大原利明算法点竄指南点竄術算額絵馬堂

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1 算額

2 大原利明算法点竄指南点竄術算額絵馬絵馬絵馬堂

3 甲一乙二丙三丁四戊五己六庚七辛八壬九癸十十五二十百千

4 甲乙丙丁傍書法関孝和点竄術松永良弼甲甲甲甲甲乙甲甲乙甲乙甲乙甲乙丙丁戊 a + b 2 c 甲 a a 二乙 2 b b 2 小 c c

5 SOLVING SANGAKU Rules of Tenzan Jutsu In the Sanpo Tenzan Shinan, Ōhara includes a set of rules instructing the reader how to manipulate the symbols and lines of the tenzan jutsu system to solve a variety of geometrical problems. We have examined and translated the introduction section of the Sanpo Tenzan Shinan to produce the following summary of the rules of tenzan jutsu. Figure 2. Original text from the Sanpo Tenzan Shinan explaining the convert rule 自乘 Self Multiplication The self multiplication rule 自乘 is a means of squaring. For example, applying the rule to 2a produces 4a2. When there are two terms, squaring and expansion of the square occurs. For example, applying the rule to a + b produces a2 + 2ab + b2. 括之 Put Together The put together rule 括之 is used to combine elements in various ways. Firstly it can be used to combine terms listed separately with different lines in the tenzan system to bring them under one line. For example, say we have the following addition. The rule is also used for combining like terms in the operation equivalent to factorisation. 解之 Splitting

6 a(b + c) (a b) + (a c) y + c = x x = a + b a + b x y + c = a + b 遍省過乗 3x(a + b + c) 6ab = 0 x(a + b + c) 2ab = 0 同加異減 +2ab + 2ab 4ab 2ab 2ab 4ab +a 2 a 2 (a + b) 2 (a b) 2 a 2 + 2ab + b 2 a 2 + 2ab b 2 (+a 2 a 2 ) (+2ab + 2ab) (+b 2 b 2 ) 變換開平方商變換 = a 2 b 2 = (a + b)(a b) a a( a + x)( a x) ( a + x)( a x) = 1 a a x a + x 1 = ( 2 1)( 2 + 1) = ( 2 1) 2 ( 2 + 1) 2 = ( 3 2)( 3 + 2) = ( 5 2)( 5 + 2) 2 = ( 2 1) 2 ( 2 + 2) 2 = ( 3 1)( 3 + 1) 2 = ( 2 1)( 2 + 2) = ( 2 2) 2 ( 2 + 1) 2 4 = ( 5 1)( 5 + 1)

7 2 = ( 2 2)( 2 + 2) = ( 3 2)( 3 + 1) 2 2 = ( 2 2)( 2 + 1) 乗除括之加減括之 a2 a2 + 2a + b b b + 2ab b + b2 b a 2 + ab + b 2 ab 2ab ab a 2 + 2ab ab + b 2 (a + b) 2 ab a 2 + 2ab + b 2 a 2 + 4ab 2ab + b 2 (a b) 2 + 4ab

8 片山日子神社今有如圖方内隔斜容甲乙円各二個只云甲円径一寸問乙円径幾何甲乙甲乙答云乙円径五分八厘五毛乙術日置二個開平方以域二個余乘甲円径得乙円径合問甲乙

9 2r+2 2r 2P T +2r P T = t(1 + 2) 2 t = r = (2 2)r = r 2 + 1

11 今有如圖方内隔斜容甲乙圓各二個只云甲圓径一寸問乙圓径幾何甲乙甲乙答云乙圓径五分八厘乙五毛術日置二個開平方以域二個余乘甲圓径得乙圓径合問甲乙甲乙方斜解日置乙乘方斜率加甲乙和

12 乙甲乙乙二商 b 2 2 b 甲乙和 a b + a + b = x 方斜寄左又置甲乘方斜率加甲甲甲甲二商 a 2 2 a 甲 a a = x 方斜與寄左相消求矩合乙二商 b 2 2 b 2

13 甲乙和 a b + a + b 甲二商 a 2 2 a 甲合矩 a a = 0 異域括之二乙商加一 2 b + 1 b( 2 + 1) 二甲商合矩變換 2 a a 2 = 0 二乙商加一 2 b + 1 b( 2 + 1)

14 二二二甲商商商加去一一合矩 a a( 2(( 2 1)( 2 + 1))) 遍省過乗得乙 b 二二甲商商去一合矩 a a( 2( 2 1)) = 0 解之乙 b 甲 a a( 2( 2 1)) 2a + a 2

15 二甲商 2 a a 2 = 0 合矩定如定例求得乙径歸除式乙二甲商得乙径式甲 2 a a 2a + 2a = b 故絶答術則如左 2r a 2t b a + 2a = b( 2 + 1) + a 2 b = a (2 2)a 2 1 2( 2 1) b = ( 2 + 1)( 2 1) a

16 2( 2 1) b = a 1 b = (2 2)a 2a ( 2 + 1)( 2 + 1) b( 2 + 1) 2a = 0 b( 2 + 1) 2a( 2 + 1)( 2 1) = 0 b 2a( 2 1) = 0 2a + 2a = b 2a 2a = b (2 2)a = b a x a + x 算法天生法指南会田安明藤田定資 2 b = a 2 + 1

17 会田安明算法天生法指南平山諦学術を中心とした和算史上の人々富士短期大学出版部防衛庁防衛年鑑刊行会防衛年鑑防衛年鑑刊行会大原利明算法点竄指南早稲田大古典籍総合データベース

18 下平和夫日本人の数学和算河出書房新社

0 0 = 1 0 = 0 1 = = 1 1 = 0 0 = 1

0 0 = 1 0 = 0 1 = = 1 1 = 0 0 = 1 0 0 = 1 0 = 0 1 = 0 1 1 = 1 1 = 0 0 = 1 : = {0, 1} : 3 (,, ) = + (,, ) = + + (, ) = + (,,, ) = ( + )( + ) + ( + )( + ) + = + = = + + = + = ( + ) + = + ( + ) () = () ( + ) = + + = ( + )( + ) + = = + 0

大原利明 算法点竄指南 点竄術 算 額 絵馬堂

大原利明算法点竄指南点竄術算額絵馬堂