Appendix 1

This is a sub-page of a longer article, The classical Chinese version of Horner’s method: Technical considerations.

Appendix 1: Polynomials in some classical Chinese mathematical texts, 7th–14th century

This sample is taken from available translations of the texts. The root given in the text is shown in boldface.

Jigu suanjing 緝古算經, ca. 648 CE

Section references are to the translation by Lim & Wagner (2017).

Section	Equation	Real roots
3.2.3.1	x³ + 170x² + 7,166²/₃ x = 1,677,666²/₃	70
3.2.3.2	x³ + 1,620x² + 850,500x = 146,802,375	135
3.2.3.4	x³ + 276x² + 19,184x = 633,216	24
3.2.3.5	x³ + 840x² = 4,459,000	–833.58 –76.42 70
3.3.3.2	x³ + 5,004 x² + 1,169,953¹/₃ x = 41,107,188¹/₃	–4,756.20 –278.80 31
3.3.3.3	x³ + 3,298²/₃₁ x² + 2,474,941²⁹/₃₁ x = 23,987,761,548¹²/₃₁	1,920
3.4.3	x³ +62x² + 696x = 38,448	18
3.4.4.2	x³ + 594x² = 682,803	–592.05 –34.95 33
3.4.4.3	x³ + 693x² + 42,471x = 683,665.488	–623.08 –83.12 13.2
3.5.3.2	x³ + 1,728x² + 746,496x = 7,644,119,040	1,440
3.6.3	x³ + 135x² + 4,550x = 285,000	30
3.6.4	x³ + 315x² + 32,400x = 435,888	12
3.7.3.1	x³ + 15x² + 66x = 360	3
3.7.3.2.3	x³ + 18x² + 108x = 1,512	6
3.8.3	x³+ 90x² = 839,808	72
3.9.3	x³ + 30x² + 264x = 20,304	18
3.9.4	x³ + 162x² + 8,748x = 215,784	18
3.10.3	x³ + 60.392 x² + 911.998x = 32,369.022	15.5
3.11.3	x³ + 17 ³⁷/₈₉ x² + 99 ¹⁴/₈₉ x = 6,429²⁷/₈₉	13
3.12.3	x³ + 6.8x² + 15.2x = 4,289.6	14
3.13.3	x³ + 35x² + 441x = 5,145	7
3.14.3	x³ + 35x² + 441x = 5,145	7
3.15	x³ + 18⁹/₂₀ x² = 6,754¹²⁹/₅₀₀	14⁷/₂₀
3.16	x³ + 3¹/₁₀ x = 1,313,783¹/₁₀	108¹/₂
3.17	x³ + 2³/₄ x² + ²/₅₀ x = 812,591⁵⁹/₁₂₅	92²/₅

Shushu jiuzhang 數書九章, 1247 CE

Libbrecht (1973: 209–211) gives a list of the equations solved in this book. It uses fractional approximations instead of decimals; in these cases I give the approximation, its value using decimals, and the correct root.

Equation	Roots
x² + 82,655x – 2,269,810,000 = 0 *	-104,397.0827 21,742 ¹⁰⁴²⁶/₁₂₆₁₄₀ = 21,742.0827
x² + 2x – 399 = 0	–21 19
9x² + 5,100x – 322,500 = 0	–624.0841 57 ⁸⁵³/₂₀₄₅ = 57.4171 ≈ 57.4175
528,381x² + 360,099,600x –18,933,652,500 = 0	–730.5639 49^20276319/_412406319 = 49.0492 ≈ 49.0489
16x² + 192x – 1,863.2 = 0	–18.3471 6.35 ≈ 6.34706
36x² + 360x – 13,068.8 = 0	–24.6983 14.7 ≈ 14.6983
0.5x² – 152x –11,552 = 0	–62.9605 366 ⁴¹²/₄₂₉ = 366.9604 ≈ 366.9605
6x² + 234x –2,600 = 0	–48.0234 9.0234
–x⁴ + 763,200x² – 40,642,560,000 = 0	–840 –240, 240 840
–x⁴ + 15,245x² – 6,262,506.25 = 0	–121.7477, –20.5548 20 ^1298025/_2362256 = 20.5495 ≈ 20.5548 121.7477
–x⁴ + 1,534,464x² – 526,727,577,600 = 0	–1008 –720 720 1008
x¹⁰ +15x⁸ + 72x⁶ – 864x⁴ –11,664x² – 34,992 = 0	–3 3

* On this text see ‘Qin Jiushao’s curious formula for the area of a banana leaf’.

Ceyuan haijing 側圓海鏡, 1248 CE

These are the equations from this book which are treated by Chemla (1982). The left column gives the Problem number in the volume ‘Appendice III’.

	Equation	Roots
III-9	–2x² + 28,800 = 0	120
III-13	x⁴ – 654x³ + 106,929x² + 22,472,640x –1,955,119,680 = 0	–152 72
V-12	0.5x³ –1,200x² +427,200x –40,320,000 = 0	168.7371 240 1,991.2629
V-13	2x⁴ –1,200x³ +319,200x² +36,720,000x –7,344,000,000 = 0 (Corrected from Chinese text, p. 13)	–153.5510 120
V-13	0.5x⁴ –600x³ +319,200x² –240,480,000x +64,800,000,000 = 0	360 907.1019
V-14	–4x³ + 3600x² –1,256,400x + 105,840000 = 0	120
V-14	–0.5x³ –88,200x – 55,080,000 = 0	360
VI-18	4x³ –1,280x² + 270,080x – 20,889,600 = 0	120
VI-18	–0.5x³ + 320x² –135,040x + 20,889,600 = 0	240
VII-1	–x² + 204x + 8,640 = 0	–36, 240
VII-1	–x² + 102x + 2,160 = 0	–18 120
VII-1	–2x² + 204x + 4,320 = 0	–18 120
VII-1	x² – 19,044 = 0	–138 138
VII-2	–2160x⁴ + 444,960x³ – 10,628,820,000x + 717,445,350,000 = 0	–143.7799 81
VII-2	–x⁴ +8,640x² + 652,320x + 4,665,600 = 0 (Corrected from Chinese text, p. 33)	–7.9921 120
VII-2	–2x⁴ + 604x³ + 17,280x² – 8,553,244x + 401,067,842 = 0 (Corrected from Chinese text, p. 36)	–127.2601 289
VII-2	–2x⁶ –714x⁵ –62,165x⁴ – 2,220,302x³ + 82,926,816x² + 1,725,602,816x + 51,336,683,776 = 0 (Corrected from Chinese text, p. 39)	–254.2681 34
VIII-15	–70.4375x² –6,198.5x + 25,921 = 0	–92 4
XI-18	–x⁴ –1,406x³ –511,907x² –4,730,640x + 10,576,065,600 = 0	120
XII-1 (56)	16x² –328,960x + 26,214,400 = 0	80 20 480
XII-7	–289x² + 462,400 = 0	–40 40
XII-9	–0.5x² –680x + 192,000 = 0	–1,600 240

Yang Hui suanfa 楊輝算法, 1275 CE

These are taken from Lam Lay Yong (1977).

Page	Equation	Roots
71, 146	-3x² + 228x - 4,320 = 0	36 40
72, 146	-x² + 312x - 6,912 = 0	24 288
72, 146	-8x² + 312x - 864 = 0	3 36
73, 147	7x² - 9072 = 0	–36 36
74, 147	x² + 200x - 8,225 = 0	–235 35
75, 147	6x² + 48x - 390 = 0	–13 5
75, 144	-5x⁴ + 52x³+ 128x² - 4,096 = 0	4 12.056
75–76, 147	-x² + 36x - 180 = 0	6 30
76, 147	4x² - 144 = 0	–6 6

Shoushi li 授時曆 astronomical system, 1280 CE

On the Shoushi li see especially Sivin 2009. On its ptoto-trigonometry see Qian Baocong 1932: 148–151; Martzloff 1997: 328–335; Sivin 2009: 66–67; Needham 1959: 39, 108–110.

In the conversion of ecliptic to equatorial coordinates by a proto-trigonometric method this system uses an approximation for the sagitta of an arc given the diameter and the length of the arc:

s⁴ + (d²–ad)s² – d³s + d²a²/4 ≈ 0

where

d = diameter
a = length of arc
s = sagitta

The system uses this approximation with d = 121.75 and 0 < a < 91.3125. In this range the equation has two roots, shown here:

The correct root is the smaller of the two.

Siyuan yujian 四元玉鑑, 1303 CE

These are taken from Guo Shuchun et al. (2006).

Page	Equation	Roots
42, 43, 47, 48	x⁵ – 9x⁴ – 81x³ +729x² –3,888 = 0	-8.8439 -2.1372 3 6.6143 10.3668
50, 53, 55	x² – 2x – 8 = 0	-2 4
66, 69	x⁴ – 6x³ + 4x² + 6x –5 = 0	-1 1 5
84, 85	4x² – 7x – 686 = 0	-12.25 14
86, 87	181x⁸ – 22,868x⁶+ 278,926x⁴ – 818,100x² + 253,125 = 0	-10.6332 -1.9816 -0.5916 -3 0.5916 1.9816 3 10.6332
88, 89	–x⁸ – 70x⁶ + 6,479x⁴ – 186,624x² + 1,679,616 = 0	-4 4
90, 91	x⁴ – 12x³ – 54x² + 140x + 1,525 = 0	5 14.5547
90	–x³ + 2x² + 100x – 200 = 0	-10 2 10
92, 93	–9x³ + 59x² – 123x + 8,532 = 0	12
94, 95	27x³ – 600x² + 3325x – 130,000 = 0	25
94, 95	4x³ – 51x² + 132x – 29,088 = 0	24
96, 97	100x⁴ – 3,960x³ + 39,204x² – 81x – 944,055 = 0	-4.0698 9 10.7958 23.8740
98, 99	–x² + 200x – 4,071 = 0	23 177
98	x⁴ – 4x³ – 4,249x² + 4,494,400 = 0	40 54.1629
100, 101	x² – 36x + 315 =0	15 21
102, 103	5x⁴ – 3x³ – 12x² – 9x – 503,334	-17.6960 18
102, 103	x⁶ – 2x² –x – 46,578 = 0	-5.9997 6
104, 105	x⁶ + 18x⁵ + 99x⁴ + 162x³ + 74x² – 9x – 155,805 = 0	-11.3018 5
106, 107	x⁶ – 4x⁵ + 14x⁴ – 20x³ + 26x² – 5x – 78,414 = 0	-5.7099 7
108	x⁵ – 28x⁴ + 295x³ – 1,386x² + 2,450x – 7,800 = 0	12
108, 109	18x⁹ – 6x⁸ + 72x⁶ +x⁴ –9x² – 12268 = 0	2
110, 111	16x¹⁰ – 64x⁹ + 160x⁸ – 384x⁷+ 512x⁶ – 544x⁵ + 465x⁴ + 126x³ + 3x² –4x – 177,162 = 0	-1.9728 3
112, 113	x⁴ – 30x³ + 499x² – 5,320x + 22,178 = 0	8.2898 13
114, 115	5x³ – 13x² – 5x – 13,875 = 0	15
116, 117	x⁴ – 26x³ – 467x² + 8,300x + 97,440 = 0	-14.1047 -10.7276 24 26.8323
118, 119	x³ + 3x² – 50x – 4,064 = 0	16
120, 121	x² – 17x – 200 = 0	-8 25
122, 123	x⁴ – 17x³ + 98x² – 255x + 189 = 0	1.1638 9
124, 125	–49x⁶ – 495x⁵ + 5,371.75x⁴ + 278,768x³ + 843,296x² – 7,023,616x – 133,448,704 = 0	8 16.6759
126, 127, 129	5x⁴ + 19x² – 106,416 = 0	-12 12
129, 130, 131	21x⁷ + 40x⁶ – 277x⁵ – 425x⁴ + 908x³ + 1,009x² + 236x – 190,656 = 0	4
132, 133	–x⁴ + 2,046x² + 2,520x – 447,525 = 0	-41.5460 -16.6317 15 43.1777
134, 135	9x⁴ – 48x³ + 80x² – 115,200 = 0	-9.3543 12
136, 137	3x³ – 321x² + 4,072x + 12,816 = 0 *321 appears to be an error in the original text for 231	-2.6011 17.9130 91.6880
	3x³ – 231x² + 4,072x + 12,816 = 0	-2.7146 36 43.7146
138, 139	–x⁴ + 74x³ – 214x² + 420x – 33,831 = 0	9 70.9735
140, 141	11x² – 12x – 128 = 0	-2.9091 4
142, 143	2x³ + 2x² –27x – 11,826 = 0	18