瀟湘館112 / 數學及古代數學 / 《測圓海鏡》之角差及虛差等式說﹝諸差4﹞

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《測圓海鏡》之角差及虛差等式說﹝諸差4﹞

2021-02-24  瀟湘館112

測圓海鏡之角及虛差等式諸差4

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。本文著重十五勾股形之角差虛差之相關等式

關鍵詞:極差旁差角差虛差次差

《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,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》之精注意勾股定理成立,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。

本文主要談及十五勾股形有三邊相差之等式,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 (c1b1)(c1a1) = (a1 + b1c1)2

本文取自《測圓海鏡‧卷一‧諸差》。筆者有以下文涉及〈諸差〉

測圓海鏡大差差小差差等式﹝諸差1

測圓海鏡髙差旁差極雙差等式﹝諸差2

測圓海鏡極差等式﹝諸差3

本文乃以上三文之延續。閱讀本文宜注意角差旁差次差之定義

以下為有關”及相關之等式:

角差內加旁差為二髙差內減旁差即二平差也內加明二差而半之得極差內減明二差而半之則虛差也內加極差則通差內減極差則虛差也

以虛差減於明和為明二股共以虛差加於和為明二勾共也又副置二和共上加次差而半之即明二股共減次差而半之即明二勾共也二股共以髙差為之較二勾共以平差為之較

以下為各條目之証明:

角差內加旁差為二髙差

據《測圓海鏡》所云,“髙股平勾差”是為“角差”。

股:b6 = = (a1 + b1c1) 勾:a8 = = (a1 + b1c1)

髙股平勾差 = b6a8 = (a1 + b1c1) –(a1 + b1c1)

= (a1 + b1c1)[]

= (a1 + b1c1)

以上是為角差”或稱為“逺差”。

旁差”又傍差”,據《測圓海鏡》所云,“*二差較”是為傍差

明差 = b14a14 = (c1a1)( a1 + b1c1)[]

* = b15a15 = (c1b1)( a1 + b1c1) []

旁差 = 二差 = 明差*

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

= ( a1 +b1c1)[][(c1a1) – (c1b1)]

= (a1 + b1c1)(b1a1)

=(a1 + b1c1)

以上之式是為“旁差

角差內加旁差,即:

(a1 + b1c1) + (a1 + b1c1)

= (a1 + b1c1)[(b1 + a1) + (b1a1)]

= (a1 + b1c1) × 2b1

=(b1a1)(a1 + b1c1) #

髙差”即髙勾髙股差 = b6a6在勾股形天日旦 6 日山朱 7

髙勾髙股差= (a1 + b1c1) – (a1 + b1c1)

= (a1 + b1c1)( – 1)

= (a1 + b1c1)(b1a1)

二髙差= 2 ×(a1 + b1c1)(b1a1) = (b1a1)(a1 + b1c1) #

比較兩式可知相同,所以角差內加旁差 = 二髙差

內減旁差即二平差也

本條指角差內減旁差,即:

(a1 + b1c1) – (a1 + b1c1)

= (a1 + b1c1)[(b1 + a1) – (b1a1)]

= (a1 + b1c1) × 2a1

=(b1a1)(a1 + b1c1) #

平差平弦上勾股較。

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

= (a1 + b1c1)(1 –)

=(a1 + b1c1)(b1a1)

兩個平差= 2 ×(a1 + b1c1)(b1a1) =(a1 + b1c1)(b1a1) #

比較兩式,可知角差內減旁差 = 二平差

內加明二差而半之得極差

據《測圓海鏡》所云,*二差共名次差”,又名近差又名戾音列

明差指明勾與明股之差。

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

= (c1a1)( a1 + b1c1)[]

=(c1a1)(b1 c1 + a1)(b1a1)

**勾與*股之差。

* = 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,故*二差共次差

角差內次差角差內二差,即:

(a1 + b1c1) + (a1 + b1c1)(2c1a1b1)

= (a1 + b1c1)[(b1 + a1) +(2c1a1b1)]

= (a1 + b1c1) × 2c1

半之”即 × ,即得 (a1 + b1c1)(b1a1) #

極差”指皇極勾股較

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

= (a1 + b1c1)(b1a1) #

比較兩式,可知角差內二差而半之 = 極差

內減明二差而半之則虛差也

本條指角差內減明二差,即:

(a1 + b1c1) – (a1 + b1c1)(2c1a1b1)

= (a1 + b1c1)[(b1 + a1) – (2c1a1b1)]

= (a1 + b1c1) × 2(a1 + b1c1)

= (a1 + b1c1)2

半之”即 × ,即得 (a1 + b1c1)2 #

”指太虛勾股在勾股形月山泛 13

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

= (c1b1)(c1a1)[]

=(c1b1)(c1a1)(b1a1)

= (a1 + b1c1)2 #

注意等式 (c1b1)(c1a1) = (a1 + b1c1)2

所以角差內減明二差而半之 = 虛差

內加極差則通差

極差”指皇極勾股較在勾股形日川心 12

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

= (a1 + b1c1)[]

= (a1 + b1c1)(b1a1)

角差內加極差,即:

(a1 + b1c1) + (a1 + b1c1)(b1a1)

= (a1 + b1c1)(b1a1)[(a1 + b1) + c1]

= (a1 + b1c1)(b1a1)(a1 + b1 + c1)

= [(a1 + b1)2c12](b1a1)

= [a12 + b12+ 2a1b1c12](b1a1)

= (b1a1) × 2a1b1

= b1a1 #

上式是為通差所以角差內加極差 = 通差在勾股形天地乾 1

內減極差則虛差也

本條指角差內減極差,即:

(a1 + b1c1) – (a1 + b1c1)(b1a1)

= (a1 + b1c1)(b1a1)[(a1 + b1) – c1]

= (a1 + b1c1)(b1a1)(a1 + b1c1)

= (a1 + b1c1)2(b1a1)

=(c1b1)(c1a1)(b1a1) #

”指太虛勾股 =(c1b1)(c1a1)(b1a1) #﹝見前條﹞。

比較答案兩式可知相等,所以角差內減極差 = 虛差

以虛差減於明和為明二股共

已知“=(c1b1)(c1a1)(b1a1)

明和”即明弦勾股和 = b14 + a14

明弦勾股和= (c1a1)(b1 c1 + a1) +(c1a1)(b1 c1 + a1)

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

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

以虛差減於明和,即:

(c1a1)(b1 c1 + a1)(a1 + b1) – (c1b1)(c1a1)(b1a1)

= (c1a1)(b1 c1 + a1)(a1 + b1) –(a1 + b1c1)2(b1a1)

= (b1 c1 + a1)[(c1a1)(a1 + b1) – (a1 + b1c1)2(b1a1)]

= (b1 c1 + a1)[(c1a1)(a1 + b1) – (a1 + b1c1)(b1a1)]

= (b1 c1 + a1)[c1a1 + c1b1a12a1b1– (b12a12c1b1 + c1a1)]

= (b1 c1 + a1)(c1a1 + c1b1a12a1b1b12 + a12+ c1b1c1a1)

= (b1 c1 + a1)(2c1b1a1b1b12)

= (b1 c1 + a1)(2c1b1a1b1b12)

=(b1 c1 + a1)(2c1a1b1) × b1

= (b1 c1 + a1)(2c1a1b1) #

已知日南﹝又﹞:b14 =(c1a1)(b1 c1 + a1)

山東﹝又﹞:b15 = (c1b1)(a1c1 + b1)

二股共,即:

(c1a1)(b1 c1 + a1) + (c1b1)(a1c1 + b1)

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

= (b1 c1 + a1)(2c1a1b1) #

比較兩式,可知以虛差減於明和 = 二股共

以虛差加於和為明二勾共也

已知=(c1b1)(c1a1)(b1a1)

”即*弦上勾股和 = b15 +a15

b15 + a15 = (c1b1)(a1c1 + b1) +(c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)(+)

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

虛差加於,即:

(c1b1)(c1a1)(b1a1) + (c1b1)(a1c1 + b1)(b1 + a1)

= (a1c1 + b1)2(b1a1) + (c1b1)(a1c1 + b1)(b1 + a1)

=(a1c1 + b1)[(a1c1 + b1)(b1a1) + (c1b1)(b1 + a1)]

=(a1c1 + b1)(b12a12c1b1 + c1a1 + c1b1 + c1a1b12a1b1)

=(a1c1 + b1)(– a12 + 2c1a1a1b1)

=(a1c1 + b1)(2c1a1b1) × a1

=(a1c1 + b1)(2c1a1b1) #

已知南月勾﹝又﹞:a14 = (c1a1)(b1 c1 + a1)

東川勾﹝又﹞:a15 =(c1b1)(a1c1 + b1)

二勾共,即:

(c1a1)(b1 c1 + a1) + (c1b1)(a1c1 + b1)

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

=(a1c1 + b1)(2c1a1b1) #

比較答案兩式,可知相等,所以以虛差加於= 二勾共

又副置二和共上加次差而半之即明二股共

本條之“二和共”指二和共

已知“明和”即明弦勾股和 = b14 +a14在勾股形日月南 14,即:

 b14 + a14= (c1a1)(b1 c1 + a1) +(c1a1)(b1 c1 + a1)

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

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

*和”即*弦上勾股和 = b15 +a15 在勾股形山川東 15,即:

b15 + a15 = (c1b1)(a1c1 + b1) +(c1b1)(a1c1 + b1)

= (c1b1)(a1c1 + b1)(+)

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

二和共,即:

(c1a1)(b1 c1 + a1)(a1 + b1) + (c1b1)(a1c1 + b1)(b1 + a1)

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

=(a1 + b1c1)(2c1a1b1)(a1 + b1)

又已知次差 = (a1 + b1c1)(2c1a1b1)

二和共上加次差,即:

(a1 + b1c1)(2c1a1b1)(a1 + b1) +
(a1 + b1c1)(2c1a1b1)

= (a1 + b1c1)(2c1a1b1)[(a1 + b1) + (b1a1)]

= (a1 + b1c1)(2c1a1b1) × 2b1

= (a1 + b1c1)(2c1a1b1)

半之”即 × ,即得 (b1 c1 + a1)(2c1a1b1) #

已知二股共 = (b1 c1 + a1)(2c1a1b1) #

比較兩式,可知二和共上加次差而半之 = 二股共

減次差而半之即明二勾共也

本條指二和共上減次差,即:

(a1 + b1c1)(2c1a1b1)(a1 + b1) –
(a1 + b1c1)(2c1a1b1)

= (a1 + b1c1)(2c1a1b1)[(a1 + b1) – (b1a1)]

= (a1 + b1c1)(2c1a1b1) × 2a1

= (a1 + b1c1)(2c1a1b1)

半之”即 × ,即得 (b1 c1 + a1)(2c1a1b1) #

已知二勾共 =(a1c1 + b1)(2c1a1b1) #﹝見前條﹞

比較兩式,可知二和共上次差而半之 =

二股共以髙差為之較

已知二股共 = (b1 c1 + a1)(2c1a1b1)﹝見前條﹞。

髙差”指髙弦上勾股較在勾股形月川青 8 川地夕 9

髙弦上勾股較= b6a6 = (a1 + b1c1) – (a1 + b1c1)

= (a1 + b1c1)(– 1)

= (a1 + b1c1)(b1a1)

依《測圓海鏡》所云,(a1 + b1c1)(b1a1)
 (b1 c1 + a1)(2c1a1b1) 式之“”,含因子 (b1a1) 之式是為“”,即將 (b1 c1 + a1)(2c1a1b1) 式之因子 (2c1a1b1) 更換成因子 (b1a1),有此關係, 遂說成 (a1 + b1c1)(b1a1)
(b1 c1 + a1)(2c1a1b1) 式之“”。

二勾共以平差為之較

已知二勾共 =(a1c1 + b1)(2c1a1b1)﹝見前條﹞

平差”指平弦上勾股較。

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

= (a1 + b1c1)(1 –)

=(a1 + b1c1)(b1a1)

依上條之定義,(a1 + b1c1)(b1a1) (a1c1 + b1)(2c1a1b1) 式之“”。

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