大成算経巻之十六(權術)について (数学史の研究)

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1 ( ) (Yasuo Fujii) Seki Kowa Institute of Mathematics, Yokkaichi Univeresity ( ) 3) ( 1) ( ) ( 4) ( ) 1 ( ) 1 ( ) $\triangleright\backslash$, 2

2 66 O 1 2,3,4, $\mathscr{d}$ 2 ( ) ( ) ( ) ( ) ( ) ( ) ( ) [ ] ( ) ( ) ( ) ( ) ( ) ( $)$

3 looo ( $)$

4 $-\ovalbox{\tt\small REJECT} 4$ 68 ( ) O $\text{ ^{}\prime}$ ( ) ( ) O 05 O O ( ) $=$03, $=$ $0.3\cross 10=3$ 21 $21\div 20=1.05$ 2 $03$ $1.35\cross 10=13.5$ 105 $10.5\div 20=0.525$ 3 $03$ $0.825\cross 10=8.25$ 1575 $15.75\div 20=0.7875$

5 $1.0875\cross 10=10.875$ $13.125\div 20= $ $ \cross 10=9.5625$ $ \div 20= $ $a_{n}arrow a$ $b_{n}arrow b$ $10a+20b=24a=b+O.3a=1b=0.7$ $b_{0}=0$ $a_{1}=0.3$ $b_{1}=(24-10a_{1}) \div 20=\frac{24}{20}-\frac{3}{20}=\frac{21}{20}$ $a_{2}=b_{1}+0.3=- \underline{24}\underline{3}+\underline{3}b_{2}=(24-10a_{2})\div 20=\frac{24}{20}-\frac{24}{20\cdot 2}+\frac{3}{20\cdot 2}-\frac{3}{20}$ $a_{3}=b_{2}+0.3= \frac{24}{20}-\frac{24}{20\cdot 2}+\frac{3}{20\cdot 2}$ $\frac{3}{20}+\frac{3}{10}$ $b_{3}= \frac{24}{20}-\frac{24}{20\cdot 2}+\frac{24}{20\cdot 2^{2}}-\frac{3}{20\cdot 2^{2}}+\frac{3}{20\cdot 2}-\frac{3}{20}$ $b= \frac{24}{20}\{1-\frac{1}{2}+\frac{1}{2^{2}}-\cdots\}-\frac{3}{20}\{1-\frac{1}{2}+\frac{1}{2^{2}}-\cdots\}$ $= \frac{24}{20}\frac{1}{1+\frac{1}{2}}+\frac{3}{20}\frac{1}{1+\frac{1}{2}}=\frac{24}{20}\frac{2}{3}-\frac{3}{20}\frac{2}{3}=\frac{7}{10}$ $a=b+0.3=1$ O OO O ( ) ( )2 ( )1 6 7O

6 $\div$ (70) $(16)=4.375\cdots$ 1 $=( )\cross 16\div 20=12.5$ $=( )4.375\div 2= $ $= = $ $\ovalbox{\tt\small REJECT}=\frac{ }{12.5}=0.6125$ $+$ $=$ $-\ovalbox{\tt\small REJECT}=( )\cross 16\div 20=12.01$ $=( )4.9875\div 2= $ $= = $ $\frac{\text{ }}{(R\overline{E}}=\frac{ }{12.01}= $ $+$ $=$ $a_{1}= \frac{70}{16}$ $7 \ovalbox{\tt\small REJECT} b_{1}=(20-a_{1})\frac{16}{20}$ $c_{1}=(16+b_{1}) \frac{a_{1}}{2}$ $d_{1}$ $=$ 70( $c_{1}$ ) $\text{ _{}\ovalbox{\tt\small REJECT}}=\frac{d_{1}}{b_{1}}=e_{1}$ $a_{2}=a_{1}+e_{1}b_{2}=(20-a_{2}) \frac{16}{20}c_{2}=(16+b_{2})a_{2}\div 2d_{2}=70-c_{2}$ $( \text{ _{}\ovalbox{\tt\small REJECT}}=\frac{d_{2}}{b_{2}}=e_{2}a_{3}=a_{2}+e_{2}$ $c_{2}= \frac{\{16+(20-a_{1})\frac{16}{20}\}a_{1}}{2}=\frac{16a_{1}+16a_{1}-\frac{16}{20}a_{1}^{2}}{2}=\frac{32a_{1}-\frac{16}{20}a_{1}^{2}}{2}$ $d_{2}=70(a)-c_{1}$ $e_{1}= \frac{d_{2}}{b_{2}}=\frac{140(2a)-(32a_{1}-\frac{16}{20}a_{1}^{2})}{2(20-a_{1})\frac{16}{20}}=\frac{35(a/2)\cross 5-40a_{1}+a_{1}^{2}}{2(20-a_{1})}$ $a_{2}=a_{1}+e_{1}= \frac{35(a/2)\cross 5-a_{1}^{2}}{2(20-a_{1})}$ $a_{i+1}= \frac{35\cross 5-a_{i}^{2}}{2(20-a_{i})}$ $5-a_{i+1}= \frac{25-10a_{i}+a_{i}^{2}}{2(20-a_{i})}=\frac{(5-a_{i})^{2}}{2(20-a_{i})}$ $ 5-a_{i+1} = \frac{ 5-a_{i} ^{2}}{ 2(20-a_{i}) }<\frac{1}{2} 5-a_{i} ^{2}arrow 0(a_{i}>a_{1}=\frac{70}{16})$ $a_{n}arrow 5$ $\frac{(16+b)a}{2}=70$ $b:(20-a)=16$ : 20 $b=(20-a) \frac{16}{20}$ $a^{2}-40a+175=0$ $(a-5)(a-35)=0a=5$ 1

7 -51) O O O OOO O $\frac{\pi}{4}=0.7854$ $\mathscr{d}$ $=5- \frac{ }{10}= =2.573$ $=\sqrt{431}$ ( $\mathscr{c}$ 1 $=2.5$ $=\sqrt{4\cross 25\cross 75}=v\sqrt{75}= $, $= $ 1 $\mathscr{d}$ $=$ 2 $ \div = =2.459$ $= $ $= $ 2 $\mathscr{d}$ $=$ 3 $ \div = = $ $c 0=\frac{R}{2}=r$, $S_{0}= \frac{\pi r^{2}}{2}$, $a_{0}=r=2r$ $A$ $r- \frac{\frac{\pi r^{2}}{2}-a}{r}=c_{1}c_{0}-\frac{s_{0}-a}{a_{0}}=c_{1}s_{1},$ $a_{1}$ $c_{1}- \frac{s_{1}-a}{a_{1}}=c_{2}s_{2},$ $a_{2}$ $= \frac{1}{4}$ ( $-2$ { $\cross$ ) } $A= \frac{1}{4}\{sr-(r-2c)a\}a=\sqrt{r^{2}-(r-2c)^{2}}=\sqrt{4c(r-c)}$

8 72 $s=r$ arcsin $\frac{a}{r}=r\arccos\frac{r-2c}{r}$ $S_{0}= \frac{1}{4}\{s_{0}r-(r-2c_{0})a_{0}\}=\frac{\pi rr}{4}=\frac{\pi r^{2}}{2}$ $S_{1}= \frac{1}{4}\{s_{1}r-(r-2c_{1})a_{1}\},$ $s_{1}=r \arccos\frac{r-2c_{1}}{r},$ $a_{1}=\sqrt{4c_{1}(r-c_{1})}$ $f(c)=a$, $f(c_{0})=s_{0}$ $\frac{f(c_{0})-f(c)}{c_{0}-c_{1}}=a_{0}$, $\frac{f(c_{i})-f(c)}{c_{i}-c_{i+1}}=a_{i}$ $f(c)= \frac{r^{2}}{4}\arccos\frac{r-2c}{r}-\frac{1}{4}(r-2c)\sqrt{4c(r-c)}$ $( \arccos\frac{r-2c}{r}) =\frac{1}{\sqrt{c(r-c)}}$ $\{(R-2c)\sqrt{c(R-c)}\} =-2\sqrt{c(R-c)}+\frac{(R-2c)^{2}}{2\sqrt{c(R-c)}}$ $f(c) = \frac{r^{2}}{4\sqrt{c(r-c)}}-\frac{1}{2}\{-2\sqrt{c(r-c)}+\frac{(r-c)^{2}}{2\sqrt{c(r-c)}}\}=2\sqrt{c(r-c)}=a$ $\frac{f(c_{i})-f(c)}{c_{i}-c_{i+1}}=f(c_{i})=a_{i}$ $c_{i}- \frac{f(c_{i})-f(c)}{a_{i}}=c_{i+1}$ 1 1 $a$,, $c$ $x$ $=16cx$, $=4c^{2}+a^{2}$, $=$ $(($ $-8c^{2})a+$ $)^{2}$, $= c^{8}$ $+81$ 5 $A= c^{1} c^{6}$ $ c^{4}$ $ c^{2}$ 4 $=$ $B= c^{4}$ $\cross$ $A-B=0$ $=$

9 73 $B$ $c^{4}$ $c$ ( ) ( ) $B= c^{2}$ $\cross$ $=$ $c$ 2 $(c^{2})$ $=2.5$ $=$ $=2.459$ $=$ $30\div 5=6$, 6 $\cross 21=126$ ( ) REJECT})$ $(-\ovalbox{\tt\small $a$ $b$,, $S,$ $S=18$, ( ) $h$ $\sqrt{a}$ $x$ $h-x=\sqrt{b}$ $(h-x)^{2}=b$

10 74 $25-10x+x^{2}=b$ $a\cross$ $=2S=25x^{2}-10x^{3}+x^{4}$ $2S=36$ $-36+25x^{2}-10x^{3}+x^{4}=0$ $(x-3)(x+1)(x^{2}-8x+12)=0$, $x=3,$ $a=9,$ $b=4$ ( ) (( ), ( )) $b$ $a$,, $c,$ $c=37$, $S,$ $S=210$ $c^{2}-4s=37^{2}-4\cross 210= =529$ $S= \frac{1}{2}$ ab, $c^{2}-4s=(a^{2}+b^{2})-2ab=(b-a)^{2}$ $\sqrt{529}=b-a=23$ $a+(b-a)=a+23=b$, $ab=a^{2}+23a=2s$ $2S=420$ $ a+a^{2}=0$ $(a-12)(a+35)=0$, $a=12,$ $b=35$

11 75 ( ) 1 (3) 73 ( ) $a$ $b$,, $S$, $a-3=b$ $b^{2}=9-6a+a^{2}$ $a^{2} \frac{3}{4}+$ $=S,$ $9-6a1.75a^{2}$ :, ( $\frac{3}{4}$ ) $S=73$ $-64-6a^{2}+1.75a^{2}=0$ $(a-8)(1.75a+8)=0,$ $a=8,$ $b=5$ $+$ $+$ $n$, $b,$ $b=11$ $d,$ $d=2$, $a,$ $a=75$, $n(n-1)d \frac{1}{2}=-n+n^{2}$ $a-(-n+n^{2})=bn,$ $75+n-n^{2}$ $bn=11n$ $-75+10n+n^{2}=0$ $(n-5)(n+15)=0,$ $n=5$ $b,$ $b=11$, $d,$ $d=2$, $a= \frac{n\{2b+(n-1)d}{2}$ $a=bn+n(n-1),$ $75-(-n+n^{2})=$ lln

12 $R$, $D$, $a,$ $a=8$, c, c $=2$, $\frac{c^{2}+(\frac{a}{2})^{2}}{c}=d,$ $\frac{2^{2}+4^{2}}{2}=10$ $=D-2c,$ $10-2\cross 2=6$ $(6+R)^{2}=36+12R+R^{2}$ $(D-R)^{2}$ $=(10-R)^{2}-(36+12R+R^{2})=64-32R=R^{2}$ \copyright, $R^{2}$ $64-32R-R^{2}=0$ $R=8(V5-2)= $ $D^{2}=(D-2c)^{2}+a^{2},$ $a^{2}-4dc+4c^{2}=0,$ $D= \frac{4c^{2}+a^{2}}{4c}$ $\{(D-2c)+R\}^{2}+R^{2}=(D-R)^{2}$, $R^{2}+4(D-c)R-4c(D-c)=0\cdots$ $R=2\sqrt{D(D-c)}-2$ (D c) $=$ 10 $\cross$ 2 $\sqrt{}$ $\cross$ $=$ 8 $($ $-2)= $ $R^{2}+4 \{\frac{c^{2}+(\frac{a}{2})^{2}}{c}-c\}r-4c\{\frac{c^{2}+(\frac{a}{2})^{2}}{c}-c\}=0$ $R^{2}+ \frac{a^{2}}{c}r-a^{2}=0,$ $cr^{2}+a^{2}r-a^{2}c=0$ $R= \frac{a(\sqrt{a^{2}+4c^{2}}-a)}{2c}=2(\sqrt{80}-8)=8(\sqrt{5}-2)$

13 V O ( ) 8 ( ) $\cross$ $= $, $=\sqrt{2}$ O $5^{3}\cross $ $= $ $a$ $( \sqrt{2}a)^{3}-8\cross(\frac{\sqrt{2}}{2}a)^{3}\frac{1}{6}=2\sqrt{2}a^{3}-2\sqrt{2}a^{3}\frac{1}{6}=\frac{5\sqrt{2}}{3}a^{3}$, $= \frac{5\sqrt{2}}{3}= $ ( )

14 78 $=6$, $=7$, $a$, $h$, $S$ $\cross a=$ $\cross h$ $\cross ( a$) $a=2s\cross$ $6a^{2}$ $2s\cross$ $=1176$ $a^{2}=0$ $a=\sqrt{196}=14$ $h= \frac{\sqrt{3}}{2}a=\frac{6}{7}a=$ a [1] [2] [3] ( ) 2011 [4] 2011