14 : n (1) n a n a n (2) a n n (1) 1 (n 1) a n 1 2 (n 2) a n 2 2 n 3 a n = a n 1 + a n 2 a 1 = 1 a 2 = 2 (2) a n = a n 1 + a n 2 ( ) ( a n = 1 1

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1 14 : n 1 1 (1) n a n a n () a n n (1) 1 (n 1) a n 1 (n ) a n n 3 a n = a n 1 + a n a 1 = 1 a = () a n = a n 1 + a n ( ) ( a n = ) n+1 ( 5 1 ) n a n = a n 1 + a n ( a 1 = 1 a = ) a 1 = 1 a =

2 (1) (Rhind Mathematical Papyrus) ( 1 pp.1011) a d 1 {(a + d) + (a + 3d) + (a + 4d)} = a + (a + d) 7 a + (a + d) + (a + d) + (a + 3d) + (a + 4d) = 100 a = 5 3 d = ( 1 p.11)

3 () (Pujagìrac (Pythagoras) : 57? 497?)) (Loukianos (Lucianos) : 10?180?) ( 7 pp.9091)?? (Nikìmaqoc (Nicomachus) : 0?10?) (Introductio Arithmetica) ( p.35) ( ) 1 1 ( ) ( ) ( ) ( ) (trðgwnoi) (tetrĺgwnoi) (pentĺgwnoi) (áxĺgwnoi) (áptĺgwnoi) n 1 n(n + 1) 3

4 ( 7 pp.9899) (EÎkleÐdhc (Eukleides : Euclid) : 300 ) (StoiqeÐwsic) 9 8 ( 3 pp.007) A B G D E Z 3 B 4 G 7 Z A B G D E Z A A B A A B A A A B A A B B B G D B D Z 4 G A B G A B G A A B G A A B G A B B G G G D E Z G Z 7 4

5 ( 3 pp.45) 9 35 A A BG D EZ BG EZ A BH ZJ HG A EJ A BG D A B H G D E L K J Z ZK BG ZL D ZK BG ZJ BH JK HG EZ D D BG BG A D ZL BG ZK A ZJ EZ ZL LZ ZK ZK ZJ EL LZ LK ZK KJ ZJ KJ ZJ EL LK KJ LZ ZK JZ KJ GH ZJ A LZ ZK JZ D BG A GH A EΘ BG A {a n } a 1 : a = a : a 3 = a 3 : a 4 = = a n : a n+1 (a a 1 ) : a 1 = (a n+1 a 1 ) : (a 1 + a + + a n ) a 1 + a + + a n = a 1(a n+1 a 1 ) a a 1 1 a r a 1 + a + + a n = a(arn a) = a(rn 1) ar a r

6 (>Arqimădhc (Archim ed es) : 87? 1) (Quadrature of the Parabola) 1 4 ( 4 p.418) ( 3) 1 4 {a n} (a 1 +a + +a n )+ 1 3 a n = 4 3 a 1 (a 1 + a + + a n ) + 1 { ( 3 a n = a1 1 1 ) n } ( ) n a = 4 3 a1 4 ( ) n 1 a ( ) n 1 1 a 1 = a1 (On Spirals) ( 8 p.1) a 1 a a 3 a n a 1 n (n+1)(a n ) +a 1 (a 1 +a + +a n ) = 3{(a 1 ) +(a ) + +(a n ) } ( 10) {a n } a d = a (> 0) a k = a + (k 1)a = ka (n + 1)(na) + a(a + a + + na) = 3{a + (a) + + (na) } a (n + 1)n + ( n) = 3( n ) n = 1 { (n + 1)n + ( n) } 3 = 1 {(n + 1)n + 1 } 3 n(n + 1) = 1 n(n + 1)(n + 1) 3 3{(a 1 ) + (a ) + + (a n 1 ) } < n(a n ) < 3{(a 1 ) + (a ) + + (a n ) } 3{ (n 1) } < n 3 < 3( n ) 1

7 (3) 3 ( : ) 9 3 ( 5 p.118) < > ( ) ( ) ( ) ( ) ( ) 1 ( 5 pp ) ( ) ( ) 3 1 ( ) 7

8 < > ( ) ( ) ( 403 1) ( ) 5 : 4 : 3 : : 1 5a 4a 3a a a 5a + 4a + 3a + a + a = ( )a = 5 ( ) = 15 15a = 5 a = ( 5 p.10) 3 4 < > 4 4 a 5 50 (5 = 50 ) 5 8

9 (4) ( Aryabhaṭa : 47?550?) ( Aryabhaṭ ya : 499?) ( pp ) 19 ( ) ( ) ( ) ( ) {a n } a 1 = a d p + 1 p + n n S p,n {( ) } { ( ) } n 1 n 1 S p.n = + p d + a n = n a + + p d p + 1 a p+1 = a + pd p + n a p+n = a + (p + n 1)d S p,n = n {(a + pd) + (a + (p + n 1)d)} = n (a + nd d + pd) = n (a + n d 1 ) { ( ) } n 1 d + pd = n a + + p d p = 0 p + 1 p + n p + 1 a p+1 = a + pd d n S p,n = S p+n S p 1 ( ) ( ) ( ) ( ) ( p.104) ( ) ( ) ( ) ( ) ( ) ( pp ) 1 n k j = k=1 j=1 n k=1 k(k + 1) n k n(n + 1)(n n) = k=1 = n(n + 1)(n + ) = n(n + 1)(n + 1) 9 = (n + 1)3 (n + 1) n { n(n + 1) k 3 = k=1 }

10 (L l avat : 1150 ) (Bh askara : 1114?1185?) 117 ( ) 119 ( ) ( p.) ( ) ( p.) 117 ( ) ( p.) ( p.7) 11 ( ) ( ) a 1 = a d n S n = S a n = (n 1) d + a = a + (n 1)d = (a n + a) = a 1 + a n = 1 {a + (n 1)d} S = n = n {a + (n 1)d} 1 ( ) ( p.7) 13 ( pp.78)

11 ( p.8) ( ) ( p.9) ( ) ( ) ( ) ( ) ( p.9) 14 a = S (n 1)d n ( ) S 1 d = n a n 1 18 n = 1 ( ) d ds + d a a + d S = n na = S n(n 1)d n(n 1)d {a + (n 1)d} = na + n(n 1)d = S na 14 (n 1)d = S n a 1 S = na + n(n 1)d dn + (a d)n S = 0 n = (a d) ± (a d) 4 d ( S) ( d ) = 1 d a + d ± 1 4 (a d) + ds a + d < 1 4 (a d) + ds ( ) ( ) ( ) ( ) ( pp.970) n = 1 ( ) = 1 ( ) = 1 (38 ) = 18 11

12 ( ) ( p.70) 130 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r n a = r = ( ) (1) ( 1) 3 10 = S = ( ) (3 1) = ( ) 3 1 = S = a(rn 1) r n r = (3 5 ) = (3 3 4 ) = { 3 (3 ) } (3 ) 3 (3 ) { 3 (3 ) } 1 ( 1 ) ( p.71) 18 1

13 (5) (Fibonacci : 1174?150?) (Liber Abaci : 10 ) ( 9 pp ) 1 1 ( ) 1 1 ( ) ( ) f n f n 1 f n f n = f n 1 + f n (n 3) {f n } {f n } 13

14 ( ) F 1 = 1 F = 1 (F n ) (i) n m p F n F m F p (ii) F 1 + F + F F n = F n = AB AP : PB = PB : AB AB AP : PB AP = 1 PB = x 1 : x = x : (1 + x) x = x + 1 x = 1 ± : 8 (Khufu : ( B.C.553?-B.C.530?)) m ( m) m ( 1.57) ( ) (å Łkroc kaì mèsoc lìgoc) ( 3) 30 ( 3 p.117 p.145) 5 1 : ( ) ABCD AB 1 E BC F CB E EA PQFB BF 1 AP : PB DC : CF D C E A B P F Q ( 5) ( I) 1979 ( 54) 3 ( ) 1971 ( 4) 4 ( ) ( 9) 197 ( 47) 5 ( ) ( ) 1980 ( 55) ( ) ( 1) 1980 ( 55) 7 Ivor Thomas(transl.) Greek Mathematical Works I Harvard U.P.(Loeb Classical Library) 1939(1991) 8 T. L. Heath The Works of Archimedes Dover 1953(00) 9 L. Sigler Fibonacci's Liber Abaci Springer J. ( ) 005 ( 17) 11 I A ( 1) 1997 ( 9) ( 1) 14

) 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 =

) 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 = 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