andothers

atSam

arqandI he

tablein

thenj

isfor

the

beginningof

801Y

azdigird

Section

1T

heE

quationof

lime

i)erivationof

111,1.1

aboveS

c1

51

This

isa

detailedexplanation

ol’hosto

caiculatethe

ditfhienuem

ceotrue

andapparent

solartim

eIn

thezi,

liguresin

usedto

mm

detlb

ean

ho

imcr

aloar alieters

embedded

inthe

tablesare

cited

f95r1

3(A

S65

5S

etion2.

With

hit

igureexplained

cgorou.1

nt

denew

oher

ar

si

kenoperation

rt

&c

ivrefi

f.95v:l$A

S68v

SSeebon

Iat

Frimot

Ill,1;3.)

III1

(1T11m

li

I&

&cer

(n

The

much

more

com

p1u

ate

ltolem

aicm

odel.

tc‘

ci1m

enlike

tTeamlent.

Forthe

eccentnert,the

auth

or

acuep

tP

iole

ms

I?a

tle

tth

eep

ic’tc1c

radiushe

takes5

16,47,rounded

ottfrom

theesulls

uiu

sst

iilu

nar

eclipse

observationsreported

inthe

pre

face

tothe

ui

Ihis\a

iue5p

iaec

iP;olem

aie.

iS

To

buttressan

operationhe

citesa

pro

pos’llo

rlfro

mM

enlaos

u”his

ture

forderiving

Ptolems

interpolation1’

hcnssthi,

epc

iih

cci

ion

al

positionsof

mm

nm

ax

imu

nI

med

iate

Itis

fren

ntr&

stthat

Iuss

i1n

spcc

i

mention

ofto

01kr

dclx

[2in

0

Copernicus,

maT

aimal

thI

‘1san

ce

itr

Ncv’

irem

ai

f98r20

(AS

)(

Section4

11c

Ionuitudei

tit’t’l,cn

u(R

ele

vant

to111.1

2ahos

eIn

thissectio

nth

eP

tclenielnodel

ftcthe

mw

sic

ep

2In

ct

is

meticulously

derived,w

ithtables

tc’ssnc

theresuits

ofmu

rm,daeem

tmputazm

on.A

sfor

parameters.

theauthom

asc

ep

tsP

tole

ms

secceatm

ms

mm

iP

152:45.

and

6:0for

Saturn,Jupiter.

andM

ars

resOC

t05

cEhut

fbr

\e.

ihe

eec

5hased

Oil

new

observ

ations,

he

sassin

steadct

&m

holem

aicI

10

40

shi

taKes

the

Ptolem

aic6,3(

20,and

13U

nIupm

tm,

ipu

isci

Ifor

Mars

40;18(4

1

f99

e1

8(

SI

abee

if

esant1

mm1

ILa

mmito

f3

1)(A

S67v

1)C

hapter2. G

eometric

Proofs

of the

Operations

inT

his1

reatiseFor

iiino

dorms

nih

,I

in

FS

Ki’cvi:in

Kh

sK

hqa

fI00v:11

(AS

71v;18)Section

6.L

atitudesof

theSuperior

Planets

(Relevant

toIII,I

above)No

numerical

computations

appearin

thissection.

f101

v19

(AS

72v:4)Section

7,L

atitudesof

theInferior

PlanetsR

elevantto

111,14

.The,

threelatitude

components

ofeach

ofthese

two

planetsrequire

longexplanations,

againw

ithoutactual

computation

beingshow

nT

hediffering

resultsof

Ptolem

aicand

more

recentobservations

arecited.

f. 104s‘6

(AS

74r19)Section

8,D

etermination

ofPlanetary

Longitudes

andL

atitudesby

aM

ethodO

riginated1w

theA

uthor(R

elevantto

111,18

This

isK

ashi’sderivation

ofhis

own

method

of

computing

planetarylongitudes

andlatitudes

Itdeals

with

allfive

planetssim

ultaneouskO

nlyfor

Venus

arethe

lengthynum

ericalcalculation

displayed.

flO

8v:20(A

S77r:15)

Section9.

Determ

inationof

Retrogradations

andForw

ardM

otions(R

elevantto

1ll,l;5,)T

hetheorem

ofA

polloniusfor

locatinga

retrogradestation

iscited

andapplied

The

Ilkhãniobservations

ledto

parameters

forM

arsand

Venus

which

differfrom

thoseof

theA

lmagest

Ilencetw

osets

oftables

arepresented

onefrom

theIlkhãni

nj

theother

fromthe

Alniagest.

f,ll3i’15

(AS

80r’6)Section

10P

lanetar)Sectors

(Supp1em

entato

Ill1’5

aboveSee

[1°],pp

247253.)D

efinitionsof the

sectors, deferentandepicvclic

aregiven.

There

aredirections.

with

proof,for

calculating.‘1hese

arein

generalterm

sw

ithno

particularplanet

named,

andno

actualcalculations

displayed.

f.1l4v16

(AS

8lr:l1)

Section11.

PlanetaryD

istancesfrom

theC

enterofthe

Universe

(Suppl.to

IlLL

5.)T

hisis

abrief

statement

concerningthe

layoutof

thedistance

tables,K

ãshIalludes

toa

proofalready

given.Perhaps

hew

asthinking

ofa

tracthe

wrote

calledSE//urn

(zi-S

arn

a’

(The

Heavenly

ladder)on

planetarydistances.

lierefers

specificallyto

ittwo

sectionslater.

fI15r1

(AS

81r’18)Section

12D

etermination

of theI

engthof

Half D

ayhghtand

ofthe

Ascendent

fromthe

Tim

eof D

ay(C

fIL

19

aboveA

formal

proofis

givenonly

forthe

determination

ofthe

ascendent

f.I16r8

(AS

82r.3)Section

13D

etermination

ofI .unai

I.

l.oso.S

upIcmentarv

to

JILL

12.)T

hisis

adetailed

expositiono

moon

isto

ene

heshadoss

cornA

iththose

oh‘iii

toK

ashihi

f.119r:l1A

S8

r5cc

10

(Su

pp

lemen

tato

111

The

sectun

tderive

hreevisib

leclim

ae

Nux

isth

dotof

theal

coordinatesthe

longitudeof

this

lentand

thel

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aiitude.

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Ithis,

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theJ

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(AS

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upplementary

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100,200

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782783

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Solar

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maxim

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Giving

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f132v

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andthe

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the

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theargum

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entriesconsist

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sexagesimal

digits.T

hereis

onetable

forboth

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planets,but

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secondtable

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Mars.

fl4lv

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thefise

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its

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ofthe

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ofthe

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Jupiterand

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thereare

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minutes,

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northernand

asouthern

functioneach

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ableof

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ofV

enusand

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Three

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each.Ihe

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thefirst

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with

am

aximum

entryof

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heother

two

aretypical

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sixty,

f140r

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Com

ponentsof

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theargum

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i140v

(AS

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andprecision

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isthe

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forV

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hastw

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many

columns

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them

aximum

entr

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is45.

f141r

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omponents

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helayout

andprecision

ofthis

tableis

identicalw

iththat

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theanalogous

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above.

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Retrograde

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orw

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Longitudes

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manuscript

ForV

enusthe

formtakes

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completely

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page,f.

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thefirst

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ism

issing.T

hesecond

andthird.

112r

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areidentical

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Office

version.This

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acase

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Apparently

KäshI

calculatedtrue

longitudesalong

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sand

columns,

leavingthe

rectangularspaces

tohe

filledin

byinterpolation

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completing.

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andthe

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20,..,

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Itisalso

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theindia

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was

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inthe

interim.

[l56v

(AS

l14v)T

able

ofthe

Distances

ofthe

Planets

fromthe

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aretables

forthe

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thesingle

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aly:0.

5.10,

...

360°.For

them

oon,the

entriesfor

theequation

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takea

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radiusas

onefor

allthe

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ofunctions

aretabulated:

thegreatest

distance,and

theequation,

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ountto

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moon,

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fI 57r

(AS

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Table

ofthe

Solar

Distance

fromthe

Earth

andT

ablesof

interpolationF

unctionsfor

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lanetsFor

allthe

functions,the

argument

islongitude

measured

fromapogee

0,5.

10,360°.

Forthe

sunonly,

ithaving

onlyone

equation,the

unitfor

theentries

isa

sixtiethof

thesolar

deferentradius,

carriedto

two

fractionalsexagesim

alplaces.

inthe

caseof

them

oonand

thefive

planets,they

havingtw

oequations.

their

distancesfrorr

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forcarl

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ep

allases

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ts

areintegers.

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abe

ned

ssiththe

ta1e

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..

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calculatingplanetary

distances

[able

forS

implif

ingInterpolation

Argum

entsare

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2..

roundedto

aninteger

This

tableis

a’

oftables

thrdot

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planetary

S.7,l

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‘t.4seen,

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to8

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RtIK

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hereare

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tables,one

f:rt

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theother

ICr

hm

a1a

tone

argument

ism

1,2.

310.

2tja

50m

inuieso

aftin

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ateof

advancein

minutes

perhour

l’orthe

sun1 s

:24

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them

oon2345,

23:50,23,55,

..,

32,25E

ntrie.nie

rrVn

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torthe

objectto

traversein

minutes

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.ritd

10

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fI62x.

(AS

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Forito

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calendari

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0’the

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ofttc

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The

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10,20,

30

f163r

(AS

121ri

Table

ofan

hours

%olar

rindI

unarI r

nd

‘nt

I1°’

ction%and

Oppositio

ns

Forvalues

ofthesolar

orIunto

anomals

of2,.

10,..3

ein

istravel

ofthe

sun,and

thehourly

travelofthe

moon

isow

enboth

onthe

coltrw,i‘

itemjc

node.

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‘.

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Ki’\rm

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During

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valuesof

theadjusted

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10,,3

60

theradius

ofthe

lunardisk,

bothfor

solarand

lunareclipses,

andthe

shadowradius

(forthe

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column

eachfor

solarlongitudes

fromthe

apogeeof

I,2

3,..,

6zodiacal

signs)’all

entriesare

toseconds

ofarc.

f.16

3r(A

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lr)T

ableof

Ad

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Lunar

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inA

ltitude

Forlunar

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of’0,

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90,

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secondsgive

thelunar

parallaxand

itsequation

Form

odifyingthe

equationthere

isan

interpolationfunction.

toseconds,

theargum

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which

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5,10

360of

theadiusted

lunaranom

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ff. l63v,164r(AS

121v,122r)L

un

arE

clipseT

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Forlunardailvratesof

Il50

,l2

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12;301450°,atiu

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deso

f0,1

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70m

inutesof

arc,four

functionsare

tabulated,linear

andareal

digits,and

times

oftotality

andim

mersion

allto

minutes

ff.164v,165r(A

S122v,l23r)

Tab

lesof

IntervalsB

etween

True

andA

pparentC

onjunctionand

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inL

atitudeFour

tabularform

shave

beenruled,

forlocalities

of

atitude20

3040,

and50°,butofthese

onlythe

onefor

300contains

entriesFor

noonand

1,23

4,5,6

and6

57hours

beforenoon

andafler

noon,andfor

eachzod.ac

alsign

thereare

tiseentries

thenum

berof

hoursbetw

eentrue

andapparent

coniunction,and

thelunar

parallaxin

latitude,both

tom

inutes

f.l65v(A

S123v)

Solar

Eclipse

Table

Forlunar

dailyrates

of’11,50,

12,10,12,30,

l450

‘.and

forlunar

latitudesof

0,2,

434

minutes,

threefunctions

were

tohave

beentabulated:

linearand

arealdigits,

andtim

esof

imm

ersion.H

owever,

thecolum

nfor

arealdigits

hasbeen

leftblank.

‘Theother

two

havebeen

carnedto

minutes

f.l66r(A

Sl2

4r

Visibility

Tables

forth

eM

oonand

the

Five

Planets

Forthe

moon

thereis

atable

oftheequation

ofsetting,

tom

inutes,for

thesecond,

third,fourth,

andfifth

climates.,for

lunarlatitudes

of1. 2

,3,4

,5’,

andfor

eachzodiacal

signFor

thefive

planetsthere

aretables

ofthearc

ofvisihilitv,to

minutes.

forthe

third

1 (ag

f167r(AS

125rA

Star

Table

Eigh’-four

starsare

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each,longitude

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tm

inuiec,m

agnitu

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temperam

ent(ra

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oftransit,

maxim

umaltitude,

mean

sine,adius

eddiam

etciin

”to

aoic

n‘

is:ng

andsetting

ascensions,degree

ofrising..

a.rOuineni

Nuiiniuth.

ajirnut1’q

ul

rat

raim

uth

.distance

between

stars,easocresi

1in

andine’,natinn

((Idec

ter

‘‘or

,,a

f167’s

.24(A

S126r

6)C

hapter1,

Rules

ftrthe

Opei

ation’

andourti

ims

INs

Idisappearanci

atftit

IiN

ri

infenoplanets

iiad

ito

tI

disappearincre

sng

hian’

It

I’166v(\

164rah

kof

the(

antis

iirkni

a

‘Ihiis

thu

Ion

A,

)0

.9r

A(I

measured

frorhi

apogeethe

en

tiic

si

Itsr.

onds,‘it’

solarm

eanlongitude

measured

ft”niiN

apoget

(AS

125rT

able

ofP

recessionalM

otionT

hereare

entries,to

thirds,hoT

1.2.

3Y

azdigirdyear

.and1.

2..

.12

iontlir

Ide

sto

the

32

ES.

KiN

NI

Section

1D

etermination

ofStellar

[)eclinationsT

herule

isa

complicated

sequencecom

mencing

with

thedeterm

inationof

aquantity

/arcS

in[Sin

AC

osw

here1

iscelestial

latitudeK

ãshithen

givesthree

similar

oxpressionsD

clast

ofw

hichis

therequired

declination,ô

There

follows

analternative

rule,of

comparable

length

f168r

14(A

S126r

22

Section2

Determ

inationofthe

Ascension

ofT

ransitT

hisam

ountsto

rightascension

inthe

modern

senseA

firstrule

isarcC

os[(S

in//C

osó

)R1

,w

heref

isthe

quantityfound

inthe

precedingsection.

Three

alternativerules

follow.

f.l68v10

(AS

126

22)Section

3.D

etermining

theC

ulminating

Altitude

ofa

StarT

herules

ofthissection

followim

mediately

fromthe

equivalentof

theexpression

h,,,

(90-4

)ó

where

Jistands

foraltitude.

flfl8v;21(A

S127r2)

Section4.

Determ

inatiorof

theM

eanSine

This

may

hedefined

asm

Sin2[S

in(4

ô)+

Sin

()]

Anum

berof

equivalentexpressions

areproved,

suchas

mSan

Co

sô

(os

RT

hem

eansine

isapplied

belowin

determining

theculnvnating

alttud,

o‘a

star

f169r:5

(AS

127r12)

Section5

Determ

mation

ofthe

Iquation

ofH

alfD

a1ightand

thelen

gth

ofH

alfD

aylight(a’)

Halfdaylight

isthe

algebraicsum

ofaquadrant

plusthe

equationof

halfdaylight

rulesfor

which

sserederived

aboveH

erethe

authorproves

thatd

arcV

ers[Sinh,,

mSinj

togetherw

ithother

relatedexpressions.

f169v:17

(AS

127r:24)Section

6.D

etermining

theA

scendentand

Descendent

ofa

StarT

herules

obtainedhere

involvem

anipulationsw

iththe

obliqueascension

functions.

f169v:4(A

S127v

lOt

Section7.

Conversion

fromA

ltitudeto

Azim

uthT

hesection

beginsw

iththe

equivalentof

theexpression

expressionarsS

inl(O

SC

osto

producean

auxiliarsquantits

sshichi

usedto

turnout

aairu

Cee

nda

thirdT

healgebraic

sumo

fthe

secondand

thudauxiliars

quan

irnu.

th.ln

rcd

altitude

[I70r:

17A

S128r:

18Section

QD

eiconinalionof

(Thliqru

\cu’r

“sIii

aG

ivenI,ocalityT

hestandard

method

forcalcuiatinn

ohkqueascensions

haairs

adsbeau

ins enin

111,1,7and

Ill,24

ahoseK

ashihas

thusbeen

presentedssith

anungoitanits

toexhibit

hisversatilit

which

hedoes

Itse\poundlng

acom

pletolsdifhrenr

method

usingthe

latitudeofvisib

r[m

ate,nom

’iedright

as<10

idlbs

tan,at

mcli

rm

uch

longertha

tnethod

1170

128sS

ctionI)

iso\aC

SI

hesectio

{A

ithtu

ii.ins

agiien

ac

on

ca

gsec

rN

nitr

‘i,‘c

an

117

r8

4S

ction

11icaiN

(CC

AS

S‘n

on

tofind

th)ni’

sunat

ths

oil

es1,’hen

whence

(pd

C

Forthe

mu

chm

oredifficult

problemo

flo

ng

itudas

Disht

hassso

‘olitions.

fineis

tochose

alocality

ofknow

nlo

ng

itude

and

(liTanne

thata

ia,ua,

[u

se

beobsers

edfrom

thatplace

andalso

from

thelocalits

otunknossn

Cuaitad

eher

thetim

edifference

betseen.say,

the

middle

otthe

eclipseat

the

Issop

(aaa

olieu

conserted

iPtO

degrees,is

therequired

longitudinaldifference

hetsseenth

etsv

o’a

iac

‘s[h

ire

hniq

ue

hadpreio

uslv

beenapplied

tsee131

1T

hesecond

method

assum

esas

hut

ssnth

e1n

itud

Na

C‘s

andthe

hSin

forcalcu

atingthe

Iac

at

therising

mitu

IcI

mini

origin

f170r

5A

2‘

Sectio

n8

l)eO

nn

eta

rtto

dis

gte

niv

i

asat

thetu

de

(toit

tas

old

a’in

C,I

te

34

ES

Kii1

)yK

,si’sK

hiqanñi

great circledistance

between

them,

1hePythagorean

theoremis

thenapplied

togive

the

longitudedifference.

Itis

astonishingthai

anyonew

ithK

shi

sintim

ateknow

ledgeof

thegeom

etryof

thesphere

would

propoundsuch

aschem

e.It

contrastsw

ithB

iruni’s

admirable

solutionof

thesam

eproblem

in[3]

f.17lv:9(A

Sl29v6)

Section12

Determ

inationofthe

Distance

Betw

eenF

soStars

After

disposingof

specialcases, the

generalproblem

isattacked.

Itis

solvedb

thesuccessive

determination

offour

quantitiesw

hichare

partsof

variousspherical

triangles.O

nthe

same

pageis

along

marginal

glossby

Ulugh

Beg

himself

discussing

aspectsof

theproblem

This

pageis

reproducedin

facsimile

in[30],w

hichalso

describesthe

contentsof the

gloss,In

itI I lugh

Beg

remarks

correctlythat

thesection

canbe

improved

byadding

toit one

more

specialcase

fhishe

doesadding

tothe

rule

aproof

of

itw

iththe

aidofa

figure.T

husthe

glosssupplies

strongevidence

thatthe

astronomer

princew

asa

highlycom

petentm

athematician

andastronom

eras

ssellas

an

administrator

(seealso

[26])A

tthe

correspondingplace,

p2(15,

inth

(airocopy

the

onlything

inthe

margin

isa

copyof the

statement

inthe

IndiaO

fficem

anuscriptthat

thesultan

hasadded

agloss,

butthe

actualgloss

doesnot

appear.

f,172s2

(AS

l3Or.22)

Section13

Determ

inationof

theL

ocalM

eridian

Tw

om

ethodsare

described.O

neis

toobserve

thealtitude

ofa

celestialboth

of

laiown

coordinates,and

simultaneously

tom

arkthe

directionofthe

shadowit

causesa

gnomon

tocast

By

methods

previouslypresented

calculatethe

azimuth

fthe

shadow

fromthe

altitude.From

theazim

uthlay

offthe

cardinaldirections

Alternatielv

calculatebeforehand

fora

particularday

thealtitude

forsshich

the

sunw

illhae

ieroaltitude

The

shadowdirection

atthat

time

will

hethe

east-west

line.

fl72s

23(A

S130v

19)Section

14D

etermining

theA

ngleB

etween

theM

eridianand

Any

Horizontal

Line

Assum

ingthe

tssoto

hein

thesam

eplane,

froma

pointon

theline

drawa

perpendicularto

them

eridianT

henthe

lengthof

thisperpendicular

inunits

suchthat

thehypotenuse

ofthe

resultingtriangle

issixty,

isthe

(medieval’l

sineif

thtrequired

angle.

f173r:7

(AS

130v.25)Section

15D

etermination

oftheQ

ibla

The

Muslim

,sshen

making

hisfive

dailypraaersi’aees

toward

Mccca.

Sott

isof

practicalim

portancethat

heknoss

theangle

betsseenhis

meridian

andthe

greatcircle

connectinghis

stationw

ithM

eccaH

encesolutions

ofthis

problemof

theobla,

,it

iscalled

proliferatcI

‘nss

dl’s

Kash’i dispose.

of.scveral

pecic a,

sI

ihcem

itio

12above

toobtain

thedistance

fromth

ecis

enlocality

1051

‘cca

ofhe

lawo

fsines

thens

ifticto

‘odu’thu

requ,‘d

cm

u

ulim

aely

frt

tIe

bhiI

rr

oI

II

f1 74r:9

AS

13h5

1C

haptei2.

Piu

of

ofthe

Abm

eO

perit

otis

Secion

I)lina

usit1

A‘ci

otis‘It

sit

1is

hasa

n,le,u

,nc

rIs

Theorem

sare

invokedsshicn

ctahl

sha

laiionsSet

is‘

Sc’

triangle,or

pairsof

them

5v

.lfA

a,3_

,,

to

Here

areproots

thr15

,l

onthe

mendian

planeshow

n

h’n

gnando

I

1’. 176s.ll

AS

133v:5)S

ection.\zim

u’h1m

mA

ltitude

11s

pr‘sces

IV, 1

7ib

cIc’rc’

‘ Lo

pcri

Vt’

ecifigtI

isn

et

il

Section

4A

ltitude1m

mA

iimitS

‘thiambI

iiji

sol

dh

icc

eo

i,.,o

fX‘iI

I’. l 79r: 19Section

6D

etermination

otIns

cisc

Ssu

‘imo

n

Jhe

s ‘nbcc

pyin

cii

45‘cu

tto

hass

Len

5’

theco

di

eeo

n5

F

manuscript

notedthe

omission

amidto

pic

dS

eton

(s5

of threesides

off

I 35v

Again

iiithe

set‘

itt1

pr‘en

ul

sI

iec

sSc

tions ppi

ittitm

orm

method

stems

2w

rigit

spheiaca

(‘ru

mn

t4’

ii

and4

abose“so

ssbc

‘A

,’.ssh

pro

tectio

n’

tiles

figure

‘1 hsurtS,c

u.a’thc

phc’“

sfor

thetm

nit

f.l77v’18(AS

i34v:iI)

‘Ih’sis

ne

in‘rst

inIV

,a

so.

Nos

entirelyon

thesurface

of

thelic

r

:m

portp

lan

e

toI

thrule

gisentic

0)

ti

f178s

8A

S115

lIi

Stom

nat

cits

101

hits

ispa

of

ei

aV

oS

p1p

t

Rule

ofFour

totw

opairs

ofspnemm

c’afnuht

Irmangie’

1k’L

h)rsm

.‘

assti

onlcb

rfirs

quadrantarcs

som

iiinecessark

totdd

arrangemnc”itc

iüt

nct,a’

s‘te

nths

argumen’

isn

on

eftl

oI

‘10

i‘te

nhe

came

ite

eeLio

nhe

arp

a1.

r‘d

edth

rtti

a

IS

Ki’v

simK

c

procedureof

Section10

above.T

o,justifíthe

nghtascensioninversion

theR

uleo

fF

ouris

invokedw

itha

pairof

sphericaltriangles

1heresult

isused

toset

upa

secondrelatio

n,

alsovalidated

bythe

Rule

of

Pourhe

secondrelation

ilds

thelecired

inverseoblique

ascension.

f179v23

(AS

135v19)Section

7D

etermination

oftheD

istanceB

etween

Two

StarsT

hisconsists

ofproofsfor

therules

givenin

IV.l

12above

They

involverepeated

applicationsof

theR

uleof

Four.U

niqueto

thisproblem

isthe

largenum

berof

no

ntrivial

specialcases

As

aresult,

theauthor

foundit

necessaryto

presentten

figuresin

thissingle

section.

flS

lr13

(AS

136v21)

Section8,

Azim

uthof

theQ

iblaThe

matenal

of(V

.1,15is

provedhere

with

theaid

oftheubiluit

usR

uleo

four

Notice

thatChapter

2om

itsall

mention

ofSections

5,6,11,

13,and14

ofChapter

1.K

ãshiclaim

ingthat

them

aterialthey

containis

self-esidentT

htsincludes

Section11,

where

much

ofw

hathe

saysis

wrong

f183r

2(AS

137v13jT

RE

AT

ISE

V.

DE

TE

RM

1A

TIO

NO

FT

HE

AS

CE

ND

EN

TF

RO

MV

AR

IOL

SC

OM

BIN

A[iO

NS

OF’

OB

SE

RV

ED

QLA

NT

[[IES

(The

contentsof

thistreatise

havebeen

describedin

271lo

castthe

birthhoroscope

ofa

person,the

astrologerm

ustknow

theas

endent(tall

)at

theinstant

ofthe

hirin,i ne

ascendentis

thepoint

onthe

eclipticcrossing

theeastern

horizorit

thetim

eand

placeofthe

birth.)

Intro

ductio

n:

A1

echnicalV

ocabularyT

hisis

notso

much

alist

ofdefinitions

asa

setof

descriptionsof

observationalinstrum

ents,particularly

therotating

triquetum.

butalso

includingthe

handquadrant.

them

uralquadrant

(lthna),the

cupclepsydra,

andthe

plumb

line.H

owever,

thereare

alsodefinitions:

thatof

thearrangem

entsine

(hSin,jayb

i

tartbi

c/dr ir,24l,

p45)

hourangle

totaland

partialparallax

flR3vi3

(AS

138r1)

Chap

terl,R

ule

sfor(a

rrvm

gO

utO

pera

tion

Section

1V

ariousT

echniquesfor

Determ

iningthe

Ascendent

The

operationssketched

areall

observational,em

pliying

theinstrum

entspreviously

described,also

anyhands’

verticalw

all‘Ihe

subsequentm

athematical

reductionsare

detailedin

thefollow

ingsections.

fI 84s

AS

8vI

vi

olP

roofsn

tiQ

0,1

(1,1,.ent

Ialtitude

andts

culrnmt

andthe

star,,asccnsiot

ascensions

f185r(A

S139r)

Table

ofV

enus(C

fV

.2,2Forthe

‘anthe

art’r

ForV

etu,s

acidtoe

moo

scord

f,185vA

zimuth

(Iroofs

areit

Fromthe

azimuth

cakunto

thealtia,dr

e’riaino

inthe

precedingsection

fI86r

1(A

S1 39s

41Sec

or4

Dererm

unungthe

\NL

’S’r

Cast

bsa

(inomon

ProofsTi

Fortsi

nationsat

ertd

issitu

thotherw

istisoich

sen

hit

rhtain

Hfii

liarci

1’trigonon

ett

uriC

f186s

$(A

S140r

71Sett

51‘rte’rn

the\

or“

Iiltcatinr

iSD

irectionof

aShadoss

‘Theshadow

lengthnot

hem

,,m

rneduatc’lsn\a1

;,’io‘‘ii

uris

o\pcd1erttsare

described,either

toconstiuct

thehorizontal

shadoirand

‘‘ui’’

gths,or

toobtain

theazim

uthT

henuse

Sertto

i,‘

orI

toI

ndthe

aseenr

(I87r’ó

(S

Or’23)

ic\sc

)rivno

ll,r

yT

0,

tnt

Ie

VTi

151‘(ii I,’

ideit

ofth

cint

thtast

identus

ob

t

lint1

Altitu

iiri

sour

ang

iI

,‘

-(irned

rioht

Maxim

umP

arallaxin

ltitud

c(to

theu

n1

oo

n,

and

iseat

iilet

‘roesol

to

Ltie

nuie

ec

ii.i’di

lt’cno

apoceeu1h

areI

Aitt

from

Iho

isOpro

ecuas

‘‘i‘

‘,

‘inhun

theK

hadoss

IIa.

4(1

‘it

leselor

uhedI

in

two

ligso

ri‘lit

tnT

heoth

iyhe

simu

applyS

ection2

abase

ut

aP

easurvent

nil

38(I

FS

KIN

NI

)K

[aqaru

f.I 87r: 17(A

S140s’. 11)

Section7.

Extracting

theA

scendentby

Marking

theShadois

of

theT

opof

aW

all,the

Top

Being

Horizontal

This

method

isludicrously

impractical.

Once

them

easurements

havebeen

made.

theym

aybe

reducedby

analgebraic

rulew

hichthe

authorclaim

she

hasproduced,

butit

would

spoilthe

funfor

theuser

ifhe

were

toldT

hesolution

isleft

tohim

.T

hechallenge

was

met

byM

.-Th

Deharnot

andpublished

in[13f.

f189r:18(A

S142v101

Section8.

Determ

iningthe

Ascendent

fromthe

Shados

ofan

InaccessibleO

bjectT

hisseem

sto

hesolved

with

apractical

andclever

constructioninvolving

sstring

anda

rulerIt

isfully

described(in

1271, pp131132

with

afigure

therehe

ngnone

inthe

zIj

I190r

3(A

Sl43r

14)Section

9.l)eterm

inationot

theA

scendentA

tterthe

Fapse

01Four

,JuhanY

earsN

owthe

astrologeris

invitedto

applythe

factthatfour

Julian(or

Yazchgird)

yearsare

veryclose

tofour

solaryears,

sothat

foursuch

yearsafter

thenativity

thesituation

replicatesitself

Ifit

iscloudy,

neverm

indw

aitanother

fouryears

andtD

again

f.190r.11(A

S143r:20)

Section10.

Determ

inationofthe

Ascendent

When

Tw

oStars

Have

theSam

eA

zimuth

(Proofsin

V,25)

An

observercan

determine

ssithno

otherinstrum

entthan

aplum

bline

ishenthe

givencondition

issatisfied.

How

ever,to

determine

theascendent

fromthis,

involsesthe

successivetrigonom

etriccom

putationof

fiveauxiliary

quantities.

flQiv

:l(A

Sl44r:20)

SectionII

Determ

inationof

theA

scendentby

theU

seof

Ciepsvdras

(fanevent

occursw

henthe

skyis

clouded,theobserver

may

measure

with

aw

aterclock

thetim

eelapsed

untilthe

weather

clears,then

observeand

calculatethe

ascendent.T

hetechnique

described,counting

successivecupfuls

ofw

ater.seem

scrude

fl9lv

:l4(A

S144v

6)Section

12.D

etermination

of theA

scendentW

hena

StarIs

in

theM

eridianor

onthe

Ho

rizo

n

Ifthelum

inaryis

thesun,

anapplication

ofnormed

rightascensions

sulTi‘Cs

Forother

objectsreduction

becomes

more

elaborate

f.I 92r,23(A

S1 44v26)

Section13

Determ

inationof

theA

scendentfor

[Torion

fthe

Afternoon

Evening

andM

orningPrayers

TheM

uslimp

raertim

asire

definedastrunornicalls

someone

remem

beredthat

ab

ioS

cided

with

them

ue,’iiecalculation

ofthe

ascendent

f.I92v.10(A

S145r

1(1Section

lO

nthe

Determ

inationol

tIn,S

cendentfrom

theN

amiidãrs

(Dem

onstrationin

S.i.(If

thetim

eof

thenaiL

rt‘s

knosvnunIv

approxinmatel\

.sere

techniquescalled

nam

,iär,

tLatin

unm

nalu

’ishi

‘Sriurported

toenahlc

an‘determ

ination

193v5

5v25

tei

I1 he

prx

edurei

tp

uponappro

atte

time

otthe

i

f1 94s

(AS

I 46r.4I

(‘haptenetrie

Proofs

ofthe

Section1

Determ

inationo

he\scendent

hornlb

sS

tIlL

The

rulesprosed

herears

rkedout

fromplane

cni’i

ultimately

Indianorigin

fI95r: 16

tAS

146s15>

Section

2D

etermination

ofthe

.aoilude

fromthe

Observed

Apparent

Altitude

l-Ierethe

parallaxtheori

orx

‘sentedupon

isinchthe

:uhlsS

Si

isbased

P195v

20iA

SI47r

16tSection

i)rim

natronot

thIion

(Prrvf3

oI’\1,3

1o

thedem

onstrations, Iii

lbs.u

rlice

casetl

Tule

ofI our

S47s

2)ion

of

1tan

ccanH

ot1

4

i.eidea

rsthat

rfw

ou

ldenable

f193rkS

145r23iU

heNonici

heprocedure

isbased

hirtls

Iccendentat

thei

cones-pton

theascend

convergentIc

alivealgo

po

Julae

IaP

e

Is’rm

esthe

Sanestulate

thlunt

rt

inand

eons‘is

inap

p

din

gto

heinstant

ofrittitude

atnation

aS

ing

(Sc

sat

the

eratmns

fur\

1,2ni’

prohahlsot

f19(r

Shadow

ii’the

zimu

th

thetanuent

noino

40

EK

iNsID

Ys}

cn

f. 196v:23(A

S148r:5)

Section5.

Determ

inationof

theA

scendentfrom

Tw

oStars

Having

theSam

eA

zimuth

(ProoGfor

V,l1O

.)T

hedem

onstrationruns

thruugha

successiunof

sixtngunom

Ltiic

equations

involvingapplications

of thelaw

ofsmm

andthe

secondcase

of thesubstitution

theorem

(shikiim

ughnit.

F197v:13

(AS

148v13)

Section6

Difficulties

With

theN

arniidärof

Herm

es

Adetailed

descriptionof

thealgorism

.

f. 198v(A

S149v:H

TR

EA

TIS

EV

I.O

NT

HE

RE

MA

ININ

GA

ST

RO

LO

GIC

A[

OP

ER

AT

ION

S

Introduction:F

echnicalG

lossaryT

hefirstdefinition

of thelist

isthat

ofincidi

itthorizon

(ufuqihadiih),

aconcept

much

appliedin

astrology,particularly

inthe

PersianzIjes

Fora

givenstar,

itis

thegreat

circlethrough

thestar

andthe

northand

southpoints

onthe

localhorizon

Most

of the

remaining

definitionsinvolve

operationsw

iththe

incidenthorizon.

Of

these,tw

o

introducethe

important

conceptsof tasyir

(Greek

uphesis)and

fardar(1

atm.

fIrduria)

f. l99r:6lA

S149s.23)

(‘hapter1.

Rules

forthe

Operations

Section1.

Determ

inmg

theL

atitudeof

IncidentIlorizon

This,

consistentlyw

ithother

latitudes,is

definedas

thedistance

ofthe

incident

honronfrom

thecelestial

northpole

The

rightascension

anddeclination

ofthe

given

starbeing

known

thedeterm

inationinvolves

asuccession

of verbalrules,

applications

oftheR

uleof Jour

andthe

sinetheorem

,the

calculationof

fourauxiliars

quantities,the

star’shorizon

coordinates,and

eventuallythe

desiredlatitude

ofincident

horizon.A

partialproof,

usinga

figure,is

tohe

foundin

Chapter

2,at

ff2O3r,8

-204r17

The

sectionalso

givesan

alternativem

ethodof calculating

thelatitude,as

longand

complicated

asthe

first

f199v:23(A

S150v:8)

Section2.

Determ

intngthe

Verified

Ascension

(maiäia

rnu.cuhha;T

heverified

ascensionis

theintersection

of theincident

horizonw

iththat

halfo’

theeclestial

equatorw

hichis

nearerthe

givenstar

Tw

orules

aregiven

forcalculatinp

it. employing

suchquantities

asthe

equationof daylight

Here

also,as

inthe

precedinp

section.a

partialproof

isgiven,

at204r:l8-23.

f.200r11

(AS

150\.2zSection

3l)cterm

rninthe

Protsetiono

faxs

matarrn

uish

uã1)i’tw

O.

smu

ts;

iine

ico

rn

icu

sepa‘

‘ii

iiiC

lancenes

aresaid

tob

rspect

‘x6

awt

ay

projectits

rayupo

hei.

.I

tonitw

aeh

toA

zby

Fltm

:in

[5],pp.

1377-1303,

abstractedmor

readersot

Inglish

in.

orurn

methods

atedescribed

in[16].

Inthe

zilK

ãshiconi:ncncesw

itha

piocedurebc

thutesto

Ptolernsas

doesB

irUni

The

ttare

If’ren

roccestot

nit

1:1to

ive

nuters

gwenabov

hsibj

ci

p11

ds:

itheir

icisc

shoreK

Iippl

enas

onso,’

icihoi

nK

ãshrthen

describesi

ccond

rnetnod:s h

.hho

..:ihj.c..

,oLhe

astrolocers(uhkäynn’ãni,

T’hisinvohes

tfeascension

ofthe

lanets

iran,it

,.::1

15d

ascensionsand

normed

rightiscensions

/81

4

.2(O

v(A

IIc.

Oct

(atm

asrhe

tassfrofanx

persor:s

ana

otthe

ccliiI

betat

tun

nomnts

relatedto

important

events‘n

hislith.

inparticular

thelength

oflife

a.

t32

1lIn

thedeterm

inationof

theendpoi

ticof

thetas

iideser

P‘d

ini

.r

incones

pi01

ncidenth

cphi

tmc.’

:1

f201

v1

AS

151v53

Sec0

vi

SO

nN

atestv

intihiT

hereare

many

anetmes

ot:rm

hbut

eachs

cenemaice

ivthe

eclipticat

constantsp

cethc’

positionat

schechhe

pointa::

eunit

Itim

efi)

a,

as

trtd

A‘

nitfor

nesign

pciyear

Fle

ear

n.

generatingpoint

isdetern

tedP

om

;o

nt

Ibusastrologer,

anim

portantsear’

n’rhrcoirnience’.

itth’

ecJi.

‘cru

atthe

instantof

hisbirth

(heposition

ot’‘he

bhit’

i’

heles

cdtc

rdicatem

dm

mlt

Iia,

‘I

attrrh.ies

ofp

anss

rc’:

mh

entered

f.201s’lO(A

Sl52r’O

<Se

tim,

61)::

Nat:s

itsI’m

durs:

‘a“

rs,\]o

ned

tothe

Plarm.

‘hisccli

vi

les1

oi.

cc

centutyJevish

asirokper

omd

otIm

oh‘r

1ha

eis

idcinto

periodsof

years.each

mb

acv

oneof

aset

ortim

ecebesai

h:ec

Ihaanr:hures

tothe

sunten

years.to

Venus

:glrt‘ears

PM

ere:v

thi‘

‘cii.iI

mon

nine.Saturn

pointm

ovingalone

othe

udof

some

vol‘S

cml

Is

orIi.

or’

ca

nsu

Cin

,..an

cop

edby

thecliii

heCuer:t

time

scas‘v.

‘aiimhiSt

42

.tS

KIN

NID

iK

aqarn/

eleven,Jupiter

twelve,

andM

arsseven.

To

theascending

anddescending

lunarnodes

threeand

two

searsrespectively,

sothat

thelife

spantotals

seventr-fiveyears.

The

setcanalso

hearranged

asa

cyclein

theorder

ofdecreasingsize

ofsuccessiveorbits, om

ittingthe

nodes.So

Saturnw

illbe

followed

byJupiter,

thenM

ars,and

soon,

returningeventually

toSaturn.

The

complicated

manner

byw

hichthe

scheme

isapplied

toindividual

nativesis

describedbelow

inthe

explanationof

thetable

onf.208v

(see[34],p.

62).T

hetable

isfrom

theIlkhãniZ

ij,T

heauthor

remarks

thatsom

eastrologers

prefera

differentschem

e,in

which

theintervals

totala

lifespan

ofninets-four

yearsT

hisis

thesum

ofintervals

givenin

thetext,

butK

ãshiasserts

thatth

eadd

toninety-

eight.1

hereis

notable

Oarthesecond

schernr..

f20

l,23

(AS

l5r2

3)

Section7

On

thc[asyirs

lntiha’s,Fardars

andC

yclesC

onnectedw

iththe

Horo

Scope

ofthe

lJniserseA

nalogouslyto

theIiOa

ofaperson,

theen

tirspan

ofthe

universttaken

tobe

aw

orld-yearof

360,000years,

was

thoughtto

hedivisible

intoa

complicated

systemof

periodsA

ccordingto

BIruni,

in[51, this

doctrineof

world

indicatorsw

asassem

bledhI

Abfl

Ma

shar(A

ihurnasar,c.

850),the

most

famous

astrologerofthe

Middle

Ages

fhem

iddleof

thespan,

saidto

coincidew

iththe

Flood,is

infact

theIndian

Kalivuga

era(17

February,-3101

AH

)T

hreesets

ofindicators

areassum

e&w

orldtasyirs,

intiha’s,and

fardárs.For

eachofthese

thereare

foursubdivisions

them

ighty(a

:arn,fem

,rlrnrA

, thebig

(ak/vu’,fern.

kubra’), them

iddle(uw

sat,fem

.w

usta’),and

thesm

all(asghar,

fern..cughrd

j.T

hesubdivisions

ofthe

world

tasyirare

calledqism

a(share

orponion)

thedivisions

ofthe

intihã’sand

fardärsretain

thesenam

es.In

additionto

these,atf202v:23.

acycle

tdawr,

p1. udwãfr)

of4,590years

isdefined,

made

upby

summ

ingthe

mighty

astrologicalgifts

(atãvu)

ofthe

planetsthus:

thesun

1461,V

enus11

51,M

ercury480,

moon

520,Saturn

265,and

Mars

284(cf

[34],p

28).A

llthe

indicatorspass

throughthe

firstpoint

ofA

riesat

thebeginning

ofthe

Kalivuga,

eachat

itsow

nrate

moving

throughthe

zodiacalsigns.

Foreach

thereis

atable

inthe

collectionoftables

which,

asusual,follow

sC

hapter2

(ff203v

213r)T

herate

atw

hicheach

indicatoradvances

isgiven

belowin

thedescription

ofthe

tablesshow

ingthe

zodiacalpositions

ofeach

asfunctions

oftim

e

f203r:8(A

S153v

I)C

hap

ter2,

Proofs

ofthe

Above

Operations

There

areno

numbered

sectionsT

hechapter

beginsw

itha

long(203r

9aO

4r,171partial

proof of therules

forcalculating

thelatitude

ofincident

horizonw

hichare

giverin

199r6-

99v:22.Follow

ingthis

isamu

chshorter

(204rl8-23)partial

prootof tbe

rtilesfor

determining

theverified

ascensiongiven

in199v

23200r

10A

tthis

staheK

ãshistates

thatthe

determination

ofthe

projectionsof

therays

sobvous

asare

thr

restof

theoperations

ofthe

treatise,anct

thechanter

Ion

.

f.20S

(AS

154v)T

ableof

Portions

ofthe

Days

ofthe

searthe

I asir

ofthe

Indicato

rof

Nativities,

One

Solar

sear(for)

One

Ascensional

I)sgreeT

heindicator

heretabulated

IIcirnedat

f200

I1

nio

sciro’p

theecliptic

atthe

rateof

onedegree

periropical

ear0f

length/n

”.I ‘t

Odays

Other

referencesin

thezij

toa

yearim

ps

oneof

thislenszth

unlessa

.1‘r.

at

ear-lengthis

specifiedI

The

main

tablegiser

om

rnutesand

secondsi

theam

ounttne

indicatorw

illhas

em

mcd

finnti

eheoinn

noo

ihison

ann’‘inning

ofd

act

ofthat

searH

encethe

functioni

t’di4/.

innded

tosec

ondsof arc

mal

1u”clliar3.1

InsInun

1s

rat,ssh

usedh

nan

cithe

Ka’h

sathis

(thi

II

tabslated

isrnfn;

tifor

a1

‘.

The

minutes

areexpressed

ino

nth.

ofthirts

da

scools’

1:’‘

hours

(Table

of)P

ortionsof

Tasvir

Secondsin

Days

ot’thc

\ear

Forthissecond

tablethe

enires

are‘seconds’

ixtrcthsot

s.teth

,of

thesam

es

asabove

Sothe

entriesare

now

expressedin

daysand

hours

f2O

6v

5AS

155v)T

ableofth

Iis

ndicatoradvancc

itco

rnletes

arevolution

Tort

cm

aintahi’

ftc/iI)

Tbfu

c’ci

iisphees

IheJ

Ux

l’iI0

(It)

t(h)

06

(S

55r)(iah

leoh

sgives

succt’ssiss

age

syirliisu

trsixtieths

ofh

earolson

iss’rsa

rv[a

svir

inM

ossh

liptiLat

oonstoi)

le‘o

rIf

/0

it

I)sisnianner

that

44

i:S

Ki’i’iy

Kash

,hcS

n

displayedin

minutes

ofarc

f,207r(A

SI56r)

Table

ofthe

Anniversary

(enters

Tasyir,

Obtained

forD

ays,to

Be

Added

tothe

Ascensions

ofthe

Transfer

Ascendent

Enthe

courseofa

yearthis

tasyiris

totravel,

atconstant

speed,a

revolutionplus

theexcess

ofrevolution

r87,15

startingfrom

thevernal

point([201

rI 0-22)

To

obtainthe

increment

perday,

thew

holedistance

isdivided

bythe

yearlength.

fheupshot

isthat

after294

dayshave

elapsedthe

tasyirw

illcom

pletea

revolution,and

beagain

atits

startingpoint

At

theend

ofthe

yearit

will

beat

r

The

functiontabulated

inthe

main

tabieis

/‘d,

(il-I)1rt360”);v

,for

d1,2

.3.

.366

days.T

heau\illar3

tableg

ies

hourlyincrem

entsIt

isproduced

bythe

function(h/24)(r+

360”)yfor

h1,2

324

hours

f207v

(AS

156v)T

ableof

theA

nnualIntihã’

ibistable

sprcadsout

onesign,

30,over

thesolar

year.So

thefunction

tabulate

j1d(d-1)(30’!yi,

ford

1,2

.3,

366days.

The

auxiliarytable

givesthe

advanceper

hourH

encethe

functionis

fiJi)(h-I)24(30

”y

,for

hI.

2.3.

,24hours.

f208r(A

S157r)

Table

ofthe

Monthly

lntihä’For

thisintiha’

them

otionin

oneyear

isthirteen

signsor

390’(f2O

ls6-8)

Soits

motion

indays

will

hegisen

bythe

functienf(’d,I

(d1

)(3

9O”y

ford

1,2,

3,,366

daysT

heauxiliary

functiongives

them

otionin

hoursso

itsfunction

is/(h;

(h/24)(390u

)for

h1,2,

74hours

f208v

(AS

l57v)T

ableof

theN

ativityF

’ardãrT

hem

aintable

onthis

pagehas

fourteencolum

nsindicated

byparallel

linesbelow

thetitle

blockC

olumn

1gives

theyear

numbers:

1,2,3,

ofa

daytime

birth.A

tyear

31,how

ever, thebottom

ofthe

pagehas

beenreached,

andyear

32appears

atthe

lopof

Colum

n6.

The

sequenceof successive

yearnum

bersresum

es. reaching63

at thebottom

ofthis

column.

Fora

thirdtim

ethe

sequenceresum

es,now

atC

olumn

1],com

mencing

with

64and

terminating

with

year75

Seventy-fiveyears

isthe

maxim

imlife

spanof

anative

inthe

cycledescribed

int201v

10above

Sincethe

yearslisted

inC

olumn

Iare

fordaytim

ebirths

thesu

thenbeing

visible,thecycle

istaken

ascom

mencing

with

thesun.

Inthe

cycle,the

number

ofyears

allottedto

thesun

isten

Ilencethe

firstten

veerof

theeu’is

lord.This

isindicated

in(

olumi

iis

theisord

“em-vnttr’rr

sshia

takesup

aoluinn

tIcnum

irour’hi

theord

f‘audi

‘nusr

noSo

oiran

athessora”V

er

itonC

r

continuesw

ithM

er’uiy

sthatr

finishiT

hesears

oflordship

ofthiten

ai”engplani

ssuccessively

inC

olumns

hand

11,their

names

yearsassigned

tothe

ascendinuirnole

andru,

completes

theentries

fordiurnal

birthsFor

nocturnalbirths

thes

cc

mm

rnencesu

9dc

rng

which

tm

onis

lordtu

atthe

topI a

nc’tum

aly

attuou

saturnon

though

hye

ein

alreadym

eni

In

areem

pty.T

hererem

ainsthe

desr’rraon

olthe

ISO

ir’ss/m

rA

iiesr

are

theses

enplanets

proper,om

ittingthe

nodecIhe

associates.:rs

pome

I(0

exertm

inorinfluences

onthe

lifeof

thenatruic

durmir

cpansof

time

ishr.: Ir

iren

Ushorter

thanthe

spansof

Integerrears

of

them

ajurLordshnps

[hesare

noii.:’

ribsof

thelatter

These

aredesignated

inC

iiiin

s4,

anUs

uis”

,‘ehcruplicutm

‘nd

er

The

dtpt

not

thsrd

ship

cm

ttI

,bet

,tc

I

cm

paredW

ia

siates

loW

Irita

oft

systt mI as

beenft

Ir

as

dich

sic

tyes

rrrr.lan

ruo

thre’A

rabicIc

cire

midd

isone

adrilpl

nunier,I

2.3.

.,

6E

achof

thes.:a

in.:naS

appealse

uid

aAs

athe

spanvearu

assignedto

thelordship

ofeach

planetIhe

letteron

eitheruiJu

.ridie

numera

islire

terminal

(notthe

initial)letter

ofa

planetA

particulartilan.:t

,navhe

designatedin

more

thanone

compartm

ent.hut

repetitionnesaur

occursinn.:

art‘once,

andins

ariablsin

alouer

compartm

entT

heord

”is

insaruahi

t’at

of

1.:eon

rhosize

I orexam

plehe

spant

1up

eno

ve

meri’r

,1

irS,

heg

un

nsu

rf

dirrail

birthr

Ut’

spt

‘Uasso

Sr

U0

7d

three

JenI

‘s

Ihe

ml

\1

ccid

antM

ahr

ua

Ca

As

asr

aI

aple.t

arca

rdshpans

f8

r

wam

Iui

thenirs’n

II

olu

rnns

c

inthe

main

tableis

hiscon

aIe

dr’scthe

am

lirebase

thesun

as<ills

ina

reetu,tI.:

r cnnI

adg

Ic

rdi

trr

theinc

inC1

2r

aridso

onappeai

endingw

iththe

cudingnode

This

‘esears

-through

>eecddoss

rdo

um

nt

tnair

isol mini

fitthe

irsso

olonii’

(I)N

i

1.1

P1

46I-

FS

KN

oui

foreach

of

theseven

associatesY

ears7175

ofdiurnalbirth

ruledby

thenodes,

haveno

associatesC

olumns

5and

10are

empty

Asm

allauxiliary

tableoccupies

thespace

availableunder

Colum

ns1114

of them

aintable.

These

columns

haveonly

twelve

entriesand

donot

reachthe

bottomof

theofthe

pageT

histable

givesthe

lengthsof

12

37

seventhsof

astandard

yeary

expressedin

months

ofthirty

daysdays

andhours

Ihehours

arecarried

tothree

fractionalsexagesim

alplaces

rhefunction

ofthetable

isto

facilitatethe

computation

ofthefractions

ofyears

inthe

spansofthe

associates

f709r

(AS

l58r)l’aN

eof(

velesof

theN

ativityY

earsIbis

page,in

bothcopies

ofthem

anuscripthas

thetitle

blockand

borderonly

The

interiorcontains

noentries

ff209v210v(AS

158v159v)T

ableof

the‘lasyirs,

Intiha”s,andthe

Mighty

Eardar

Betw

eenT

hree

andF

ourH

undredM

alikiT

herates

ofadvancefor

them

ightyw

orldtasyn

thebig,

thelittle

andth

small,

areP

1000C

100,

l’lOand

1’P

respectively(f

202r6)A

sfor

theintiha

s, forthe

same

fourcategories,their

ratesarc

1’l0

00

1/100,

II0

,and

I/

respectively,w

heres

standsfor

zodzaal

czgn(f

202r912)

Ihem

ightyfard3r

advancesatthe

rateof

‘360”(f

202r: 14)A

llnine

oftheseindicators

setoutfrom

thefirst

pointof

Aries

atthe

epoh

ofthe

Flood(K

aliyugi)T

hethree

pagesof

thistable

showthe

positionsthey

havereached

bythe

beginningsof

Maliki

years301

(1379A

ll)302

303,402

foreach

of these

datesthe

entriesgive

thepositions

of

theW

orldtasyirs

rhem

ightyin

zodiacalsigns

anddegrees

tothree

fractionalsexagesim

alplacesT

hebig,

insigns

and&

greestu

two

seAagesm

alplacts

The

middle

toone

placeU

hesm

all, tointeger

degreesW

oildm

tihas.

The

mighty,

insigns

anddegrees

toseconds

ofa

c.

Es

Thm

gh

C

Ilr(A

sien

Mt

inH

undreds,T

housanIs

aA

stheitl

ise

11

00

)90(0

00)(1

ah

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fardaii

S

f21

v

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Inxplaint

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thedef

niion

oth

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in,

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the

iI

I

At

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[auiis

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ce

kinter

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ableof

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otionof

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earsT

heargum

entof

thistable

isyears,

,w

hichrun

throughthe

set1

2,3

,..,59,

60,120,

180,240,

300,360,

representedin

abjadnorvplace

valuenum

eralsT

hefunction

tabulatedis

/)0

5‘y

y12

The

entriesare

inablad

sexagesimals

The

maxim

umentry

inthe

tableis

/(360)30’

azodiacal

signA

propertofthe

mighty

fardar(f

202r14)is

thatit traverses

asign

in360

yearslk

nce

tis

clearthat

thetable

givesthe

motion

of

thefardar

indegrees

peryear

Ifthear

ument

isa

number

greaterthan

60and

lessthan

360,subtract

successivesixties

fromit

tofind

arem

ainderless

than60,

Tind

theentry

inthe

tablehaving

therem

ainderas

argument

Thenum

berfrom

which

therem

ainderhas

beensubtracted

will

hea

mem

berof

theset

60,120

180,240

100L

ookup

itstabular

value,and

addit

tothe

valueprey

otisly

found

‘fableof

Positions

ofthe

Big

Fard

arl’he

argument

ofthistable

isthe

set1,2,3

....78

years,represented

innon-place

valueabjad

numerals

The

entriesin

thetable

arenum

bers,each

displayedas

asequence

fsigns,

degrees.and

minutes.

allin

abjadform

.If

plottedas

agraph,

itw

ouldappear

asa

sequenceof

straightline

segments

ofcontinually

decreasinglengths.

hutcontinually

increasingslopes.

‘Ihisbroken

linestarts

atthe

origin,and

endsw

iththe

78year

periodat

aheight

of

elevensigns

(11X

300-

33

00

)

To

explainthe

complicated,

notto

sayperverse,

systemunderlying

thecom

putations,is

necessaryto

statea

propertyofthe

number

78It

isthe

sumofthe

firsttw

elvepositive

integers‘l’hat

islf2

13

1278

Com

mencing

fromiero,

thefirsttw

elveentries

havea

constantncrem

enof

112

30°

12230’

.H

encef(2)

210”,t(3)

5(

aadso

on

until/(13)10,

Now

however

thesign

tobe

addedis

spit

elevenw

ays,30’

112,44’

tohe

addedsuccessively

tothe

next elevenentries

So/(14)

12

44

0and

,24)

20’O

heailded

to

.0ean

’O

ut

aused

do”M

alikifo

addto

ittw

oless

than‘7

ff.212v,213r(A

Slh

lvl62ri

1able

ofthe

Middle

andm

air

tarsand

theA

ssociatesntss

two

pagescontain,

hastinning

do

nthe

sideo

nonplac

vsre

abtadnun

compartm

entsth

t,a

totalI

m,m

eralsth

kindpro

The

colam1eadir

gsre

3T

heascending

node,8

Mars,

and9

\enus

Sow

hichJupiter

was

lord.T’nt

eatnesof

Colum

n171

96i

heexpected

that theentre

atthe

tmi ran

2ssould

P5‘

‘1entry

171has

beendisplaced

11)com

partments

down,

andapjw

tix

sA

tthispoint

thesequence

ofentries

acincreasing

Psone

resim

thebottom

of’thecolum

n.H

erethe.

ISi

enrrsis

23a\V

Cifl

1

236w

ouldappear

atthe

topof

Colum

n3

Iiis

indeedat

thePeon

uC

olumn

2,not

3.So

Colum

n2

conm

encesssrth

2H

’‘o

sshaI’

.ni

addedbelow

,until

allthe

emp

to

mpartm

enrt

ore

t]leiargum

ent10.

with

entryN

t’his

curiouspractice

ol’an

completing

itselfat

itsossa

tOfl.

toaeie

ris’ic01

tiPthe

.ena’table

Matin

gon

toC

olumn

3t

IP1

246s

insertedin

Colu

oppositear

at

eat

20,an

aI

From

thorthe

atrit’sinc

301E

ntras.

throughIn

olm

entcom

partntens

0t

proones

oh

hotot

fthe

ofadjacententries

Within

theentire

360year

spandeterm

inedby

thetw

oentries

thereis

asingle

combination

ofsign

andplanet

givenin

theupper

ofthe

tssorow

sso

designated.T

hisis

validfor

thew

holeinterval

The

differencein

theepochs

ofthe

Floodand

ofthe

Maliki

calendar(15

March,

1070A

D),

additionof

3000,and

theinitial

tabularentry

of261

insurethat

theresult

will

becorrect

Next

thesign

issplit

tenssa

.‘i

tean

increment

ofthe

nextten

arguments,

andso

on‘Tie

tableends

ati

Sincethis

indicatorcom

mences

atthe

Floodepoc’

‘‘

remform

ingoperation

sothat

thetable

stillproduce

corT

ect

resi.iits

‘oh”years

To

thisend,

itis

prescribedtha

,grsena

Maliki

eat

theiser

years,then

subtractfront

thesum

successIve78’s

untilInc

re.m,oud,

Enter

thetable

with

thisas

argument

it’sIL

tfirst

ii

loomol

the)per

hasd

ssith

itoSaiiher

inI

islargest

75in

tIPH

aloe

Ott.

ode,during

iit

wo

uld

thet’ase.

forait’utitent

Iii

Lontinues

toor

expectedthat

slum

rt,hut

His

cC‘is

eo

nes

ars

a.ir

sop

posite

tolereroluryri

i,uinnsin

the

it’to

lottedasern

en

trs

thetop

Sentry

‘nfl

tw

ith

t‘ntrt

thusfar

mode

ithi

top[firs

imeni

5to

ilan

ittat

of

it,op

37

E.S

KE

\vE

DY

Ksn

Kh

q:m

iZ5

entry374,

tow

hichsuccessive

onesare

addeduntil

theem

ptycom

partments

arefilled,

andthe

column

iscom

pleteIn

thism

annerthe

entiretable

iscom

pleted.T

hesuccessive

increasesin

thedisplacem

ent oftheentries

are.10, 9

3,12,

13,11

2,and

8,‘I hese

arethe

divisionsof

thelife

of

anative

utilizedin

thenativity

fardar(see

f2Olv

10)N

otethat

them

aximum

entryin

thetable

is770

(954

675).

Because

ofthe

displacements

oftheentries,

thisdoes

notappear

atthe

endofthe

table.It

isin

thelast

column,

oppositeargum

ent68

To

usethe

table,given

aM

alikiyear

greaterthan

770,the

textprescribes

thatsuccessive

675’she

subtracteduntil

therem

ainderis

770or

less.Find

therem

ainderas

anentry

inthe

table.T

heplanet

named

atthe

headof

theC

olu

mn

inw

hichthe

entryis

foundis

thelord

ofthatyear

inthe

middle

fardar.T

hesecond

tablehas

todo

with

thesm

allfardãr

Ittakes

uptw

oadditional

columns

at theopposite

sideofthe

pagefrom

theargum

entT

hefirst

ofthese

isdivided

intonine

longcom

partments.

Lach

ofthese

compartm

entsin

successioncontains

thenam

eof

theplanet

which

appearsin

thefirst

tableat

thehead

of

eachcolum

nThe

lengthofeach

compartm

entcorresponds

tothe

number

ofyears

attributedto

eachplanet

inthe

nativityfardar.

These

spansof years

adduplo

75,w

hichis

thenum

berof

rows

inthe

firsttable.

Sothe

lower

boundaryof

thefirst

compartm

ent,that

of thesun,

isan

extensionof

theline

justbelow

argument

10,and

soon.

‘I’hesecond

column

ofthe

tablediid

esinto

seventhseach

ofthe

sevencom

partments

ofthefirst

column

which

appertainto

theseven

planetsproper.

Each

ofthese

small

compartm

entscontains

eitherone,

orm

oreoften

three,A

rabicletters.

To

uscthis

table,given

aM

alikiyear

findas

describedabuve

thecorrepunding

entryin

thefirst

table‘This

entryw

illhe

oppositea

particularlong

compartm

entin

Colum

n1

ofthesecond

tableI/he

planetnam

edin

thiscom

partment

isthe

lordofthe

small

fardarcurrent during

thegiven

yeaiT

hesam

eentry

will

beopposite

aparticular

small

compartm

entin

Colum

n2

ofthe

secondtable

One

ofthe

Arabic

lettersin

thecom

partment

designatesthe

associateof

thesm

allfardãr

duringthe

givenyear.

The

method

byw

hichthis

isarrived

at isvery

complicated,

andis

sketchedin

thedescription

ofthe

tableof

thenativity

fardãrat

f.208v.

At

thebottom

of6213r

isanote

statingthat

thebook

was

completed

duringthe

year81611(1413/4

Al).)

andthe

copyin

theyear

905Fl

(14991500

AD

).here

isno

othercolophon

IA

sTer

Aahoe,

75’ihi’s

Itt

pp..4../19

2A

l.B

inii,

Aita

/aa

tf1

itl

As

rologv)tr

asR

Rphi,

c.uac

3.A

PB

rum,

7’ahda

ata

atlash

ihan

at.asa

thetext

publishedby

P.C

I3uiakositt .1idjath

ma

hhI

a’C

airo.1962

Russian

transIatioiand

cominentars

bR

exhan,Izhranni ‘e

I ‘r,vcd

‘ama

1,/1n

’ocia

Ak’idem

irSSR

,T

ashkent.1966

English

‘ranclationhr

.Jamii

Ci

.C(

oordmacs

(77ito

,ti/-B

ain

‘aitd

ki

ai

4mtAm

Ami,

Beirut,

1967p

rtedas

Vol

‘fliseric

mm.j

Sezgn,Fran

IM

ainns

oft

ea

slai

Smr

b1d

Birdn

.cii

lid1

ros

aas

Vol.

27o

ths

esl,sgrip/m

ri,li

Main:

Publicatis

ofthe

lrmtmt)Y

theIlist

fS

rah,.ls

4.A

l-Birunion

‘/rancitst’tthnhFi

ilziataqart

Ittat qiq

ma

hJ

5

byM

Saffouriand

A.

Jt’ramssith

acom

mentars

b\‘

S.

University

of l3eirut,l95Q

.R

eprintedas

Vol

33of

theseries

si

and

Astronom

y,edited

hri’uat

Seigui,

Frankfurtam

y1amn

Institutefor

theITistorv

ofA

rahiIslam

icS

an

I’)S

5B

ai’r’i,AlQ

‘‘

Cc

iF/i

1!,isuds.C

6B

outne-Lee

csfr

Px

7lerin

vanD

SItuare

iiim

vetC

83enno

vanD

ala

SR

en

aci

Musts

ai’,id.C

alendarin

IasiC7

i-iIlk

an‘.

Cm

isii

ir

Isiamisehen

liasefl,Sch

attmn,

II

.PP

1115

9.L

eRoy

FD

oggettand

Flradie\L

Schacler

“Isoar

Cm107(1994),

pp.388-40310

A1—

Kãshi,

GhivSth

al—Dir’

himsi’rd

‘ii/a/ha

7”.sal

edR

,enfel’d,$

aand

AP

1k.’ich

as/

rIa

11A

lK

ashi,A

tic/

inn

/ited

anted

FI

hrome

zhr

Ja

aha

dA

ssermsc

all3

rimIi

121

SK

ennisI

Auiiai

,ma.‘

‘Vs

S

BB

IIO

GR

API45’

/lethodlh

.1toi

.7 31

4cal

editici/

ara

hm

sa.

lsigakovis

.dhu

\auk

Ishekskoi

“mm

ina(ion‘l

55’

vrisersits

at‘Iitr’d

hrI

sofA

iii

(1rpr

aci

IrankliI

‘cience,1

3at’

translatedsned’,

Am

ericantin

.C

i’/mcm

aizP

icationsof

the

atiese-F

/LP

;,

(uIit’

7,,rsm

rus,

Sin

Ch

13.\

iS(tW

195ck

e5

idI

m,s:iii

ES

Kio

’ir)

K

Astron

my

,(entaurus,7(1960),

pp.134-140R

eprintedin

131],pp

144-15013.

1.S.

Kennedy

andM

Th

Deharnot

‘Al-K

ashi’sim

practicalM

ethodof

1)etermining

theSolar

Altitude”,

,Journalfor

theH

ntoiyof

Arabic

Sczencc3(1979),

pp.219-22714.

F.S.

&M

H.

Kennedy

Geographical

(oordznatecof

localities,from

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Sources,

institutfür

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rabischlslan’nschen

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Frankfurt

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ain,1987

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S&

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11K

ennedy,“A

l-Küshi’s

Ueographical

Table”,

Transactions

oft/u

Am

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hilosophicalSociety,

77,Part

71987>.

pp1-45

16.F.

S.K

ennethand

Haiganush

Krikorian-Preisler.

“The

Astrological

Doctrine

of

Projectingthe

Ravs,”A

l-Ahhath,

25(1972),

pp.3

-I4reprinted

in31],

pp.372-383.17.

F.S.

Kennedy,

“Parallax‘[heory

inIslam

icA

stronomy”,

isis,47(1956),

pp.33-53.reprinted

in[31]

pp.164-18418.

F.S.

Kennedy,

ihe

Planetary

Equatorium

oJam

shk

lC

/hiyälhal-I)m

al-Ka,chi.

Princeton

University

Press,1960.

19.F.

S.K

enneth“T

heSasanian

Astronom

icalH

andbookZ

il-iShah

andthe

Astrological

Doctrine

ofT

ransit(M

amarr).

Journalo/th

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merican

Or:entai

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pp.246-262;reprinted

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S.K

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Medieval

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ondon,1962

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reprintedin

[31],pp.522-525.

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ennedy,“R

amifications

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orld-Year

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((

tehindu

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FS

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egA

PP31

1)v

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insn

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Kt.ni

c’ii

thei

ci

Am

ee

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S“

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eugehauer..1

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volumes.

hu

tw

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aaiio

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34D

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ndon

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1.Jnivcrsty

ofl,ondon,

196835

Ptolemy’s

Alm

agest.P

iolen

aar,JfrncJ”r,’,

0r

r”,,,,

KM

anitius,2

so13,

‘l’euhnecL

e’pr

o.163

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bji’l-Qãsim

Qurh6nI

A,is/rin

pram

I’vai

‘1‘cc”

‘.

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Mark

aN

ashrl)arvshg5hi

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ino

ne

tw

tothe

iecot,d

prin

tr

S37.

FI

Ra

epA

Ui.] tin

iii/

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,pn

t.V

AN

ew‘fork

I38

-1R

Rem

efh

Cei.’iSoi

in.‘/

rr

/cc’0

i‘

1101)

1983054

ap4

I‘r’lt

40.V

thV

Antiocnc’ii

y

ici

ncr,1989

41Ju

anS

eit

.r,u’iat

9’190cr,

orri

ci

dci3uc

nastra.s

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