调日法

何承天调日法原理

${\displaystyle {\frac {a}{b}}<{\frac {a+c}{b+d}}<{\frac {c}{d}}}$

${\displaystyle {\frac {a}{b}}<{\frac {ma+kc}{mb+kd}}<{\frac {c}{d}}}$ ，其中 m,k 为正整数。

${\displaystyle f_{0}={\frac {a}{b}}}$ 为弱率，${\displaystyle f_{1}={\frac {c}{d}}}$ 为强率。

${\displaystyle f_{2}={\frac {a+c}{b+d}}}$

${\displaystyle f_{3}={\frac {a+c+a}{b+d+b}}\ }$

${\displaystyle f_{3}={\frac {a+c+c}{b+d+d}}\ }$

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

${\displaystyle mbd_{1}=kdd_{2}}$

${\displaystyle m(bx-a)=k(c-dx)}$

${\displaystyle x={\frac {ma+kc}{mb+kd}}}$

应用

朔望月

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

{\displaystyle {\begin{aligned}{\frac {49k}{17m}}&={\frac {1173}{27}}\\{\frac {k}{m}}&={\frac {1173\times 17}{49\times 27}}&\approx {}15.07\ldots \end{aligned}}}

${\displaystyle {\frac {1\times 9+15\times 26}{1\times 17+15\times 49}}={\frac {399}{752}}}$

727年唐朝天文学家一行在《大衍历》中用同样的弱率和强率求得 ${\displaystyle {\frac {1613}{3040}}}$

圆周率约率和密率

${\displaystyle \pi \approx 3.1416}$ ，先只考虑小数部分，根据${\displaystyle {\frac {1}{8}}<0.1416<{\frac {1}{7}}}$ ，用调日法进行计算：

{\displaystyle {\begin{aligned}d_{1}&=0.1416-0.125&=0.0166\\d_{2}&=0.142857-0.1416&=0.001257\end{aligned}}}

{\displaystyle {\begin{aligned}{\frac {7k}{8m}}={\frac {0.0166}{0.001257}}\\{\frac {k}{m}}={\frac {8\times 0.0166}{7\times 0.001257}}&\approx {}15.09\ldots \end{aligned}}}

${\displaystyle 3+{\frac {1\times 1+1\times 15}{1\times 8+15\times 7}}=3+{\frac {16}{113}}={\frac {355}{113}}}$

黄金分割与斐波那契数列

${\displaystyle \varphi ={\frac {{\sqrt {5}}+1}{2}}\approx 1.6180339887...}$

${\displaystyle {\frac {1}{1}},{\frac {2}{1}},{\frac {3}{2}},{\frac {5}{3}},{\frac {8}{5}},{\frac {13}{8}},{\frac {21}{13}},{\frac {34}{21}},{\frac {55}{34}},{\frac {89}{55}},{\frac {144}{89}},{\frac {233}{144}},{\frac {377}{233}},{\frac {610}{377}},{\frac {987}{610}},{\frac {1597}{987}},{\frac {2584}{1597}},{\frac {4181}{2584}}}$

其他

• √2=1.4142135623 ~=${\displaystyle {99 \over 70}}$
• √3=1.7320508075 ~=${\displaystyle {71 \over 41}}$
• √5=2.2360679775 ~=${\displaystyle {199 \over 89}}$
• √10=3.162277660 ~=${\displaystyle {117 \over 37}}$
• ${\displaystyle {\sqrt[{12}]{2}}}$ =1.059463094~=${\displaystyle {107 \over 101}}$
• e=2.718281828 ~=${\displaystyle {2721 \over 1001}}$
• 普朗克常数 ~=${\displaystyle {53 \over 8}}$ x10-34
• 万有引力常数 G~=${\displaystyle {227 \over 34}}$ x10-11
• 阿伏伽德罗常量~=${\displaystyle {241 \over 40}}$ x1023
• 玻尔兹曼常数~=${\displaystyle {29 \over 21}}$ x10-23

参考文献

1. ^ 中国古时将天文数据的小数部分的分母称为“日”，“调日术”即是调节分母的意思。
2. ^ 吴文俊 主编 《中国数学史大系》第四卷 123页，ISBN7-300-0425-8/O
3. ^ 傅海伦编著 《中外数学史概论》 第四章 刘徽的割圆术 51页 科学出版社，ISBN978-7-03-018477-1
4. ^ 吴文俊 主编 《中国数学史大系》第四卷 125页，ISBN7-300-0425-8/O