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

* There are some mysteries here. According to Libbrecht (1973: 108), the root given for this equation in the book is 10,871, which is exactly one-half of the integer part of the root actually given in the book. Note also a typographical error of Libbrecht (1973: 209), an extra zero on the constant term of the equation. Further, the calculation given in the Chinese text for the area of a “banana leaf-shaped field” is clearly incorrect, and has inconsistent dimensions, and there is no obvious way of repairing it. The original text of the problem, in the Yijiatang congshu 宜家堂叢書 edition, is in juan 9, pp. 14b–15a, ctext.org/library.pl?if=en&file=83425&page=70

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 + (d2ad)s2d3s + 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 – 2x2x – 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