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1 : π ( )

2 TOSM kanie/tosm/ HP kanie/agora/ kanie/ kanie/jaise/ V.I. ( ) 1985 V.I (1997.1),

3 (1793) [7] 7 1, = = + 6 9, 9 6 = , 6 3 = 0 3 (1) = = = = = = /4 3

4 ( (1748) 38) /4 1 1/ = = = /1 = = = (4000/400) = =

5 3 : (1) ( ) () q 0 + p 1 q 1 + p q + p 3 q p 1 /q 1, p /q, p 3 /q 3,... p k = 1 i q i q 0 [q 0 ; q 1, q, q 3, q 4, ] ( ) ( ) (1) ( p i = 0 ) α = = = = = α = α 1 = (α 1)(α + 1) = 1 5

6 1 α 1 = 1 + α α = α α = (1 + 1 α = α ) 1+α ( α = α ) 1 1+α α (3) = (157) [1; 1, 1, 1, 1, ] (4) = = = γ γ = γ + 1 β = 3 β = 3 3 = [1; 1,, 1,, 1,, ] = β 1 = (β 1)(β + 1) = β 1 = 1 + β β = β β = (1 + 1+β ) = β 3 6

7 β = β = β 1 + β β β > 0 [β] {β} 0 < {β} < 1 1/{β} > 1 {β} 1 + β β = β = 1 + (1 + 1+β ) = β 1 + β = β = β 1+β 5 n 30 n 5 = [; 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4, 4,...] 6 = [;, 4,, 4,, 4,, 4,, 4,, 4,, 4,, 4,, 4,,...] 7 = [; 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4, 1, 1, 1, 4,...] 8 = [; 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4, 1, 4,...] 10 = [3; 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6, 6,...] 11 = [3; 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6, 3, 6,...] 1 = [3;, 6,, 6,, 6,, 6,, 6,, 6,, 6,, 6,, 6,...] 13 = [3; 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6, 1, 1, 1, 1, 6,...] 14 = [3; 1,, 1, 6, 1,, 1, 6, 1,, 1, 6, 1,, 1, 6, 1,, 1, 6,...] 7

8 15 = [3; 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6, 1, 6,...] 17 = [4; 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8, 8,...] 18 = [4; 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8, 4, 8,...] 19 = [4;, 1, 3, 1,, 8,, 1, 3, 1,, 8,, 1, 3, 1,, 8,...] 0 = [4;, 8,, 8,, 8,, 8,, 8,, 8,, 8,, 8,, 8,...] 1 = [4; 1, 1,, 1, 1, 8, 1, 1,, 1, 1, 8, 1, 1,, 1, 1, 8,...] = [4; 1,, 4,, 1, 8, 1,, 4,, 1, 8, 1,, 4,, 1, 8,...] 3 = [4; 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8, 1, 3, 1, 8,...] 4 = [4; 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8, 1, 8,...] 6 = [5; 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10, 10,...] 7 = [5; 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10, 5, 10,...] 8 = [5; 3,, 3, 10, 3,, 3, 10, 3,, 3, 10, 3,, 3, 10,...] 9 = [5;, 1, 1,, 10,, 1, 1,, 10,, 1, 1,, 10,, 1, 1,, 10,...] 30 = [5;, 10,, 10,, 10,, 10,, 10,, 10,, 10,, 10,...] 5, 6, 3 δ = δ 4 = 3 = δ = δ, δ 1 = 6 = δ = δ 3 +δ + δ δ 1 + δ 3 δ = δ = δ 3 = δ = δ 3 = δ = δ = δ + δ = δ = δ 1, 1, 1, 4 8

9 x = 1 a + 1 a + 1 a + 1 a +..., y = 1 a + 1 b + 1 a + 1 b + 1 a +... x = 1 a + x, y = 1 a + 1 b + y x + ax 1 = 0, ay + aby b = 0 a + 4 a a b x =, y = + 4ab ab = a b + 4b/a b x a = x = a +4 a = +4 = 1 a = 1 x = a +4 a = x + 1 = 5+1 a = n x = 4n +4 n = n + 1 n n y a = 1, b = n y = b + 4b/a b = 4n + 8n n = n + n n 3, 8, 15, [q 0 ; q 1, q, q 3,...] k q 1, q,..., q k 1, q k = q 0 ( ) 4. 3 = [1; 3, 1, 5, 1, 1, 4, 1, 1, 8, 1, 14, 1, 10,, 1, 4, 1,, 3,, 1, 3, 4, 1, 1,, 14, 3, 1...] 3 3 = [1;, 3, 1, 4, 1, 5, 1, 1, 6,, 5, 8, 3, 3, 4,, 6, 4, 4, 1, 3,, 3, 4, 1, 9

10 4, 9, 1, 8...] 3 4 = [1; 1, 1,,, 1, 3,, 3, 1, 3, 1, 30, 1, 4, 1,, 9, 6, 4, 1, 1,, 7,, 3,, 1, 6, 1...] 3 5 = [1; 1,,, 4, 3, 3, 1, 5, 1, 1, 4, 10, 17, 1, 14, 1, 1, 305, 1, 1, 1, 1, 1, 1,,, 1, 3,...] 3 6 = [1; 1, 4,, 7, 3, 508, 1, 5, 5, 1, 1, 1,, 1, 1, 4, 1, 1, 1, 3, 3, 30, 4, 10, 158, 6, 1, 1,...] 3 7 = [1; 1, 10,, 16,, 1, 4,, 1, 1, 1, 3, 5, 1,, 1, 1,, 11, 5, 1, 3, 1,, 7, 4, 1, 8, 8...] 3 9 = [; 1,, 18, 1, 1, 1, 1, 4, 1, 1, 4, 1, 9, 1,, 19, 1,,, 1, 3,, 1, 3, 1,, 1,, 1...] 3 10 = [; 6,, 9, 1, 1,, 4, 1, 1, 1, 1, 1, 1, 57, 4,, 16, 1, 1, 1, 1, 9, 6,, 3, 1, 1, 1, 1...] 3 11 = [; 4,, 6, 1, 1,, 1,, 9, 88,, 1,, 1, 8, 1, 1, 3, 4, 1, 7, 1, 40, 1, 1, 36,, 3, 1...] 3 1 = [; 3,, 5, 15, 7, 3, 1, 1, 3, 1, 1, 96, 7,, 6, 3, 36, 1, 17, 5,, 4, 9, 4, 9, 1, 3,, 34...] 3 13 = [;, 1, 5, 1, 1, 43, 3,, 1, 1, 3, 10, 7, 1, 1,, 0, 3, 1, 3, 9, 1, 6, 1, 1,, 1,,...] 3 14 = [;,, 3, 1, 1, 5, 5, 9, 6, 1, 1, 1, 54, 1,, 1, 1, 3,, 1, 5, 3, 37,, 0, 1, 1, 3, 3...] 3 15 = [;, 6, 1, 8, 1, 10, 8, 1, 1, 719, 4,, 5,,, 3, 3,, 1, 46, 4,, 11,, 1, 3, 11,, 1...] 3 16 = [; 1, 1, 1, 10, 18, 1, 6, 1, 1, 1,,, 4, 1, 6, 1,, 1, 1, 1, 1, 1, 3, 1, 8, 1, 1, 1, 5...] 3 17 = [ : 1, 1, 3, 138, 1, 1, 3,, 3, 1, 1, 07, 1,,, 1, 1, 1, 1,, 4, 9, 1,, 4, 1, 1, 3, 4...] 3 18 = [; 1, 1, 1, 1, 1, 3,, 1,,,, 4, 64,,, 1,, 1,, 1, 4, 4, 1, 1, 1,,, 1, 16...] 3 19 = [; 1,, 63, 1,,,, 1, 95,, 1, 1,, 7, 4,, 3, 1,, 3, 17, 1, 4, 1, 3, 1, 4, 4, 1...] 3 0 = [; 1,, 1, 1, 154, 6, 1, 1, 1, 6, 31, 1, 15, 8, 3, 1, 10, 3,, 1, 1, 17, 1,, 77, 4, 1, 4, 8...] 3 1 = [; 1, 3, 6, 1, 3, 17, 1, 7, 3, 3, 11,, 9, 1, 3, 1, 3, 1,,, 6,, 1, 10

11 0, 1, 4,, 10, 43...] 3 = [; 1, 4, 19,,,,,, 9, 56, 35, 49, 39, 4,, 56, 1, 97,, 11, 1, 5, 1,, 1, 1, 1,, 1...] 3 3 = [; 1, 5,,, 7, 1, 16, 4, 1, 8, 10, 7, 1, 4, 5, 1,,, 3, 1, 1, 1, 1, 3, 7, 1, 1, 1, 1...] 3 4 = [; 1, 7, 1, 1, 1, 1, 13, 1, 10, 4, 6, 1, 1, 1, 1, 1,, 1,, 1, 1, 1, 1, 1, 1, 1, 7, 1, 6...] 3 5 = [; 1, 1, 6, 4, 1,,,, 5, 1, 1, 4, 1,, 1, 3, 1,, 3, 3, 610, 3, 10, 1, 14, 1, 5, 1, 1...] 3 6 = [; 1, 5, 1, 1, 1, 39, 1, 1, 1, 4, 4, 13, 93, 3, 17, 3, 1, 85, 1, 3, 5, 1, 1, 8, 1, 6, 1,, 1...] 3 8 = [3; 7, 3, 40, 1, 10, 1, 1, 1, 13, 1,,, 1, 7,,, 63, 1, 1,, 1, 5, 3, 3, 1, 1, 1, 11, 4...] 3 9 = [3; 13, 1, 4, 1, 4,,,, 3, 1, 1,, 1, 1, 4, 1, 3,, 3, 8, 7,,, 1,, 8, 1, 3, 1...] 3 30 = [3; 9, 3, 13, 1, 9, 1,, 5, 4, 1, 1, 3, 1, 18, 3,, 4, 5, 3, 4, 1,,,, 1, 3, 1, 3, 79...] Mathematica 4.3 (5) e =

12 (6) e 1 e + 1 = (7) π = e (e 1)/(e + 1) (e 1)/(e + 1) tanh(1/) (31) e A., p.130 π (1770a) 7 (1963) 968 π/4 π (1655) (8) π = (1655 1, pp J. (1685) p ( ) ( )... 1 πr : (r) = π : 4 = 1 : 4/π 1

13 (9) π 4 = π 4 =

14 5 5.1 () k k (10) q 0 + p 1 q 1 + p q p k q k () k k , 1, 3, 5 3, 8 5, 13 8, 1 13, 34 1, 55 34, 89 55, , , 3, 7 5, 17 1, 41 9, 99 70, , , , , , 1, 5 3, 7 4, 19 11, 6 15, 71 41, 97 56, , 36 09, ,... e 1, 3 1, 8 3, 11 4, 19 7, 87 3, , , , ,... π 3 1, 7, , , , , /153 < 3 < 1351/780 π /7( ) 355/113 ( (480 ) ( )) q k ( q k = 1 ) 5. (9) π π (7) (7) (11)

15 ±1 (1) = 1 + / (13) = [6] I p j, q j j 0 j j 0 (14) p j q j 1 () α 5.3 π ( ) tan x (1768) tan x x = 0 tan x = sin x/ cos x sin x, cos x tan x = sin x cos x = x x3 /6 + x 5 / x / + x 4 /4... = x 1 x / + x 4 / x /6 + x 4 /10... x tan x = x 1 x /3 x 4 / x /6 + x 4 / = x 1 x 1 x / /3 x /

16 x tan x = x 1 x 3 x... 5, 7, (15) tan x = x 1 x 3 x 5 x 7 x 9... = 1 1 x 1 3 x 1 5 x 1 7 x... 0 (1794) ( (1768, 1770a) (1794)) x(x 0) tan x x = m/n (10) (6.31) tan m n = m/n 1 m /n 3 m /n 5 m /n 7... = m n m 3n m 5n m 7n... p 1 = m, p i = m (i ); q i = (i 1)n (i 1) m n (14) i 0 arctan y x = arctany y = tan x π = 4arctan1 3 16

17 6 Archimedes of Syracuse, 87?-1. ( ) 9 (,1981) 1 ( ) ( ) 9 (,1980) 1, 1 11 ( ) John Wallis, ( ) BA ( ) I (164-5) I (1648) (1649- ) 1955 ( ) (1660) (1669) (1653) (1687) Leonhard Euler, (7-41) (41-66) (66-83)

18 ( ) ( ) 13 Gregorius XIII, (1539) (1565) (157-85) (158..4) (1585) Tsu Ch ung -Chih, ( ). (46) (478) ( π /7 355/113 ) Diophantos = Diopantus, 46?-330?(00?-84?) (5 6 ) = Leonardo da Pisa, Leonardo Pisano (= Fibonacci), 1170?(1174?)-150. (1 ) 100 (Liber Abaci, 10) 10 0 Zephirum 10 ( ) 14 Lord William Brouncker, (166-77) 18

19 ( 64-67) Adolf Hurwitz, F. (1884-9) (189) e (1853) (1901) R. ( ) Rafael Bombelli, (Alessandro Rufini) P.F. (157,199) Adrianus Metius, ( ) (Adriaen Anthonisz=Adriaan Metsue, ) π < π < = (5 ) (1584) = Eucleides = Euclid of Alexandria = EYKΛEI OY, 330?-75?(365?-300?). 19

20 Gottfried Wilhelm Leibniz, (1666) (1667) (1666) XIV 3 (1673) Joseph-Louis Lagrange, ( ) (1755) ( 66-86) 16 (1787) (1794) ( 1 ) Johann Heinrich Lambert, e, π (1766) (1770) Adrian Marie Legendre, ( ) ( ) (1816-) ( ) 40 [1] (94 95 ) (1996),

21 [] ( 95) (1996),-55. [3] (1997),-37. [4] 96 6 ( 96 ) (1997), [5] (1998). [6] E. G. ( ) (1997 ) Analysis by Its History, Springer Verlag(1996), by E.Hairer & G.Wanner. [7] ( ) EYKΛEI OY : ΣTOIXEIA. ( 148), J.L. ( ) L Dover 3 1

17 ( :52) α ω 53 (2015 ) 2 α ω 55 (2017 ) 2 1) ) ) 2 2 4) (α β) A ) 6) A (5) 1)

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