《測圓海鏡》圓城圖之太虛弦﹝6﹞等式說

測圓海鏡圓城圖之太虛弦6﹞等式

上傳書齋名:瀟湘館112  Xiāo XiāngGuǎn 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,本文主要談及太虛弦、極弦、大差弦相關之等式。

關鍵詞:太虛弦、極弦、大差弦、虛弦

《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》之〈諸雜名目〉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》之精髓。注意勾股定理成立,即  ai2 + bi2 = ci2

《測圓海鏡》之〈諸弦〉篇涉及諸勾股形之斜邊,本文重點在於証明太虛弦之等式,而諸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﹞說〉、〈《測圓海鏡》圓城圖之極弦及諸弦篇﹝2﹞說〉、〈《測圓海鏡》圓城圖之邊弦及相關弦﹝3﹞說〉、〈《測圓海鏡》圓城圖之黃廣弦及髙弦﹝4﹞說〉及〈《測圓海鏡》圓城圖之髙弦及平弦﹝5﹞等式說〉。

有關以 a1b1c1aibici 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”。

第 13 點為“泛”﹝一作“水”﹞,第 7點為“朱”,第 12 點為“心”,第 8 點為“青”。

注意圓徑為 a1 + b1c1,見上圖之東南西北圓,圓徑式經常出現於十五勾股形之三邊。

以下為與太虛弦有關之等式:

太虛弦: 加入極弦為極和。極弦內去之即明

二弦共。再去之則明大差

小差併也。加於大差弦即黃廣弦。加於小差弦即黃長弦。內去明勾則

股。 加明勾為圓徑內少虛黃

股共。加入明股為明和

股共。減於明股即明較內去

股。加入明弦為極股。減於明弦為明大差

小差內少個

弦。加於明和即兩個虛弦一個髙差共也。減於明和即髙差也。內去

勾即明勾

較共。又為

股平差共。加於

勾即

和明勾共。加於

股為二虛弦內少明勾。又為圓徑內少虛黃明勾共。內減

股即明勾。

注意月山太虛弦﹝簡稱太虛弦,在勾股形月山泛 13﹞:

c13 =

(c1b1)(c1a1),見上圖。

以下為各條目之証明:

加入極弦為極和。

極弦,即皇極弦﹝在勾股形日川心 12﹞。

已知皇極弦 = c12 =

(a1 + b1c1) 。

太虛弦 + 皇極弦 = c13 + c12 =

(c1b1)(c1a1) +

(a1 + b1c1)

=

(a1 + b1c1)2+

(a1 + b1c1)

=

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

=

(a1 + b1c1)(a1 + b1) 。

“極和”即皇極勾股和。

皇極勾股和= b12 + a12 =

(a1 + b1c1) +

(a1 + b1c1)

=

(a1 + b1c1)[

+

]

=

(a1 + b1c1)(b1 + a1)。

比較兩式,可知皇極弦加入極弦為極和。

極弦內去之即明

二弦共。

“極弦內去之”指從皇極弦減去太虛弦。

皇極弦 – 太虛弦 = c12c13

c12c13 =

(a1 + b1c1) –

(c1b1)(c1a1)

=

(a1 + b1c1) –

(a1 + b1c1)2

=

(a1 + b1c1)[c1 – (a1 + b1c1)]

=

(a1 + b1c1)(2c1a1b1)。

已知明弦﹝在勾股形日月南 14﹞:c14 =

(c1a1)(b1c1 + a1)。

又已知

弦﹝在勾股形山川東 15﹞:c15 =

(c1b1)(a1c1 + b1)。

“明

二弦共”即明弦 +

弦 = c14 + c15

c14 + c15 =

(c1a1)(b1c1 + a1) +

(c1b1)(a1c1 + b1)

=

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

=

(a1 + b1c1)(2c1a1b1)。

比較兩式,可知極弦內去太虛弦 = 明

二弦共。

再去之則明大差

小差併也。

“再去之”指極弦內去太虛弦,再減太虛弦,或曰上條答案再減太虛弦。

(a1 + b1c1)(2c1a1b1) –

(c1b1)(c1a1)

=

(a1 + b1c1)(2c1a1b1) –

(a1 + b1c1)2

=

(a1 + b1c1)[(2c1a1b1) – (a1 + b1c1)]

=

(a1 + b1c1)(3c1 – 2a1 – 2b1) 。

“明大差”即明弦﹝在勾股形日月南 14﹞勾弦較。

明弦勾弦較 = c14a14

=

(c1a1)(b1c1 + a1) –

(c1a1)(b1c1 + a1)

=

(c1a1)(b1c1 + a1)[

– 1]

=

(b1c1 + a1)(c1a1)(c1a1)

=

(b1c1 + a1) (c1a1)2

小差”即

弦﹝在勾股形山川東 15﹞上股弦較。

弦上股弦較 = c15b15

c15b15=

(c1b1)(a1c1 + b1) –

(c1b1)(a1c1 + b1)

=

(c1b1)(a1c1 + b1)[

– 1]

=

(c1b1)(a1c1 + b1)(c1b1)

=

(c1b1)2(a1c1 + b1) 。

“明大差、

小差併”即以上兩式之和:

(b1c1 + a1)(c1a1)2 +

(c1b1)2(a1c1 + b1)

=

(b1c1 + a1)[(c1a1)2 + (c1b1)2]

=

(b1c1 + a1)[c12 – 2c1a1 + a12 + c12– 2c1b1 + b12]

=

(b1c1 + a1)[3c12 – 2c1a1 – 2c1b1]

=

(a1 + b1c1)(3c1 – 2a1 – 2b1)。

所以極弦內去太虛弦,再減太虛弦= 明大差 +

小差。

加於大差弦即黃廣弦。

“加於大差弦”指太虛弦加以大差弦。

已知天月大差弦﹝簡稱大差弦﹞:c10 =

(c1a1)。

太虛弦 + 大差弦 = c13 +c10

c13 + c10 =

(c1b1)(c1a1) +

(c1a1)

=

(c1a1)[

(c1b1) + 1]

=

(c1a1)(c1b1 + a1)

=

(c1a1)( – b1 + c1 + a1)

=

(a1b1c1b1 + c12a12)

=

(a1b1c1b1 + b12)

=

(a1 + b1c1) 。

已知天山黃廣弦﹝簡稱黃廣弦﹞:c4 =

(a1 + b1c1) 。

比較兩式,可知太虛弦+ 大差弦 = 黃廣弦。

加於小差弦即黃長弦。

“加於小差弦”指太虛弦 + 小差弦。

山地小差弦﹝簡稱小差弦﹞:c11 =

(c1b1) 。

太虛弦 + 小差弦 =

(c1b1)(c1a1) +

(c1b1)

=

(c1b1)[

(c1a1) +1]

=

(c1b1)(c1a1 + b1)

=

(c1b1)( – a1 + c1 + b1)

=

(a1b1c1a1 + c12b12)

=

(a1b1c1a1 + a12)

=

(a1 + b1c1) 。

月地黃長弦﹝簡稱黃長弦﹞:c5 =

(a1 + b1c1) 。

比較兩式,可知太虛弦+小差弦 = 黃長弦。

內去明勾則

股。

“內去明勾”指太虛弦內減明勾﹝在勾股形日月南 14﹞。

南月勾﹝又稱明勾﹞:a14 =

(c1a1)(b1c1 + a1)。

太虛弦 – 明勾 =

(c1b1)(c1a1) –

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)2

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)[

(b1c1 + a1) – (c1a1)]

=

(b1c1 + a1)(c1b1c12+ c1a1c1a1 + a12)

=

(b1c1 + a1)(c1b1b12)

=

(c1b1)(a1c1 + b1)。

已知

股﹝在勾股形山川東 15﹞:b15 =

=

(c1b1)(a1c1 + b1)。

所以太虛弦 – 明勾 =

股。

加明勾為圓徑內少虛黃

股共。

明勾式見前。先算出:

太虛弦 + 明勾 =

(c1b1)(c1a1) +

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)2 +

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)[

(b1c1 + a1) + (c1a1)]

=

(b1c1 + a1)(c1b1c12+ c1a1 + c1a1a12)

=

(b1c1 + a1)(c1b1c12+ 2c1a1a12) 。

“黃”即黃方。“虛黃”即太虛黃方。

已知虛黃 = 太虛弦三事較 = 太虛弦和較 = b13 + a13c13

–       c13 + b13 + a13

= –

(c1b1)(c1a1) +

(c1b1)(c1a1) +

(c1b1)(c1a1)

=

(c1b1)(c1a1)(– c1 + b1 + a1)

=

(a1 + b1c1)2(– c1 + b1 + a1)

=

(a1 + b1c1)2(a1 + b1c1)

=

(c1a1)(c1b1)(a1 + b1c1) 。

注意等式 (c1b1)(c1a1) =

(a1 + b1c1)2

已知

股:b15 =

=

(c1b1)(a1c1 + b1)。圓徑 = a1 + b1c1

虛黃 +

=

(c1b1)(a1c1 + b1) +

(c1a1)(c1b1)(a1 + b1c1)

=

(c1b1)(a1c1 + b1)[

+

(c1a1)]

=

(c1b1)(a1c1 + b1)(b1 + 2c1 – 2a1)。

圓徑內少虛黃

股共,即:

(a1 + b1c1) –

(c1b1)(a1c1 + b1)(b1 + 2c1 – 2a1)

= (a1 + b1c1)[1–

(c1b1)(b1 + 2c1 – 2a1)]

=

(a1 + b1c1)(2a1b1c1b1 – 2c12 + 2c1a1 + b12 + 2c1b1– 2a1b1)

=

(a1 + b1c1)(– 2c12+ 2c1a1 + b12 + c1b1)

=

(a1 + b1c1)(– c12a12b12+ 2c1a1 + b12 + c1b1)

=

(a1 + b1c1)(– c12a12 + 2c1a1 + c1b1) 。

所以太虛弦+ 明勾 = 圓徑內少虛黃

股共。

加入明股為明和

股共。

已知日南股﹝又稱明股在勾股形日月南 14﹞:b14 =

(c1a1)(b1c1 + a1)。

太虛弦 + 明股 = c13 +b14

=

(c1b1)(c1a1) +

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)2 +

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)[

(b1c1 + a1) + (c1a1)]

=

(b1c1 + a1)(c1b1c12+ c1a1 + c1b1a1b1)

=

(b1c1 + a1)(2c1b1c12+ c1a1a1b1) 。

“明和”即明弦勾股和 = b14 + a14

b14+ a14  =

(c1a1)(b1c1 + a1) +

(c1a1)(b1c1 + a1)

=

(c1a1)(b1c1 + a1)[

+

]

=

(c1a1)(b1c1 + a1)(a1 + b1) 。

已知山東股﹝又稱

股﹞:b15 =

(c1b1)(a1c1 + b1)。

因此明和

股共

=

(c1a1)(b1c1 + a1)(a1 + b1) +

(c1b1)(a1c1 + b1)

=

(a1c1 + b1)[

(c1a1)(a1 + b1) + (c1b1)]

=

(a1c1 + b1)(c1b1 + c1a1a12a1b1 + b1c1b12)

=

(a1c1 + b1)(2c1b1 + c1a1a12a1b1b12)

=

(a1c1 + b1)(2c1b1 + c1a1c12a1b1) 。

比較兩式可知相同,所以太虛弦 + 明股 = 明和

股共。

減於明股即明較內去

股。

明股 – 太虛弦 = –c13 + b14

= –

(c1b1)(c1a1) +

(c1a1)(b1c1 + a1)

= –

(b1c1 + a1)2 +

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)[ –

(b1c1 + a1) + (c1a1)]

=

(b1c1 + a1)( –c1b1+ c12c1a1 + c1b1a1b1)

=

(b1c1 + a1)(c12c1a1a1b1) 。

“明較”即明弦勾股較 = b14a14

b14a14 =

(c1a1)(b1c1 + a1) –

(c1a1)(b1c1 + a1)

=

(c1a1)(b1c1 + a1)[

]

=

(c1a1)(b1c1 + a1)(b1a1) 。

已知山東股﹝又稱

股﹞:b15 =

(c1b1)(a1c1 + b1)。

明較內去

股,即明較 –

股,即:

(c1a1)(b1c1 + a1)(b1a1) –

(c1b1)(a1c1 + b1)

=

(b1c1 + a1)[

(c1a1)(b1a1) – (c1b1)]

=

(b1c1 + a1)[c1b1c1a1a1b1 + a12c1b1 + b12]

=

(b1c1 + a1)[ – c1a1a1b1 + c12 ] 。

比較答案兩式,可知明股– 太虛弦 = 明弦勾股較。

加入明弦為極股。

“加入明弦”指太虛弦加以明弦。

已知明弦﹝在勾股形日月南 14﹞:c14 =

(c1a1)(b1c1 + a1)。

太虛弦 + 明弦 = c13 + c14

=

(c1b1)(c1a1) +

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)2 +

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)(b1c1 + a1 + c1a1)

=

(b1c1 + a1) × b1

=

(a1 + b1c1)。

已知日心股﹝又稱皇極股,簡稱“極股”﹞:b12 =

(a1 + b1c1)。

比較答案兩式,可知太虛弦 + 明弦 = 極股。

減於明弦為明大差

小差內少個

弦。

“減於明弦”指明弦 – 太虛弦,即c14c13

(c1a1)(b1c1 + a1) –

(c1b1)(c1a1)

=

(c1a1)(b1c1 + a1) –

(b1c1 + a1)2

=

(b1c1 + a1)[(c1a1) – (b1c1 + a1)]

=

(b1c1 + a1)[ c1a1b1 + c1a1)]

=

(b1c1 + a1) (2c1 – 2a1b1)。

“明大差”指明弦﹝在勾股形日月南 14﹞勾弦較=c14a14

c14a14 =

(c1a1)(b1c1 + a1) –

(c1a1)(b1c1 + a1)

=

(c1a1)(b1c1 + a1)[

– 1]

=

(b1c1 + a1)(c1a1)(c1a1)

=

(b1c1 + a1) (c1a1)2

又“

小差”指

弦﹝在勾股形山川東 15﹞上股弦較 = c15b15

c15b15=

(c1b1)(a1c1 + b1) –

(c1b1)(a1c1 + b1)

=

(c1b1)(a1c1 + b1)[

– 1]

=

(c1b1)(a1c1 + b1)(c1b1)

=

(c1b1)2(a1c1 + b1) 。

山川

弦﹝簡稱

弦﹞:c15 =

(c1b1)(a1c1 + b1)。

“明大差、

小差內少個

弦”指前兩式相加減去第三式,即:

(b1c1 + a1) (c1a1)2 +

(c1b1)2(a1c1 + b1)

(c1b1)(a1c1 + b1)

=

(b1c1 + a1)[(c1a1)2 + (c1b1)2c1(c1b1)]

=

(b1c1 + a1)(c12 – 2c1a1 + a12 + c12– 2c1b1 + b12c12 + c1b1)

=

(b1c1 + a1)(c12 – 2c1a1 + a12c1b1 + b12 )

=

(b1c1 + a1)(2c12 – 2c1a1c1b1)

=

(b1c1 + a1) (2c1 – 2a1b1)。

所以太虛弦減於明弦 = 明大差

小差內少個

弦。

加於明和即兩个虛弦一个髙差共也。

“明和”指明弦﹝在勾股形日月南 14﹞勾股和。

明弦勾股和=b14 + a14 =

(c1a1)(b1c1 + a1) +

(c1a1)(b1c1 + a1)

=

(c1a1)(b1c1 + a1)[

+

]

=

(c1a1)(b1c1 + a1)(a1 + b1) 。

“加於明和”即太虛弦 + 明和,即:

(c1b1)(c1a1) +

(c1a1)(b1c1 + a1)(a1 + b1)

=

(b1c1 + a1)2 +

(c1a1)(b1c1 + a1)(a1 + b1)

=

(b1c1 + a1)[c1(b1c1 + a1) + (c1a1)(a1 + b1)]

=

(b1c1 + a1)[c1b1c12+ c1a1 + c1a1 + c1b1a12a1b1]

=

(b1c1 + a1)[2c1b1c12+ 2c1a1a12a1b1]。

已知虛弦﹝在勾股形月山泛 13﹞:c13 =

(c1b1)(c1a1)。

“髙差”指髙弦﹝在勾股形天日旦 6 或日山朱7﹞上勾股較。

髙弦上勾股較= b6a6 =

(a1 + b1c1) –

(a1 + b1c1)

=

(a1 + b1c1)(

– 1)

=

(a1 + b1c1)(b1a1) 。

“兩個虛弦一个髙差共”即:

(c1b1)(c1a1) +

(a1 + b1c1)(b1a1)

=

(a1 + b1c1)2 +

(a1 + b1c1)(b1a1)

=

(a1 + b1c1)[

(a1 + b1c1) +

(b1a1)]

=

(a1 + b1c1)[2c1b1 + 2c1a1 – 2c12 + b12a1b1]

=

(a1 + b1c1)[2c1b1 + 2c1a1c12a12b12 + b12a1b1]

=

(a1 + b1c1)[2c1b1 + 2c1a1c12a12a1b1]。

所以太虛弦+ 明和 = 兩个虛弦 + 髙差。

減於明和即髙差也。

“明和”即明弦﹝在勾股形日月南 14﹞勾股和。

明弦勾股和=b14 + a14 =

(c1a1)(b1c1 + a1) +

(c1a1)(b1c1 + a1)

=

(c1a1)(b1c1 + a1)[

+

]

=

(c1a1)(b1c1 + a1)(a1 + b1) 。

“減於明和”即明和 –太虛弦,即:

(c1a1)(b1c1 + a1)(a1 + b1) –

(c1b1)(c1a1)

=

(c1a1)(b1c1 + a1)(a1 + b1) –

(b1c1 + a1)2

=

(b1c1 + a1)[(c1a1)(a1 + b1) – c1(b1c1 + a1)]

=

(b1c1 + a1)(c1b1 + c1a1a12a1b1c1b1 + c12c1a1)

=

(b1c1 + a1)(– a12a1b1 + c12)

=

(b1c1 + a1)(b12a1b1)

=

(a1 + b1c1)(b1a1) 。

“髙差”即髙弦﹝在勾股形天日旦 6 或日山朱7﹞上勾股較。

髙弦上勾股較= b6a6 =

(a1 + b1c1) –

(a1 + b1c1)

=

(a1 + b1c1)(

– 1)

=

(a1 + b1c1)(b1a1) 。

比較兩式,可知明和 –太虛弦= “髙差”即髙弦上勾股較。

內去

勾即明勾

較共。

已知東川勾﹝又稱

勾,在勾股形山川東 15﹞:

a15 =

(c1b1)(a1c1 + b1)。

“內去

勾”指太虛弦 –

勾,即:

太虛弦 –

勾=

(c1b1)(c1a1) –

(c1b1)(a1c1 + b1)

=

(a1c1 + b1)2

(c1b1)(a1c1 + b1)

=

(a1c1 + b1)[

(a1c1 + b1) – (c1b1)]

=

(a1c1 + b1)(c1b1 + c1a1c12c1a1 + a1b1)

=

(a1c1 + b1)(c1b1c12 + a1b1) 。

已知南月勾﹝又稱明勾,在勾股形日月南 14﹞:a14 =

(c1a1)(b1c1 + a1)。

較”指

弦上勾股較。

弦上勾股較 = b15a15

=

(c1b1)(a1c1 + b1) –

(c1b1)(a1c1 + b1)

=

(c1b1)(a1c1 + b1)(

)

=

(c1b1)(a1c1 + b1)(b1a1) 。

明勾

較共,即:

(c1a1)(b1c1 + a1) +

(c1b1)(a1c1 + b1)(b1a1)

=

(a1c1 + b1)[(c1a1) +

(c1b1)(b1a1)]

=

(a1c1 + b1)(c1a1a12 + c1b1c1a1b12+a1b1)

=

(a1c1 + b1)(– a12+ c1b1b12+ a1b1)

=

(a1c1 + b1)(– c12+ c1b1 + a1b1) 。

比較兩式,可知太虛弦–

勾 = 明勾 +

較。

又為

股平差共。

已知山東股﹝又稱

股﹞:b15 =

(c1b1)(a1c1 + b1)。

“平差”指平弦上勾股較。

平弦上勾股較 = b8a8 =

(a1 + b1c1) –

(a1 + b1c1)

=

(a1 + b1c1)(1 –

)

=

(a1 + b1c1)(b1a1)

=

(b12a12c1b1 + c1a1)

=

[(b1a1)(b1 + a1) – c1(b1a1)]

=

(b1a1)(b1 + a1c1)。

股平差共 =

(c1b1)(a1c1 + b1) +

(b1a1)(b1 + a1c1)

=

(a1c1 + b1) [

(c1b1) +

(b1a1)]

=

(a1c1 + b1)( c1b1b12 + a1b1a12)

=

(a1c1 + b1)(– c12+ c1b1 + a1b1)。

與上條目答案式比較,可知太虛弦–

勾 =

股 + 平差。

加於

勾即

和明勾共。

已知

勾:a15 =

(c1b1)(a1c1 + b1)。

“加於

勾”指太虛弦 +

勾。即:

=

(c1b1)(c1a1) +

(c1b1)(a1c1 + b1)

=

(b1c1 + a1)2 +

(c1b1)(a1c1 + b1)

=

(b1c1 + a1)[

(b1c1 + a1) + (c1b1)]

=

(b1c1 + a1)(c1b1 + c1a1c12+ c1a1a1b1)

=

(b1c1 + a1)(c1b1 + 2c1a1c12a1b1) 。

和”即

弦上勾股和。

弦上勾股和 = b15 +a15

b15 + a15 =

(c1b1)(a1c1 + b1) +

(c1b1)(a1c1 + b1)

=

(c1b1)(a1c1 + b1)(

+

)

=

(c1b1)(a1c1 + b1)(b1 + a1) 。

已知南月勾﹝又稱明勾﹞:a14 =

(c1a1)(b1c1 + a1)。

和明勾共

=

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

(c1a1)(b1c1 + a1)

=

(a1c1 + b1)[

(c1b1)(b1 + a1) + (c1a1)]

=

(a1c1 + b1)(c1b1 + c1a1b12a1b1+ c1a1a12)

=

(a1c1 + b1)(c1b1 + 2c1a1b12a1b1a12)

=

(a1c1 + b1)(c1b1 + 2c1a1c12a1b1)。

比較答案兩式,可知太虛弦+

勾 =

和 + 明勾。

加於

股為二虛弦內少明勾。

已知山東股﹝又稱

股﹞:b15 =

(c1b1)(a1c1 + b1)。

“加於

股”即太虛弦 +

股,即:

=

(c1b1)(c1a1) +

(c1b1)(a1c1 + b1)

=

(b1c1 + a1)2 +

(c1b1)(a1c1 + b1)

=

(b1c1 + a1)[

(b1c1 + a1) + (c1b1)]

=

(b1c1 + a1)(c1b1 + c1a1c12+ c1b1b12)

=

(b1c1 + a1)(2c1b1 + c1a1c12b12)。

已知虛弦 =

(c1b1)(c1a1),二虛弦 =

(c1b1)(c1a1) 。

明勾:a14 =

(c1a1)(b1c1 + a1)。

二虛弦內少明勾即:

二虛弦 – 明勾 =

(c1b1)(c1a1) –

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)2

(c1a1)(b1c1 + a1)

=

(b1c1 + a1)[

(b1c1 + a1) –

(c1a1)]

=

(b1c1 + a1)(2c1b1 – 2c12+ 2c1a1c1a1 + a12)

=

(b1c1 + a1)(2c1b1 – 2c12+ c1a1 + a12)

=

(b1c1 + a1)(2c1b1c12a12b12 + c1a1 + a12)。

=

(b1c1 + a1)(2c1b1c12b12 + c1a1)。

所以太虛弦+

股 = 二虛弦內少明勾。

又為圓徑內少虛黃明勾共。

注意圓徑為 a1 + b1c1,見上文。

已知“虛黃” = 太虛弦三事較 = 弦和較 = b13 + a13c13

–       c13 + b13 + a13

= –

(c1b1)(c1a1) +

(c1b1)(c1a1) +

(c1b1)(c1a1)

=

(c1a1)(c1b1)(a1 + b1c1) ﹝見前﹞。

又已知明勾﹝在勾股形日月南 14﹞:a14 =

(c1a1)(b1c1 + a1)。

虛黃明勾共=

(c1a1)(c1b1)(a1 + b1c1) +

(c1a1)(b1c1 + a1) ]

=

(c1a1)(a1 + b1c1)[

(c1b1) +

]

=

(c1a1)(a1 + b1c1)[2c1 – 2b1 + a1]

=

(a1 + b1c1)(2c12 – 2b1c1 + c1a1 – 2a1c1 + 2a1b1a12)

=

(a1 + b1c1)(2c12 – 2b1c1a1c1 + 2a1b1a12)。

圓徑內少虛黃明勾共

= (a1 + b1c1) –

(c1a1)(a1 + b1c1)[2c1 – 2b1 + a1]

= (a1 + b1c1) [1 –

(c1a1)(2c1 – 2b1 + a1)]

=

(a1 + b1c1)[2a1b1 –(2c12 – 2b1c1a1c1 + 2a1b1a12)]

=

(a1 + b1c1)[2a1b1 –2c12 + 2b1c1 + a1c1 – 2a1b1 + a12]

=

(a1 + b1c1)[ –2c12 + 2b1c1 + a1c1 + a12]

=

(a1 + b1c1)[ –c12a12b12 + 2b1c1 + a1c1 + a12]

=

(a1 + b1c1)[ –c12b12+ 2b1c1 + a1c1]。

比較上式與前式,可知太虛弦+

股 = 圓徑內少虛黃明勾共。

以下為《測圓海鏡細草》原文:

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