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 = 1 1 + ) n+1 ( 5 1 ) n+1 5 5 1 0 3 a n = a n 1 + a n ( a 1 = 1 a = ) 1 3 5 8 13 1 a 1 = 1 a = 1 1 1 3 5 1
(1) (Rhind Mathematical Papyrus) ( 1 pp.1011) 40 100 5 3 1 7 5 1 5 1 3 17 1 1 1 1 0 100 0 0 1 3 100 1 3 3 38 1 3 17 1 9 1 1 0 1 10 3 1 3 1 1 0 100 5 1 1 3 9 1 1 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 = 55 55 4 10 10 1 8 ( 1 p.11)
() (Pujagìrac (Pythagoras) : 57? 497?)) (Loukianos (Lucianos) : 10?180?) ( 7 pp.9091)?? (Nikìmaqoc (Nicomachus) : 0?10?) (Introductio Arithmetica) ( p.35) 3 3 1 ( ) 1 1 ( ) ( ) ( ) ( ) 3 1 3 (trðgwnoi) (tetrĺgwnoi) (pentĺgwnoi) (áxĺgwnoi) (áptĺgwnoi) 1 3 10 1 n 1 n(n + 1) 3
( 7 pp.9899) 1 3 10 15 1 8 3 45 55 1 4 9 1 5 3 49 4 81 100 1 5 1 35 51 70 9 117 145 1 15 8 45 91 10 153 190 1 7 18 34 55 81 11 148 189 35 (EÎkleÐdhc (Eukleides : Euclid) : 300 ) (StoiqeÐwsic) 9 8 ( 3 pp.007) 9 8 3 4 7 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
( 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 1 1 1 3 18 54 1 48 1458 5
(>Arqimădhc (Archim ed es) : 87? 1) (Quadrature of the Parabola) 1 4 ( 4 p.418) 4 1 3 4 3 ( 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 } 4 1 1 + 1 ( ) n 1 1 3 a 1 4 4 = 4 3 a1 4 ( ) n 1 a 1 + 1 ( ) n 1 1 a 1 = 4 3 4 3 4 3 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 + (1 + + + n) = 3(1 + + + n ) 3 3 1 + + + n = 1 { (n + 1)n + (1 + + + 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{1 + + + (n 1) } < n 3 < 3(1 + + + n ) 1
(3) 3 ( : ) 9 3 ( 5 p.118) < > ( ) ( ) ( ) ( ) ( ) 1 ( 5 pp.118119) ( ) ( ) 3 1 ( ) 7
< > ( ) ( ) ( 403 1) ( ) 5 : 4 : 3 : : 1 5a 4a 3a a a 5a + 4a + 3a + a + a = (5 + 4 + 3 + + 1)a = 5 (5 + 4 + 3 + + 1) = 15 15a = 5 a = 1 3 5 5 0 15 10 5 5 0 15 10 15 15 15 15 5 15 4 ( 5 p.10) 3 4 < > 4 4 a 5 50 (5 = 50 ) 5 8
(4) ( Aryabhaṭa : 47?550?) ( Aryabhaṭ ya : 499?) ( pp.103104) 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.104105) 1 n k j = k=1 j=1 n k=1 k(k + 1) n k n(n + 1)(n + 1 + 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 }
(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) 5 1 13 10
14 1 18 ( 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 18 18 ( ) ( ) ( ) ( ) ( pp.970) n = 1 ( ) 30 + 3 3 + = 1 ( 1440 + 4 3 + 1 ) = 1 (38 ) = 18 11
( ) ( p.70) 130 ( ) ( ) ( ) ( ) ( ) ( ) ( ) ( ) r n a = r = 3 10 3 10 10 5 4 1 0 1 1 10 5 4 1 0 59049 43 81 9 3 ( ) (1) ( 1) 3 10 = 59049 S = (59049 1) (3 1) = (59049 1) 3 1 = 59048 S = a(rn 1) r n r 1 3 10 = (3 5 ) = (3 3 4 ) = { 3 (3 ) } 1 3 3 (3 ) 3 (3 ) { 3 (3 ) } 1 ( 1 ) ( p.71) 18 1
(5) (Fibonacci : 1174?150?) (Liber Abaci : 10 ) ( 9 pp.404405) 1 1 ( ) 1 1 ( ) ( 1 1 1 ) 1 1 3 1 3 1 5 1 1 3 4 f n f n 1 f n f n = f n 1 + f n (n 3) {f n } 1 3 4 5 1 3 5 8 13 1 7 8 9 10 11 1 34 55 89 144 33 377 1 377 {f n } 13
( ) F 1 = 1 F = 1 (F n 1 1 3 5 8 13 ) (i) n m p F n F m F p (ii) F 1 + F + F 3 + + F n = F n+ 1 1 + 5 = 1.18033988 AB AP : PB = PB : AB AB AP : PB AP = 1 PB = x 1 : x = x : (1 + x) x = x + 1 x = 1 ± 5 1 + 5 5 : 8 (Khufu : ( B.C.553?-B.C.530?)) 14.59 m ( 138.74m) 30.37 m ( 1.57) ( 51.84 ) (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 1 1977 ( 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 003 10 J. ( ) 005 ( 17) 11 I A ( 1) 1997 ( 9) 1 004 ( 1) 14