0 sin cos tan 3 θ θ y P c a r sin θ = a c = y r θ b C O θ x cos θ = b c = x r tan θ = a b = y x ristarchus >rðstarqoc) : 30? 30?) PerÐ megejÿn kai aposthmĺtwn HlÐou kai Selănhc : On the Sizes and istances of the Sun and Moon) 8 0 7) 9 : 3 43 : 6 5) 08 : 43 60 : 9 7) pp.50754),39,000 km,756 km 3,474 km 49,597,870 km 384,400 km : r = ) R) R = 60 R = 3438 Sin α = R sin α) Cos α = R cos α)
) Euclid Eukleides : EÎkleÐdhc) : 300 ) StoiqeÐwsic) p.4647) ΑΒΓ ΒΑΓ ΓΑ Β ΒΓ ΑΓ ΓΑ Α G G G G G G G G G G G G G G G G ΑΒΓ ΒΑΓ = θ ΒΓ = a ΑΓ = b = c Α = c cos80 θ) = c cos θ a = b + c + b c cos θ) = b + c bc cos θ 3 ΑΒΓ ΒΓ Α ΑΓ ΓΒ ΓΒ Β G 3 ΑΒΓ = θ ΒΓ = a ΑΓ = b = c 3
) Claudios Ptolemaios Ptolemaĩoc) : 85?65?) Ptolemy) 7 5 lmagest) Majhmatik hc SuntĹxewc) magiste Hipparchus VIpparqoc) : 90? 0?) α Crd α 0 30 80 30 α α Crd α ) R = 60 Crd 80 ) 0 Crd 60 ) 60 Crd α Crd α = R sin α 3 p.0) C C E C + C = C CE S E = C E C = CE E S C C = E 3
3 pp.0) ) C O = α CO = β = Crd α C = Crd β C = Crdα β) = Crd80 α) C = Crd80 β) O Crd80 α) Crd β + Crdα β) R = Crd80 β) Crd α Crdα β) = Crd80 β) Crd α Crd80 α) Crd β R = Crd80 β) Crd α Crd80 α) Crd β 0 C C OC = CO = α/ E = CF CE O E F F = R E)/ = R )/ C S CF C = F C = RR ) Crd α ) = RR Crd80 α)) = 600 Crd80 α)) Crd Crd80 α) = 4R Crd α) = 4400 Crd α) C O E O = α OC = β = E = Crd α C = Crd β C = Crdα+β) = Crd80 α) CE = Crd80 β) C = Crd80 α+β)) 4 CE C E + E C = CE E C = CE C E 0 Crd80 α + β)) = Crd80 α) Crd80 β) Crd β Crd α Crd80 α + β)) = Crd80 α) Crd80 β) Crd α Crd β 0 4
) R = 60 i) 60 80 Crd 60 = 60 Crd 80 = 0 ii) 90 90 R iii) 36 7 Crd 90 = R = 60 84.858374 O O E O E = EF OF = Crd 36 F = Crd 7 OF F 3 9 0 OE = 5 R EF = E = R 5 OF = EF OE = R = 30 5 ) 0 5 F = OF + O F = OF = Crd 36 = 30 5 ) 37.080393 F = Crd 7 = 30 0 5 70.534308 F O E R = 30 0 5 iv) Crd 08 = 4400 Crd 7 ) = 4400 30 0 5) = 4400 9000 5) = 5400 + 800 5 97.080393 Crd 0 = 4400 Crd 60 ) = 4400 60 = 0800 = 60 3 03.9304845 Crd = Crd7 60 ) = Crd 0 Crd 7 Crd 08 Crd 60 0 = 60 3 30 0 5 5400 + 800 5 60 0.5434559 5
v) 8 Crd 44 = 4400 Crd 36 ) = 4400 {30 5 )} = 4400 9006 5) = 9000 + 800 5 4.67896 ) Crd 8 ) = Crd 36 = 600 Crd 44 ) = 60 0 9000 + 800 ) 5 35.3930867 vi) Crd 8 = 60 0 9000 + 800 ) 5 8.773580 30 = 60 Crd 30 = Crd 60 = 600 Crd 0 ) = = 60 3 3.058854 Crd 45 = Crd 90 = 600 Crd 90 ) = = 60 45.9088 45 = 90 600 60 3) 600 60 ) 6 3.5 Crd.5 Crd 68 = 4400 Crd ) 4400.5434559) 9.346744 Crd 6 = Crd = 600 Crd 68 ) 600 9.346744) 6.803477 Crd 74 = 4400 Crd 6 ) 4400 6.803477) 9.8355447 Crd 3 = Crd 6 = 600 Crd 74 ) 600 9.8355447) 3.43380 Crd 77 = 4400 Crd 3 ) 4400 3.43380) 9.9588790 Crd.5 = Crd 3 = 600 Crd 77 ) 600 9.9588790).570754 Crd 3 Crd.5.047676 6
Kanìnion t wn ân kôklú eîjei wn) 3 pp.69) /30 /30 0 30 0 3 5 0 50... 0 50 0 50 64 0 8 49 56 0 0 8 37 30 34 5 0 50 64 30 8 54 5 0 0 8 0 0 5 40 0 50 65 0 8 58 5 0 0 8 4 30 37 4 0 48 65 30 9 6 0 0 7 48 3 0 3 8 8 0 48 66 0 9 6 0 0 0 7 3 3 30 3 39 5 0 48 66 30 9 0 6 0 0 7 5 4 0 4 6 0 47 67 0 9 3 44 0 0 6 59 4 30 4 4 40 0 47 67 30 9 7 3 0 0 6 4 5 0 5 4 4 0 46 68 0 9 0 34 0 0 6 6 5 30 5 45 7 0 45 68 30 9 3 47 0 0 6 0 6 0 6 9 49 0 44 69 0 9 6 5 0 0 5 54 6 30 6 48 0 43 69 30 9 9 49 0 0 5 37 7 0 7 9 33 0 4 70 0 9 3 37 0 0 5 0 7 30 7 50 54 0 4 70 30 9 35 7 0 0 5 4 8 0 8 5 0 40 7 0 9 37 49 0 0 4 48 8 30 8 53 35 0 39 7 30 9 40 3 0 0 4 3 9 0 9 4 5 0 38 7 0 9 4 9 0 0 4 4 9 30 9 56 3 0 37 7 30 9 44 36 0 0 3 58 0 0 0 7 3 0 35 73 0 9 46 35 0 0 3 4 0 30 0 58 49 0 33 73 30 9 48 6 0 0 3 6 0 30 5 0 3 74 0 9 50 9 0 0 3 9 30 0 30 74 30 9 5 43 0 0 53 0 3 36 0 8 75 0 9 53 0 0 0 36 30 3 3 50 0 7 75 30 9 54 7 0 0 0 3 0 3 35 4 0 5 76 0 9 55 38 0 0 3 3 30 4 6 6 0 3 76 30 9 56 39 0 0 47 4 0 4 37 7 0 77 0 9 57 3 0 0 30 4 30 5 8 38 0 9 77 30 9 58 8 0 0 4 5 0 5 39 47 0 7 78 0 9 58 55 0 0 0 57 5 30 6 0 56 0 5 78 30 9 59 4 0 0 0 4 6 0 6 4 3 0 3 79 0 9 59 44 0 0 0 5 6 30 7 3 9 0 0 79 30 9 59 56 0 0 0 9... 80 0 0 0 0 0 0 0 0 7
Crd 6 = 6 9 49 Crd 6 = 6 + 9 60 + 49 3600 6.330778 sin θ = Crd θ = Crd θ R 0 sin 45 = 0 Crd 90 = 84 + 5 0 60 + 0 ) 0.7070648 3600 Crd 90 84 5 0 sin 45 0.7070678 α 60 3 30 0 5 5400 + 800 5 60 0 8 60 0 9000 + 800 ) 5 3 36 8 46 9 30 60 3 3 3 30 36 30 5 ) 37 4 55 45 60 45 55 9 60 60 60 0 0 7 30 0 5 70 3 3 90 60 84 5 0 08 5400 + 800 5 97 4 56 0 60 3 03 55 3 44 9000 + 800 5 4 7 37 80 0 0 0 0 Crd 5 Crd.5 Crd 75 Crd 50 sin 60 8
3) ryabhaṭa : 476?550?) 499 50 ) ryabhaṭ ya) 4 33 30550) ) 6 pp.9800) 9cd ) ) ) ) ) ) ) ) ) ) ) R H Sin α = R sin α) jy a) O α H P H jy ardha, jiba) k aṣṭha, c apa) HP iṣu, sara) R = 3438 9cd 6 O = 60 = R = 3438 OH = 30 H = = R Sin 30 = R = 79 9
I P OP = OX = R P H = OI = Sin α OH = IP = Sin90 α) = Cos α O a a H K Q X Sin90 α) = Cos α = R Sin α) = 89844 Sin α) Sin 60 = Sin90 30 ) = 3438 Sin 30 ) = 79 3 977.395338 P K = KX = Sin α HX = R Cos α P X = HX + P H = R Cos α) + Sin α) Sin α R Cos = P K = α) + Sin α) = R Cos α ) ) Sin α + = 3438 Cos α ) ) Sin α + Sin 90 = R = 3438 Cos 90 = 0 Sin 45 = Sin 90 = 3438 Cos 90 ) ) Sin 90 + = 3438 0 43.03337 ) ) 3438 + = 79 + 79 = 79 Sin 30 = 79 Cos 30 = Sin 60 = 79 3 ) Sin 5 = Sin 30 3438 Cos 30 ) Sin 30 = + = 3438 79 ) ) 3 79 + = 79 3 = 79 3 ) 889.8987706 0
) ) ) ) n K i+ N i+ K i N i i+ i+ C i i i+ H i+ S i+ = Sin α i+ i H i S i = Sin α i T i+ = S i+ S i O 0 H i+ H i M i+ M i i S i T i = S i S i i R 4 n 0 n n i i+ = i i+ = a i OM i+ = O i K i = α i i O i+ = α i+ α i i+ OM i+ = α i + α i+ α i C i i+ = 80 α i+ α i ) C i+ i = 90 90 α i+ = α i+ + α i α i = 90 α i+ α ) i C i+ i = i+ OM i+ i i+ C i+ C : i i+ = OM i+ : O i+ = α i+ + α i S α i i+ OM i+ T i+ = i+ C = i i+ OM i+ = a O i+ R OM i+ i i+ i OH i i : i i+ = i H i : O i S M i M i+ = i = i i+ O i i H i = a R S i T i T i+ = a R OM i a R OM i+ = a R OM i OM i+ ) = a a R M im i+ = R ) Si T i T i+ a ) = S i R T i T i+ S i = T T S T i+ = T i T T ) S i S
T i 6 p.93) 0 4 ) ) ) ) 5 4 9 5 0 05 99 9 83 74 64 54 4 3 9 06 93 79 65 5 37 7 T = 5 T = 4 T T = 5 4 = T i+ = T i S i S 7 T i+ = S i+ S i S = T + S 0 = 5 + 0 = 5 S 4 90 4 3.75 Sin 3.75 = 5 0 Sin 7.5 = S = T + S = 4 + 5 = 449 0 T 3 = T 3 = T S = 4 449 S 5 = 4995.00444444 5 Sin.5 = S 3 = T 3 + S = 4995 + 449 = 50976 67.00444444 5 5 50976 5 T 4 = T 3 S 3 = 4995 = 087999 S 5 5 5065 0 Sin 5 = S 4 = T 4 + S 3 = 087999 + 50976 5065 5 890.066469 9.0047 = 45057599 5065 0 S = 5 T =4 S = 449 T 3= S 3 = 67 T 4=9 S 4 = 890 T 5=5 3.75 Sin 90 = 3438 Sin 30 = 79 S i T i
4) al- r un : 973048?) bu Rayḥ an Muḥammad ibn ḥmad al- r un al-q an un al-mas` ud f -l-hay'a wa-l-nuj um : 030 ) 030 sulṭ an ) Mas` ud : 03004) 3 ) ) ) /3) Crd.5 4 9 3 3 4 = Ya`q ub al-sijz ) 9 9 3 i) 9 3 ii) 8 iii) 30 i) 8 pp.349375) r 3 0 3 r 9 3 9 9 3
a) 9 0 4 b) 9 0 0 5 6 4 9 Z E W E L Z L M G H T ) EZ E L ) L = LE + EZ ) E EZ + E = rchim ed es >rqimădhc) : 87? ) 3 EM E = EM ) G E W Z H T 9 EZ E L L = LE + EZ E EZ LE = LE = LM M M = EZ EL 3 EL = 60 LE = LM E = EZ = x M = EZ E = EZ + EM = x + ) = E EZ + E = x + )x + = x + x + ) E + G = G G = E = G = EZ = x = {G G )/E} = x ) = x 4 x + x 0 x + x + = x 4 x + x 3 3x = 0 3 x 3 4
3 x 3 3x = 0 Gerolamo Cardano : 50576) + 3 y 3 x = 3 + 3 y = x 3 3x x x.8793854578677 ; 5, 45, 47,3 0.34796355.53088886 + x) m = m C n x n m C n = n=0 x = 3 i + 3 3 i) = n=0 3 C n 3 i ) n = 3 i ) 9 mm )m ) m n )) n! n 5 3n 4) ) 3 n n! n=0 ) 5 3 i + 8 3 i ) 3 0 43 m = 3 3 i ) n 3 i ) 4 + = + 3 + 3 3 i) + 3 3 i) 3 i i x E 9 3 = x + x + x.8793854578677 ) ) 9 : 3 = E : = : 6.44747809778 3 E = y y : 3 ) = y : 3 = : 6.44747809778 y 0.684040866533747 9 0 ; 4,, 3, 4, 55 0.68404073 3 3 4 ummah at al-awt ar) 0 d = d 3 = d d + d ) 4 = 5 = d m + ) ) d 5d m = 6 d 4 5
6 = d 8 = d 0 = m 7 9 ) d w 4 d ) w 4 4 ) 3 9 4 0 4 0 ;,, 49, 5, 35 0.074530547 9 y 40 y = sin 0 0.684040866533747 0.68404073 = sin 0.5 0.0745307099674787 840 al-khw arizm : 850 ) Sin θ 3 ) ) 9 ) 97 47) ) 97 46) 3 ) ) 958 33) 4. ) 97 46) 5 977 5) 6 ) ) 980 55) 7 55) 993 5) 8 ) ) 987 6) 9 977 5) 6