Chapter 1 Shíbié Zájí: Section I 1.1 SBZJ I.1-2: Zhu za ming mu 1.1.1 Text 1.1 Tiān zhīyúrìyǔrìzhīyúxīn tóng. 1.2 Xīn zhīyúchuān yǔ chuān zhīyúdètóng. 2.1.1 Rìzhīxīn yū rìzhīyúshān tóng, 2.1.2 gǔ yǐshān zhī chuān wéi xiǎo chā. 2.2.1 Chuān zhīxīn yū chuān zhīyúyuètóng, 2.2.2 gǔ yǐyuèzhīrìwéi dà chā. 1.1.2 Translation 1.1 AK = KO,(c 6 = b 12 ). 1.2 ON = NB,(a 12 = c 9 ). 2.1.1 KO = KM,(b 12 = c 6 ). 2.1.2 Therefore MN = c 15 = KN KM = KN KO and is called xiǎo chā. 2.2.1 NO = NL, (a 12 = c 8 ). 2.2.2 Therefore LK = c 14 = NK NL = NK NO and is called dà chā. 1.1.3 Notes (1.1) and (2.1.1) The easiest explanation of AK = KO is by joining AK and noting that KOA = OAC = OAB = OAK = KAO. There is, however, no evidence that the ancient Chinese, before the translation of Euclid s Elements, reasoned with angle measurement in right triangles. We shall seek an explanation using the out-in principle which LIU Hui used in his commentary of Jiuzhang Suanshu.
2 Shíbié Zájí: Section I A K O B C The equality AK = KO would follow from the congruence of the right triangles AKP and POT, where T, though not indicated in the circular city diagram, is the point where the incircle touches the hypotenuse AB. This congruence may not be immediately obviously. If, however, we swap the positions of the hypotenuse and gǔ of triangle KOT, this will result in a right triangle with hypotenuse along the line KB, gǔ along KJ, and with gōu equal to OT, the radius of the incircle. This must be the right triangle KMR, which clearly is also congruent to AKP. From this we have 1.1 and 2.1. A A K P K P T O I M R O I B J C B J C (a) (b) (2.1.2), (2.2.2) and the use of dà chā and xiǎo chā The use of the terms dà chā (big difference) and xiǎo chā (small difference) is somewhat ambiguous in CYHJ. Section II of SBZJ is on the five sums and five differences (wu huówujiào). The five differences are b a c a c b (a + b) c c (b a) gōu gǔ chā gōu xiān chā gǔ xiān chā xiān huó chā xiān chā chā chā dà chā xiǎo chā The use of dà and xiǎo is explained by c a>c b. These terms therefore apply to specific right triangles. This reference triangle is understood when it is the main one. (2.1.2) and (2.2.2) establish c 15 = MN and c 14 = LK as the xiǎo chā
1.2 SBZJ I.3: Zhu za ming mu 3 and dà chā of 12, but simply name them as xiǎo chā and dà chā, with implicit reference to 1. There are also occasions when dà chā and xiǎo chā refer to b d and a d. (Give explicit references). These terms are also used for two of the right triangles. 10 is named dà chā gōugǔ. It is the right triangle whose gǔ is equal to dà chā (c a), i.e., b 10 = c a. Likewise, 11 is named xiǎo chāgōugǔ, with a 11 = c b. 1.2 SBZJ I.3: Zhu za ming mu 3.1 Ming gōu zhuān gǔxiǎng dé, míng wéi nèi luu, qiu xū jī. 3.2 Ming gǔ zhuān gōu xiǎng dé, míng wéi wài luu, qiu xū jī. 3.3 Xū gōu xū gǔxiǎng dé, míng xū luu, qiu xū jī. 1.2.1 Translation 3.1 a 14 b 15 is called nèi luu, to find xū jī. 3.2 b 14 a 15 is called wài luu, to find xū jī. 3.3 a 13 b 13 is called xū luu, to find xū jī. 1.2.2 Notes a 15 b 14 b 13 a 13 a 14 b 15
4 Shíbié Zájí: Section I 1.3 SBZJ I.4: Zhu za ming mu 4.1 Fán gōu gǔ huójíxiān huāng huó. 4.2 Fán dà chājígǔhuāng jiào. 4.3 Fán xiǎo chājígōu huāng jiào. 1.3.1 Translation 4.1 a + b = c + d. 4.2 c a = b d. 4.3 c b = a d. 1.3.2 Notes (1) Fán means whenever. A statement beginning with fán (or dà fán) conveys universality. These formulae were given by LIU Hui in his commentary on Problem IX.16. 1 A O I B J C Figure 1.1: a + b = c + d (2) Huāng means yellow. Ancient manuscripts of Jiuzhang Suanshu and Zhoubi Suanjìng presumably were accompanied by colored diagrams. Traditionally certain elements were named by colors. The side length of a yellow square is called huāng fāng zhī miàn, or simply huāng fāng. In the present context, huāng refers to the diameter of the inscribed circle, which presumably was colored yellow. 1.4 SBZJ I.5: Zhu za ming mu 1.4.1 Text 1 LIU Hui gave at least three formulae: (i) b (c a)=d, (ii) (a+b) c = d, (iii) 2(c a)(c b)= d 2, which, along with a (c b) =d easily follow from the solution of Problem IX.12.
1.4 SBZJ I.5: Zhu za ming mu 5 5.1.1 Gaōgǔpíng gōu chā míng jiǎo chā, yòu míng yuǎn chā. 5.1.2 Cǐ shùjígaōpíng èr chāgòng yě. 5.1.3 Yòu wéi míng huó zhuān huójiào yě. 5.2.1 Míng zhuān èr chāgòng míng cìchā, yòu míng jín chā, yòu míng lièhuó. 5.2.2 Cǐ shùyòu wéi míng dàchā zhuān xiǎo chā jiaòyě. 5.3.1 Gōu yuán chāzhīgǔ, gǔyuán chāzhīgōu xiǎng bìng míng hùn dòng huó. 5.3.2 Cǐ shùyòu wéi yī jìng yī xūxiān gòng yě. 5.4.1 Míng zhuān èr chājiào míng bàng chā. 5.4.2 Cǐ shùyòu wéi gaō píng èr chājiào, 5.4.3 yòu wéi jíshùāng chānèi jiān xū huó, 5.4.4 yòu wéi jíxiān nèi jiān chéng jíng yě. 5.5.1 Xū chābùjíbàng chāmíng cuò chā. 5.5.2 Cǐ shùyòu wéi dà chāchānèi qù jiǎo chā, 5.5.3 yòu wéi jíchānèi qù èr zhīpíng chā, 5.5.4 yòu wéi cī chānèi qùxiǎo chāchā, 5.5.5 yòu wéi míng qǔ zhuān gōu gòng qù èr zhīmíng gōu yě. 5.6 Xū chābàng chāgòng míng cuòhuó. 1.4.2 Translation 5.1.1 Jiǎo chā (yuàn chā) := b g a p 5.1.2 = (b g a g )+(b p a p ) 5.1.3 = (a 14 + b 14 ) (a 15 + b 15 ). 5.2.1 Cì chā (jín chā, lièchā) := (b 14 a 14 )+(b 15 a 15 ) 5.2.2 = (c 14 a 14 ) (c 15 b 15 ). 5.3.1 Hùn tóng huó := b 11 + a 10 5.3.2 = d + c 13. 5.4.1 Bàng chā := (b 14 a 14 ) (b 15 a 15 ) 5.4.2 = (b g a g ) (b p a p ) 5.4.3 = (c 12 a 12 )+(c 12 b 12 ) (a 13 + b 13 ) 5.4.4 = c 12 d. 5.5.1 Cuò chā := bàng chā (b 13 a 13 ) 5.5.2 = (b 10 a 10 ) jiǎo chā 5.5.3 = (b 12 a 12 ) 2(b p a p ) 5.5.4 = emphcìchā (b 11 a 11 ) 5.5.5 = b 14 + a 15 2a 14. 5.6 Cuò huó := (b 13 a 13 )+ bàng chā.
6 Shíbié Zájí: Section I 1.4.3 Notes (5.1) (1) The right triangles 6 (shǎng gaō) and 7 (xià gaō) being congruent, they are simply referred to as (gaō) and denoted by g when the position are not important in the context. Likewise, a g for a 6 = a 7 etc. The same applies to the pair of congruent triangles 8 (shǎng píng) and 9 (xià píng). They are denoted by p with a p for a 8 = a 9 etc. (2) The segments b g = b 6 and a p = a 9 are situated at the corners of the circular city diagram. Their difference is named yuǎn (distant) or jiǎo (corner). This is given by the quartic form f 70 =(p 2 2pq 2q 2 )(p 2 +2pq q 2 ). (5.3) Yuán means circle. Here it is taken as the diameter d of the inscribed circle. Gōu yuán chā zhīgǔ is the gǔ on the right triangle which has the difference a d (gōu yuán chā)asgōu. Since a d = c b = (xiǎo chā) = a 11, this is b 11. Similarly, gǔ yuán chāzhīgōu is a 10. The sum b 11 + a 10 is called hùn dòng huó. This is also the sum of d and c 13. (5.4) (1) Shuāng means pair. Jí shùāng chā is the pair of differences c 12 a 12 and c 12 b 12.in 12. (2) Bàng chā is given by the quartic form f 69 =(p 2 2pq q 2 ) 2. What is the relevance of these definitions? 1.5 SBZJ I.6: Zhu za ming mu This important section contains a list of 10 formulae of the form f i f j = k r 2 for a constant k. The list is exhaustive except for f 10 f 36 = r 2 which can be written as Is this somewhere in CYHJ? d 13 = d 2 a + b + c.
1.5 SBZJ I.6: Zhu za ming mu 7 1.5.1 Text 6.1 Fán dàxiǎo chāxiāng chéng wéi bàn duàn jíng mì. 6.1 Dàchāgōu xiǎo chāgǔxiāng chéng yìtóng. 6.2 Xū gōu chéng dàgǔdébàn dùan jíng mì. 6.2 Xūgǔchéng dàgōu yì tóng shǎng. 6.3 Biān gǔ zhuān gǔ xiāng chéng dé bàn jíng mì. 6.3 Míng gǔdǐgōu xiāng chéng yìtóng shǎng. 6.4 Huāng gǔang gǔhuāng cháng gōu xiāng chéng wéi jíng mì. 6.5 Gaō gǔpíng gōu xiāng chéng dé bán jíng mì. 6.6 Míng xiān míng gǔ bìng yú zhuān xiān zhuān gōu bìng xiāng chéng dé bán jíng mì. 6.6 Míng xiān míng gōu bìng yú zhuān xiān zhuān gǔbìng xiāng chéng yì tóng shǎng. 6.7 Gaō xiān bìng xiān xiāng chéng wéi yì dùan huāng jíjī. 6.8 Míng gōu zhuān gǔxiāng chéng bèi zhīwéi yī dùan tài xū jī. 6.8 Míng gǔ zhuān gōu yì tóng. 1.5.2 Translation 1.5.3 Exegetical notes 1.5.4 SBZJ I.6.1 6.1 b 10 a 11 = 1 2 d2 6.1 a 10 b 11 = 1 2 d2 6.2 a 13 b 1 = 1 2 d2 6.2 b 13 a 1 = 1 2 d2 6.3 b 2 b 15 = r 2 6.3 a 14 a 3 = r 2 6.4 b 5 a 4 = d 2 6.5 b g a p = r 2 6.6 (b 14 + c 14 )(a 15 + c 15 )=r 2 6.6 (a 14 + c 14 )(b 15 + c 15 )=r 2 6.7 c g c p = a 12 b 12 6.8 2a 14 b 15 = a 13 b 13 6.8 =2b 14 a 13 The use of fán suggests reading this as (c a)(c b) = 1 2 d2 for an arbitrary right triangle. 2 This, as we mentioned, had been given by LIU Hui. 3 However, in the circular city diagram, dàchā as c a is also equal to dàchāgǔ b 10, 4 and xiǎochā as c b is also equal to xiǎochāgōu a 11. 5 This explains I.6.1. Along this line, RUǍN 2 See notes on I.4. 3 See notes on SBZJ I.4. 4 b 10 = b d = c a.
8 Shíbié Zájí: Section I Yuán remarked that the product of Dàchā gōu a 10 and xiǎochā gǔ b 11 is also the same. This gives I.6.1. (c-a)(c-b) = b 10a 11 = 1 2 d2 a 10 b 11 = 1 2 d2 1.5.5 SBZJ I.6.2 1.5.6 SBZJ I.6.3 1.5.7 SBZJ I.6.4 1.5.8 SBZJ I.6.5 1.5.9 SBZJ I.6.6 c 14 + b 14 = c 2 a 2, c 15 + a 15 = c 3 b 3. See the section on wu huo wu jiao. 6 Now, c 2 a 2 = b 2 b (c a) and c 3 b 3 = a 3 a (c b). Therefore, (c 14 +b 14 )(c 15 + a 15 )= b 2 a 3 (c a)(c b) = 1 ab 2 2r2 = r 2. Note: It is easy to see that b 2 a 3 = 1ab since the complement of b 2 2 a 2 in ab is equal to 1 = 1 ab. 2 5 a 11 = a d = c b. 6 b 14 + c 14 = b 12 + c 14 r = c 6 + c 14 r = c 10 r = c 10 + a 12 (a 12 + r) =c 10 + a 12 a 2 = c 10 + c 8 a 2 = c 2 a 2.
1.5 SBZJ I.6: Zhu za ming mu 9 a 13 b 1 = 1 2 d2 b 13 a 1 = 1 2 d2 b 2a 15 = r 2 a 14 a 3 = r 2
10 Shíbié Zájí: Section I b 4a 5 = d 2 b g a p = r 2 6 14 15 9 b 15 +c 15 = a p a14 +c 14 = b g
c g a 1.5 SBZJ I.6: Zhu za ming mu 15 11 1.5.10 SBZJ I.6.7 b 14 13 b 15 12 a 14 c p 9 c g c p =b 12 a 12 2a 14 b 15 = a 13 b 13 = 2a 15 b 14 1.5.11 SBZJ I.6.8