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2 4 202 9 202 9 6

3................................................... 3.2................................................ 4.3......................................... 6.4....................................... 7.5........................................ 0.6................................................ 3 5 2.............................................. 5 2............................................... 5 2..2............................................ 5 2..3....................................... 6 2..4.......................................... 6 2.2............................................. 6 9 3................................................. 9 3.2............................................ 9 3.3.............................................. 20 3.4................................................ 2 22 4................................................. 22 4.2............................................. 22 4.3......................................... 23 4.4................................................ 24 26

2 Contents. Noboru NAKANISHI: Sums of the Series Involving cosech 2. Tohru MUTO: The Origin of the Geometry and the Word of 3. Tohru MUTO: The Value of π in Ancient Babylonia 4. Tadashi YANO: The Value of π in Ancient Babylonia (Appendix) 5. Tadashi YANO: Editorial Comments

3 Sums of the Series Involving cosech Noboru Nakanishi 2. sin cos cosec sec sinh cosh sinh (2n ) sinh[(2n )π] = log 2 (..) 8.3 log e (+i)πn e πn (..2) II [] p.69 ( ) n n sinh(nπ) x 4 + n 4 = π ( 2x 2 2π 2 x 2 ) cosh( 2πx) cos(. (..3) 2πx) x 4 x ( ) n n sinh(nπ) = = 0.079577... (..4) 4π x = 0 cosh( 2πx) cos( 2πx) = 2π 2 x 2( + 23 π 4 x 4 6! + 25 π 8 x 8 0! ) + (..5) [ 2 3 ( 2 4πx 4 6! π4 x 4 5 ) ] ) + 0! 26 (6!) 2 π 8 x 8 + = ( π3 3π4 360 260 x4 +. (..6) ; nbr-nak@trio.plala.or.jp 2 Professor Emeritus, Kyoto University

4 ( ) n n 3 sinh(nπ) = π3 = 0.086283..., (..7) 360 ( ) n n 7 sinh(nπ) = 3π7 = 0.0865592... (..8) 453600 n = / sinh π = 0.0865895... (..) (/8) log 2 = 0.08664339... [] p.82 n sinh(nx) = π2 2x 4x 2 (x > 0) (..9) x 2e x /(2x).4.2 2 K(k) 0 E(k) dt ( t2 )( k 2 t 2 ), (.2.) 0 dt k2 t 2 t 2 (.2.2) k 0 k k k 2 K(k) K K(k ) K K(k) E(k) K E(k) + KE(k ) KK = π/2 4 ϑ 0 (u, τ) + 2 ( ) n q n2 cos(2nπu), ϑ (u, τ) 2 ϑ 2 (u, τ) 2 ( ) n q (n+ 2 )2 sin[(2n + )πu], n=0 q (n+ 2 )2 cos[(2n + )πu], n=0 ϑ 3 (u, τ) + 2 q n2 cos(2nπu). (.2.3)

5 τ ik /K q e iπτ = e πk /K ϑ 0 (u, τ) = Q 0 [ 2q 2n cos(2πu) + q 4n 2 ], ϑ (u, τ) = 2Q 0 q 4 sin(πu) [ 2q 2n cos(2πu) + q 4n ], ϑ 2 (u, τ) = 2Q 0 q 4 cos(πu) [ + 2q 2n cos(2πu) + q 4n ], ϑ 3 (u, τ) = Q 0 [ + 2q 2n cos(2πu) + q 4n 2 ]. (.2.4) Q 0 ( q2n ) u = 0 ϑ (0, τ) = 0 ϑ 0 ϑ 2 ϑ 3 ϑ 0 = + ( ) n q n2 = ( q 2n ) ( q 2n ) 2, ϑ 2 = 2 q (n+ 2 )2 = 2q 4 n=0 ( q 2n ) ( + q 2n ) 2, ϑ 3 = + 2 q n2 = ( q 2n ) ( + q 2n ) 2 (.2.5) k k = ϑ 2 2 ϑ3 2, k = ϑ 0 2 ϑ 2 3, K = π 2 ϑ 2 3. (.2.6) kk = π 2 ϑ 2 2, k K = π 2 ϑ 2 0 (.2.7) sn(u, k) ϑ (u/2k, τ) k ϑ 0 (u/2k, τ), k ϑ cn(u, k) 2 (u/2k, τ) k ϑ 0 (u/2k, τ), dn(u, k) k ϑ 3(u/2K, τ) ϑ 0 (u/2k, τ). (.2.8) snu (.2.) 2 sn 2 u + cn 2 u = dn 2 u + k 2 sn 2 u = u = 0 u = K sn0 = 0, cn0 = dn0 = ; snk =, cnk = 0, dnk = k (.2.9) d d snu = cnu dnu, du d cnu = snu dnu, du du dnu = k2 snu cnu (.2.0)

6.3 (..) (..) I [2] p.72 (2n ) sinh[(2n )(K /K)π] = 8 log( k2 ) (.3.) k = / 2 K = K (..) (..) [ ] (.3.) S S = 2 = 2,odd,odd,odd n sinh[n(k /K)π] e n(k /K)π n( e 2n(K /K)π ) n m=,odd e mn(k /K)π (.3.2) mn = N /n = m/n N ( N m σ(n) S = 2 = = m N N=,odd N= N= N σ(n)e N(K /K)π N σ(n)[(e (K /K)π ) N ( e (K /K)π ) N ] e (K /K)π σ(n) dx x N. e (K /K)π (.3.3) σ(n) σ(n)x N nx n = x n (.3.4) N= S = = e (K /K)π e (K /K)π = log dx [ log( x n ) nx n x n ] e (K /K)π e (K /K)π ( ) n e (K /K)πn. e (K /K)πn (.3.5) n n 2m S = log m= + e (K /K)π(2m ) e = log (K /K)π(2m ) m= + q 2m. (.3.6) q2m

7 q = e (K /K)π (.2.5) (.2.6) (.3.) ( q 2n ) 2 ( + q 2n ) 2 = ϑ 0 ϑ 3 = k (.3.7) + q 2n q 2n = k /4 = ( k 2 ) /8 (.3.8) S = log k /4 = 8 log( k2 ) (.3.9).4 [2] p.72-p.723 (.3.) 3 x (K /K)π S S 5 S 6 ( ) n sinh[(n 2 )x] = k π K, n sinh(nx) = K[K E(k)], π2 ( ) n n sinh(nx) = k2 KE(k) π2 π 2 K 2. (.4.) S 5 (..9) S 6 k = / 2 = k K = K x = π E(k ) = E(k) 2KE K 2 = π/2 (..4) S 3 S 3 S 4 S 2 n= n= n= n= cosh(nx + a) = 2 [ 2a π Kdn π K ], cosh(nx) = 2 π K, ( ) n cosh(nx) = 2 k 2 K = 2 π π k K, cosh[(n 2 )x)] = 2k π K. (.4.2) S 3 [2] x π S 4 k 4 k 2 3 cosech sech sinh cosh S k

8 q = e x (.2.7) (.2.5) 2 S = 4 S 2 = 4 (S 3 ) = 4 (S 4 ) = ( ) n q n 2 q 2n = kk ( 2π = q (n+ )2) 2 2, n=0 q n 2 kk ( = + q2n 2π = q (n+ )2) 2 2, n=0 q n + q 2n = K 2π 4 = ( q n2 + q n2) 2, ( ) n q n + q 2n = k K 2π 4 = ( ( ) n q n2 + ( ) n q n2) 2 (.4.3) 2 S 4 S 2 m= ( ) n q 2nm n m+ 2 (.4.4) ( ) n = ( ) n2 S 3 S 4 q S 5 S 6 E(k) q 2 (S 5 + S 6 ) =,odd,odd n sinh(nx) = k2 π 2 K2 = 4 ϑ 4 2. (.4.5) nq n ( q 2n = q (n+ )2) 4 2 n=0 (.4.6) (.4.) (.4.2) [3] p.9-p.92 22 sn(u, k) = 2π kk cn(u, k) = 2π kk dn(u, k) = q n 2 (2n )πu sin, q2n 2K q n 2 (2n )πu cos, + q2n 2K q n nπu cos + q2n K. π 2K + 2π K (.4.7)

9 u = 0 u = K (.2.9) ( ) n q n 2 q 2n = kk 2π, q n 2 kk = + q2n 2π, q n + q 2n = K 2π 4, ( ) n q n + q 2n = k K 2π 4. (.4.8) (.4.3) S S 2 S 3 S 4 (.2.0) (.2.9) (2n )q n 2 q 2n = kk2 π 2 = 4 ϑ2 2ϑ 2 3, ( ) n (2n )q n 2 + q 2n = kk K 2 π 2 = 4 ϑ2 2ϑ 2 0 (.4.9) 2n sinh[(n 2 )(K /K)π] = 2kK2 π 2, ( ) n (2n ) cosh[(n 2 )(K /K)π] = 2kk K 2 π 2 (.4.0) u = 0 u = K log sn(u, k) = log 2K π πu + log sin 2K 4 q n nπu n( + q n sin2 ) 2K, log cn(u, k) = log cos πu 2K 4 q n nπu n( + ( ) n q n sin2 ) 2K, q 2n (2n )πu log dn(u, k) = 8 (2n )( q 2(2n ) sin2 ) 2K u = K u K (.4.),odd,odd,odd q n n( + q n ) = 4 log 2K π, q n n( q n ) = 4 log 2k K π, q n n( q 2n ) = log k 8 (.4.2) (.3.) 2

0.5 q ) S S 2 (.4.3) S S 2 n=0 ( ) n q n q 2n+ = q n ( + q 2n+ = q n(n+)) 2 n=0 n=0 (.5.) (.4.4) m=0 (q 2 ) m (q 2 ) 4m+ m=0 (q 2 ) 3m+2 ( (q 2 ) 4m+3 = n=0 ) (q 2 ) n(n+) 2 2 (.5.2) [ ] q 2 m=0 n=0 (q 2 ) 4mn+m+n m=0 n=0 (q 2 ) 4mn+3m+3n+2 (.5.3) N=0 a N (q 2 ) N 2 N=0 b N(q 2 ) N, (.5.2) N=0 c N(q 2 ) N N a N b N = c N m n a N N = 4mn+m+n (m, n) b N N = 4mn+3m+3n+2 (m, n) c N N = 2 m(m+)+ 2n(n+) (m, n) a N 4N + = (4m+)(4n+) (m, n) b N 4N + = (4m+3)(4n+3) (m, n) c N 4N + = (m + n + ) 2 + (m n) 2 (m, n) 30 N a N b N c N N 0 2 3 4 5 6 7 8 9 0 a N 2 2 2 2 2 3 2 2 2 2 b N 0 0 0 0 2 0 0 2 0 0 c N 2 2 2 0 3 2 0 2 2 4N + 5 3 2 3 7 3 7 5 2 29 3 37 4 N 2 3 4 5 6 7 8 9 20 a N 4 2 2 2 2 4 2 2 2 3 b N 2 0 2 0 0 2 0 2 2 c N 2 2 0 2 4 0 2 0 4N + 3 2 5 7 2 53 3 9 6 5 3 3 23 73 7 3 4 N 2 22 23 24 25 26 27 28 29 30 a N 4 2 2 2 2 4 2 2 4 2 b N 0 0 2 0 0 4 0 0 2 c N 4 2 0 2 2 0 2 2 2 4N + 5 7 89 3 3 97 0 3 5 7 09 3 3 2 3 2

4k + (+) 4k + 3 ( ) (+) (+) ( ) ( ) (+) (+) ( ) ( ) 4N + 2 (+) ( ) a N b N 4N + 4N + = l j= qν j j ( ) q j 4N + d (d, (4N + )/d) d q j (ν d ) j ( ν j ) l j= (ν d) j d (+) d ( ) (d, (4N + )/d) ( ) P l j= (ν d) j + a N b N l j= (ν j + ) ν j ν j a N b N = a N b N = 0 4N + = k j= pµj j l j= qνj j (p j (+) q j ( ) k (+) a N b N j= (µ j + ) ν j a N b N = k j= (µ j + ) a N b N = 0 c N 4 Wikipedia M 2 m 2 + n 2 r(m) = 4 d M,odd M d ( ) (d )/2 (.5.4) 4 m n m n 0 2 0 = 0 M = 4N + ( ) (d )/2 d ( ) k + (+) j= (µ j +) 4N + ( ) k ν j j= (µ j + ) 0 c N = a N b N 2) S 3 S 4 (.4.3) S 3 q n + q 2n = ( q n2 + q n2) 2. (.5.5) [ ] q q N a N b N 2 c N a N = b N + c N m= ( ) m q n(2m ) = [ ( ) (M )/2] q N (.5.6) N= 4 Hardy-Wright [4] Theorem 278(p.34) m 2 + n 2 = (m + in)(m in) M N

2 a N (+) M ( ) M N = 2 λ k j= pµ j j l j= qν j j p j (+) q j ( ) ν j a N = k j= (µ j + ) a N = 0 b N λ ν j 0 c N ν j a N = k j= (µ j + ) a N = 0 (.5.5) n 0 λ λ 0 a N = b N + c N S 3 + 4 q n ( + q 2n = n= q n2) 2 (.5.7) 3) S 5 + S 6 n (.4.6) n=0,m=0 ( ) (2n + )q (2n+)(2m+) 4 = q (2n+)2 /4. (.5.8) [ ] (.5.8) q N N A N B N A N N σ(n) B N 4N 4 N = A = 4 = 2 + 2 + 2 + 2 B =. N = 3 A 3 = + 3 = 4 2 = 3 2 + 2 + 2 + 2 B 3 = 4 C = 4 N = 5 A 5 = + 5 = 6 20 = 3 2 + 3 2 + 2 + 2 B 3 = 4 C 2 = 6 N = 7 A 7 = + 7 = 8 28 = 3 2 + 3 2 + 3 2 + 2 = 5 2 + 2 + 2 + 2 B 7 = 4 + 4 = 8 N = 9 A 9 = + 3 + 9 = 3 36 = 3 2 + 3 2 + 3 2 + 3 2 = 5 2 + 3 2 + 2 + 2 B 9 = + 2 = 3 N = A = + = 2 44 = 5 2 + 3 2 + 3 2 + 2 B = 2 N = 3 A 3 = + 3 = 4 52 = 7 2 + 2 + 2 + 2 = 5 2 + 3 2 + 3 2 + 3 2 = 5 2 + 5 2 + 2 + 2 B 3 = 4 + 4 + 6 = 4 n=0 B N. 5 Wikipedia 5 Hardy-Wright [4] Theorem 386(p.45) + 4 X 2 + 8 X m= ( ) n x 2n X x 2n = x n2 2 n= (2m )x 2m X x 2m + 8 m= (4m 2)x 4m 2 x 4m 2

3 M 4 0 r 4 (M) = 8σ (4) (M) σ (4) (M) 4 M N σ (4) (4N) = σ(2n) 2N N 2 σ(2n) = 3σ(N) r 4 (4N) = 24σ(N) 4N 4 (0 ) 4 4 4 4 r 4 (N) = 8σ (4) (N) = 8σ(N) 0 4 2 4 B N 24σ(N) = 8σ(N) + 6B N B N = σ(n) = A N 4) (.4.9) (.4.9) 2 q (2n )q n [ 2 q 2n = q (n+ )2] 2 2[ + 2 q n2] 2 n=0 (.5.9) [ ] n=0 (2n + )q n+ 2 q 2n+ = n=0 m=0 (2n + )q 2 (2n+)(2m+) (.5.0) N=0 A Nq N/2 N = (2n+)(2m+) (n 0, m 0) A N = σ(n) [ ] q (2n+)2 2 [ 4 4 n= n= q n2] 2. (.5.) N=0 B Nq N/2 2N = (2n + ) 2 + (2n 2 + ) 2 + (2n 3 ) 2 + (2n 4 ) 2 2 2 4C 2 B N = 4 r 4(2N) B N 4 C 2 (.5.) 4 N r 4 (2N) = 8σ (4) (2N) = 8σ(2N) = 24σ(N) B N = σ(n) = A N.6 (..3) (.4.7) (.4.) (..3) sinh

4 [5] 200 dx log log x = 0 0 0.5447... [3] p.574 0 dx e x log x = 0 6 [] II [2] I 99 [ ] [3] 983 [ ] [4] G. H. Hardy and E. M. Wright, An Introduction to the Theory of Numbers (Sixth Edition, Oxford University Press, 2008) [5] 988 3 23 6 x =

5 The Origin of the Geometry and the Word of 7 Tohru Muto 8 2. (562 633) ( 2.. (Christopher Clavius 538 62) Euclidis Elementorum libri XV (574) Commentaria In Euclidis Elementa Geometria 574 9 3 6 2..2 Aleni 5 625 magnitude 5 7 ; mutoh.ab@wine.ocn.ne.jp 8 Historian of Mathematical Thought 9

6 2 2 a : b c a : b = c : d d a : b = e : c e 2..3 0 ( ) Elementorum Euclidis Elementorum 3 6 geometria geo jihe giho geometria ( ) 856 885 (854 92) 2..4 ( ) ( ) 4 (87) 5 (872) 6 2.2 0

7 2000 a b L a b a + b a 2 b 2 = (a b)(a + b) a 2 + b 2 = c 2 a 2 = (c b)(c + b) a = 3, c b =, c + b = 9 a = 3, b = 4, c = 5 a = 5, c b =, c + b = 25 a = 5, b = 2, c = 3 (rect triple) 3 3.25 0 2 30 0 40 0 3 20 5cm, 0cm, 20cm 25 250cm 80cm 3.25 9 0.790 2 484 425 r [2r( /9)] 2 = 4 (8/9) 2 r 2 4 (8/9) 2 = 3.60 3.6 2 r 4r 2 π = 3.4 π/4 = 0.785 0.790

8 γɛoµɛτρια γɛoµɛτρια γɛoµɛτ ρια 2 2 (zerlegungsgleich) 2 2 3 (202.8.3) 3 3

9 The Value of π in Ancient Babylonia 4 Tohru Muto 5 3. 60 0 50 [ ] [ ] 60 50 0 50 0 500 8 20 [ ] 0 00 40 500 8 20 00 40) 450 7 30 ) 400 50 [ ] [ ] [ ] 60 400 450 3.2 60 a 50 b) 450 S R T AC O ABC O AB h 3. πr 2 2T R cos T 50 360 2 2πR 2T = 60 (3.2.) 360 30R 25R sin 2 T = 450 2R sin T = 50 (3.2.2) = 450 (3.2.3) 5 R 2 (R sin T ) 2 = 6R 90 R 2 080R + 23725 = 0 R = 33.82, T = 48.9 400 4 ; mutoh.ab@wine.ocn.ne.jp 5 Historian of Mathematical Thought

a 20 S b A B h R O T C 3.: 20 3.3 6 T 60 3 60 80 R 30 3R 3 3 = 7 4 52.5 S S = 3 πr2 2 52.5 R 2 = 900 575 4 506.25 (3.3.) 7 2 52.5 50 3R = (7/4)R = 50 R = 200 7 (3.3.2) (3.2.) R 200/7, T = 60 π 2πR (2T/360) = 60 (3.2.) π 3 60 7/400 63/20 3.5 S 400 450 506.25 S (/3)(63/20)R 2 (/2) 50 (R/2) 6000/7 2500/7 500 (3.3.3) (3.2.3) 450 ( ) 0 50 500 (3.3.4) S π 3.5 63/20 π R 200/7 500 6 7

2 3.4 3.5 (202.8.20)

22 The Value of π in Ancient Babylonia (Appendix) 8 Tadashi Yano 4. 9 20 [] 2 a b S S = b(a b) (4..) S S = b (2a b) (4..2) 7 22 (4..2) (4..) 23 20 4.2 a b (4..) (4..2) (4..) a b (4..2) 2a b 7 π 8 yanotad@earth.ocn.ne.jp 9 20 2 [2] 22 (4..2) 23

23 a, b π R 3 = 7 4 b = 7 4 R (4.2.) a = 2π 3 R (4.2.2) a b = 2a b 7 a b = (απ + β)r (4.2.3) 2a b = (γπ + δ)r 7 (4.2.3) π π = δ β α γ (4.2.4) α = 2 3 β = 7 4 γ = 7 4 3 δ = 4 π = 63 = 3.5 (4.2.5) 20 [3] 4.3 20 (4.., (4..2) 20 20 a b b = 7 4 R (4.2.) a = 2π 3 R (4.2.2) a b = 8π =.96 (4.3.) 2 π = 3.4 a/b.2.2 a b =.2 = 6 5

24 5a = 6b (4.3.2) (4.3.2) 5a 6b = 0 7 (2a b) = (2a b + 5a 6b) 7 = a b (4..) (4..2) 20 4.4 (4..) 20 20 (4..) (4..2) (4..) [2] (3.2.3) (4..2) [2] (3.2.),(3.2.2) 20 a a = 2π 3 R (4.2.2) T = 60 b b = 2R sin T = 3R (4.A.) [2] (3.2.3) S [2] 3. S = πr 2 20 360 h 2 b, h = R 2 R = 3 πr2 br (4.A.2) 4 S = 4 (2a b)b 3 3 = 7 4 S = (2a b)b (4..2) 7

25 2 (4..) (4..2) (4..) (4..2) 4.3 (4..2) (4..) (4..2) (4..) 7 (2a b) = a 0 + a (a b) + a 2 (a b) 2 + a 3 (a b) 3 + (4.A2.) a b a 0, a, a 2, a, b a 2 = a 3 = = 0 (4.A2.) a = b a 0 = a 7 (4.A2.) b 7 = a 2a 2 (a b) 3a 3 (a b) 2 (4.A2.2) (4.A2.2) a = b a = 7 7 (2a b) = a 7 + (a b) (4.A2.3) 7 (4.A2.3) (4.A2.3) a b 7 (2a b) = a b 5a = 6b a b = 6 5 (4.A2.4) (4.A2.5) (4.2.),(4.2.2) a/b.96.2 4.3 (202. 8. 29) [] [2] 9-2 [3] Kazuo Muroi Mathematics Hidden Behind the Practical Formulae of Babylonian Geometry, The Empirical Dimension of Ancient Near Eastern Studies, edited by G.J.Selz, (20), 49-57.

26 202 9 2 4 latex latex latex latex latex latex

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