调日法

何承天调日法原理

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

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

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

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

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

应用

朔望月

${\displaystyle {\frac {9}{17}}\ }$.
${\displaystyle {\frac {26}{49}}\ }$.
${\displaystyle {\frac {35}{66}}\ }$.
${\displaystyle {\frac {61}{115}}\ }$.
${\displaystyle {\frac {87}{164}}\ }$.
${\displaystyle {\frac {113}{213}}\ }$.
${\displaystyle {\frac {139}{262}}\ }$.
………
${\displaystyle {\frac {347}{654}}\ }$.
${\displaystyle {\frac {373}{703}}\ }$.
${\displaystyle {\frac {399}{752}}\ }$.

727年唐朝天文学家一行大衍历法中用同样的弱率和强率求得更精确的 ${\displaystyle {\frac {1613}{3040}}\ }$

${\displaystyle {\frac {9}{17}}\ }$.
${\displaystyle {\frac {26}{49}}\ }$.
${\displaystyle {\frac {35}{66}}\ }$.
${\displaystyle {\frac {61}{115}}\ }$.
………………
${\displaystyle {\frac {1561}{2942}}\ }$.
${\displaystyle {\frac {1587}{2991}}\ }$.
${\displaystyle {\frac {1613}{3040}}\ }$.

闰周问题

${\displaystyle {4 \over 11}}$
${\displaystyle {7 \over 19}}$
${\displaystyle {11 \over 30}}$
${\displaystyle {18 \over 49}}$
${\displaystyle {25 \over 68}}$
……………………
${\displaystyle {130 \over 353}}$
${\displaystyle {137 \over 372}}$
${\displaystyle {144 \over 391}}$

近点月

${\displaystyle {56 \over 101}}$.
${\displaystyle {5 \over 9}}$.
${\displaystyle {61 \over 110}}$.
${\displaystyle {117 \over 211}}$.
……………………
${\displaystyle {356 \over 642}}$.
${\displaystyle {361 \over 651}}$.
${\displaystyle {417 \over 752}}$.

${\displaystyle {356 \over 642}}$.
${\displaystyle {361 \over 651}}$.
${\displaystyle {417 \over 752}}$. 何承天近点月。
……………………
${\displaystyle {14621 \over 26359}}$.
${\displaystyle {14626 \over 26368}}$.
${\displaystyle {14631 \over 26377}}$. 祖冲之的近点月。

圆周率约率和密率

${\displaystyle {3 \over 1}}$为弱率以${\displaystyle {4 \over 1}}$为强率，用调日法进行计算：

${\displaystyle {3 \over 1}}$= 3 周髀算經》周三径一
${\displaystyle {4 \over 1}}$= 4
${\displaystyle {7 \over 2}}$= 3.5
${\displaystyle {10 \over 3}}$= 3.3333333333
${\displaystyle {13 \over 4}}$= 3.25
${\displaystyle {16 \over 5}}$= 3.2
${\displaystyle {19 \over 6}}$= 3.1666666667
${\displaystyle {22 \over 7}}$= 3.1428571429 祖冲之约率
${\displaystyle {25 \over 8}}$= 3.125
${\displaystyle {47 \over 15}}$= 3.1333333333
${\displaystyle {69 \over 22}}$= 3.1363636364
${\displaystyle {91 \over 29}}$= 3.1379310345
${\displaystyle {113 \over 36}}$= 3.1388888889
${\displaystyle {135 \over 43}}$= 3.1395348837
${\displaystyle {157 \over 50}}$= 3.14 刘徽圆周率：徽率
${\displaystyle {179 \over 57}}$= 3.1403508772
${\displaystyle {201 \over 64}}$= 3.140625
${\displaystyle {223 \over 71}}$= 3.1408450704
${\displaystyle {245 \over 78}}$= 3.141025641
${\displaystyle {267 \over 85}}$= 3.1411764706
${\displaystyle {289 \over 92}}$= 3.1413043478
${\displaystyle {311 \over 99}}$= 3.1414141414
${\displaystyle {333 \over 106}}$= 3.141509434
${\displaystyle {355 \over 113}}$= 3.1415929204 祖冲之密率

黄金分割与斐波那契数列

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

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

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. ^ 中國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