《測圓海鏡》之極差等式﹝諸差3﹞
《測圓海鏡》之極差等式﹝諸差3﹞
上傳書齋名:瀟湘館112 Xiāo XiāngGuǎn 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。本文著重皇極勾股較﹝即“極差”﹞之相關等式。
關鍵詞:極差、旁差、角差、蓌和、蓌差
《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》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。
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”。
注意圓徑為 a1 + b1 – c1,見上圖之東南西北圓。
本文主要談及十五勾股形有關三邊相差之等式,其中部分等式曾在“五和五較”等式中出現,可參閱筆者相關之文章。
注意等式 (c1 – b1)(c1 – a1) =
(a1 + b1 – c1)2。
本文取自《測圓海鏡‧卷一‧諸差》。筆者有以下文涉及〈諸差〉:
《測圓海鏡》之大差差、小差差等式﹝諸差1﹞
《測圓海鏡》之髙差、旁差、極雙差等式﹝諸差2﹞
本文乃以上二文之延續。
以下為有關“極差”及相關之等式:
極差內加旁差為大差差。內減旁差為小差差也。內加虛差即角差。內減虛差即次差也。倍極差為大差差小差差共,則倍旁差為之較。倍極弦為大差弦小差弦共,倍極差為之較。以極差為明差平差共,則以蓌差為之較。以極差為髙差
差共,則以蓌和為之較。副置蓌和上加蓌差而半之即旁差也。減蓌差而半之則虛差也。極差內減二之平差得蓌差。
以下為各條目之証明:
極差內加旁差為大差差。
“極差”指皇極勾股較﹝在勾股形日川心 12,可參閱上兩圖﹞。
已知皇極勾股較 = b12 – a12=
(a1 + b1 – c1) –
(a1 + b1 – c1)
=
(a1 + b1 – c1)[
–
]
=
(a1 + b1 – c1)(b1 – a1) 。
“旁差”又名“傍差”,據《測圓海鏡》所云,“明
二差較”是為傍差,此亦為“旁差”定義。
明差 = b14 – a14 =
(c1 – a1)(b1 – c1 + a1) –
(c1 – a1)(b1 – c1 + a1)
=
(c1 – a1)( a1 + b1 – c1)[
–
]。
差 = b15 – a15 =
(c1 – b1)(a1 – c1 + b1) –
(c1 – b1)(a1 – c1 + b1)
=
(c1 – b1)( a1 + b1 – c1) [
–
]。
旁差 = 二差較 = 明差 –
差
=
(c1 – a1)(a1 + b1 – c1)[
–
] –
(c1 – b1)( a1 + b1 – c1) [
–
]
=
( a1 +b1 – c1)[
–
][(c1 – a1) – (c1 – b1)]
=
(a1 + b1 – c1)(b1 – a1)
=
(a1 + b1 – c1) 。
以上之式是為“傍差” 。
極差內加旁差,即:
(a1 + b1 – c1)(b1 – a1) +
(a1 + b1 – c1)
=
(a1 + b1 – c1)(b1 – a1)[c1 + (b1 – a1)]
=
(a1 + b1 – c1)(b1 – a1)[b1 – a1+ c1]
=
(b1 – a1)[b1 – (c1 – a1)][b1 + (c1– a1)]
=
(b1 – a1)[b12 – (c1 – a1)2]
=
(b1 – a1)[b12 – c12 – a12 + 2c1a1]
=
(b1 – a1)[b12 – a12 –b12 – a12 + 2c1a1]
=
(b1 – a1)(2c1a1 – 2a12)
=
(b1 – a1)(c1 – a1) #。
“大差差”指大差上勾股較,勾股較即勾股差﹝在勾股形天月坤 10﹞。
大差上勾股差 = b10 – a10 = (c1 – a1) –
(c1 – a1)
= (c1 – a1)(1 –
)
=
(c1 – a1)(b1 – a1) #。
比較兩式可知相同,所以極差內加旁差 = 大差差。
內減旁差為小差差也。
指極差內減旁差,即極差 –旁差,即:
(a1 + b1 – c1)(b1 – a1) –
(a1 + b1 – c1)
=
(a1 + b1 – c1)(b1 – a1)[c1 – (b1 – a1)]
=
(a1 + b1 – c1)(b1 – a1)[– b1 + a1 + c1]
=
(b1 – a1)[a1 – (c1 – b1)][a1 + (c1– b1)]
=
(b1 – a1)[a1 – (c1 – b1)][a1 + (c1– b1)]
=
(b1 – a1)[a12 – (c1 – b1)2]
=
(b1 – a1)[a12 – c12 – b12 + 2c1b1]
=
(b1 – a1)[a12 – a12 –b12 – b12 + 2c1b1]
=
(b1 – a1)[ – 2b12 + 2c1b1]
=
(b1 – a1)(c1 – b1) #。
“小差差”指小差﹝在勾股形山地艮 11﹞上勾股較。
小差上勾股較 = – (c1 – b1) +
(c1 – b1)
= (c1 – b1)(
– 1)
=
(c1 – b1)(b1 – a1) #。
比較答案兩式可知相等,所以極差內減旁差 = 小差差。
內加虛差即角差。
“虛差”指太虛勾股較﹝在勾股形月山泛 13﹞。
太虛勾股較 = b13 – a13 =
(c1 – b1)(c1 – a1) –
(c1 – b1)(c1 – a1)]
= (c1 – b1)(c1 – a1)[
–
]
=
(c1 – b1)(c1 – a1)(b1 – a1)。
極差內加虛差,即:
(a1 + b1 – c1)(b1 – a1) +
(c1 – b1)(c1 – a1)(b1 – a1)
=
(a1 + b1 – c1)(b1 – a1) +
(a1 + b1 – c1)2(b1 – a1)
=
(a1 + b1 – c1)(b1 – a1)[c1 + a1 + b1 – c1]
=
(a1 + b1 – c1)(b1 – a1)[a1 + b1]
=
(a1 + b1 – c1) #。
據《測圓海鏡》所云,髙股平勾差名為“角差”。
髙股:b6 =
=
(a1 + b1 – c1), 平勾:a8 =
=
(a1 + b1 – c1)。
髙股平勾差 = b6 – a8 =
(a1 + b1 – c1) –
(a1 + b1 – c1)
=
(a1 + b1 – c1)[
–
]
=
(a1 + b1 – c1) #。
以上是為“角差”或稱為“逺差”。
比較答案兩式,可知極差內加虛差 = 角差。
內減虛差即次差也。
本條指極差內減虛差﹝極差與虛差見前條﹞,即:
(a1 + b1 – c1)(b1 – a1) –
(c1 – b1)(c1 – a1)(b1 – a1)
=
(a1 + b1 – c1)(b1 – a1) –
(a1 + b1 – c1)2(b1 – a1)
=
(a1 + b1 – c1)(b1 – a1)[c1 – a1 – b1 + c1]
=
(a1 + b1 – c1)(b1 – a1)[2c1 – a1 – b1] #。
據《測圓海鏡》所云,明
二差共名次差,又名近差,又名戾﹝音列﹞和。
明差指明勾與明股之差。
明差 = b14 – a14 =
(c1 – a1)(b1 – c1 + a1) –
(c1 – a1)(b1 – c1 + a1)
=
(c1 – a1)( a1 + b1 – c1)[
–
]。
差指
勾與
股之差。
差 = b15 – a15 =
(c1 – b1)(a1 – c1 + b1) –
(c1 – b1)(a1 – c1 + b1)
=
(c1 – b1)( a1 + b1 – c1)[
–
]。
二差共 = 明差 +
差
=
(c1 – a1)( a1 + b1 – c1)[
–
] +
(c1 – b1)( a1 + b1 – c1) [
–
]
=
( a1 +b1 – c1)[
–
](c1 – a1 + c1 – b1)
=
(a1 + b1 – c1)(2c1 – a1 – b1) #。
上式是為次差,故明
二差共得次差。
倍極差為大差差小差差共。
“極差”指皇極勾股較。
皇極勾股較= b12 – a12 =
(a1 + b1 – c1) –
(a1 + b1 – c1)
=
(a1 + b1 – c1)(b1 – a1)。
倍極差= 2 ×
(a1 + b1 – c1)(b1 – a1) =
(a1 + b1 – c1)(b1 – a1) #。
“大差差”指大差上勾股較 =
(c1 – a1)(b1 – a1) 。
“小差差”指小差上勾股較 =
(c1 – b1)(b1 – a1)。以上兩式見前條。
大差差小差差共,即:
(c1 – a1)(b1 – a1) +
(c1 – b1)(b1 – a1)
= (b1 – a1)[
(c1 – b1) +
(c1 – a1)]
=
(b1 – a1)[b1(c1 – b1) + a1(c1 – a1)]
=
(b1 – a1)[b1c1 – b12+ a1c1 – a12]
=
(b1 – a1)[b1c1 + a1c1 – c12]
=
(b1 – a1)c1(a1 + b1 – c1)
=
(a1 + b1 – c1)(b1 – a1) #。
比較答案兩式可知相等,所以倍極差 =大差差 + 小差差。
則倍旁差為之較。
“傍差”=
(a1 + b1 – c1) ﹝見前﹞。
倍旁差 = 2 ×
(a1 + b1 – c1) =
(a1 + b1 – c1) #。
大差差小差差較,即:
(c1 – a1)(b1 – a1) –
(c1 – b1)(b1 – a1)
= (b1 – a1)[ –
(c1 – b1) +
(c1 – a1)]
=
(b1 – a1)[– b1(c1 – b1) + a1(c1 – a1)]
=
(b1 – a1)[– b1c1 + b12+ a1c1 – a12]
=
(b1 – a1)[(b1 – a1)(b1 + a1) – c1(b1 – a1)]
=
(b1 – a1)2[b1 + a1 – c1]
=
(a1 + b1 – c1) #。
比較答案兩式可知相等,所以倍旁差 =大差差小差差較。
倍極弦為大差弦小差弦共。
已知皇極弦﹝日川﹞:c12 =
(a1 + b1 – c1) 。
倍極弦= 2 ×
(a1 + b1 – c1) =
(a1 + b1 – c1) #。
已知大差弦﹝在勾股形天月坤 10﹞=c10 =
(c1 – a1) 。
小差弦﹝在勾股形山地艮 11﹞= c11 =
(c1 – b1) 。
大差弦小差弦共= c10 + c11 =
(c1 – a1) +
(c1 – b1)
= c1[
(c1 – a1) +
(c1 – b1)]
=
[a1(c1 – a1) + b1(c1 – b1)]
=
[a1c1 – a12+ b1c1 – b12]
=
[a1c1 + b1c1 – c12]
=
(a1 + b1 – c1) #。
比較答案兩式可知相等,所以倍極弦 = 大差弦 + 小差弦。
倍極差為之較。
“極差”指皇極勾股較。
皇極勾股較= b12 – a12 =
(a1 + b1 – c1)(b1 – a1)。
倍“極差”= 2 ×
(a1 + b1 – c1)(b1 – a1) =
(a1 + b1 – c1)(b1 – a1) #。
大差弦小差弦較 = c10– c11 =
(c1 – a1) –
(c1 – b1)
= c1[
(c1 – a1) –
(c1 – b1)]
=
[a1(c1 – a1) – b1(c1 – b1)]
=
[a1c1 – a12 – b1c1 + b12]
=
[(b1 + a1)(b1 – a1) – c1(b1 – a1)]
=
(b1 – a1)(a1 + b1 – c1) #。
比較兩式可知相同,所以倍“極差”= 大差弦小差弦較。
以極差為明差平差共。
極差即皇極勾股較 =
(a1 + b1 – c1)(b1 – a1) #。
“明差”指明弦勾股較﹝在勾股形日月南 14﹞。
明弦勾股較=b14 – a14=
(c1 – a1)(b1 – c1 + a1) –
(c1 – a1)(b1 – c1 + a1)
=
(c1 – a1)(b1 – c1 + a1)[
–
]
=
(c1 – a1)(b1 – c1 + a1)(b1 – a1) 。
“平差”指平弦上勾股較﹝在勾股形月川青 8 或川地夕 9﹞。
平弦上勾股較 = b8– a8 =
(a1 + b1 – c1) –
(a1 + b1 – c1)
=
(a1 + b1 – c1)(1 –
)
=
(a1 + b1 – c1)(b1 – a1) 。
明差平差共,即:
(c1 – a1)(b1 – c1 + a1)(b1 – a1) +
(a1 + b1 – c1)(b1 – a1)
=
(a1 + b1 – c1)(b1 – a1)[
(c1 – a1) + 1]
=
(a1 + b1 – c1)(b1 – a1)[(c1 – a1) +a1]
=
(a1 + b1 – c1)(b1 – a1) #。
比較答案兩式可知相等,所以極差 = 明差 + 平差。
則以蓌差為之較。
明差平差較,即:
(c1 – a1)(b1 – c1 + a1)(b1 – a1) –
(a1 + b1 – c1)(b1 – a1)
=
(a1 + b1 – c1)(b1 – a1)[
(c1 – a1) – 1]
=
(a1 + b1 – c1)(b1 – a1)[(c1 – a1) – a1]
=
(a1 + b1 – c1)(b1 – a1)(c1 – 2a1) #。
據《測圓海鏡》所云,虛差不及傍差名“蓌差”,此即“蓌差”之定義。
已知虛勾 = a13 =
(c1 – a1)(c1 – b1),虛股 = b13 =
(c1 – a1)(c1 – b1)。
虛差 = b13 – a13
=
(c1 – a1)(c1 – b1)–
(c1 – a1)(c1– b1)
= (c1 – a1)(c1– b1)[
–
]。
虛差不及傍差,即傍差 – 虛差,即:
(a1 + b1 – c1) – (c1 – a1)(c1 – b1)[
–
]
=
(a1 + b1 – c1) –
( a1 + b1 – c1)2
=
(a1 + b1 – c1)[c12 – 2a1b1 – (b1 – a1)[(b1 + a1) – c1]]
=
(a1 + b1 – c1){(b1 – a1)2 –(b1 – a1)[(b1 + a1) – c1]}
=
(a1 + b1 – c1)(b1 – a1){b1 – a1– [(b1 + a1) – c1]}
=
(a1 + b1 – c1)(b1 – a1){b1 – a1– b1 – a1 + c1}
=
(a1 + b1 – c1)(b1 – a1)(c1 – 2a1) #。
以上之式是為“蓌差”。
比較兩式可知相同,所以明差平差較 = 蓌差。
以極差為髙差、
差共。
極差即皇極勾股較 =
(a1 + b1 – c1)(b1 – a1) #﹝見前條﹞。
“髙差”指髙弦上勾股較﹝在勾股形天日旦 6 或日山朱7﹞。
髙弦上勾股較= b6 – a6 =
(a1 + b1 – c1) –
(a1 + b1 – c1)
=
(a1 + b1 – c1)(
– 1)
=
(a1 + b1 – c1)(b1 – a1) 。
“
差”指
弦上勾股較﹝在勾股形山川東 15﹞。
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) 。
髙差
差共,即:
=
(a1 + b1 – c1)(b1 – a1) +
(c1 – b1)(a1 – c1 + b1)(b1 – a1)
=
(a1 + b1 – c1)(b1 – a1)[1 +
(c1 – b1)]
=
(a1 + b1 – c1)(b1 – a1)[b1 + (c1 – b1)]
=
(a1 + b1 – c1)(b1 – a1) #。
比較兩式可知相同,所以極差 =髙差 +
差。
則以蓌和為之較。
本條之“較”指髙差
差較,即:
=
(a1 + b1 – c1)(b1 – a1) –
(c1 – b1)(a1 – c1 + b1)(b1 – a1)
=
(a1 + b1 – c1)(b1 – a1)[1 –
(c1 – b1)]
=
(a1 + b1 – c1)(b1 – a1)[b1 – (c1 – b1)]
=
(a1 + b1 – c1)(b1 – a1)(b1 – c1 + b1)
=
(a1 + b1 – c1)(b1 – a1)(2b1 – c1) #。
據《測圓海鏡》,蓌和即傍差 + 虛差,即:
(a1 + b1 – c1) + (c1 – a1)(c1 – b1)[
–
]
=
(a1 + b1 – c1) +
( a1 +b1 – c1)2
=
(a1 + b1 – c1)[c12 – 2a1b1 + (b1 – a1)[(b1 + a1) – c1]]
=
(a1 + b1 – c1){(b1 – a1)2 + (b1 – a1)[(b1 + a1) – c1]}
=
(a1 + b1 – c1)(b1 – a1){b1 – a1 + [(b1 + a1) – c1]}
=
(a1 + b1 – c1)(b1 – a1){b1 – a1 + b1 + a1 – c1}
=
(a1 + b1 – c1)(b1 – a1)(2b1 – c1) #。
以上是為“蓌和”之式。
所以髙差
差較 = “蓌和”。
副置蓌和上加蓌差而半之即旁差也。
此處之“副置”疑指另外其他之算法。
蓌和上加蓌差,即:
(a1 + b1 – c1)(b1 – a1)(2b1 – c1) +
(a1 + b1 – c1)(b1 – a1)(c1 – 2a1)
=
(a1 + b1 – c1)(b1 – a1)[(2b1 – c1) + (c1 – 2a1)]
=
(a1 + b1 – c1)(b1 – a1)(2b1 – 2a1)
=
(a1 + b1 – c1)(b1 – a1)2。
“半之”即除以 2 =
(a1 + b1 – c1)(b1 – a1)2 # 。
又已知“傍差”=
(a1 + b1 – c1) #﹝見前條﹞。
比較兩式可知相同,所以蓌和上加蓌差而半之即旁差。
減蓌差而半之則虛差也。
蓌和上減蓌差,即:
(a1 + b1 – c1)(b1 – a1)(2b1 – c1) –
(a1 + b1 – c1)(b1 – a1)(c1 – 2a1)
=
(a1 + b1 – c1)(b1 – a1)[(2b1 – c1) – (c1 – 2a1)]
=
(a1 + b1 – c1)(b1 – a1)(2b1 + 2a1 – 2c1)
=
(a1 + b1 – c1)(b1 – a1)(a1 + b1 – c1)
=
(a1 + b1 – c1)2(b1 – a1)。
“半之”即除以 2 =
(a1 + b1 – c1)2(b1 – a1),
但
(a1 + b1 – c1)2(b1 – a1) =
(c1 – b1)(c1 – a1)(b1 – a1) #。
注意等式 (c1 – b1)(c1 – a1) =
(a1 + b1 – c1)2。
“虛差”指太虛勾股較﹝在勾股形月山泛 13﹞。
太虛勾股較 = b13 – a13 =
(c1 – b1)(c1 – a1)(b1 – a1) #。
比較兩式可知相同,蓌和上減蓌差而半之 = 虛差。
極差內減二之平差得蓌差。
極差﹝在勾股形日川心 12﹞即皇極勾股較 =
(a1 + b1 – c1)(b1 – a1)。
“平差”指平弦﹝在勾股形月川青 8 或川地夕 9﹞上勾股較。
平弦上勾股較 = b8– a8 =
(a1 + b1 – c1)(b1 – a1) 。
二之平差即2 ×
(a1 + b1 – c1)(b1 – a1) =
(a1 + b1 – c1)(b1 – a1) 。
極差內減二之平差得:
(a1 + b1 – c1)(b1 – a1) –
(a1 + b1 – c1)(b1 – a1)
=
(a1 + b1 – c1)(b1 – a1)[
– 1]
=
(a1 + b1 – c1)(b1 – a1)(c1 – 2a1) #。
從上題可知“蓌差”=
(a1 + b1 – c1)(b1 – a1)(c1 – 2a1) #。
比較兩式,可知極差內減二之平差得蓌差。
以下為《測圓海鏡細草》原文: