) 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 =

Size: px

Start display at page:

Download ") 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 ="

なつきはかまや
5 years ago
Views:

1 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 : : 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)

2 ) 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

3 ) Claudios Ptolemaios Ptolemaĩoc) : 85?65?) Ptolemy) 7 5 lmagest) Majhmatik hc SuntĹxewc) magiste Hipparchus VIpparqoc) : 90? 0?) α Crd α 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

4 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

5 ) R = 60 i) Crd 60 = 60 Crd 80 = 0 ii) R iii) 36 7 Crd 90 = R = O O E O E = EF OF = Crd 36 F = Crd 7 OF F OE = 5 R EF = E = R 5 OF = EF OE = R = 30 5 ) 0 5 F = OF + O F = OF = Crd 36 = 30 5 ) F = Crd 7 = F O E R = iv) Crd 08 = 4400 Crd 7 ) = ) = ) = Crd 0 = 4400 Crd 60 ) = = 0800 = Crd = Crd7 60 ) = Crd 0 Crd 7 Crd 08 Crd 60 0 =

6 v) 8 Crd 44 = 4400 Crd 36 ) = 4400 {30 5 )} = ) = ) Crd 8 ) = Crd 36 = 600 Crd 44 ) = ) vi) Crd 8 = ) = 60 Crd 30 = Crd 60 = 600 Crd 0 ) = = Crd 45 = Crd 90 = 600 Crd 90 ) = = = ) ) Crd.5 Crd 68 = 4400 Crd ) ) Crd 6 = Crd = 600 Crd 68 ) ) Crd 74 = 4400 Crd 6 ) ) Crd 3 = Crd 6 = 600 Crd 74 ) ) Crd 77 = 4400 Crd 3 ) ) Crd.5 = Crd 3 = 600 Crd 77 ) ) Crd 3 Crd

7 Kanìnion t wn ân kôklú eîjei wn) 3 pp.69) /30 /

8 Crd 6 = Crd 6 = sin θ = Crd θ = Crd θ R 0 sin 45 = 0 Crd 90 = ) Crd sin α ) ) Crd 5 Crd.5 Crd 75 Crd 50 sin 60 8

9 3) ryabhaṭa : 476?550?) ) ryabhaṭ ya) ) ) 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 = cd 6 O = 60 = R = 3438 OH = 30 H = = R Sin 30 = R = 79 9

10 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) = Sin α) Sin 60 = Sin90 30 ) = 3438 Sin 30 ) = 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 + = ) ) = = 79 Sin 30 = 79 Cos 30 = Sin 60 = 79 3 ) Sin 5 = Sin Cos 30 ) Sin 30 = + = ) ) = 79 3 = 79 3 )

11 ) ) ) ) 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 = 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

12 T i 6 p.93) 0 4 ) ) ) ) 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 S Sin 3.75 = 5 0 Sin 7.5 = S = T + S = = T 3 = T 3 = T S = S 5 = Sin.5 = S 3 = T 3 + S = = T 4 = T 3 S 3 = 4995 = S Sin 5 = S 4 = T 4 + S 3 = = S = 5 T =4 S = 449 T 3= S 3 = 67 T 4=9 S 4 = 890 T 5= Sin 90 = 3438 Sin 30 = 79 S i T i

13 4) al- r un : ?) 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 = Ya`q ub al-sijz ) i) 9 3 ii) 8 iii) 30 i) 8 pp ) r r

14 a) b) 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

15 3 x 3 3x = 0 Gerolamo Cardano : 50576) + 3 y 3 x = y = x 3 3x x x ; 5, 45, 47, x) m = m C n x n m C n = n=0 x = 3 i 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 i ) m = 3 3 i ) n 3 i ) 4 + = i) i) 3 i i x E 9 3 = x + x + x ) ) 9 : 3 = E : = : E = y y : 3 ) = y : 3 = : y ; 4,, 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

16 6 = d 8 = d 0 = m 7 9 ) d w 4 d ) w 4 4 ) ;,, 49, 5, y 40 y = sin = sin al-khw arizm : 850 ) Sin θ 3 ) ) 9 ) 97 47) ) 97 46) 3 ) ) ) 4. ) 97 46) ) 6 ) ) ) 7 55) 993 5) 8 ) ) 987 6) ) 6

( ( 3 ( ( 6 (

( ( 3 ( ( 6 ( ( ( ( 43037 3 0 (Nicolas Bourbaki (Éléments d'histoire des athématiques : 984 b b b n ( b n/b n b ( 0 ( p.3 3500 ( 3500 300 4 500 600 300 (Euclid (Eukleides : EÎkleÐdhc : 300 (StoiqeÐwsic 7 ( 3 p.49 (