Title 角術への三角法の応用について ( 数学史の研究 ) Author(s) 小林, 龍彦 Citation 数理解析研究所講究録 (2001), 1195: 165-175 Issue Date 2001-04 URL http://hdl.handle.net/2433/64832 Right Type Departmental Bulletin Paper Textversion publisher Kyoto University
1195 2001 165-175 165 2000 8 21 $\sim 22$ *) 1 $(2R)_{\text{}}$ $(2r)_{\text{}}$ 1 1 $(a)$ $(S)_{\text{}}$ $(L)$ 1 ( ) ( 2:1712 ) 3 ( ) 1) 2 18 ( 13 :1763 $-$ 2:1831) (1789) ( 19 :1743 $-$ 4 :1807) 3(1803) 2 4 ( ) 10 $*)$ $\mathrm{c}$ 12 12680003
166 2) (1804) 3) 5 (1822) 4) 10 (1827) 5) \ o) ( :1804 ) 7) ${ }$, ( 3:1783 $-$ 4:1871) (1818) (1830) 3 46 8) 80 2 3-., ( ) $\square \square$ ( + ) 2. $3^{\cdot}$ 3 11(1726) 3-1 24.5\mathrm{c}\mathrm{m}_{\text{}}$ 73 ( ) 16 2 15.5 $\cross ( : )
167 6 8 0.5 45 30 11 ( ) ( ),, 1804 3 1 7 (1824) 41 3 3 (1820) 3-2 16
168 ( ) - ( 7 :1810 ) 5 $\vdash$ 7 $+$ $+$ $000$ $-+$ $+$ +7 $+$ $000$ $\overline{\tau}$ $0$ 7 7 $(a_{7})$ $a7^{=}4$ 3 5 273 7 $(2R)$ 10 360 \div 14 $=25.7\cdots$ 7 60 25 42 AB $=a_{7_{\text{}}}$ OC $=r$ $\angle$ AOC 25 42 26 $-$ 25 30 $=04305111$ 04383711 $=000786$ 12 ( $=25$ 42-25 30 ) 30 000786 $\cross 12\div 30=0003144$
$0003144=04352271$ 169 26 04383711 $(*)$ 25 42 1)${ }$ $a_{\overline{/}}=$ 25 42 $\cross 2R$ $=04352271\cross 10$ $=4.352\underline{271}$ 3 4336590845 $(*)$ (?), 1 1-7, 4 1 27 $\cross 19\mathrm{c}\mathrm{m}_{\text{}}$ 86 ) 25 42 043366
170 1 2 O 45 15 4 7 6 8 7.. 28 34 6 9, 1) $(2R)$ 10 3 $(R)_{\text{}}$ $(r)$ $(a_{3})$ 1 180 $\div 3=60$ $(r)$ $2r=2R$ s 60 $0$ $\gamma=r\sin 30^{\mathrm{o}}$ $=5\cross 0.5=2.\mathit{5}$ $=R$ $(a3)$ $a_{3}=2r\cos 60^{\mathrm{O}}$ $=10\cross 0.86603=8.6603$ 2) $(2R)$ 10 4 $(2R)_{\text{}}$ 2 $(r)$ $(a_{4})$ 180 \div 4 $=4\mathit{5}$ $(a_{4})$ $a_{4}=2r\sin 45\circ$
171 $=10\cross 0.70711=7.0711$ $a_{4}=2r\cos 450$ $=10\cross 0.70711=7.0711$ $=2$ $=2R$ $2R=(a_{4}/2)/\sin 450$ $\text{ ^{}\ovalbox{\tt\small REJECT}_{\text{}}}\ovalbox{\tt\small REJECT}$ \tau -, - i $\nearrow\backslash$ $\backslash \sqrt[\backslash ]{}\backslash ^{\backslash }\text{}$ \nearrow \yen 2 $(2R)$ $2R=2a_{4}$ sin45 $0$ 3) $(2R)$ 10 5 $(a_{5})$ $(L_{2})$ $(r)$ 180 $\div \mathit{5}=36$ $(r)$ $r=r\cos 36$ $=5\cross 0.80902=4.0451$ $(a_{5})$ $a_{5}=2r\sin 36^{\mathrm{o}}$ $=10\cross 0.58779=5.8779$ $(L_{2})$ $L_{2}=2R\cos 72^{\mathrm{O}}$ $=0.80902\cross 10=8.0902$ 4) $(2R)$ 10 $(a_{6})$ 6 $(L)$ $(r)$ c 180 $\div 6=30$ $(a_{6})$ $a_{6}=2r\sin 30^{\mathrm{o}}$ $=10\cross 0.50000$ $=5$ (r) $r=r\cos 30^{\mathrm{o}}$ $=\mathit{5}\cross 0.86603=4.3301\mathit{5}$ $(L)$ $L=2R\cos 60^{\mathrm{O}}$ $=10\cross 0.86603=8.6603$
172 5) $(2R)$ 10 7 $(a_{:}-)$ $(L_{2})_{\text{}}$ $(L_{3})$ $(_{f}\cdot)$ $-\neg$ 180 $\div 7=25.7$ 25 42 $(a_{7})$ $\mathit{0}_{\overline{j}}=2r$ sin25 $0$ $42 $ $ 10\cross 0.43130$ $=4.313$ $(r)$ $r=r\cos 25\circ 42 $ $=\mathit{5}\cross 0.90221$ $=4.51105$ $(L_{2})$ ( $L_{2}=2R$ cos51 $024 $ $=10\cross 0.78152=7.8152$ $(L_{3})$ $L_{3}=2R$ cos77 $- 06 $ $=10\cross 0.97476\cross=9.7476$ 8 6) $(2R)$ 10 $(a_{8})$ $(r)$ -, $(L_{2})_{\text{}}$ $(L_{3})$ 180 \div 8 $=22.5$ 22 30 $(a_{8})$ $a_{8}=2r\sin 22\circ 30 $ $=10\cross 0.38268=3.8268$ $(r)$ $rarrow-r\cos 22^{\mathrm{O}}$ $30 $ $=5\cross 0.92388=4.6194$ $(L_{2})$ $L_{2}=2a_{8}$ cos22 o30 $=0.92388$ $\mathrm{x}2\cross 3.8268=7.0711$ $L_{2}=2R$ cos45 $0$ $(L_{3})$ $L_{3}=2R\cos 670$ $30 $ $=10\cross 0.92388=9.2388$ 7) $(2R)$ 10 9 $(a_{9})$ $(L_{2})_{\text{}}$ $(L_{3})_{\text{}}$ $(L_{4})$ $(r)$
173 $(a_{9})$ 180 $\div 9=22$ $a_{9}=2r$ sin20 o $=10\cross 0.34202=3.4202$ (r) $r=r\cos 20$ $=5\cross 0.93969=9.3969$ ( 2 $L$ $L_{2}=2R\sin 40^{\mathrm{o}}$ $=10\cross 0.64279=6.4279$ ( 3 $L$ $L_{3}=2R\cos 60^{\mathrm{O}}$ $=10\mathrm{X}0.86603=8.6603$ $(L_{4})$ $L_{4}=2R\cos 80^{\mathrm{O}}$ $=10\cross 0.93969=9.3969$ 8) $(2R)$ 10 10 $(a_{10})$ i3 $(L_{2})_{\text{}}$ $(L_{3})_{\text{}}$ $(L_{4})$ $(r)$ $(a_{9})$ 180 $\div 10=18$ $a_{10}=2r$ sm18 $0$ $=10\cross 0.30902=3.0902$ (r) $r=r\mathrm{c}\mathrm{o}\mathrm{s}\mathrm{l}8$ $=5\cross 0.95106=4.7\mathit{5}53$ $(L_{2})$ $L_{2}=2R\sin 36^{\mathrm{o}}$ $=10\cross 0.58779=5.8779$ $(L_{3})$ $L_{3}=2R\cos \mathit{5}4^{\mathrm{o}}$ $=10\cross 0.80902=8.0902$ $(L_{4})$ $L_{4}=2R\cos 72^{\circ}$ $=10\cross 0.95106=9.5106$,,
$\mathrm{c}$ } $\llcorner.-$ 174 11 1840 57 4 $(a_{n})$ 17 18,, 4(1631) 4 ) $n$ $\mathrm{c}$, (1723) 21 3 10 $(S)$ $a,$, $S=na_{n}r/2$ ( $3\leqq n$ ) $r$ $(L)$ 1 10 4 $11$ ) 9(1877) $12$ ) 10 18 $-\text{}$
Sato, $693-175 1) $\mathrm{k}\mathrm{e}\mathrm{n}^{1}\mathrm{i}_{\mathrm{c}\mathrm{h}\mathrm{i}}$ On the Theory of Regular Polygons in Traditional Japanese Mathematics: Reconstruction of the Process for the Calculation of the Degree of Kaihoshiki Appearing in the Taisei Sankei by SEKI and TAKEBE Brothers, Historia Scientarum, Vol.8 No. 1, 1998, pp.71-85. $3_{\text{}}$ 2 11 55 pp.977-982. 3 21173. 4 20417. 5 1713. $-\text{_{}}$ 6 $-$ 62 P.77. $-\text{_{}}$ 7 $-$ p.92. $-\text{_{}}$ 8 $-$ p.80. 9 20 \nu -26. $\cdot$ 10 pp.3 3$.710.11) 62 pp.142-147. 11 $-\text{_{}}$ 12 $-$ pp.143-148.