新符号学 (第5/6页)

\frac{a}{b}}=d_{1}}以及{\dispystyle{\frac{c}{d}}-x=d_{2}}{\dispystyle{\frac{c}{d}}-x=d_{2}}

    如果有正整数m,k满足：{\dispystyle{\frac{kd}{mb}}={\frac{d_{1}}{d_{2}}}}{\dispystyle{\frac{kd}{mb}}={\frac{d_{1}}{d_{2}}}}

    那麽就有：{\dispystylex={\frac{ma kc}{mb kd}}}{\dispystylex={\frac{ma kc}{mb kd}}}

    证明如下：由条件可得

    {\dispystyle{\begin{aligned}bd_{1}&=bx-a\\dd_{2}&=c-dx\end{aligned}}}{\dispystyle{\begin{aligned}bd_{1}&=bx-a\\dd_{2}&=c-dx\end{aligned}}}

    而根据{\dispystyle{\frac{kd}{mb}}={\frac{d_{1}}{d_{2}}}}{\dispystyle{\frac{kd}{mb}}={\frac{d_{1}}{d_{2}}}}又有

    {\dispystylembd_{1}=kdd_{2}}

    代入上面的两个关系式可得：

    {\dispystylembx-a=kc-dx}

    2

    解关於x的一元一次方程就有结果：

    {\dispystylex={\frac{ma kc}{mb kd}}}

    应用编辑

    何承天调日法被同时代和後代数学家如赵爽，祖冲之，一行等运用。

    朔望月编辑

    何承天将{\dispystyle{\frac{9}{17}}=0.529412...\}{\dispystyle{\frac{9}{17}}=0.529412...\}作为朔望月零数部分的弱率，以{\dispystyle{\frac{26}{49}}=0.530612...\}{\dispystyle{\frac{26}{49}}=0.530612...\}作为朔望月零数部分的强率。运用调日法，最後得到{\dispystyle{\frac{399}{752}}\}{\dispystyle{\frac{399}{752}}\}，根据他的观测数值0.530585，首先计算d1,d2

    {\dispystyle{\begin{aligned}d_{1}&=0.530585-0.529412&=0.001173\\d_{2}&=0.530612-0.530585&=0.000027\end{aligned}}}{\dispystyle{\begin{aligned}d_{1}&=0.530585-0.529412&=0.001173\\d_{2}&=0.530612-0.530585&=0.000027\end{aligned}}}

    寻找满足以下关系的m,k值：

    {\dispystyle{\begin{aligned}{\frac{49k}{17m}}&={\frac{1173}{27}}\\{\frac{k}{m}}&={\frac{1173\times17}{49\times27}}&\approx{}15.07\ldots\end{aligned}}}{\dispystyle{\begin{aligned}{\frac{49k}{17m}}&={\frac{1173}{27}}\\{\frac{k}{m}}&={\frac{1173\times17}{49\times27}}&\approx{}15.07\ldots\end{aligned}}}

    可以令m,k=1,15

    2

    从而得到：

    {\dispystyle{\frac{1\times9 15\times26}{1\times17 15\times49}}={\frac{399}{752}}}{\dispystyle{\frac{1\times9 15\times26}{1\times17 15\times49}}={\frac{399}{752}}}

    727年唐朝天文学家一行在大衍历法中用同样的弱率和强率求得{\dispystyle{\frac{1613}{3040}}}{\dispystyle{\frac{1613}{3040}}}

    闰周问题编辑

    南北朝数学家祖冲之熟悉调日术，他以{\dispystyle{\frac{4}{11}}}{\dispystyle{\frac{4}{11}}}为弱率，以{\dispystyle{\frac{7}{19}}}{\dispystyle{\frac{7}{19}}}为强率，通过调日法得到{\dispystyle{\frac{144}{391}}}{\dispystyle{\frac{144}{391}}}

    近点月编辑

    何承天以{\dispystyle{\frac{56}{101}}}{\dispystyle{\frac{56}{101}}}为弱率，以{\dispystyle{\frac{5}{9}}}{\dispystyle{\frac{5}{9}}}为强率，用调日法求得近点月为{\dispystyle{\frac{417}{752}}}{\dispystyle{\frac{417}{752}}}。祖冲之也得到高JiNg度的数值{\dispystyle{\frac{14631}{26377}}}{\dispystyle{\frac{14631}{26377}}}

    圆周率约率和密率编辑

    祖冲之求圆周率约率和密率的方法已失传。有学者认为他用刘徽割圆术求得圆周率的约率和密率；也有学者认为祖冲之有可能用何承天的调日法求得圆周率的约率和密率的分数表示式[2]。祖冲之对调日法是熟悉的，他自己就用过调日法改进何承天近点月{\dispystyle{\frac{417}{752}}}为更加JiNg确的{\dispystyle{\frac{14631}{26377}}}

    取{\dispystyle\pi\approx3.1416}，先只考虑小数部分，根据{\dispystyle{\frac{1}{8}}<0.1416<{\frac{1}{7}}}，用调日法进行计算：

    2

    {\dispystyle{\begin{aligned}d_{1}&=0.1416-0.125