Title 初期和算にみる Archimedean Spiral について ( 数学究 ) Author(s) 小林, 龍彦 Citation 数理解析研究所講究録 (2000), 1130: 220-228 Issue Date 2000-02 URL http://hdl.handle.net/2433/63667 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University
1130 2000 220-228 220 Archimedean Spiral 1, ( ) 39 40 41 Archimedean Spiral Archimedean Spiral ( $\mathrm{b}.\mathrm{c}$. 287-212) 1 P $P$ 1) Spiral Spiral ( 49 ) Archimedean Spiral Archimedean Spiral 2, Archimedean Spiral Archimedean Spiral $r=a\theta^{o}$... (1) $n=1$ Archimedean iral 2) 3 $S p$ $O$ 0 $O$ $P$ 3) 1) 1/3 2) $P$ $P_{1}$ $OP_{1}$ $O$ $Q$ $OQ$ $OP_{1}$ P
$\mathrm{p}$ 221 ( 20 : 1592 ) 4) $5\rangle$ ( ) : ( :1659 ) ( :1664 ) ( :1666 ) : ( :16 73 ) : 3 6) 3, Archimedean Spiral Archimedean Spiral 7) Archimedean $s_{pl}^{\mathrm{v}_{i\gamma a}}$ Archimedean Spiral $8$ ) $\circ$ Archimedean iral $S p$ H\Gamma f[ t $9$ ) $\circ$
$0\rangle$ $\text{}$ $\circ$ $(\overline{\text{}^{}r})$ 222 [ ] ( H\star f# \star ji {. $2_{=}$ $3$ 2 $+$ 2} /3 (2) ( ) E\star f\beta 1 Archimedean Spiral 1 $n$ $n$ ) \mu n (4 Archimedean Spiral (1) $n<1$ Hf\mu \beta $11$ ) Hj\star I 39 40 41 Hi\star # 40 ff\mbox{\boldmath $\lambda$} 41 40 39 40 12) 40 $0$ ( -+ 0 + $0$ -+ + 3 $0$
.. 1^{\circ}\Delta$... 223 $0$... $(i7)$ 13) (7) $=355^{2}=126,026$ $(\triangleleft )=3\cdot 113^{2}=3\cdot 1132$ $(\eta)=12\cdot 113^{2}=12\cdot 1132$ $\pi=355/113=\pi_{2}/$ \tau 1 [ ] $(2 R)$ 8 (A) 3 1 $(m)$ (1) [] 7 2 72406 [ ] $P$ () $(2 \cdot \mathrm{a})^{2}+$ $(2R)2=4p2$... (3) $\{\pi_{2}(22r)^{2}+3\cdot\pi 12.\mathit{1}i\mathrm{z}^{2}\}4p^{2}$... () 12. $D\mathit{1}^{\mathit{2}}$ $\pi 12$... () 1 $I$ $2=\{\pi_{2}(22R)^{2}+3\cdot\pi 12. m^{arrow?}\}4p^{2}/12$. $Dl^{\mathit{2}}$ $\pi (4) (4) (4) (4) 1 $2=$ $\underline{\pi_{2}42r2p2}$ 3. $\pi 12$. $m^{\mathit{2}}$ $+p^{2}$ (5) $\pi=\pi\underline{\overline{\ovalbox{\tt\small REJECT}}}/\pi_{1}$ (5) $(2 \pi R)^{2}$ 1 $2_{=}-$ $()^{2}\underline{p}+p^{2}$. 3 $m$ (6) (6)
224 $\mathit{1}=72$. 72406687 $\cdots 14$ ) $\pi=355/113_{\text{}}$ $p=5$ $\mathit{1}=$ 7 2 72406 (6) 2 3 ( ( 3 $\angle$ AOB 4 $\angle$ AOB 3 $(m)$ 3 $\pi $p_{\text{}}$ 4 4 =2 \mathrm{r}\cross$ $(P/m)$ (6) 40 Archimedean Spiral $5$ ) 39 1 $0$ 39 - +$\nearrow\gamma_{\text{}}$ + $+$ 0 + $+$ o 16) [ ] (2 r) 2 $(2 R)$ 8 2 $(m)$ 1 R\rho \star # $7\rangle$ 1 (1) $0$ ] -
225 [ ] 40 $\pi=\pi_{2}/\chi_{1}$ 17)] $\{\mathit{1} \pi_{1}-(2r+2\mathrm{r})\pi_{2}\}\mathrm{z}_{1}2m^{2}$ $=J^{2}\sim 12m^{2}$ $(_{\backslash }\mathit{1}_{1})$ $\{(2x)2\pi_{2}+23\mathit{1}D^{2}\pi_{1}\}2$ (2 r) 2 $=l1212m^{2}$... Ei\star I (1) $(2 R)^{22}\pi_{2}+3\mathit{1}22\pi 122\}$ $(2R)^{2}=\mathit{1}\mathrm{z}^{\mathit{2}}12m^{2}$... - (+ ) $\}$ 2, 1=2 4. 2 1 4 39 2347881998414 2950435785600 13 20572291 988256 12 $\neq 7116523129315200\mathit{1}$ $+$ $132095417940416400=0$ 4 $=$ 1 93. 35809048... $-30$. 52623208 85. 83227497... $-23$. $00041657\cdots$ 4 2 :1256 ( :1257) 1 4 1 4 2 85. 83227497...
$8\rangle$ $\text{}$ 226 4, EfrL\beta Archimedean Spiral 1 6 $\ovalbox{\tt\small REJECT}$ 1 : 1 3 ( ) 1 Efr4D H\not\in 1 2 5 $0333$ 19) 2 : 10 2 7 7 $03575$ 20) 2 = (6) $+$ 3 : 2 3 1 8 5 82227 21) 39 1 4 $J$ 2347881998414-2950435785600 $-2057229188256\mathit{1}^{2}+- 711652312935200\mathit{1}$ +13209541794041640 $0=0$ 93. 35809049 4 : 4 1 2 2 7 6 3 4719 22) 2 $3)_{\mathrm{O}}$ 5 : 1 1 2 1 2 3 6 1 52 24) 40 (6) 6 : 1 5 7 1 2 1 2 5 2 3 766
$\text{}$ 198 \mathrm{p}-\text{}$ pp. 227 5 2 3 1 1 (6) 4, 39 40 $\mathrm{e}^{\star}i\mathbb{i}$ 25) $\# t\mathrm{d}\div$ Archimedean Spiral (2) Archimedean Spiral E\mu fi 3 #\star f# $26$ ) 27) $\pi$ 1) =L : 19 2) $0_{\text{}}$ : Jl 9 8 3) $\mathrm{e}.\mathrm{j}$.dijkstrerhuis $\mathrm{c}.\mathrm{d}\mathrm{i}\mathrm{k}\mathrm{s}\mathrm{b}0\mathrm{o}\mathrm{n}\mathrm{l}^{\mathrm{p}_{\mathrm{n}}}\mathrm{n}\mathrm{c}\mathrm{e}\iota_{\mathrm{o}}\mathrm{n}$ Archimedes,Translated by $0\not\in University Press, 1987,$\mathrm{p}\mathrm{p}$ 4) : 1990 p. 929. 2 8. 264-277 5) 6) (1 714 1783) 6 (1769) 5 $\mathrm{p}.317$ 7) 4 \mbox{\boldmath $\zeta$}... Rt $\ovalbox{\tt\small REJECT}\hslash$ 8) [L 7 $ $ ([ $\mathrm{p}.317$) $\text{}$ 2 9) $\mathrm{p}$ P. 128 129 $6_{\mathrm{o}}$ 10) 2 pp. 195 19 11)
1998 228 12) :1256 13) 41 45 T 39 3 14) 40 $\mathit{1}=63$. 27127 1556 15) :1256 16) P. 6 :1256 17) 18) 19) $03333288\cdots$ 20) $03574872\cdots$ 21) 82227498 22) 4718789 23) Yoshimasa Michiwaki, Tatsulio Kobayashi: On ffie Resemblance Problems of $Lilava\hslash$, Chiu-Chang Suan-Shu and $\mathrm{w}\mathrm{a}\mathrm{m}$, 32 1 1987 pp. 105 106 24) 520292 $\ovalbox{\tt\small REJECT}_{\lambda 1}^{\wedge}$ 25) 41 6 26) : ( $4_{\text{}}$ 106 10 ) pp. 63 74 27) : ( \ 158 1998 pp. 1 $-14$ )