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《測圓海鏡》勾股形山川東﹝15﹞之五和五較說

2021-01-05  瀟湘館112

測圓海鏡勾股形山川東15五和五較

上傳書齋名:瀟湘館112  Xiāo Xiāng Guǎn 112

何世強 Ho Sai Keung

提要:《測圓海鏡》乃金‧李冶所撰,其書之“圓城圖式”含十四勾股形,連同原有之大勾股形共十五勾股形。此等勾股形三邊形成一系列之恆等式,本文主要談及第15勾股形山川東關之等式

關鍵詞:山川東

《測圓海鏡》乃金‧李冶所撰,書成於 1248 年,時為南宋淳祐八年。該書卷一“圓城圖式”主要討論與十五勾股形相關之等式,本文介紹其部分等式並作出証明。

本文所引用之勾股式源自“圓城圖式”之十五勾股形,a1b1c1 乃最大勾股形天地乾之勾、股及弦長。故 a1b1c1 又稱為大勾﹝地乾﹞、大股﹝天乾﹞及大弦﹝天地﹞。

《測圓海鏡》之〈五和五較〉篇涉及一系列之勾股恆等式,所有恆等式皆與十五勾股形有關。十五勾股形中最大者為天地乾,其三邊勾股弦分別以 a1b1c1 表之,其餘十四勾股形三邊勾股弦則分別以 aibici 表之,其中 1 < i 15。但 aibici 均可以 a1b1c1 表之,此乃《測圓海鏡》之精注意勾股定理成立,即  ai2 + bi2 = ci2

有關以 a1b1c1 aibici 之式可參閱筆者另文〈測圓海鏡》“圓城圖式”之十二勾股弦算法〉。

以下左為“圓城圖式”右為“圓城圖式十五句股形圖”

本文談及之勾股形乃“山川東”﹝又稱為*”﹞,亦即以下兩圖帶紅色之二勾股形山川東之斜邊 c15 山川”是為*,其直角為 15,以 15 之位置為 “東”,其勾與股分別為  a15川東b15山東注意 15 之點亦可表示山川東勾股形。唯“*”之義未詳。

注意以下之山川東勾股形之位置

注意圓徑為 a1 + b1c1,見上圖之東南西北圓。

以下為山川東勾股形之三事﹝三事,三邊之長也﹞:

東川勾﹝又﹞:a15 = (c1b1)(a1c1 + b1)

山東﹝又﹞:b15 = (c1b1)(a1c1 + b1)

山川﹝簡﹞:c15 =(c1b1)(a1c1 + b1)

山川東勾股形之三事和或較亦可以以 a1b1c1 表之。山川東”乃上圖之最小之勾股形,亦為十五勾股形中之最後者。

若一勾股形之 = c = a = b,則以下為五和五較:

(1)      勾股和:a + b

(2)      勾股較:ba

(3)      勾弦和:a + c

(4)      勾弦較:ca

(5)      股弦和:b + c

(6)      股弦較:cb

(7)      弦較和:c + (ba) ﹝較指勾股較,和指弦與勾股較之和﹞

(8)      弦較較:c – (ba) ﹝第一較字指勾股較,第二較字指弦與勾股較之較﹞

(9)      弦和和:(a + b) + c ﹝第一和字指勾股和,第二和字指弦與勾股和之和。又稱為三事和﹞

(10)      弦和較:(a + b) – c ﹝第一和字指勾股和,第二較字指弦與勾股和之較。又稱為三事較﹞

以下為與勾股形山川東 (15)﹞有關五和五較之等式:

*弦上勾股和即小差內減*其較則虛勾內減*弦也勾弦和即底勾內減小差股又為底股底弦差其較則半個虛黃方也股弦和即平勾其較則平勾內少二個*股也弦較和即虛勾其較則小差上股弦較也三事和即勾圓差其較則太虛上股弦較又為虛勾內減虛黃方也

以下為各條目之証明:

*弦上勾股和即小差內減*

*弦上勾股和 = b15 +a15

b15 + a15 = (c1b1)(a1c1 + b1) +(c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)(+)

= (c1b1)(a1c1 + b1)(b1 + a1)

已知小差 = c1b1

小差內減*= c1b1(c1b1)(a1c1 + b1)

= (c1b1)[1–(a1c1 + b1)]

= (c1b1)[2a1b1c1a1 + c12c1b1]

= (c1b1)[2a1b1c1a1 + b12 + a12c1b1]

= (c1b1)[(b1 + a1)2 c1(a1 + b1)]

= (c1b1)(b1 + a1)(b1 + a1c1)

比較兩式,可知*弦上勾股和 = 小差內減*

其較則虛勾內減*弦也

其較”指*弦上勾股較。

*弦上勾股= b15a15

= (c1b1)(a1c1 + b1) – (c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)()

= (c1b1)(a1c1 + b1)(b1a1)

已知泛山勾﹝又太虛﹞:a13 = (c1b1)(c1a1)

虛勾內減* = a13c15 = (c1b1)(c1a1) –(c1b1)(a1c1 + b1)

= (c1b1)[(c1a1) –(a1c1 + b1)]

= (c1b1)[2a1c1 – 2a12a1c1 + c12c1b1]

= (c1b1)[a1c1a12 + b12c1b1]

= (c1b1)[(b1a1)(b1 + a1) – c1(b1a1)]

= (c1b1)(b1a1)(b1 + a1c1)

所以*弦上勾股= 虛勾內減*

勾弦和即底勾內減小差股

勾弦和”指*弦上勾弦和 = c15 +a15

c15 + a15 = (c1b1)(a1c1 + b1) + (c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)[ + 1]

= (c1b1)(b1 + a1c1)[c1 + a1]

= (c1b1)[b1 – (c1a1)][c1 + a1]

= (c1b1)(b1c1 + b1a1c12 + a12)

= (c1b1)(b1c1 + b1a1b12)

= (c1b1)(c1 + a1b1)

已知北地勾﹝簡勾﹞:a3 = a1(a1 + b1c1) = (a1b1 + c1)

日北﹝簡股﹞:b3 = = (a1b1 + c1)

日地弦﹝簡弦﹞:c3 = ( a1b1 + c1)

山艮﹝又小差股﹞:b11 =  = (c1b1)

底勾內減小差股 = (a1b1 + c1) –(c1b1)

= (a12a1b1 + a1c1 – 2b1c1 + 2b12)

= (c12a1b1 + a1c1 – 2b1c1 + b12)

= [c1(c1 + a1b1) – b1(c1 + a1b1)]

= (c1b1)(c1 + a1b1)

所以*弦上勾弦和 = 底勾內減小差股

又為底股底弦差

底股底弦差”指底弦上股弦較,即 b3a3

已知底弦上股弦較 = ( a1b1 + c1) –(a1b1 + c1)

= (a1b1 + c1)(c1b1)

所以*弦上勾弦和 = 底股底弦差

其較則半個虛黃方也

其較”指*弦上勾弦

*弦上勾弦= c15a15

= (c1b1)(a1c1 + b1) – (c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)[ – 1]

= (c1b1)(b1 + a1c1)[c1a1]

= (c1b1)(c1a1)(b1 + a1c1)

虛黃方即虛弦和較,又名太虛弦三事較。

太虛弦三事= 弦和較 = (b13 + a13) – c13 = b13 + a13c13

       c13 + b13 + a13

= – (c1b1)(c1a1) +(c1b1)(c1a1) + (c1b1)(c1a1)

=(c1b1)(c1a1)(– c1 + b1 + a1)

上式是為虛黃方

所以半個虛黃方 = (c1b1)(c1a1)(b1 + a1c1)

所以*弦上勾弦= 半個虛黃方

股弦和即平勾

股弦和”指*弦上股弦和 = c15 +b15

c15 + b15 = (c1b1)(a1c1 + b1) +(c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)[+ 1]

= (c1b1)(a1c1 + b1)(c1 + b1)

= (c12b12)(a1c1 + b1)

= (a12)(a1c1 + b1)

= ( a1 +b1c1)

已知月青﹝又上平勾﹞:a8 = = ( a1 +b1c1)

比較兩式,可知*弦上股弦和 = 勾。

其較則平勾內少二個*股也

其較”指*弦上股弦 = c15b15

c15b15= (c1b1)(a1c1 + b1) –(c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)[– 1]

= (c1b1)(a1c1 + b1)(c1b1)

= (c1b1)2(a1c1 + b1)

已知勾:a8 = = (a1 + b1c1)

b15 = (c1b1)(a1c1 + b1)

平勾內少二個*

= ( a1 +b1c1) – 2 × (c1b1)(a1c1 + b1)

= ( a1 +b1c1) – (c1b1)(a1c1 + b1)

= (a1 + b1c1)[(c1b1)]

= ( a1 +b1c1)(a12 – 2c1b1 + 2b12)

= ( a1 +b1c1)(c12 – 2c1b1 + b12)

= ( a1 +b1c1)(c1b1)2

所以*弦上股弦 = 平勾內少二個*

弦較和即虛勾

*弦上弦較和 = c15 + (b15 a15) = c15 + b15 a15

c15 + b15 a15

= (c1b1)(a1c1 + b1) +(c1b1)(a1c1 + b1)
 –(c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)[+  –]

=(c1b1)(a1c1 + b1)(c1 + b1a1)

=(c1b1)[b1– (c1a1)][ b1 + (c1a1)]

=(c1b1)[b12– (c1a1)2]

=(c1b1)[b12c12a12+ 2c1a1]

=(c1b1)[b12a12b12a12 + 2c1a1]

=(c1b1)[– 2a12+ 2c1a1]

= (c1b1)(c1a1)

已知泛山勾﹝又太虛﹞:a13 = (c1b1)(c1a1)

比較兩式,可知*弦上弦較和=

其較則小差上股弦較也

其較”指*弦上弦較較。

*弦上弦較= c15 – (b15 a15) = c15b15 + a15

c15b15 + a15

= (c1b1)(a1c1 + b1) –(c1b1)(a1c1 + b1)
 +(c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)[ –  + ]

=(c1b1)(a1c1 + b1)(c1b1 + a1)

=(c1b1)[a1– (c1b1)][ a1 + (c1b1)]

=(c1b1)[a12– (c1b1)2]

=(c1b1)[a12c12b12+ 2c1b1]

=(c1b1)[– 2b12+ 2c1b1]

= (c1b1)(c1b1)

= (c1b1)2

已知小差 = b11=  =(c1b1)

小差= c11(c1b1)

小差上股弦較= (c1b1) – (c1b1)

= (c1b1)(c1b1)

= (c1b1)2

所以*弦上弦較= 小差上股弦較

三事和即勾圓差

*弦上三事和即弦和和 = c15 + (b15 + a15) = c15 + b15 + a15

c15 + b15 + a15

= (c1b1)(a1c1 + b1) +(c1b1)(a1c1 + b1)
 +(c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)[ +  + ]

=(c1b1)(a1c1 + b1)(c1 + b1 + a1)

=(c1b1)[a1– (c1b1)][a1 + (c1 + b1)]

=(c1b1)[a12+ a1c1 + a1b1 a1c1 + a1b1c12 + b12]

=(c1b1)[2a1b1]

= c1b1

已知勾圓差 =a1 – (a1 + b1c1) = a1 a1b1 + c1 = c1b1

所以*弦上三事和 = 勾圓差

其較則太虛上股弦較

其較”指*弦上三事較。

*弦上三事較即弦和較 = (b15 + a15) – c15 = b15 + a15c15

b15 + a15c15

= –(c1b1)(a1c1 + b1) +(c1b1)(a1c1 + b1)
 +(c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)[ – +  + ]

=(c1b1)(a1c1 + b1)( – c1 + b1 + a1)

=(c1b1)(a1c1 + b1)2

= (c1a1)(c1b1)(c1b1)

=(c1b1)2(c1a1)

注意 (c1b1)(c1a1) = (a1 + b1c1)2

太虛股弦較= c13b13 = (c1b1)(c1a1) – (c1b1)(c1a1)

= (c1b1)(c1a1)[– 1]

= (c1a1)(c1b1)(c1b1)

= (c1a1)(c1b1)2

比較兩式,可知*弦上三事= 太虛上股弦較

又為虛勾內減虛黃方也

已知太虛a13 =(c1b1)(c1a1)

虛黃方”指太虛弦三事= 弦和較 = b13 + a13c13

c13 + b13 + a13= (c1a1)(c1b1)(a1 + b1c1)

虛勾內減虛黃方=(c1b1)(c1a1) –(c1a1)(c1b1)(a1 + b1c1)

= (c1b1)(c1a1)[1 –(a1 + b1c1)]

=(c1b1)(c1a1)(a1a1b1 + c1)

=(c1b1)2(c1a1)

別証法:

虛勾內減個小黃方 = a13 – (a13 + b13 c13)

= c13b13

= 太虛股弦較﹝見前﹞。

黃方”定義可參閱筆者另文。

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