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 – 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