《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﹞等式說
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,a1、b1、c1 乃最大勾股形天地乾之勾、股及弦長。故 a1、b1、c1 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 a1、b1、c1 表之,其餘十四勾股形三邊勾股弦則分別以 ai、bi、ci 表之,其中 1 < i ≦ 15。但 ai、bi、ci 均可以 a1、b1、c1 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 a1、b1、c1 表 ai、bi、ci 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 + b1 – c1,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 =
(c1 – b1)(c1 – a1),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 + b1 – c1) 。
太虛弦 + 皇極弦 = c13 + c12 =
(c1 – b1)(c1 – a1) +
(a1 + b1 – c1)
=
(a1 + b1 – c1)2+
(a1 + b1 – c1)
=
(a1 + b1 – c1)[(a1 + b1 – c1) + c1]
=
(a1 + b1 – c1)(a1 + b1) 。
“極和”即皇極勾股和。
皇極勾股和= b12 + a12 =
(a1 + b1 – c1) +
(a1 + b1 – c1)
=
(a1 + b1 – c1)[
+
]
=
(a1 + b1 – c1)(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 = c12– c13 。
c12 – c13 =
(a1 + b1 – c1) –
(c1 – b1)(c1 – a1)
=
(a1 + b1 – c1) –
(a1 + b1 – c1)2
=
(a1 + b1 – c1)[c1 – (a1 + b1 – c1)]
=
(a1 + b1 – c1)(2c1 – a1 – b1)。
已知明弦﹝在勾股形日月南 14﹞:c14 =
(c1 – a1)(b1 – c1 + a1)。
又已知
弦﹝在勾股形山川東 15﹞:c15 =
(c1 – b1)(a1 – c1 + b1)。
“明
二弦共”即明弦 +
弦 = c14 + c15。
c14 + c15 =
(c1 – a1)(b1 – c1 + a1) +
(c1 – b1)(a1 – c1 + b1)
=
(a1 + b1 – c1)[(c1 – a1) +(c1 – b1)]
=
(a1 + b1 – c1)(2c1 – a1 – b1)。
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 + b1 – c1)(2c1 – a1 – b1) –
(c1 – b1)(c1 – a1)
=
(a1 + b1 – c1)(2c1 – a1 – b1) –
(a1 + b1 – c1)2
=
(a1 + b1 – c1)[(2c1 – a1 – b1) – (a1 + b1 – c1)]
=
(a1 + b1 – c1)(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 = c14 – a14
=
(c1 – a1)(b1 – c1 + a1) –
(c1 – a1)(b1 – c1 + a1)
=
(c1 – a1)(b1 – c1 + a1)[
– 1]
=
(b1 – c1 + a1)(c1 – a1)(c1 – a1)
=
(b1 – c1 + a1) (c1 – a1)2。
“
小差”即
弦﹝在勾股形山川東 15﹞上股弦較。
弦上股弦較 = c15 – b15 。
c15 – b15=
(c1 – b1)(a1 – c1 + b1) –
(c1 – b1)(a1 – c1 + b1)
=
(c1 – b1)(a1 – c1 + b1)[
– 1]
=
(c1 – b1)(a1 – c1 + b1)(c1 – b1)
=
(c1 – b1)2(a1 – c1 + b1) 。
“明大差、
小差併”即以上兩式之和:
(b1 – c1 + a1)(c1 – a1)2 +
(c1 – b1)2(a1 – c1 + b1)
=
(b1 – c1 + a1)[(c1 – a1)2 + (c1 – b1)2]
=
(b1 – c1 + a1)[c12 – 2c1a1 + a12 + c12– 2c1b1 + b12]
=
(b1 – c1 + a1)[3c12 – 2c1a1 – 2c1b1]
=
(a1 + b1 – c1)(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 =
(c1 – a1)。
太虛弦 + 大差弦 = c13 +c10。
c13 + c10 =
(c1 – b1)(c1 – a1) +
(c1 – a1)
=
(c1 – a1)[
(c1 – b1) + 1]
=
(c1 – a1)(c1 – b1 + a1)
=
(c1 – a1)( – b1 + c1 + a1)
=
(a1b1 – c1b1 + c12 – a12)
=
(a1b1 – c1b1 + b12)
=
(a1 + b1 – c1) 。
已知天山黃廣弦﹝簡稱黃廣弦﹞:c4 =
(a1 + b1 – c1) 。
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 =
(c1 – b1) 。
太虛弦 + 小差弦 =
(c1 – b1)(c1 – a1) +
(c1 – b1)
=
(c1 – b1)[
(c1 – a1) +1]
=
(c1 – b1)(c1 – a1 + b1)
=
(c1 – b1)( – a1 + c1 + b1)
=
(a1b1 – c1a1 + c12 – b12)
=
(a1b1 – c1a1 + a12)
=
(a1 + b1 – c1) 。
月地黃長弦﹝簡稱黃長弦﹞:c5 =
(a1 + b1 – c1) 。
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 =
(c1 – a1)(b1 – c1 + a1)。
太虛弦 – 明勾 =
(c1 – b1)(c1 – a1) –
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)2 –
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)[
(b1 – c1 + a1) – (c1 – a1)]
=
(b1 – c1 + a1)(c1b1 – c12+ c1a1 – c1a1 + a12)
=
(b1 – c1 + a1)(c1b1 – b12)
=
(c1 – b1)(a1 – c1 + b1)。
已知
股﹝在勾股形山川東 15﹞:b15 =
=
(c1 – b1)(a1 – c1 + 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 =
(c1 – b1)(c1 – a1) +
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)2 +
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)[
(b1 – c1 + a1) + (c1 – a1)]
=
(b1 – c1 + a1)(c1b1 – c12+ c1a1 + c1a1 – a12)
=
(b1 – c1 + a1)(c1b1 – c12+ 2c1a1 – a12) 。
“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 + a13 – c13。
– c13 + b13 + a13
= –
(c1 – b1)(c1 – a1) +
(c1 – b1)(c1 – a1) +
(c1 – b1)(c1 – a1)
=
(c1 – b1)(c1 – a1)(– c1 + b1 + a1)
=
(a1 + b1 – c1)2(– c1 + b1 + a1)
=
(a1 + b1 – c1)2(a1 + b1 – c1)
=
(c1 – a1)(c1 – b1)(a1 + b1 – c1) 。
注意等式 (c1 – b1)(c1 – a1) =
(a1 + b1 – c1)2。
已知
股:b15 =
=
(c1 – b1)(a1 – c1 + b1)。圓徑 = a1 + b1 – c1。
虛黃 +
股
=
(c1 – b1)(a1 – c1 + b1) +
(c1 – a1)(c1 – b1)(a1 + b1 – c1)
=
(c1 – b1)(a1 – c1 + b1)[
+
(c1 – a1)]
=
(c1 – b1)(a1 – c1 + b1)(b1 + 2c1 – 2a1)。
圓徑內少虛黃
股共,即:
(a1 + b1 – c1) –
(c1 – b1)(a1 – c1 + b1)(b1 + 2c1 – 2a1)
= (a1 + b1 – c1)[1–
(c1 – b1)(b1 + 2c1 – 2a1)]
=
(a1 + b1 – c1)(2a1b1 – c1b1 – 2c12 + 2c1a1 + b12 + 2c1b1– 2a1b1)
=
(a1 + b1 – c1)(– 2c12+ 2c1a1 + b12 + c1b1)
=
(a1 + b1 – c1)(– c12 – a12 – b12+ 2c1a1 + b12 + c1b1)
=
(a1 + b1 – c1)(– c12 – a12 + 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 =
(c1 – a1)(b1 – c1 + a1)。
太虛弦 + 明股 = c13 +b14
=
(c1 – b1)(c1 – a1) +
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)2 +
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)[
(b1 – c1 + a1) + (c1 – a1)]
=
(b1 – c1 + a1)(c1b1 – c12+ c1a1 + c1b1 – a1b1)
=
(b1 – c1 + a1)(2c1b1 – c12+ c1a1 – a1b1) 。
“明和”即明弦勾股和 = b14 + a14。
b14+ a14 =
(c1 – a1)(b1 – c1 + a1) +
(c1 – a1)(b1 – c1 + a1)
=
(c1 – a1)(b1 – c1 + a1)[
+
]
=
(c1 – a1)(b1 – c1 + a1)(a1 + b1) 。
已知山東股﹝又稱
股﹞:b15 =
(c1 – b1)(a1 – c1 + b1)。
因此明和
股共
=
(c1 – a1)(b1 – c1 + a1)(a1 + b1) +
(c1 – b1)(a1 – c1 + b1)
=
(a1 – c1 + b1)[
(c1 – a1)(a1 + b1) + (c1 – b1)]
=
(a1 – c1 + b1)(c1b1 + c1a1 – a12 – a1b1 + b1c1 – b12)
=
(a1 – c1 + b1)(2c1b1 + c1a1 – a12 – a1b1 – b12)
=
(a1 – c1 + b1)(2c1b1 + c1a1 – c12 – a1b1) 。
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
= –
(c1 – b1)(c1 – a1) +
(c1 – a1)(b1 – c1 + a1)
= –
(b1 – c1 + a1)2 +
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)[ –
(b1 – c1 + a1) + (c1 – a1)]
=
(b1 – c1 + a1)( –c1b1+ c12 – c1a1 + c1b1 – a1b1)
=
(b1 – c1 + a1)(c12 – c1a1 – a1b1) 。
“明較”即明弦勾股較 = b14 – a14 。
b14 – a14 =
(c1 – a1)(b1 – c1 + a1) –
(c1 – a1)(b1 – c1 + a1)
=
(c1 – a1)(b1 – c1 + a1)[
–
]
=
(c1 – a1)(b1 – c1 + a1)(b1 – a1) 。
已知山東股﹝又稱
股﹞:b15 =
(c1 – b1)(a1 – c1 + b1)。
明較內去
股,即明較 –
股,即:
(c1 – a1)(b1 – c1 + a1)(b1 – a1) –
(c1 – b1)(a1 – c1 + b1)
=
(b1 – c1 + a1)[
(c1 – a1)(b1 – a1) – (c1 – b1)]
=
(b1 – c1 + a1)[c1b1 – c1a1 – a1b1 + a12 – c1b1 + b12]
=
(b1 – c1 + a1)[ – c1a1 – a1b1 + 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 =
(c1 – a1)(b1 – c1 + a1)。
太虛弦 + 明弦 = c13 + c14
=
(c1 – b1)(c1 – a1) +
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)2 +
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)(b1 – c1 + a1 + c1 – a1)
=
(b1 – c1 + a1) × b1
=
(a1 + b1 – c1)。
已知日心股﹝又稱皇極股,簡稱“極股”﹞:b12 =
(a1 + b1 – c1)。
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c14 – c13:
(c1 – a1)(b1 – c1 + a1) –
(c1 – b1)(c1 – a1)
=
(c1 – a1)(b1 – c1 + a1) –
(b1 – c1 + a1)2
=
(b1 – c1 + a1)[(c1 – a1) – (b1 – c1 + a1)]
=
(b1 – c1 + a1)[ c1 – a1 – b1 + c1 – a1)]
=
(b1 – c1 + a1) (2c1 – 2a1 – b1)。
“明大差”指明弦﹝在勾股形日月南 14﹞勾弦較=c14 – a14。
c14 – a14 =
(c1 – a1)(b1 – c1 + a1) –
(c1 – a1)(b1 – c1 + a1)
=
(c1 – a1)(b1 – c1 + a1)[
– 1]
=
(b1 – c1 + a1)(c1 – a1)(c1 – a1)
=
(b1 – c1 + a1) (c1 – a1)2。
又“
小差”指
弦﹝在勾股形山川東 15﹞上股弦較 = c15 – b15 。
c15 – b15=
(c1 – b1)(a1 – c1 + b1) –
(c1 – b1)(a1 – c1 + b1)
=
(c1 – b1)(a1 – c1 + b1)[
– 1]
=
(c1 – b1)(a1 – c1 + b1)(c1 – b1)
=
(c1 – b1)2(a1 – c1 + b1) 。
山川
弦﹝簡稱
弦﹞:c15 =
(c1 – b1)(a1 – c1 + b1)。
“明大差、
小差內少個
弦”指前兩式相加減去第三式,即:
(b1 – c1 + a1) (c1 – a1)2 +
(c1 – b1)2(a1 – c1 + b1)
–
(c1 – b1)(a1 – c1 + b1)
=
(b1 – c1 + a1)[(c1 – a1)2 + (c1 – b1)2 – c1(c1 – b1)]
=
(b1 – c1 + a1)(c12 – 2c1a1 + a12 + c12– 2c1b1 + b12 –c12 + c1b1)
=
(b1 – c1 + a1)(c12 – 2c1a1 + a12 – c1b1 + b12 )
=
(b1 – c1 + a1)(2c12 – 2c1a1 – c1b1)
=
(b1 – c1 + a1) (2c1 – 2a1 – b1)。
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 =
(c1 – a1)(b1 – c1 + a1) +
(c1 – a1)(b1 – c1 + a1)
=
(c1 – a1)(b1 – c1 + a1)[
+
]
=
(c1 – a1)(b1 – c1 + a1)(a1 + b1) 。
“加於明和”即太虛弦 + 明和,即:
(c1 – b1)(c1 – a1) +
(c1 – a1)(b1 – c1 + a1)(a1 + b1)
=
(b1 – c1 + a1)2 +
(c1 – a1)(b1 – c1 + a1)(a1 + b1)
=
(b1 – c1 + a1)[c1(b1 – c1 + a1) + (c1 – a1)(a1 + b1)]
=
(b1 – c1 + a1)[c1b1 – c12+ c1a1 + c1a1 + c1b1 – a12 – a1b1]
=
(b1 – c1 + a1)[2c1b1 – c12+ 2c1a1 – a12 – a1b1]。
已知虛弦﹝在勾股形月山泛 13﹞:c13 =
(c1 – b1)(c1 – a1)。
“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= b6 – a6 =
(a1 + b1 – c1) –
(a1 + b1 – c1)
=
(a1 + b1 – c1)(
– 1)
=
(a1 + b1 – c1)(b1 – a1) 。
“兩個虛弦一个髙差共”即:
(c1 – b1)(c1 – a1) +
(a1 + b1 – c1)(b1 – a1)
=
(a1 + b1 – c1)2 +
(a1 + b1 – c1)(b1 – a1)
=
(a1 + b1 – c1)[
(a1 + b1 – c1) +
(b1 – a1)]
=
(a1 + b1 – c1)[2c1b1 + 2c1a1 – 2c12 + b12 – a1b1]
=
(a1 + b1 – c1)[2c1b1 + 2c1a1 – c12 – a12 –b12 + b12 – a1b1]
=
(a1 + b1 – c1)[2c1b1 + 2c1a1 – c12 – a12 – a1b1]。
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 =
(c1 – a1)(b1 – c1 + a1) +
(c1 – a1)(b1 – c1 + a1)
=
(c1 – a1)(b1 – c1 + a1)[
+
]
=
(c1 – a1)(b1 – c1 + a1)(a1 + b1) 。
“減於明和”即明和 –太虛弦,即:
(c1 – a1)(b1 – c1 + a1)(a1 + b1) –
(c1 – b1)(c1 – a1)
=
(c1 – a1)(b1 – c1 + a1)(a1 + b1) –
(b1 – c1 + a1)2
=
(b1 – c1 + a1)[(c1 – a1)(a1 + b1) – c1(b1 – c1 + a1)]
=
(b1 – c1 + a1)(c1b1 + c1a1 – a12 – a1b1 – c1b1 + c12– c1a1)
=
(b1 – c1 + a1)(– a12 – a1b1 + c12)
=
(b1 – c1 + a1)(b12 – a1b1)
=
(a1 + b1 – c1)(b1 – a1) 。
“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= b6 – a6 =
(a1 + b1 – c1) –
(a1 + b1 – c1)
=
(a1 + b1 – c1)(
– 1)
=
(a1 + b1 – c1)(b1 – a1) 。
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 =
(c1 – b1)(a1 – c1 + b1)。
“內去
勾”指太虛弦 –
勾,即:
太虛弦 –
勾=
(c1 – b1)(c1 – a1) –
(c1 – b1)(a1 – c1 + b1)
=
(a1 – c1 + b1)2 –
(c1 – b1)(a1 – c1 + b1)
=
(a1 – c1 + b1)[
(a1 – c1 + b1) – (c1 – b1)]
=
(a1 – c1 + b1)(c1b1 + c1a1 – c12 – c1a1 + a1b1)
=
(a1 – c1 + b1)(c1b1 – c12 + a1b1) 。
已知南月勾﹝又稱明勾,在勾股形日月南 14﹞:a14 =
(c1 – a1)(b1 – c1 + a1)。
“
較”指
弦上勾股較。
弦上勾股較 = b15 – a15
=
(c1 – b1)(a1 – c1 + b1) –
(c1 – b1)(a1 – c1 + b1)
=
(c1 – b1)(a1 – c1 + b1)(
–
)
=
(c1 – b1)(a1 – c1 + b1)(b1 – a1) 。
明勾
較共,即:
(c1 – a1)(b1 – c1 + a1) +
(c1 – b1)(a1 – c1 + b1)(b1 – a1)
=
(a1 – c1 + b1)[(c1 – a1) +
(c1 – b1)(b1 – a1)]
=
(a1 – c1 + b1)(c1a1 – a12 + c1b1 – c1a1 – b12+a1b1)
=
(a1 – c1 + b1)(– a12+ c1b1 – b12+ a1b1)
=
(a1 – c1 + 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 =
(c1 – b1)(a1 – c1 + b1)。
“平差”指平弦上勾股較。
平弦上勾股較 = b8– a8 =
(a1 + b1 – c1) –
(a1 + b1 – c1)
=
(a1 + b1 – c1)(1 –
)
=
(a1 + b1 – c1)(b1 – a1)
=
(b12 –a12 – c1b1 + c1a1)
=
[(b1 – a1)(b1 + a1) – c1(b1 – a1)]
=
(b1 – a1)(b1 + a1 – c1)。
股平差共 =
(c1 – b1)(a1 – c1 + b1) +
(b1 – a1)(b1 + a1 – c1)
=
(a1 – c1 + b1) [
(c1 – b1) +
(b1 – a1)]
=
(a1 – c1 + b1)( c1b1 – b12 + a1b1 – a12)
=
(a1 – c1 + 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 =
(c1 – b1)(a1 – c1 + b1)。
“加於
勾”指太虛弦 +
勾。即:
=
(c1 – b1)(c1 – a1) +
(c1 – b1)(a1 – c1 + b1)
=
(b1 – c1 + a1)2 +
(c1 – b1)(a1 – c1 + b1)
=
(b1 – c1 + a1)[
(b1 – c1 + a1) + (c1 – b1)]
=
(b1 – c1 + a1)(c1b1 + c1a1 – c12+ c1a1 – a1b1)
=
(b1 – c1 + a1)(c1b1 + 2c1a1 – c12 – a1b1) 。
“
和”即
弦上勾股和。
弦上勾股和 = b15 +a15 。
b15 + a15 =
(c1 – b1)(a1 – c1 + b1) +
(c1 – b1)(a1 – c1 + b1)
=
(c1 – b1)(a1 – c1 + b1)(
+
)
=
(c1 – b1)(a1 – c1 + b1)(b1 + a1) 。
已知南月勾﹝又稱明勾﹞:a14 =
(c1 – a1)(b1 – c1 + a1)。
和明勾共
=
(c1 – b1)(a1 – c1 + b1)(b1 + a1) +
(c1 – a1)(b1 – c1 + a1)
=
(a1 – c1 + b1)[
(c1 – b1)(b1 + a1) + (c1 – a1)]
=
(a1 – c1 + b1)(c1b1 + c1a1 – b12 – a1b1+ c1a1 – a12)
=
(a1 – c1 + b1)(c1b1 + 2c1a1 – b12 – a1b1– a12)
=
(a1 – c1 + b1)(c1b1 + 2c1a1 – c12 – a1b1)。
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 =
(c1 – b1)(a1 – c1 + b1)。
“加於
股”即太虛弦 +
股,即:
=
(c1 – b1)(c1 – a1) +
(c1 – b1)(a1 – c1 + b1)
=
(b1 – c1 + a1)2 +
(c1 – b1)(a1 – c1 + b1)
=
(b1 – c1 + a1)[
(b1 – c1 + a1) + (c1 – b1)]
=
(b1 – c1 + a1)(c1b1 + c1a1 – c12+ c1b1 – b12)
=
(b1 – c1 + a1)(2c1b1 + c1a1 – c12 – b12)。
已知虛弦 =
(c1 – b1)(c1 – a1),二虛弦 =
(c1 – b1)(c1 – a1) 。
明勾:a14 =
(c1 – a1)(b1 – c1 + a1)。
二虛弦內少明勾即:
二虛弦 – 明勾 =
(c1 – b1)(c1 – a1) –
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)2 –
(c1 – a1)(b1 – c1 + a1)
=
(b1 – c1 + a1)[
(b1 – c1 + a1) –
(c1 – a1)]
=
(b1 – c1 + a1)(2c1b1 – 2c12+ 2c1a1 – c1a1 + a12)
=
(b1 – c1 + a1)(2c1b1 – 2c12+ c1a1 + a12)
=
(b1 – c1 + a1)(2c1b1 – c12 – a12 – b12 + c1a1 + a12)。
=
(b1 – c1 + a1)(2c1b1 – c12 – b12 + 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 + b1 – c1,見上文。
已知“虛黃” = 太虛弦三事較 = 弦和較 = b13 + a13 – c13。
– c13 + b13 + a13
= –
(c1 – b1)(c1 – a1) +
(c1 – b1)(c1 – a1) +
(c1 – b1)(c1 – a1)
=
(c1 – a1)(c1 – b1)(a1 + b1 – c1) ﹝見前﹞。
又已知明勾﹝在勾股形日月南 14﹞:a14 =
(c1 – a1)(b1 – c1 + a1)。
虛黃明勾共=
(c1 – a1)(c1 – b1)(a1 + b1 – c1) +
(c1 – a1)(b1 – c1 + a1) ]
=
(c1 – a1)(a1 + b1 – c1)[
(c1 – b1) +
]
=
(c1 – a1)(a1 + b1 – c1)[2c1 – 2b1 + a1]
=
(a1 + b1 – c1)(2c12 – 2b1c1 + c1a1 – 2a1c1 + 2a1b1 – a12)
=
(a1 + b1 – c1)(2c12 – 2b1c1 – a1c1 + 2a1b1 – a12)。
圓徑內少虛黃明勾共
= (a1 + b1 – c1) –
(c1 – a1)(a1 + b1 – c1)[2c1 – 2b1 + a1]
= (a1 + b1 – c1) [1 –
(c1 – a1)(2c1 – 2b1 + a1)]
=
(a1 + b1 – c1)[2a1b1 –(2c12 – 2b1c1 – a1c1 + 2a1b1 – a12)]
=
(a1 + b1 – c1)[2a1b1 –2c12 + 2b1c1 + a1c1 – 2a1b1 + a12]
=
(a1 + b1 – c1)[ –2c12 + 2b1c1 + a1c1 + a12]
=
(a1 + b1 – c1)[ –c12 – a12 –b12 + 2b1c1 + a1c1 + a12]
=
(a1 + b1 – c1)[ –c12 – b12+ 2b1c1 + a1c1]。
比較上式與前式,可知太虛弦+
股 = 圓徑內少虛黃明勾共。
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