瀟湘館112 / 數學及古代數學 / 《測圓海鏡》勾股形日月南﹝14﹞五和五較說

分享

   

《測圓海鏡》勾股形日月南﹝14﹞五和五較說

2021-01-03  瀟湘館112

測圓海鏡勾股形日月南14五和五較

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》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,本文主要談及第14勾股形日月南關之等式

關鍵詞:日月南

《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。

《測圓海鏡》之〈五和五較〉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。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〉。

《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

以下左為“圓城圖式”右為“圓城圖式十五句股形圖”

本文談及之勾股形乃“日月南”﹝又稱為“明”,因日月為明之故也﹞,亦即以下兩圖帶棗紅色之二勾股形日月南之斜邊“日月”是為明,其直角為 14,以 14 之位置為 “南”,其勾股與弦分別為  a14月南b14日南 c14日月。注意 14 之點亦可表示日月南勾股形。

注意以下之日月南勾股形之位置

注意圓徑為 a1 + b1c1,見上圖左之東南西北圓。

以下為日月南勾股形之三事﹝三事,三邊之長也﹞:

南月勾﹝又﹞:a14 = (c1a1)(b1 c1 + a1)

日南﹝又﹞:b14 = (c1a1)(b1 c1 + a1)

日月為明弦﹝簡﹞:c14 = (c1a1)(b1 c1 + a1)

日月南勾股形之三事和或較亦可以以 a1b1c1 表之。

勾股形之 = c = a = b,則以下為五和五較:

(1)      勾股和:a + b

(2)      勾股較:ba

(3)      勾弦和:a + c

(4)      勾弦較:ca

(5)      股弦和:b + c

(6)      股弦較:cb

(7)      弦較和:c + (ba) ﹝較指勾股較,和指弦與勾股較之和﹞

(8)      弦較較:c – (ba) ﹝第一較字指勾股較,第二較字指弦與勾股較之較﹞

(9)      弦和和:(a + b) + c ﹝第一和字指勾股和,第二和字指弦與勾股和之和。又稱為三事和﹞

(10)      弦和較:(a + b) – c ﹝第一和字指勾股和,第二較字指弦與勾股和之較。又稱為三事較﹞

以下為與明勾股形日月南 (14)﹞有關之等式:

明弦勾股和即大差股內減明弦其較則明弦內減虛股也勾弦即髙股其較則髙股內少二之明勾也股弦和即邊股內減大差勾又為邊勾邊弦差其較則半個虛黃方也弦較和即大差上勾弦較其較則虛股也三事和即股圓差其較則太虛上勾弦較又為虛股內減虛黃方也

以下為各條目之証明:

明弦勾股和即大差股內減明弦

明弦勾股和=b14 + a14 = (c1a1)(b1 c1 + a1) +(c1a1)(b1 c1 + a1)

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

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

已知大差股 =天坤股﹝又大差股﹞:

b10 = b1 – (a1 + b1c1) = b1a1b1 + c1 = c1 a1

大差股內減明弦=b10c14

= (c1 a1) – (c1a1)(b1 c1 + a1)

= (c1 a1)[1– (b1 c1 + a1)]

=(c1a1)[2a1b1c1b1 + c12c1a1]

=(c1a1)[2a1b1c1b1 + a12 + b12c1a1]

=(c1a1)[(a1 + b1)2c1(b1 + a1)]

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

所以明弦勾股和 = 大差股內減明弦。“內減”即以前者為被減數,後者為減數。

其較則明弦內減虛股也

其較”指明弦勾股較

明弦勾股較=b14a14= (c1a1)(b1 c1 + a1) –(c1a1)(b1 c1 + a1)

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

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

已知太虛b13 =(c1b1)(c1a1)

明弦內減虛股 = c14b13 = (c1a1)(b1 c1 + a1) –(c1b1)(c1a1)

= (c1a1)(b1 c1 + a1) –(b1 c1 + a1)2

= (b1 c1 + a1)[(c1a1) – (b1 c1 + a1)]

=(b1 c1 + a1)(c12c1a1b12 + b1c1b1a1)

=(b1 c1 + a1)(a12c1a1+ b1c1b1a1)

=(b1 c1 + a1)[– a1(c1a1) + b1(c1a1)]

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

所以明弦勾股較 = 明弦內減虛股

勾弦即髙股

明弦勾弦 = c14 + a14

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

=(c1a1)(b1 c1 + a1)[+ 1]

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

=(b1 c1 + a1)(c12a12)

=(b1 c1 + a1)b12

=(a1 + b1c1)

已知天旦股﹝又上髙股﹞= b6 = = (a1 + b1c1)

所以明弦勾弦 = 股。

其較則髙股內少二之明勾也

其較”指明弦勾弦較

明弦勾弦較 = c14a14

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

=(c1a1)(b1 c1 + a1)[– 1]

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

=(b1 c1 + a1) (c1a1)2

髙股內少二之﹝即乘以2明勾 = b6 – 2 × a14

b6 – 2 × a14 = (a1 + b1c1) – 2 ×(c1a1)(b1 c1 + a1)

= (a1 + b1c1)[(c1a1)]

=(a1 + b1c1)[b12 – 2a1c1 + 2a12]

=(a1 + b1c1)[c12 – 2a1c1 + a12]

=(a1 + b1c1)(c1a1)2

所以明弦勾弦較 = 髙股內少二之明勾

股弦和即邊股內減大差勾

明弦股弦和=c14 + b14

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

= (c1a1)(b1 c1 + a1)[+ 1]

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

=(c1a1)[a1 – (c1b1)](c1 + b1)

=(c1a1)(a1c1 + a1b1c12+ b12)

=(c1a1)(a1c1 + a1b1a12)

=(c1a1)(c1 + b1a1)

已知天西股﹝簡股﹞= b2= b1(a1 + b1c1) = (c1 + b1a1)

坤月﹝又大差勾﹞= a10= = (c1 a1)

邊股內減大差勾=b2a10 = (c1 + b1a1) –(c1 a1)

= (b1c1 + b12b1a1 – 2a1c1 + 2a12)

= (b1c1 + c12b1a1 – 2a1c1 + a12)

= [c1(c1 +b1a1) – a1(c1 + b1a1)]

=(c1a1)(c1 + b1a1)

所以明弦股弦和 = 邊股內減大差勾

又為邊勾邊弦差

已知勾﹝川西﹞:a2 = (c1 + b1a1)

股﹝天西﹞:b2 = (c1 + b1a1)

弦﹝天川﹞:c2 = (c1 + b1a1)

邊弦上勾弦 = c2a2 = (c1 + b1a1) – (c1 + b1a1)

= (c1 + b1a1)(c1a1)

所以明弦股弦和 = 邊弦上勾弦較﹝即邊勾邊弦差

其較則半個虛黃方也

其較”指明弦股弦較。

明弦股弦 = c14 b14

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

= (c1a1)(b1 c1 + a1)[– 1]

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

已知虛黃方 = b13 + a13c13

= – (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。“黃方”定義可參閱筆者另文。

半個虛黃方=(c1a1)(b1 c1 + a1)(c1b1)

所以明弦股弦 = 半個虛黃方

弦較和即大差上勾弦較

明弦弦較和 = c14 + (b14a14) = c14 + b14a14

c14 + b14a14

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

=(c1a1)(b1 c1 + a1)[ + ]

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

=(c1a1)[b1 – (c1a1)][b1 + (c1a1)]

=(c1a1)[b12 – (c1a1)2]

=(c1a1)[b12 c12a12+ 2c1a1]

=(c1a1)[ – 2a12 + 2c1a1]

=(c1a1)[ – a1 + c1]

=(c1 a1)2

大差上弦勾差= c10a10 = (c1 a1) – (c1 a1)

= (c1 a1)(c1 a1)

=(c1 a1)2

所以明弦弦較和=大差上弦勾差

其較則虛股也

其較”指弦較較

明弦弦較較 = c14 – (b14a14) = c14b14 + a14

c14b14+ a14

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

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

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

=(c1a1)[a1 – (c1b1)][a1 + (c1b1)]

=(c1a1)[a12 – (c1b1)2]

=(c1a1)[a12 c12b12+ 2c1b1]

=(c1a1)[ – 2b12 + 2c1b1]

= (c1b1)(c1a1)

已舍太虛b13 =(c1b1)(c1a1)

所以明弦弦較較 = 虛股

三事和即股圓差

明弦三事和即弦和和 = c14 +b14 + a14

c14 + b14 + a14

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

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

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

=(c1a1)[a1 – (c1b1)][a1 + (c1 + b1)]

=(c1a1)[a12 +a1c1 + a1b1a1c1 + a1b1c12 + b12]

=(c1a1) × 2a1b1

= c1a1

已知股圓差 =b1 – (b1 c1 + a1) = b1b1 + c1a1 = c1a1

所以明弦三事和= 股圓差

其較則太虛上勾弦較

其較”指明弦三事較,又即弦和較。

明弦三事 = 弦和較 =b14 + a14 c14

b14 + a14 c14

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

=(c1a1)(b1 c1 + a1)[ –++]

=(c1a1)(b1 c1 + a1)( – c1 + b1 + a1)

=(c1a1)(b1 c1 + a1)2

= (c1a1)(c1b1)(c1a1)

= (c1b1)(c1a1)2

注意等式 (c1b1)(c1a1) = (a1 + b1c1)2

太虛勾弦較=c13a13= (c1b1)(c1a1) –(c1b1)(c1a1)

=(c1b1)(c1a1)[– 1]

=(c1b1)(c1a1)(c1a1)

=(c1b1)(c1a1)2

所以明弦三事 = 太虛上勾弦較

又為虛股內減虛黃方也

虛股內減個小黃方= b13 – (a13 + b13c13)

= c13a13

= 太虛勾弦較

所以明弦三事 = 虛股內減虛黃方也

以下為測圓海鏡原文:

    0条评论

    发表

    请遵守用户 评论公约

    类似文章
    喜欢该文的人也喜欢 更多

    ×
    ×

    ¥.00

    微信或支付宝扫码支付:

    开通即同意《个图VIP服务协议》

    全部>>