**納皮爾的骨頭**（英語：）是蘇格蘭數學家約翰納皮爾發明的一種用來計算乘法與除法，類似算盤的工具。由一個底座及九根圓柱（方柱）組成，可以把乘法運算轉為加法，也可以把除法運算轉為減法。更為進階的用法也可以開平方根

底座跟柱子

納皮爾的骨頭在清初传入中国，数学家梅文鼎在《梅氏丛书辑要》中最先介绍納皮爾的骨頭，梅氏称之为"筹算"。后来戴震著作《策算》也叙述了这种算法。

清代骨质算筹。中国财税博物馆藏。

簡介
--

底座左邊的數字由上而下是1-9。先將之稱為n。

每根圓柱又分為十格，下面九格有自左下而右上的斜線。最上面的格子寫著1-9的數字，姑且稱為k，代表著這根柱子的編號。編號下面的數字則是由k乘上橫排相對應的n得到的，十位數擺左上角，個位數擺右下角。

### 乘法運算

舉例說明如何用納皮爾的骨頭進行乘法運算。

**46785399**乘以**7**

1.把編號4,6, 7,8, 5,3, 9,9的圓柱依序放入底座。

2.如下圖將結果相加即得到乘積（記得要進位）。

運算方法

**46785399**乘以**96431**

1.把編號4,6, 7,8, 5,3, 9,9的圓柱依序放入底座。

2.將46785399乘以9,6, 4,3, 1分別用上述方法一條一條算出來。

3.接著用直式加法把數字加起來。

運算方法

### 除法運算

同樣舉例說明。

**46785399**除以**96431**

1.把除數（96431）編號的柱子擺入底座。

2.如下圖所示把96431乘以1-9的結果寫出來。

3. 46785399從左邊六位開始看，下面寫上小於467853但最大的96431k。

4.用467853去減96431k（此例中k=4），得到的數字寫下面。

5.把上面的數字（9）移下來，如同長除法般重複進行此動作。

6.最後得到一個比96431還小的數，為餘數。

7.如有需要，可以一直進行除法工作以得到小數點位數。

### 開平方根

用來開平方根的納皮爾的骨頭會用到第十根柱子，上有三個直排。第一排上有數字1-9的平方，第二排是數字2,4, ...18，第三排則是1-9。

|  | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | √ |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| **1** | 0/1 | 0/2 | 0/3 | 0/4 | 0/5 | 0/6 | 0/7 | 0/8 | 0/9 | 0/1     2   1 |
| **2** | 0/2 | 0/4 | 0/6 | 0/8 | 1/0 | 1/2 | 1/4 | 1/6 | 1/8 | 0/4     4   2 |
| **3** | 0/3 | 0/6 | 0/9 | 1/2 | 1/5 | 1/8 | 2/1 | 2/4 | 2/7 | 0/9     6   3 |
| **4** | 0/4 | 0/8 | 1/2 | 1/6 | 2/0 | 2/4 | 2/8 | 3/2 | 3/6 | 1/6     8   4 |
| **5** | 0/5 | 1/0 | 1/5 | 2/0 | 2/5 | 3/0 | 3/5 | 4/0 | 4/5 | 2/5   10   5 |
| **6** | 0/6 | 1/2 | 1/8 | 2/4 | 3/0 | 3/6 | 4/2 | 4/8 | 5/4 | 3/6   12   6 |
| **7** | 0/7 | 1/4 | 2/1 | 2/8 | 3/5 | 4/2 | 4/9 | 5/6 | 6/3 | 4/9   14   7 |
| **8** | 0/8 | 1/6 | 2/4 | 3/2 | 4/0 | 4/8 | 5/6 | 6/4 | 7/2 | 6/4   16   8 |
| **9** | 0/9 | 1/8 | 2/7 | 3/6 | 4/5 | 5/4 | 6/3 | 7/2 | 8/1 | 8/1   18   9 |

**求出46785399的平方根**

1.把46785399分為兩位兩位一組，46 78 53 99。（如果奇數位數，如83559，寫成8 35 59）

2.從最左邊那組，46，開始。找出比這數最小的完全平方數，是在第六橫排上的36。

3.因為是從第六橫排上挑出來的，所以答案的第一位是6。

4.找到第十根柱子上的第六橫排第二直排，12，然後把1跟2的柱子放在底座上。

5.用46減去36得到10，然後把上面下一組的78移到10的右邊。

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  | 1 | 2 | √ |
| --- | --- | --- | --- |
| **1** | 0/1 | 0/2 | 0/1     2   1 |
| **2** | 0/2 | 0/4 | 0/4     4   2 |
| **3** | 0/3 | 0/6 | 0/9     6   3 |
| **4** | 0/4 | 0/8 | 1/6     8   4 |
| **5** | 0/5 | 1/0 | 2/5   10   5 |
| **6** | 0/6 | 1/2 | 3/6   12   6 |
| **7** | 0/7 | 1/4 | 4/9   14   7 |
| **8** | 0/8 | 1/6 | 6/4   16   8 |
| **9** | 0/9 | 1/8 | 8/1   18   9 |

 | 
```
         _____________
        √46 78 53 99    = 6
         36
         --
         10 78
```
 |

7.找出比1078還要小但最大的數，在此第八橫排的1024是最大的：

```
 0/8 1/6 6/4 → 1024
```

8. 8為答案的下一位數，寫在6旁邊，然後把1078減去1024得到54。

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  | 1 | 2 | √ | *（value）* |
| --- | --- | --- | --- | --- |
| **1** | 0/1 | 0/2 | 0/1     2   1 | *121* |
| **2** | 0/2 | 0/4 | 0/4     4   2 | *244* |
| **3** | 0/3 | 0/6 | 0/9     6   3 | *369* |
| **4** | 0/4 | 0/8 | 1/6     8   4 | *496* |
| **5** | 0/5 | 1/0 | 2/5   10   5 | *625* |
| **6** | 0/6 | 1/2 | 3/6   12   6 | *756* |
| **7** | 0/7 | 1/4 | 4/9   14   7 | *889* |
| **8** | 0/8 | 1/6 | 6/4   16   8 | *1024* |
| **9** | 0/9 | 1/8 | 8/1   18   9 | *1161* |

 | 
```
         _____________
        √46 78 53 99    = 6**8**
         36
         --
         10 78
         **10 24**
         -----
            **54**
```
 |

9.把之前的12乘上10，再加上第十根柱子第八橫排第二直排的數，16，得136。

10.重複上述步驟可得到結果：

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  | 1 | 3 | 6 | √ |
| --- | --- | --- | --- | --- |
| **1** | 0/1 | 0/3 | 0/6 | 0/1     2   1 |
| **2** | 0/2 | 0/6 | 1/2 | 0/4     4   2 |
| **3** | 0/3 | 0/9 | 1/8 | 0/9     6   3 |
| **4** | 0/4 | 1/2 | 2/4 | 1/6     8   4 |
| **5** | 0/5 | 1/5 | 3/0 | 2/5   10   5 |
| **6** | 0/6 | 1/8 | 3/6 | 3/6   12   6 |
| **7** | 0/7 | 2/1 | 4/2 | 4/9   14   7 |
| **8** | 0/8 | 2/4 | 4/8 | 6/4   16   8 |
| **9** | 0/9 | 2/7 | 5/4 | 8/1   18   9 |

 | 
```
         _____________
        √46 78 53 99    = 68
         36
         --
         10 78
         10 24
         -----
            54 53
```
 |

11.再次找到比5453還要小但最大的數，為4089。5453減去4089為1364，並把3寫在68的旁邊。

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  | 1 | 3 | 6 | √ |  |
| --- | --- | --- | --- | --- | --- |
| **1** | 0/1 | 0/3 | 0/6 | 0/1     2   1 | *1361* |
| **2** | 0/2 | 0/6 | 1/2 | 0/4     4   2 | *2724* |
| **3** | 0/3 | 0/9 | 1/8 | 0/9     6   3 | *4089* |
| **4** | 0/4 | 1/2 | 2/4 | 1/6     8   4 | *5456* |
| **5** | 0/5 | 1/5 | 3/0 | 2/5   10   5 | *6825* |
| **6** | 0/6 | 1/8 | 3/6 | 3/6   12   6 | *8196* |
| **7** | 0/7 | 2/1 | 4/2 | 4/9   14   7 | *9569* |
| **8** | 0/8 | 2/4 | 4/8 | 6/4   16   8 | *10944* |
| **9** | 0/9 | 2/7 | 5/4 | 8/1   18   9 | *12321* |

 | 
```
         _____________
        √46 78 53 99    = 68**3**
         36
         --
         10 78
         10 24
         -----
            54 53
            **40 89**
            -----
            **13 64**
```
 |

12.把3乘2的結果加上之前136的十倍，得到1366。把1,3, 6,6的版子擺入底座。

13.再度重複上述步驟。現在比最底下的數小的數是第九橫排的123021。

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  | 1 | 3 | 6 | 6 | √ |  |
| --- | --- | --- | --- | --- | --- | --- |
| **1** | 0/1 | 0/3 | 0/6 | 0/6 | 0/1     2   1 | *13661* |
| **2** | 0/2 | 0/6 | 1/2 | 1/2 | 0/4     4   2 | *27324* |
| **3** | 0/3 | 0/9 | 1/8 | 1/8 | 0/9     6   3 | *40989* |
| **4** | 0/4 | 1/2 | 2/4 | 2/4 | 1/6     8   4 | *54656* |
| **5** | 0/5 | 1/5 | 3/0 | 3/0 | 2/5   10   5 | *68325* |
| **6** | 0/6 | 1/8 | 3/6 | 3/6 | 3/6   12   6 | *81996* |
| **7** | 0/7 | 2/1 | 4/2 | 4/2 | 4/9   14   7 | *95669* |
| **8** | 0/8 | 2/4 | 4/8 | 4/8 | 6/4   16   8 | *109344* |
| **9** | 0/9 | 2/7 | 5/4 | 5/4 | 8/1   18   9 | *123021* |

 | 
```
         _____________
        √46 78 53 99    = 683**9**
         36
         --
         10 78
         10 24
         -----
            54 53
            40 89
            -----
            13 64 99
            **12 30 21**
            --------
             **1 34 78**
```
 |

14.再把9加到答案當中，得到整數答案6839。

我們把題目給的數46785399用完了，但是底下還有餘數13478。這意味著6839是答案的整數部分，{\displaystyle {\sqrt {46785399}}}的實際值應當比6839大，比6840小。如果有需要繼續做下去以得到小數部分的話，下一組的兩個數字就要用00：

|  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |  |
| --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- | --- |
| 

|  | 1 | 3 | 6 | 7 | 8 | √ |
| --- | --- | --- | --- | --- | --- | --- |
| **1** | 0/1 | 0/3 | 0/6 | 0/7 | 0/8 | 0/1     2   1 |
| **2** | 0/2 | 0/6 | 1/2 | 1/4 | 1/6 | 0/4     4   2 |
| **3** | 0/3 | 0/9 | 1/8 | 2/1 | 2/4 | 0/9     6   3 |
| **4** | 0/4 | 1/2 | 2/4 | 2/8 | 3/2 | 1/6     8   4 |
| **5** | 0/5 | 1/5 | 3/0 | 3/5 | 4/0 | 2/5   10   5 |
| **6** | 0/6 | 1/8 | 3/6 | 4/2 | 4/8 | 3/6   12   6 |
| **7** | 0/7 | 2/1 | 4/2 | 4/9 | 5/6 | 4/9   14   7 |
| **8** | 0/8 | 2/4 | 4/8 | 5/6 | 6/4 | 6/4   16   8 |
| **9** | 0/9 | 2/7 | 5/4 | 6/3 | 7/2 | 8/1   18   9 |

 | 
```
         _____________
        √46 78 53 99    = 6839.
         36
         --
         10 78
         10 24
         -----
            54 53
            40 89
            -----
            13 64 99
            12 30 21
            --------
             1 34 78 00
```
 |

這些步驟可以一直重複使得到的答案無限逼近正解。

最後，如果要求平方根的數是小數的話，將數字分組的時候整數部分跟小數部分要分開分組；整數部分如常，小數部分則是方向相反。

例如數字54782.917的分組情況為：

5 47 82. 91 7

接著便可開平方。

参考文献
----

1. ↑  吴文俊主编<《中国数学史大系》第七卷 156-158页