瀟湘館112 / 數學及古代數學 / 《測圓海鏡》圓城圖之諸弦篇﹝1﹞說

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《測圓海鏡》圓城圖之諸弦篇﹝1﹞說

2021-01-10  瀟湘館112

測圓海鏡圓城圖之諸弦篇﹝1

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》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,本文主要談及各勾股形與諸弦關之等式

關鍵詞:大弦、邊弦、黃長弦

《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〈弦〉篇涉及諸勾股形之斜邊,本文重點在於証明弦之等式,弦之位置可參閱以下兩圖

所有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之勾股弦則分別以 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 i 為直角之點,亦可表一勾股形。另外勾股定理成立,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”

注意圓徑為 a1 + b1c1,見上圖之東南西北圓。至於各弦之名稱及位可參閱上兩圖。

以下為與諸有關之等式:

大弦為大勾與股圓差共又為大股與勾圓差共邊弦乃邊股平勾共又為大股內減平弦上勾股較底弦乃底勾髙股共又為大勾內加一個髙差黃廣弦為大股內減虛股又為邊股股共黃長弦乃大勾內減虛勾又為底勾明勾共髙弦乃大差弦內減明弦又為明弦虛弦共平弦乃小差弦內減又為弦虛弦共大差弦乃大股內減大差勾又為髙弦明弦共又大弦內去黃長弦小差弦為大勾內減小差股又為平弦弦共又為大內去黃廣弦

以下為各條目之証明:

大弦為大勾與股圓差共

已知大弦﹝在勾股形天地乾通弦(天地) = c1;“”指圓徑。

股圓差 = b1 – (a1 + b1c1) = c1a1

勾﹝在勾股形天地乾= a1

大勾與股圓差共 = a1 + c1a1 = c1。等號最右方是為大弦

又為大股與勾圓差共

已知大股﹝在勾股形天地乾 = b1

勾圓差 = a1 – (a1 + b1c1) = c1b1

大股與勾圓差共 = b1 + c1 b1 = c1。等號右方是為大弦

邊弦乃邊股平勾共

已知天川弦﹝簡弦﹞:c2 = (c1 + b1a1)

天西股﹝簡股﹞:b2 = b1(a1 + b1c1) = (c1 + b1a1)

勾在勾股形月川青 = a8 = = ( a1 +b1c1)

勾亦在勾股形川地夕= a9

邊股平勾共 = b2 + a8 = (c1 + b1a1) + (a1 + b1c1)

= (b1c1 + b12b1a1 + a12+ a1b1a1c1)

= (b1c1 + c12a1c1)

= (c1 + b1a1)

比較兩式,可知 = 邊股平勾共

又為大股內減平弦上勾股較

股在勾股形月川青

已知 = b8 = (a1 + b1c1) = a8 = = ( a1 +b1c1)

平弦上勾股 = b8a8 = (a1 + b1c1) –(a1 + b1c1)

= (a1 + b1c1)(1 –)

=(a1 + b1c1)(b1a1)

=(b12a12c1b1 + c1a1)

大股內減平弦上勾股較 = b1(b12a12c1b1 + c1a1)

= (2b12b12 + a12+ c1b1c1a1)

= (b12+ a12 + c1b1c1a1)

= (c12 + c1b1c1a1)

= (c1 + b1a1)

比較答案兩式可知相等,所以 = 大股內減平弦上勾股較

底弦乃底勾髙股共

底弦在勾股形日地北底弦 = c3= ( a1b1 + c1)

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

在勾股形天日旦= b6 = = ( a1 +b1c1)

底勾髙股共 = a1+ b6 = = (a1b1 + c1) + (a1 + b1c1)

= (a12a1b1 + a1c1 + a1b1 + b12b1c1)

= (c12 + a1c1b1c1)

= (a1b1 + c1)

比較答案兩式,可知相等,所以底弦 = 底勾髙股共

又為大勾內加一個髙差

大勾在勾股形天地乾已知大勾 = a1

日旦或山朱a6 = (a1 + b1c1)

天旦或日朱b6 = = ( a1 +b1c1)

髙差”即髙勾髙股差 = b6a6

髙勾髙股差= (a1 + b1c1) – (a1 + b1c1)

= (a1 + b1c1)( – 1)

= (a1 + b1c1)(b1a1)

大勾內加一個髙差 = a1 + (a1 + b1c1)(b1a1)

= [2a12 + (b12a12c1b1 + c1a1)]

= (b12+ a12c1b1 + c1a1)

= (c12c1b1 + c1a1)

= (a1b1 + c1)

比較答案兩式可知相等,所以底弦 = 大勾內加一個髙差

黃廣弦為大股內減虛股

已知天山弦﹝簡弦﹞= c4 = (a1 + b1c1)

已知大股﹝在勾股形天地乾 = b1

虛股月泛﹝又太虛= b13 = = (c1b1)(c1a1)

大股內減虛股 = b1(c1b1)(c1a1)

= [a1b1 – (c12c1a1b1c1 + b1a1)]

= [a1b1c12+ c1a1 + b1c1b1a1]

= [ – c12 + c1a1 + b1c1]

= (a1 + b1c1)

比較答案兩式,可知黃廣弦 = 大股內減虛股

又為邊股股共

已知天西股﹝簡股﹞= b2= b1(a1 + b1c1) = (c1 + b1a1)

*股在勾股形山川東

= b15 = = (a1c1a12b1a1 – 2b12 + 2b1c1)

= (c1b1)(a1c1 + b1)

邊股股共 = b2+ b15 = (c1 + b1a1) + (c1b1)(a1c1 + b1)

= (a1c1 + a1b1a12 + a1c1a12b1a1 – 2b12 + 2b1c1)

= (2a1c1 – 2a12 – 2b12 + 2b1c1)

= (2a1c1 – 2c12+ 2b1c1)

= (a1 + b1c1)

所以黃廣弦 = 邊股 +

黃長弦乃大勾內減虛勾

已知月地黃長弦﹝簡黃長弦﹞:c5 = (a1 + b1c1)

大勾 ﹝在勾股形天地乾= a1

勾﹝在勾股形月山泛 = a13 = (a1 + b1c1) –(c1 a1)

= (b1a1 + b12b1c1a1c1 +a12)

= (c1b1)(c1a1)

大勾內減虛勾 = a1a13= a1(c1b1)(c1a1)

=[b1a1 – (c12c1a1b1c1 + a1b1)]

=[b1a1c12+ c1a1 + b1c1a1b1]

=[ – c12 + c1a1 + b1c1]

= (a1 + b1c1)

所以黃廣弦 = 大勾內減虛勾

又為底勾明勾共

在勾股形日地北

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

南月在勾股形日月南

南月勾﹝又= a14= (c1a1)(b1 c1 + a1)

底勾明勾共 = a3+ a14 = (a1b1 + c1) + (c1a1)(b1 c1 + a1)

= (b1a1b12+ b1c1 + 2a1c1a12c12b1a1 + b1c1)

= (2b1c1 + 2a1c1 – 2c12)

= (a1 + b1c1)

所以黃廣弦 = 底勾+明勾

髙弦乃大差弦內減明弦

已知天日上髙弦﹝簡上髙弦﹞:c6 = ( a1 +b1c1) 弦可為 c7

大差在勾股形天月坤天月大差弦﹝簡大差弦﹞:c10 = (c1 a1)

日月為明弦﹝簡﹞:c14 =(c1a1)(b1 c1 + a1)

大差弦內減明弦 = c10 c14 = (c1 a1) –(c1a1)(b1 c1 + a1)

= (c1 a1) [1 –(b1 c1 + a1)]

=(c1 a1)(2a1b1 + c1a1)

=(c1 a1)(a1b1 + c1)

=(c1 a1)( – b1 + c1 + a1)

=(– b1c1 + b1a1 + c12a12)

=(– b1c1 + b1a1 + b12)

= ( a1 +b1c1)

所以髙弦 = 大差弦內減明弦

又為明弦虛弦共

已知 = c14=(c1a1)(b1 c1 + a1)

在勾股形月山泛弦﹝簡太虛= c13 = (c1b1)(c1a1)

明弦虛弦共 = c14+ c13

= (c1a1)(b1 c1 + a1) + (c1b1)(c1a1)

= (c1a1)(b1 c1 + a1 + 2c1 – 2b1)

= (c1a1)( b1 + c1 + a1)

= (–b1c1 + b1a1+ c12a12)

= (–b1c1 + b1a1+ b12)

= (a1 + b1c1)

所以髙弦 = 明弦 + 虛弦

平弦乃小差弦內減

已知月川上平弦﹝簡上平弦﹞= c8= (a1 + b1c1)

山地小差弦﹝簡小差= c11 = (c1b1)

山川﹝簡= c15 = (c1b1)(a1c1 + b1)

小差弦內減= c11c15 = (c1b1) –(c1b1)(a1c1 + b1)

= (c1b1) [1 –(a1c1 + b1)]

= (c1b1) [2b1a1 + c1b1]

= (c1b1) [b1a1 + c1]

= (c1b1) [– a1 + c1 + b1]

= (– a1c1 + a1b1 + c12b12)

= (– a1c1 + a1b1 + a12)

= (a1 + b1c1)

所以平弦 = 小差弦內減

又為弦虛弦共

已知 = c15 = (c1b1)(a1c1 + b1)

= c13= (c1b1)(c1a1)

*弦虛弦共 = (c1b1)(a1c1 + b1) + (c1b1)(c1a1)

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

= (c1b1)[a1c1 + b1 + 2c1 – 2a1]

= (c1b1)[–a1 + c1 + b1]

= (– a1c1 + a1b1 + c12b12)

= (– a1c1 + a1b1 + a12)

= (a1 + b1c1)

所以平弦 = 弦虛弦共

大差弦乃大股內減大差勾

已知天月大差弦﹝簡大差弦﹞:c10 = (c1 a1)

大股﹝在勾股形天地乾 = b1

大差在勾股形天月坤坤月a10 == (c1 a1)

大股內減大差勾 = b1a10= b1(c1 a1)

= [b12a1c1+ a12]

= [c12a1c1]

= (c1 a1)

所以大差弦 = 大股內減大差勾

又為髙弦明弦共

已知 = c6= ( a1 + b1c1) = c14 =(c1a1)(b1 c1 + a1)

髙弦明弦共 = c6 + c14=( a1 + b1c1) + (c1a1)(b1 c1 + a1)

=( a1 +b1c1) [1 + (c1a1)]

= ( a1 +b1c1)(b1 + c1a1)

= [b1 – (c1a1)](b1 + c1a1)

= [b12 – (c1a1)2]

= [b12c12a12 + 2c1a1]

= [ – 2a12 + 2c1a1]

= (c1 a1)

所以大差弦 = 髙弦 + 明弦

又大弦內去黃長弦

已知大弦﹝在勾股形天地乾= c1黃長在勾股形月地泉

月地黃長弦﹝簡黃長弦﹞= c5 = (a1 + b1c1)

大弦內去黃長弦 = c1c5 = c1(a1 + b1c1)

= (b1c1c1a1c1b1 + c12)

= (– c1a1 + c12)

= (c1 a1)

所以大差弦 = 大弦內去黃長弦,去,減也

小差弦為大勾內減小差股

已知小差﹝在勾股形山地艮 = c11 = (c1b1) = a1

山艮﹝又小差股﹞:b11 =  = (c1b1)

大勾內減小差股 = a1b11= a1(c1b1)

=(a12b1c1 + b12)

=(c12b1c1)

=(c1b1)

所以小差弦 = 大勾內減小差股

又為平弦弦共

已知月川上平弦﹝簡上平弦﹞ = c8= (a1 + b1c1)

山川﹝簡= c15 = (c1b1)(a1c1 + b1)

平弦弦共 = c8+ c15 = (a1 + b1c1) +(c1b1)(a1c1 + b1)

= (a1 + b1c1) [1 +(c1b1)]

= (a1 + b1c1) [1 +(c1b1)]

= (a1 + b1c1)(a1 + c1b1)

= [a1 – (c1b1)](a1 + c1b1)

= [a12 – (c1b1)2]

= [a12c12b12 + 2c1b1]

= [ – 2b12 + 2c1b1]

= (c1b1)

所以小差弦 = 平弦弦共

又為大內去黃廣弦

已知大弦=c1 = c4= (a1 + b1c1)

內去黃廣弦 = c1c4 = c1(a1 + b1c1)

= (a1c1c1a1c1b1 + c12)

= (– c1b1 + c12)

= (c1b1)

所以小差弦 = 內去黃廣弦

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