中国古代の周率(上) (数学史の研究)

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1 ( ) Calculations ofpi in the ancient China (Part I) 1 Sugimoto Toshio [1, 2] proceedings 2 ( ) ( ) 335/113 2 ( ) 3 [3] [4] [5] ( ) ( ) [6] [1] ( ) 3 $\cdots$ [6] $\sim$ ( ) $-$

2 ACBO 6 $AB\cross$ CO $\div 2=1\cross 1\div 2=0.5$ , $ \cross 1\div 2=$ , ( ) EFBOA( ) ( ) I ( ) EF G, H $O$ GHO ( GHO ) $n$ $X$ $Y$ 1 AB $=x$ 2 AC $=y$ (1) AB $=x$ (2) $AD=x/2$ (3) $AD^{2}=(x/2)^{2}=l/4$ (4) D $O^{2}=u^{2}=1^{2}-x^{2}/4$ (5) $DO=u=\mapsto^{1-/4}$ (6) CD $= 1-u$ (7) $AC^{2}=J\nearrow=l/4+v^{2}$ $6$ AO $=$ CO $=$ BO $=1$ $2 +1$ $6$ $=AD^{2}+CD^{2}$ $O$ (8) $AC=y^{r}=\mapsto^{/4+V^{2}}$ 8 6 $\cdot 2^{p}$ 1 (9) $=6\cdot$ 2 $\cross x$ [3] (1) $X_{f}$ (3) $x^{2}/4$, (5) $u$, $J^{\nearrow}$ (6) $1-u$, (7) - I

3 93 [ ] ( ) $AC^{2}=AD^{2}+CD^{2}$, $1^{2}=CD^{2}+2CD\cdot DO+DO^{2}$, $1^{2}=AD^{2}+DO^{2}$ $AD^{2}=CD^{2}+2CD$. DO CD DO $+$ $=I$ $AC^{2}=CD^{2}+2CD\cdot DO+CD^{2}=2CD\cdot(CD+DO)=2CD$ ( ) (10) $AC^{2}=2CD$ 5 [6] [3] 56 (11) $x^{2}=c$ ( ) $bx$ (12) $x^{2}+b_{x}=c$ ( ) 7 $\sqrt{075}$ $t=0.8$ 0. $75-t^{2}=0.11$ $t=0.8+x$ 0.75 $= x+x^{2}$, (13) $x^{2}+1.6x=0.11$ $x$ (14) $x=0.86+y$ (15) $y^{2}+1.72y= ^{2}= =0.0104$. $y^{2}$ $x=0.11/1.6= $. $y=0.0104/1.72= $ ( ) (16) $\sqrt{075}= \cdots$ $[ =\sqrt{3}\div 2]$ $1-;$ $.75= \cdots$ 4 (5) (8) ; (mantissa) 25 6 [3] 0.75 $\sqrt{075}= \cdots$ 7 ( ) $O$ /5

4 94 I $arrow$ 2 $D^{\text{ }}$ $\cdots$ ) : $I CD$ j LAB 1 $AD^{2}.25$ $ DO$ $1339\underline{9}46CD$ $ \underline{93}445CD^{2}$ $arrow$ $\frac{175\theta \underline{361} \cdot \theta }{+\text{ }arrow)(abad^{2}docdcd^{2}}$ $\theta+$ $\frac{j\iota \underline{736} \underline{2}.0^{\ell} \underline{613}}{\text{ ^{}rightarrow}arrow t\iota \text{ }\backslash ABAD^{l}DOCDCD^{2}}$ $7569\underline{7}\underline{0}3$ $AD^{2}=.25$ ( ) $+$ CD $=.13397$ 4596 $arrow$ $AC^{2}= $ $ \underline{1}$ $\theta^{5}45842$ $CD^{2}=.01794$ , AC $=.51763$ , 6. AC $= $ [ ] 12. $\sin(\pi/12)= $ $2I$ 25 $0^{}$ 33 $1$ $C$ $3D$ $\circ$ $312\underline{40}$ $9^{}$ AD2 DO CD $CD^{2}$ $*7990arrow 80$ $0^{2}116$ $0^{2}116$ $140\underline{7990}^{*}$ $0^{2} $ 3072 AB AD2 $*734SO$ DO $*41494\underline{708}$ $CD^{2}$ $0^{2}204$ $0^{5}10458$ $9^{6}4770$ $0^{12}273$ $0^{2}204$ $0^{5}10458$ $73^{*}.9^{6}4770$ $*$ $0^{12}273$ $\frac{0^{2} ^{5} ^{6} ^{12} }{6144abad^{l*} \underline{3}\underline{9}arrow 4DO^{*}55888\underline{08}arrow 55888CD^{2}}$ $0^{2}102$ $0^{6}2614$ $9^{6}8692$ $0^{1S}17$ $0^{2}102$ $0^{6}2614$ $79^{*}.9^{6}8692$ $*$ $0^{13}17$ $0^{2} ^{6}2614$ $9^{6}8692$ $0^{13}17$

5 95 AC $8\theta 911S4$ 3. 3 AC AC $027\underline{8813}$ AC2AC $215\underline{401}2$ $\cdot\sin(\pi/24)= $ $\cdot\sin(_{\pi}/48)= $ $\cdot\sin(\pi/96)= $ ( 10 ) $arrow(\cross 6)arrow$ AC OS AC $arrow(\cross 12)arrow$ $23S44$ $23S arrow 40$ $23S443879S$ $6144AC^{*} arrow 13942$ $0^{5}10458$ $0^{2}102$ $0^{5}10458$ $0^{2}102$ $*$. $0^{5}10458$ $0^{2}102$ $\underline{298}arrow 643$ AC212288AC $0^{6}2614$ $0^{3} $ $0^{6}2614$ $0^{3} $ $0^{6}2614$ S47 $0^{3} $

$96 0.8660254 0.86602 $5^{}$ / $5^{=0.$

6 $5^{}$ / $5^{= ^{4}}/10$ 7 I $[3]\sim[5]$ $n-1$ $S$ $n$ $T$ (17) $T $ $=T$ $+$ $(T-S)$ $T$ $T $ 1 3 I 7 ( ) $N$ $2N$ 1 AB AC 4 ( ) ( ) AOD AD $=AB/2$ DO AO CAD CD AD AC [3] ( ) 1 I (18) $<$ $< $ ( (19) $3.14^{64}/625= <$ $< =3.14^{169}/625$? [3] ( ) )

7 97 10 I DO I $\sim$ [1] ( ) 8 (425500) [3]. [6] (i) ( ) (ii) $()$ ( $(i_{v})$ ) (20) $<$ $< $ 335/113 ( ( ) ) [ 1] ( ) ( ) (20) (20) 8 7 $11\sim$ I2! $T= $ $S= $ (21) $T+$ $(T-S)= ( )$ $=$ $+$

8 98 $= $? 9 [7] (22) $n\sin(_{\pi}/n)=\pi-\pi^{3}/6n^{2}+\pi^{5}/120n^{4}-\pi^{7}/5400n^{6}+-$ (21) 7 $S=6144\sin(_{\pi}/6144)$ $= \ldots$ $T=12288\sin(\pi/12288)= $ $!$ $S= $, $T= $ (20) ( ) ( ) [1] 4 (1) $\sim(9)$ [6] $I$ AD 2 (mantissa)17 14 D $O^{2}=1-AD^{2}$ 9 14 $AC^{2}=AD^{2}+CD^{2}$ AC ) ( $0$ $0$ 9 $\cdots$ $0^{}$ $\cdots$ $\cdots$ $9^{}$ $\cdots$ (1) $\sim(9)$ $arrow$ 20 AB AC AC ( ) (22) AC 18 4 (6) (7) (8) 4 (10) $AC^{2}=2CD=2v$ (23) $AC$ $=\sqrt{2}\cdot\sqrt{cd}=\sqrt{2}\cdot\sqrt{v}$

$99 ( (8) AC (7), (6) $n$ $(n- 1)$ (1) AB ) (24) $AB$ $=AC\cdot\sqrt{1-AC^{l}/4}$ 18 ( )? $\rangle\rangle$ $arrow$ $arrow$ 3. $I3262\cdots$ ( 24 $\sin(\pi/24)$ ) 12 $\cdots$ 0.$

9 99 ( (8) AC (7), (6) $n$ $(n- 1)$ (1) AB ) (24) $AB$ $=AC\cdot\sqrt{1-AC^{l}/4}$ 18 ( )? $\rangle\rangle$ $arrow$ $arrow$ 3. $I3262\cdots$ ( 24 $\sin(\pi/24)$ ) 12 $\cdots$ ( AC ) 4 (8) (1) $2\sin(\pi/24)$ $\cdots$ ( AC) ( 4 )? (3) $AD^{2}$ (5) DO (6) CD (7) $CD^{2}$ (8) AC $\sim$ AC $\cdot 2$ 6 (9) $=6\cdot$ 2 $x_{j }$ AD $=AB/2$, D $O^{2}=1-AD^{2}$, CD $=1-$ DO ( ) 11 ( ) $n$ ( ) $n-1$ ( ) I 1 3 AC AB

10 100 2 ( ) AB AC ( ) AD2, DO, $CD^{2}$, AC ( ) (20) (18) $n$ $arrow+$ AC AC 3072 $arrow$ $arrow 6144$ AC 6144 $arrow$ AC DO $=$ $1-$ CD 9 9 $0$ $T$ $S$ CD AD2 $CD^{2}=AC^{2}-AD^{2}$, CD $=\sqrt{cd^{2}}$ (10) CD $=AC^{2}/2$ $AD^{2}$ ( ) ( ) $S=6144\sin(\pi/6144)=$ $\cdots$ $T=12288\sin(\pi/12288)= \cdots$ 12

11 101 [8] 9 $S$ $T$ $S= \cdots$ ( ) $T= \cdots$ ( ) 49152,, $U= \cdots$ $V= \cdots$ $T =T+(T-S)= \cdots$ 49152,, $=U+(U-T)= \cdots$ $V=V+(V-U)= \cdots$ $<$ $<$ $<$ $<$ $<$ $<$ ! 1 ( ) (24576 ) [1] : ( ) [2] : 355/133 ( ) [3] : : : [4] : ( ) ( ) [5] : ( ) ( ) [6] : ( ) [7] : [8] :

, 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x 2 + x + 1). a 2 b 2 = (a b)(a + b) a 3 b 3 = (a b)(a 2 + ab + b 2 ) 2 2, 2.. x a b b 2. b {( 2 a } b )2 1 =

x n 1 1.,,.,. 2..... 4 = 2 2 12 = 2 2 3 6 = 2 3 14 = 2 7 8 = 2 2 2 15 = 3 5 9 = 3 3 16 = 2 2 2 2 10 = 2 5 18 = 2 3 3 2, 3, 5, 7, 11, 13, 17, 19.,, 2,.,.,.,?.,,. 1 , 1. x 2 1 = (x 1)(x + 1) x 3 1 = (x 1)(x