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notion s iounded in thc lue €ph€mdjs ih.y pEpaed (Fat@hi d al, 1999). Ac@rding lo lheF $udies it is pointed out lhal the noonset-su$et lag atone.ould nor have been used as lhc visibilily dit.rion by rhe Babylo.ids. BabytoniaN sysren had $c tollowi.S rlohedrior rL, - moon*r tasrme rin deCrees,,S) - conninr In vdious $ud,cs lhe vrtue of$e conqant rs dedlce ftum | 7 degFes lo anyrhere &ound 2l d.8R Ao ahd koans.t,ulet td|tin >.t2 nin cr_ lr has been indicared lhar Mrslins. reatization thar rhe Eddh-Moon dhlancc laries during one comptet€ cyctc ot tlhar ph.ses and lhe hininuh moons€r lag lime considercd b, ^Ebs vaiied fiom 42 ninures for Moon ar perigee.nd 48 ninurcs for Moon at aposee Thonsh lhis -nrle of thunt. is no4 sophisricated 6 compNd ro $e one art'buled ro |hc Babylonims sli jr does nor povidc rhe complere prcrup of rhe cffons ol Arabs ad Mulin' h $c hedirwt ines exl€trir ur ofAolemaic sy{en ed rhe spherical risonodeq devetop.d b, Ambs lead to the L!n& Rrp€.es flnclion (bat depends on locatbritud. ofrheptace ofobrNadon dd lbe c.testiat to,gilude ofthe srn and lhe Mooh) 6 eartr as lO cenrury AD (Bruin t97) Babylonie crirerion h indeed lery sinplc for praclcar purPoses and is supposed to have ben cmpirical in no significdt chahgc in rhh rill
Flarively r€c.nr rimes, ho$v€r,lhc atiesr Hildu tdis litc pa,ch Sidn it ( D 5OO) hints towards thc idponance of$c Vidrh oflhc luM crc$ent (Bruii, 1977) Thus an elabootc stsreo of calcularioc involvcd in delemining the tinc of €licsr visibilitt appeds to have developed only abund AD 500. Morcexplicirnentiohsofrhcsederaited calcularions are loud ar vuios plaes in the dly Islmic litenle (Brui., 197). Ohe ol thc ealiesl Muslin Aircnomcr eho developed lnc lables fo. se lidnC the luna!crescenl s lisibilitywas Yaqub lbn Tdiq (Ke.nedy. 1968).lt has bcen eponcd in $e litcraluE (Atuin, 1977) that lbn Taliq had r.cogniad the imponance oflhc Width (w) oflhe crescenr. Thh not only shows rhat ar rhd rine rhe varying dislance ot rhc Moon had been rcaliud bul it aho mdc ir possible 10 inpov€ upon dc ncs cr*cnl vnibilily cinerion. Bruin hd Eponed th Al-Ailuni had $e Falizario. oflhe tong ind dillicult calcularions inlolved in lhe dercnnimlion tor the nev lun,r aesccm vnibitiry lnd in hG Crloro.,os " reconrmcnded rhc work ofMuhanm.d lbn crbn At B,llani l ltnlhook oJAtto"o,'t trstard to Lario by Nalino, l9|]3). 'lhe sihplc *nclioh for cadiesr vhibiliry oflunarcrescchr evotved sjnce lhe riftes of B6btlonids qas passd onb rnc Muslims drcueh Hindus wirh !e,)-' tinle inprclemenr. Molilated by the euEnic iijuhctions a.d rhe salnlgs of rn prollrr Muhamnad (PBUH) rh. Fobtem ofearliest sighring of hnar cE*enl \s lhomnehl). Invesriglredby rlrc earlJ Mrslih aslronomcGotsLhto l0'tccnluryAt). OD the bash ofreatiarion of the idjponance ofwid$ various Arab asllonomcrs conclndcd td lor celics( vjsibilil, ofcrc$.nl thc minimun equabrial $pararion offie Sun lnd the Moon varics from t00 whcn lhe crescent is widesr up to l2i whcn llr cNcc.l is n@$cst Such der,il.d calculariotr \.rc wor\cd oul.s earty as 9ri cenrur) AD by Muslinr asrrononers as$ciated wirh rhc Ab&sid coun of Al,Mahun. Fronl monesr the* {stbnoncf Al-BaIsli kncw lh,r ihc cnleria that.gc of n@, shoutd b€ moe tnan 24 hou6 (or arc sepdarion belveeo $e Sun sd lhe Mooh) isa good sldtine poinr but i A only m appnrnarion. H. b€t,.vcd LlEr lhc dcienl6hnomc6 d,d nor understohd the ph€nonenon conpteretr, According to Bruih, AtDatani,s compuradonal sork is o very elabo6re sysr€m of mrhehari€t catculalions (Bruin, | 977). t2
Page 1 and 2: ON I'HE COMPARATIYE STADY OF MATHEM
Page 3 and 4: CERTIFICATE c. incd thn th. ..ndidl
Page 5 and 6: Dedicated to my late mother Anwer S
Page 7 and 8: Abstract The pmbl.m of delemining f
Page 9 and 10: allow visibilil, ofthe cltsm wilnou
Page 11 and 12: ilf'LLf' c)'i ", ( ),1 -t z-r -t z,
Page 13 and 14: 't' f'L )) L 72' A ety { d-- w L,t
Page 15 and 16: Phyaic.l Modala & their Evolution 4
Page 17 and 18: moden (Reingold & Dschowttz, 2OO1).
Page 19 and 20: SuDpoe ('-.r",/.)!d (.,,r",t) E thc
Page 21 and 22: 5. 6, 1. E. 10. B€dTimeofvhibilil
Page 23 and 24: Thi! figue clorly 3howr th.! for hi
Page 25: and a.rc$cnr snh ldgc clone.lion $a
Page 29 and 30: Solsr Cal.nd6 m b4d on the ulrl mot
Page 31 and 32: Arnhn.tic calcndar ther. are cilh.r
Page 33 and 34: 8@rly vdied s6r lo s@n, 'fte t€en
Page 35 and 36: visul sighring epo.t must b. suppon
Page 37 and 38: visibility curve. So fd th. mod€l
Page 39 and 40: Chapter No, 2 ASTRONOMICAL ALGORITH
Page 41 and 42: O@ thc rim. of thc g.cdric biil of
Page 43 and 44: ine argunent md the explicii tin€
Page 45 and 46: Onc. lhe lincs of fting edor seltin
Page 47 and 48: L 2. L Pl&erary otbiB d. clliptic w
Page 49 and 50: BdicaUy a nhcory of pldelary ed lun
Page 51 and 52: $.t%'sr'(sfl + dI'.,) (r25) Tb vdB
Page 53 and 54: a - sdimjor uis of th. dbil l, = m@
Page 55 and 56: ,is = I +l8O! Fs= -e 12.2.2t) ond t
Page 57 and 58: F " M@'!.4ruqtof hind. - ! 6'0. ! a
Page 59 and 60: elled th. Julid Dare abb@ialed B JD
Page 61 and 62: How.r s noE @nhtlon l !'pNch ir ro
Page 63 and 64: Now orc n6ds ro c@t 6. nub.r of diy
Page 65 and 66: LMU-UMM stcFr2: alMM>2 { UW-C-1716
Page 67 and 68: DE to lh. nodotr of thc ob6*d. th.
Page 69 and 70: I. =6r4t-50'.54841+E tot84,.s12s66I
Page 71 and 72: t,-'i^,-"(t"*", \ 2t L,-ZA,s\8,+c,.
Page 73 and 74: The coftcctions for Abe@don md Nura
Page 75 and 76: FimUy the coordides ofthc Ss @ Edcd
Page 77 and 78:
Si"At = Sinpstnq +C6e,Cot6,CotH, (2
Page 79 and 80:
An |pDDxiE.r. valu. of "'*" wirh e
Page 81 and 82:
anlAt = c6(Ir)s,,{t) rz,(d)co{l) (2
Page 83 and 84:
"='"[ (289) Sinc. ihc acienl dd thc
Page 85 and 86:
9. JD of Conjunct, rh€ Julie Dat
Page 87 and 88:
h*.@rdin l€s @ c.bdat.d th. etr
Page 89 and 90:
TL dl irpde dgsitor tu d. ddroni4 m
Page 91 and 92:
In rhb ch.pter we stan wtth a b,t.f
Page 93 and 94:
wA diislaled by S. Oddz 6 Crlr of M
Page 95 and 96:
80 (r. I r)
Page 97 and 98:
for visibilny thftud rel.$ope. IlE
Page 99 and 100:
(3.2.2) (3.2.1) FDm ridcle sh, st n
Page 101 and 102:
fig. 1.2.2: G@tn r.y ofP@riv. Ag. .
Page 103 and 104:
usirg (3.2.l0) il tlr4 lat s into:
Page 105 and 106:
Sm p.r.[d lo lqldo. (difrcqE of ddr
Page 107 and 108:
sin rr'r _ rh& sn(e+a) @s(e+a) erhl
Page 109 and 110:
As mc ioned abov. dE Mulin sao&Ders
Page 111 and 112:
vi€w of $udying ihe behaliour of
Page 113 and 114:
. yomg.r ct!.c. (f8; 3J.1). Tb nioi
Page 115 and 116:
'Ihe table in App€n.tix'U shos S.
Page 117 and 118:
All lhe* conditions falolr rhe lisi
Page 119 and 120:
''if h. nns tizt b.nv.r -3,t otd 0.
Page 121 and 122:
I 4 15 't0 0 loo zlo 3b 400 ^ actua
Page 123 and 124:
All thes. figu6 show ! lend for lhe
Page 125 and 126:
In cae ofMoon nonh ofediflh: DAZ !
Page 127 and 128:
69{0. Fo(brilgb.(l (t910) hs &!.rd,
Page 129 and 130:
w. d.firc a pa@et . !r , .,visibihy
Page 131 and 132:
fhis nbdi6c.tio! i! rn oL by Fo$dnB
Page 133 and 134:
35 30 25 20 15 10 5 0 -10 . hisible
Page 135 and 136:
:i:: alF{s_{";+E:'-.-l {-: 'l Fia.
Page 137 and 138:
of Fsnivo 3igidier dwidirB froo rbd
Page 139 and 140:
(54.lYo). Howv.r, in t m of th. su.
Page 141 and 142:
e{inctton .nd sky briShtn* ds€ to
Page 143 and 144:
cxnibircd in n8 9 ir is mr.d lhnl a
Page 145 and 146:
For thimer c6c.nr rh€ ,r + s aSai
Page 147 and 148:
t32
Page 149 and 150:
tJ4
Page 151 and 152:
1lz Our ofth. 196 positiv€ sighri
Page 153 and 154:
' = ls
Page 155 and 156:
i { Fig No 4'2 2 lD addirion 10 @Nd
Page 157 and 158:
In all crloldjoE in rhis rcrk rhe c
Page 159 and 160:
@l@) of lhb wolk lb@ e 33 .M our of
Page 161 and 162:
4J SCHAEFFER'S LIMITING MAGNITUDE M
Page 163 and 164:
i = l,.cor ,+ 1i - l L I+441 (4 t.2
Page 165 and 166:
,.tdchltl = K, r ^ s + K,' x, + Ko'
Page 167 and 168:
Thdhold conExl vr| f.e.ltld. conbtt
Page 169 and 170:
TheE is only onc noc clain ofnajcd
Page 171:
t56
Page 174 and 175:
4,4 A NEW CRITERION TOR NEW CRtrSCE
Page 176 and 177:
Alttudo v.6 Sohr Oottesalon 19 13
Page 178 and 179:
l6l
Page 180 and 181:
l Calculate r-vdu. (or vp @odins ro
Page 182 and 183:
Tl|rcforc, if Fothcdnghd's criGdon
Page 184 and 185:
obsMriom bul lcs sue.senn in @mpdie
Page 186 and 187:
may 6ose that nay b. lhe ay€rlse
Page 188 and 189:
criterion or our s-value (or Q&K) c
Page 190 and 191:
fomulE. by Nunins sphdicEl 6gL at Q
Page 192 and 193:
s.lld width eould nor b. visibl.. A
Page 194 and 195:
IE it ir ma dly lh koglh of.ilsc.d
Page 196 and 197:
200 F 180 fi reo 3ro Era i roo o80
Page 198 and 199:
T.bl. 5.1.1 {Continued r8l
Page 200 and 201:
Tne nodel dcvcloped in lhis e for b
Page 202 and 203:
AII the* fout deviati.S cdes ae pho
Page 204 and 205:
Tlre modcl for rhe calculation of c
Page 206 and 207:
6.2 LUNAR CALENDAR FOR PAKISTAN Mos
Page 208 and 209:
Out of rhe 95 public obsesalioDs re
Page 210 and 211:
lhe lmd ft*e.t. C€ncrally thc crn
Page 212 and 213:
oftr\i'i$ lbir tNinun rcldirio! @!H
Page 214 and 215:
1..2) dd is ns applicarion is srudi
Page 216 and 217:
", =(^* - l- # -e4,:!.,'l) f " v, =
Page 218 and 219:
(escenr $idr varying wea$cr condili
Page 220 and 221:
APPEDNIX-I COMPUTER PROGRAM HILALOl
Page 222 and 223:
gotox(52,12):p.i {"Ase = '),sororr(
Page 224 and 225:
i(ldltohonthdayslin(lnonlhl.0)l) {
Page 226 and 227:
) gobxy(2,21);cout
Page 228 and 229:
{ fs@f(fpt'%lf yolf y.lf,&sx,&bx,&c
Page 230 and 231:
+alp3'pow(!4)/100000000.0)rPI/180.0
Page 232 and 233:
mcela=sin(!lar*PI/180.0)'sin(mdclia
Page 234 and 235:
forg nodf6c(160.lo8+390.67050284rto
Page 236 and 237:
t ning-no; , void iE_tordvoio { i(n
Page 238 and 239:
t ) iqbou
Page 240 and 241:
) nmFsinc(so 0/60 ofV l 80 0)- sin(
Page 242 and 243:
d.m=@s(ptar'PvlE00)'6(md't!prPvl80
Page 244 and 245:
) { ip nt(iprr4,""/od:%d{'/o8.3lo\t
Page 246 and 247:
APPENDI'(.U ANCIENT, MEDIEVAI, AND
Page 248 and 249:
23J
Page 250 and 251:
215
Page 253 and 254:
2]E
Page 255 and 256:
240
Page 258 and 259:
APPENDIX.III PHYSICAL MODELS 211
Page 260:
245
Page 263 and 264:
244
Page 265 and 266:
250
Page 267 and 268:
2r2
Page 269 and 270:
NB 91'! 254
Page 271 and 272:
E g t 2 I a 3 I E t c s i e : f *
Page 273 and 274:
E H I : f-1 i9 i P a E E E € g El
Page 275 and 276:
El lc a P I d t : a r' 5 5 t e 3 a
Page 277 and 278:
6 6 I a I n P F 3 E I 3 I n : 3 l :
Page 279 and 280:
APPENDIX-V F'UTURECALENDAR DATA FOR
Page 281 and 282:
: I f : t I I E 8 3 P a a B 3 E 3
Page 283 and 284:
=;::eifr ;Ci{{ i; si:€ ;e;ii i --
Page 285 and 286:
;;-i= BSxdgi:' 5i3;;n* ;i B99g .S ;
Page 287 and 288:
\ ;;g;F ;;;gis :E::sF F;i;€€ g;
Page 289 and 290:
Asbbrcok, J, 1972, "Moc rbour visib
Page 291 and 292:
4o0d AtuivNry, 1582n982". Ponrifica
Page 293 and 294:
47. H.pl@, B, 1984, Bi@rionil R.fld
Page 295 and 296:
66. India M.t@olo$/ DePdlrna! 195,
Page 297 and 298:
86. Moullon, F. R., 1914,'An Inrrcd
Page 299 and 300:
106. Roenlhd, 1952. "A History of M
Page 301 and 302:
126. ShaulGr K., w-mmBirhine.6m- !2
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