查看“︁增乘开平方法”︁的源代码

[[File:Yanghui Yongle Dadian.JPG|thumb|right|300px|永乐大典杨辉增乘开方法]]
[[File:JIA XIAN SQRT2.GIF|thumb|right|300px|贾宪增乘开平方法]]

'''增乘开平方法'''是[[北宋]]数学家[[贾宪]]发明的[[开方]][[算法]]，原收《释锁算书》一书。贾宪原作已佚，但他对数学的重要贡献，被[[南宋]]数学家[[杨辉]]引用，被抄入《[[永乐大典]]》卷一万六千三百四十四，幸得以保存下来<ref>[[吴文俊]]主编 《[[中国数学史大系]]》第五卷 第二编第一章贾宪 30-40</ref>。现存[[英国]][[剑桥大学]]图书馆。
杨辉在所著《详解九章算法》《开方作法本元》一章中作贾宪开方作法图，并说明“''杨辉详解开方本源，出《释锁算书》，贾宪用此术''”。<ref>吴文俊主编 中国数学史大系第五卷 第六编第一章杨辉 565</ref>。

==术文==
增乘开平方法，以商数乘下法递增求之。

商第一位。上商得数以乘下法为乘方。命上商除实。上商得数以乘下法入乘方。一退为廉，下法再退。

商第二位。商得数以乘下法为隅。命上商除实讫。以上商乘下法入隅，皆名曰廉。一退，下法再退，以求第三位商数。

商第三位。用法如第二位求之。

==算筹十进位制布位==
*算筹的布位。贾宪在纸上用书写方式将算筹码按[[十进位制]]布位。七万一千八百二十四分写两行：

{| border="0" width="500" align="center" style="border: 1px solid #999; background-color:#FFFFFF"
|-align="center" bgcolor="#EFEFEF"
! 七
! 一
! 八
! 二
! 四
|- align="center" bgcolor="#EFEFEF"
| 万
| 千
| 百
| 十
| 步
|}

下行的步、十、百、千、万分离出来变为算筹的'''位值'''标签；上行七一八二四成为十进位制数码。

然后将算筹码依次排在相应的位值标签步、十、百、千、万之下：
{| border="0" width="500" align="center" style="border: 5px solid #999; background-color:#FFFFFF"
|-align="center" bgcolor="#EFEFEF"
! 七
! 一
! 八
! 二
! 四
|-align="center" bgcolor="#EFEFEF"
| 万
| 千
| 百
| 十
| 步
|-align="center" bgcolor="#EFEFEF"
| [[File:Counting rod v7.png|20px]]
| [[File:Counting rod h1.png|20px]]
| [[File:Counting rod v8.png|20px]]
| [[File:Counting rod h2.png|20px]]
| [[File:Counting rod v4.png|20px]]
|}

==杨辉算草==
杨辉以七万一千八百二十四为例，列出详细算草。
算草分四行，被除数放在第二行，称为'''实'''，第一行是商，第四行为'''下法'''，第三行是廉。
将[[算筹]][[File:Counting rod v7.png|15px]]放在第四行万字之下。<math>x^2=71824</math>

{| border="0" width="500" align="center" style="border: 5px solid #999; background-color:#FFFFFF"
|- align="center" bgcolor="#EFEFEF"
|   
|   
|   
|   
|   
|  商
|- align="center" bgcolor="#EFEFEF"
| [[File:Counting rod v7.png|20px]]
| [[File:Counting rod h1.png|20px]]
| [[File:Counting rod v8.png|20px]]
| [[File:Counting rod h2.png|20px]]
| [[File:Counting rod v4.png|20px]]
| 实
|-align="center" bgcolor="#EFEFEF"
|   
|   
|   
|   
|   
| 廉 
  
|-align="center" bgcolor="#EFEFEF"
|[[File:Counting rod v1.png|icon]] 
|  
| 
| 
| 
| 下法
|}
第一步先估计第一个商数得2，将[[File:Counting rod v2.png|icon]]安置在第一行百位置之上。
将上商2乘下法1，得数2，放在第三行。命上商除实，二二得四，从[[File:Counting rod v7.png|icon]]万减去四万得[[File:Counting rod h3.png|icon]]万。

{| border="0" width="500" align="center" style="border: 5px solid #999; background-color:#FFFFFF"
|-align="center" bgcolor="#EFEFEF"
|   
|   
| [[File:Counting rod v2.png|12px]]  
|   
|   
|  商
|-align="center" bgcolor="#EFEFEF"
| [[File:Counting rod v3.png|20px]]
| [[File:Counting rod h1.png|20px]]
| [[File:Counting rod v8.png|20px]]
| [[File:Counting rod h2.png|20px]]
| [[File:Counting rod v4.png|20px]]
| 实
|-align="center" bgcolor="#EFEFEF"
| [[File:Counting rod v2.png|12px]]  
|   
|   
|   
|   
| 廉 
|-align="center" bgcolor="#EFEFEF"
|[[File:Counting rod v1.png|icon]] 
|  
| 
| 
| 
| 下法
|}
求第二个商数：将上商乘下法二一得二，加入廉二，成为[[File:Counting rod v4.png|icon]]。
将廉数[[File:Counting rod v4.png|icon]]（万）退一位成为[[File:Counting rod h4.png|icon]]（千）；将下法[[File:Counting rod v1.png|icon]]退二位。

作减根变换 <math>x=x+200</math>，得
<math>x^2+520x=4224</math>

{| border="0" width="500" align="center" style="border: 5px solid #999; background-color:#FFFFFF"
|-align="center" bgcolor="#EFEFEF"
|   
|   
| [[File:Counting rod v2.png|12px]]  
|   
|   
|  商
|- align="center" bgcolor="#EFEFEF"
| [[File:Counting rod v3.png|20px]]
| [[File:Counting rod h1.png|20px]]
| [[File:Counting rod v8.png|20px]]
| [[File:Counting rod h2.png|20px]]
| [[File:Counting rod v4.png|20px]]
| 实
|- align="center" bgcolor="#EFEFEF"
|  
| [[File:Counting rod h4.png|12px]]   
|   
|   
|   
| 廉 
|- align="center" bgcolor="#EFEFEF"
|   
|  
| [[File:Counting rod v1.png|icon]] 
| 
| 
| 下法
|}

估第二位商数得6，将新商数6乘下法1（百）得6（百）并入廉。
{| border="0" width="500" align="center" style="border: 5px solid #999; background-color:#FFFFFF"
|-align="center" bgcolor="#EFEFEF"
|   
|   
| [[File:Counting rod v2.png|12px]]  
| [[File:Counting rod h6.png|12px]]   
|   
|  商
|-align="center" bgcolor="#EFEFEF"
| [[File:Counting rod v3.png|20px]]
| [[File:Counting rod h1.png|20px]]
| [[File:Counting rod v8.png|20px]]
| [[File:Counting rod h2.png|20px]]
| [[File:Counting rod v4.png|20px]]
| 实
|-align="center" bgcolor="#EFEFEF"
|  
| [[File:Counting rod h4.png|12px]]   
| [[File:Counting rod v6.png|12px]]  
|   
|   
| 廉 
|-align="center" bgcolor="#EFEFEF"
|   
|  
| [[File:Counting rod v1.png|icon]] 
| 
| 
| 下法
|}

从实除去商6乘廉46
{| border="0" width="500" align="center" style="border: 5px solid #999; background-color:#FFFFFF"
|-align="center" bgcolor="#EFEFEF"
|   
|   
| [[File:Counting rod v2.png|12px]]  
| [[File:Counting rod h6.png|12px]]   
|   
|  商
|-align="center" bgcolor="#EFEFEF"
| 
| [[File:Counting rod h4.png|20px]]
| [[File:Counting rod v2.png|20px]]
| [[File:Counting rod h2.png|20px]]
| [[File:Counting rod v4.png|20px]]
| 实
|-align="center" bgcolor="#EFEFEF"
|  
| [[File:Counting rod h4.png|12px]]   
| [[File:Counting rod v6.png|12px]]  
|   
|   
| 廉 
|-align="center" bgcolor="#EFEFEF"
|   
|  
| [[File:Counting rod v1.png|icon]] 
| 
| 
| 下法
|}

商数6乘下法1，并入廉
{| border="0" width="500" align="center" style="border: 5px solid #999; background-color:#FFFFFF"
|-align="center" bgcolor="#EFEFEF"
|   
|   
| [[File:Counting rod v2.png|12px]]  
| [[File:Counting rod h6.png|12px]]   
|   
|  商
|-align="center" bgcolor="#EFEFEF"
| 
| [[File:Counting rod h4.png|20px]]
| [[File:Counting rod v2.png|20px]]
| [[File:Counting rod h2.png|20px]]
| [[File:Counting rod v4.png|20px]]
| 实
|-align="center" bgcolor="#EFEFEF"
|  
| [[File:Counting rod h5.png|12px]]   
| [[File:Counting rod v2.png|12px]]  
|   
|   
| 廉 
|-align="center" bgcolor="#EFEFEF"
|   
|  
| [[File:Counting rod v1.png|icon]] 
| 
| 
| 下法
|}

廉数退一位，下法退二位，估第三位商得8；商乘下法并入廉
{| border="0" width="500" align="center" style="border: 5px solid #999; background-color:#FFFFFF"
|-align="center" bgcolor="#EFEFEF"
|   
|   
| [[File:Counting rod v2.png|12px]]  
| [[File:Counting rod h6.png|12px]]   
| [[File:Counting rod v8.png|12px]]  
|  商
|-align="center" bgcolor="#EFEFEF"
| 
| [[File:Counting rod h4.png|20px]]
| [[File:Counting rod v2.png|20px]]
| [[File:Counting rod h2.png|20px]]
| [[File:Counting rod v4.png|20px]]
| 实
|-align="center" bgcolor="#EFEFEF"
|  
| 
| [[File:Counting rod v5.png|12px]]   
| [[File:Counting rod h2.png|12px]]    
| [[File:Counting rod v8.png|12px]]   
| 廉 
|-align="center" bgcolor="#EFEFEF"
|     
|  
| 
| 
| [[File:Counting rod v1.png|icon]] 
| 下法
|}

从实除去商乘廉；净除。七万一千八百二十四的平方根是二百六十八。

{| border="0" width="500" align="center" style="border: 5px solid #999; background-color:#FFFFFF"
|-align="center" bgcolor="#EFEFEF"
|      
|      
| [[File:Counting rod v2.png|12px]]  
| [[File:Counting rod h6.png|12px]]   
| [[File:Counting rod v8.png|12px]]  
|  商
|-align="center" bgcolor="#EFEFEF"
|      
|      
|       
|       
|    
| 实
|-align="center" bgcolor="#EFEFEF"
|     
|    
| [[File:Counting rod v5.png|12px]]   
| [[File:Counting rod h2.png|12px]]    
| [[File:Counting rod v8.png|12px]]   
| 廉 
|-align="center" bgcolor="#EFEFEF"
|        
|     
|     
|     
| [[File:Counting rod v1.png|icon]] 
| 下法
|}

用增乘开平方法可解得<math>x=\sqrt{71824}=\,</math>{{Root|71824|use math=yes}} 。

== 参考文献 ==
=== 引用 ===
{{Reflist|30em}}

=== 来源 ===
* [[钱宝琮]]：《增乘开方法的历史发展》，《李俨钱宝琮科学史全集》卷9 479-488页

== 参见 ==
{{Portal box|数学|中国数学史}}
* [[平方根]]
* [[贾宪]]
** [[贾宪三角形]]
** [[增乘开立方法]]

{{-}}
{{中国数学史}}

[[Category:宋朝數學]]
[[Category:数学近似]]