瀟湘館112 / 數學及古代數學 / 《測圓海鏡》圓城圖之邊弦及相關弦﹝3﹞說

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《測圓海鏡》圓城圖之邊弦及相關弦﹝3﹞說

2021-01-15  瀟湘館112

測圓海鏡圓城圖之邊及相關弦﹝3

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,本文主要談及各勾股形與邊弦底弦及其關之等式

關鍵詞:邊弦、大差弦底弦皇極弦

《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》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

《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,而弦之位置可參閱以下兩圖。筆者已有文談及此類等式名為〈測圓海鏡圓城圖之諸弦篇﹝1〉及〈測圓海鏡圓城圖之極弦諸弦篇﹝2〉。

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。

以下為與底弦有關之等式:

邊弦底弦相為大弦皇極弦共也於此並數內減大和餘為皇極弦內減圓徑也若以二弦相減餘即皇極差也此數同者最多故又為皇極弦內少個小差弦又為髙弦平弦較又為明股內少又為大差弦內少皇極弦又為次差虛差共也邊弦又為皇極股弦共又為黃廣弦弦共底弦又為皇極勾弦共又為黃長弦明弦共也以邊弦減大股餘為半徑內減平勾又為平弦內減小差勾也底弦內減大勾餘為髙股內減半徑又為大差股內減髙弦也

以下為各條目之証明:

邊弦底弦相為大弦皇極弦共也

併”,和也。天川弦﹝簡在勾股形天川西 2﹞:
c2 = (c1 + b1a1)

底弦在勾股形日地北 3 = c3= ( a1b1 + c1)

邊弦底弦相= c2 + c3 = (c1 + b1a1) + (a1b1 + c1)

= [(c1 + b1a1) + (c1b1 + a1)]

=[a1(c1 + b1a1) + b1(c1b1 + a1)]

=(a1c1 + a1b1a12 + b1c1b12 + b1a1)

=(a1c1 + 2a1b1a12 + b1c1b12)

=(a1c1 + 2a1b1c12 + b1c1)

=(a1c1 + 2a1b1a12b12 + b1c1)

=(a1c1 + 2a1b1c12 + b1c1)

已知大弦在勾股形天地乾 1=c1

皇極弦日川在勾股形日川心 12= c12 =(a1 + b1c1)

大弦皇極弦共 = c1 + (a1 + b1c1)

= c1[1 + (a1 + b1c1)]

=(2a1b1 + a1c1c12+ b1c1)

比較兩式可知相同,所以邊弦底弦相= 大弦皇極弦之和

於此數內減大和餘為皇極弦內減圓徑也

“併”指邊弦底弦之和。“大和”即通弦上勾股和 = a1 + b1圓徑之直﹝見上圖﹞。

數內減大和

= (2a1b1 + a1c1c12+ b1c1) – (a1 + b1)

= (–2a12b1 – 2a1b12 + 2a1b1c1 + a1c12c13 + b1c12)

= [–2a1b1(a1 + b1c1) + c12(a1 + b1c1)]

= (c12 – 2a1b1)(a1 + b1c1)

皇極弦內減圓徑=(a1 + b1c1) – (a1 + b1c1)

= (a1 + b1c1)(– 1)

= (a1 + b1c1)(c12 – 2a1b1)

比較兩式可知相同,所以併數內減大和 = 皇極弦內減圓徑

若以二弦相減餘即皇極差也

二弦相減”即邊弦底弦之差。

二弦相減= c2c3 = (c1 + b1a1) – (a1b1 + c1)

= [(c1 + b1a1) – (c1b1 + a1)]

=[a1(c1 + b1a1) – b1(c1b1 + a1)]

=(a1c1 + a1b1a12b1c1 + b12b1a1)

=(a1c1a12b1c1 + b12)

餘即皇極差皇極差皇極勾股較

皇極勾股較= b12a12 = (a1 + b1c1) – (a1 + b1c1)

= (a1 + b1c1)[ – ]

= (a1 + b1c1)(b1a1)

= (b12a12b1c1 + a1c1)

比較兩式可知相同,所以邊弦底弦= 皇極勾股較

此數同者最多故又為皇極弦內少個小差弦

此數同者最多”指即有多條加減式可得同一數。

已知皇極=c12 =(a1 + b1c1)

山地小差弦﹝簡小差= c11 = (c1b1)

皇極弦內少個小差弦

=(a1 + b1c1) –(c1b1)

= [(a1 + b1c1) – (c1b1)]

=[c1(a1 + b1c1) – 2b1(c1b1)]

=(c1a1 + c1b1c12 – 2b1c1 + 2b12)

=(c1a1a12b12b1c1 + 2b12)

= (b12a12b1c1 + a1c1)

比較兩式可知相同,所以邊弦底弦= 皇極弦內少個小差弦

又為髙弦平弦較

已知在勾股形天日旦 6 日山朱 7= c6 = ( a1 +b1c1)

已知平弦在勾股形月川青 8 川地夕 9=c8 = (a1 + b1c1)

髙弦平弦較= ( a1 +b1c1) –(a1 + b1c1)

= (a1 + b1c1)( –)

= (a1 + b1c1)(b1a1)

= (b12a12b1c1 + a1c1)

與前答案式比較,所以邊弦底弦= 髙弦平弦較

又為明股內少

已知= b14 == (c1a1)(b1 c1 + a1)

又已知= a15 =(c1b1)(a1c1 + b1)

明股內去*= b14a15

= (b1 c1 + a1) [(c1a1) – (c1b1)]

= (b1 c1 + a1)[b1c1a1b1a1c1 + a1b1]

= (b1 c1 + a1)[b1c1a1c1]

= (a1 + b1c1)(b1a1)

= (b12a12b1c1 + a1c1)

與前答案式比較,所以邊弦底弦= 皇極勾股較 = 明股 *

又為大差弦內少皇極弦

已知天月大差弦﹝簡大差弦,在勾股形天月坤 10﹞:c10 =(c1 a1)

皇極在勾股形日川心 12c12 =(a1 + b1c1)

大差弦內少皇極弦 = (c1 a1) –(a1 + b1c1)

= [(c1 a1) –(a1 + b1c1)]

= (2c1a1 – 2a12c1a1c1b1 + c12)

= (c1a1 – 2a12c1b1 + a12+ b12)

= (c1a1 a12c1b1 + b12)

所以邊弦底弦= 大差弦內少皇極弦

又為次差虛差共也

依《測圓海鏡》所謂“次差”,指*二差之和也,*二差指“明差”與“*”,而明差指明勾與明股之差,明勾與明股之式見前。

明差 = b14a14 = (c1a1)(b1 c1 + a1) – (c1a1)(b1 c1 + a1)

= (c1a1)( a1 + b1c1)[]

**勾與*股之差,*勾與*股之式亦見前。

* = b15a15 = (c1b1)(a1c1 + b1) – (c1b1)(a1c1 + b1)

=(c1b1)( a1 + b1c1)[]

二差共 = 明差 + *

= (c1a1)( a1 + b1c1)[] + (c1b1)( a1 + b1c1) []

= ( a1 +b1c1)[](c1a1 + c1b1)

= (a1 + b1c1)(2c1a1b1)

上式是為次差又名近差又名戾和

虛差”即太虛勾股較。

太虛勾股 = b13a13 = (c1b1)(c1a1) –(c1b1)(c1a1)]

= (c1b1)(c1a1)[]

=(c1b1)(c1a1)(b1a1)

次差虛差共,即:

 (a1 + b1c1)(2c1a1b1) + (c1b1)(c1a1)(b1a1)

= (a1 + b1c1)(2c1a1b1) + (a1 + b1c1)2(b1a1)

=(a1 + b1c1)(b1a1)[(2c1a1b1) + (a1 + b1c1)]

=(a1 + b1c1)(b1a1)c1

=(a1 + b1c1)(b1a1)

=(b12a12b1c1 + a1c1)

與前答案式比較可知相等,所以邊弦底弦= 次差虛差共

邊弦又為皇極股弦共

已知在勾股形天川西 2= c2 = (c1 + b1a1)

皇極在勾股形日川心 12股弦共,即:

 c12 + b12 = (a1 + b1c1) + (a1 + b1c1)

= (a1 + b1c1)[ + 1]

= (a1 + b1c1)(c1 + b1)

=[a1 – (c1b1)](c1 + b1)

=[a1c1 + a1b1 – (c12b12)]

=[a1c1 + a1b1a12]

= (c1 + b1a1)

比較兩式,可知邊弦 = 皇極股弦共

又為黃廣弦弦共

已知山川﹝簡弦,在勾股形山川東 15
= c15 = (c1b1)(a1c1 + b1)

天山弦﹝簡弦,在勾股形天山金 4= c4 = (a1 + b1c1)

廣弦弦共 = c4+ c15 = (a1 + b1c1) + (c1b1)(a1c1 + b1)

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

= (a1 + b1c1)(2b1 + c1b1)

= (a1 + b1c1)(b1 + c1)

= [a1 – (c1b1)](c1 + b1)

= (– c12 + b12+ a1b1 + a1c1)

= (– a12 + a1b1 + a1c1)

= (c1 + b1a1)

所以邊弦 = 黃廣弦弦共

底弦又為皇極勾弦共

已知日地弦﹝簡弦,在勾股形日地北 3﹞:c3 = ( a1b1 + c1)

皇極在勾股形日川心 12勾弦共 = c12 +a12

c12 + a12 = (a1 + b1c1) +(a1 + b1c1)

= (a1 + b1c1)[ + 1]

= (a1 + b1c1)(c1 + a1)

= [b1 – (c1a1)](c1 + a1)

 = [b1c1 + a1b1c12+ a12]

= [b1c1 + a1b1b12]

= (a1b1 + c1)

比較兩式,可知 =皇極勾弦共

又為黃長弦明弦共也

已知黃長在勾股形月地泉 5c5 = (a1 + b1c1)

在勾股形日月南 14c14 =(c1a1)(b1 c1 + a1)

黃長弦明弦共=c5 + c14 =(a1 + b1c1) + (c1a1)(b1 c1 + a1)

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

= (a1 + b1c1)(2a1 + c1a1)

= (a1 + b1c1)(a1 + c1)

= [b1 – (c1a1)](a1 + c1)

= [b1a1 + b1c1– (c12a12)]

= [b1a1 + b1c1b12]

= (a1b1 + c1)

比較兩式,可知 =黃長弦明弦共

以邊弦減大股餘為半徑內減平勾

已知在勾股形天川西 2= c2 = (c1 + b1a1)

大股在勾股形天地乾 1= b1

邊弦減大股,即:

 b1 (c1 + b1a1) = (– c12c1b1 + c1a1 + 2b12)

= (– a12c1b1 + c1a1 + b12)

= [(b1a1)(b1 + a1) – c1(b1a1)]

= (b1a1)( b1 + a1c1)

已知圓半徑= (a1 + b1c1)

上平在勾股形月川青 8 川地夕 9= a8
= = ( a1 + b1c1)

半徑內減平勾

= (a1 + b1c1) –( a1 + b1c1)

= (a1 + b1c1)[1–]

= (a1 + b1c1)( b1a1)

所以邊弦減大股 = 半徑內減平勾

又為平弦內減小差勾也

已知在勾股形月川青 8 川地夕 9c8 = (a1 + b1c1)

艮地勾﹝又小差勾,在勾股形山地艮 11= a11 = a1– (a1 + b1c1) = c1b1

平弦內減小差勾 = c8a11 = (a1 + b1c1) – (c1b1)

= [c1(a1 + b1c1) – 2b1(c1b1)]

= (c1a1 + c1b1c12 – 2b1c1 + 2b12)

= (c1a1c12b1c1 + 2b12)

= (c1a1a12b1c1 + b12)

= [(b1a1)(b1 + a1) – c1(b1a1)]

= (b1a1)( b1 + a1c1)

所以邊弦減大股 = 平弦內減小差勾

底弦內減大勾餘為髙股內減半徑

日地弦﹝簡弦,在勾股形日地北 3= c3 =(a1b1 + c1)

大勾在勾股形天地乾 1= a1

底弦內減大勾=c3a1 =(a1b1 + c1) – a1

= [c1(a1b1 + c1) – 2a12]

= [c1a1c1b1 + c12 – 2a12]

= [c1a1c1b1 + b12a12]

= [(b1a1)(b1 + a1) – c1(b1a1)]

= (b1a1)( b1 + a1c1)

已知在勾股形天日旦 6 日山朱 7
= b6 = = ( a1 +b1c1)

髙股內減半徑=( a1 + b1c1) (a1 + b1c1)

= (a1 + b1c1)(– 1)

= (a1 + b1c1)(b1a1)

底弦內減大勾= 髙股內減半徑

又為大差股內減髙弦也

在勾股形天日旦 6 日山朱7c6 = ( a1 +b1c1)

大差股天坤股﹝又大差股,在勾股形天月坤 10﹞:
b10 = b1 – (a1 + b1c1) = c1 a1

大差股內減髙弦 = (c1 a1)(a1 + b1c1)

= [2a1(c1 a1)c1(a1 + b1c1)]

= [2a1c1– 2a12c1a1c1b1 + c12]

= [a1c1a12c1b1 + b12]

= [(b1a1)(b1 + a1) – c1(b1a1)]

= (b1a1)( b1 + a1c1)

所以底弦內減大勾 = 大差股內減髙弦

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