《孫子算經卷中》 "Sun Tzŭ's Computational Classic: Volume II"
§20. Rod calculus square root: \sqrt{420000}

This section gives a worked example of the rod calculus square root algorithm.

For a fully annotated and diagrammed walkthrough of Sun Tzŭ's square root algorithm, see the previous section (§19), whose level of detail I will not be repeating here.

Translation

Chinese source text: Version A, Version B, Version C, Version D.
Unless noted otherwise, I follow the text from Version D, 《知不足齋叢書》本.

Source text Target text Notes
今有積三萬五千步。問為圓幾何。 Suppose there be area three myriad [and] five thousand paces. [We] ask, how much be [this a] circle [of circumference]?
答曰、六百四十八步一千二百九十六分步之九十六。 Answer saith: six hundred [and] forty-eight paces [and] ninety-six one thousand two hundred [and] ninety-sixths of [a] pace.
術曰、置積三萬五千步、以一十二乘之、得四十二萬為實。 Method saith: put [down the] area three myriad [and] five thousand paces, [and] multiplying it by twelve, resulteth in forty-two myriad as [the] dividend.
  • Version C has 四十二萬步 for 四十二萬.
  • In modern notation, but taking \pi \approx 3 (so that 4\pi \approx 12) as per Vol. I §5, the circumference of a circle of area A = 35000 \unit{paces}^2 is
    \begin{aligned} C &\approx \sqrt{12 A} \\ &= \sqrt{420000} \unit{paces}. \end{aligned}
    The balance of this section involves extracting the square root of 420000. For a fully annotated and diagrammed walkthrough of Sun Tzŭ's square root algorithm, see the previous section (§19), whose level of detail I will not be repeating here.
次借一算為下法。步之、超一位、至百而止。 Next borrow one rod [to] be [the] lower divisor. Step it [forth], leaping over one place, halting [when] arriving at [the] hundreds.
上商置六百於實之上。 [For the] upper quotient, put six hundred above [the] dividend.
副置六萬於實之下、下法之上、名為方法。 Subsidiarily put six myriad below [the] dividend, [and] above [the] lower divisor, [its] name being [the] upright divisor.
  • 方： upright; or square
命上商六百除實。 Name [the] upper quotient's six hundred [and] remove [this] from [the] dividend.
除訖、倍方法。 [The] removal finished, double [the] upright divisor.
方法一退、下法再退。 [The] upright divisor retreateth once; [the] lower divisor retreateth twice.
  • 再： twice; or again
復置上商四十、以次前商。 Put again [for the] upper quotient forty, to [be] next [after the] former quotient.
副置四百於方法之下、下法之上、名為廉法。 Subsidiarily put four hundred below [the] upright divisor, [and] above [the] lower divisor, [its] name being [the] incorrupt divisor.
  • 廉法： incorrupt; or side
方廉各命上商四十、以除實。 Each of [the] upright [and the] incorrupt nameth [the] upper quotient's forty, to remove from [the] dividend.
除訖、倍廉法、從方法。 [The] removal finished, double [the] incorrupt divisor, [which] followeth [the] upright divisor.
方法一退、下法再退。 [The] upright divisor retreateth once; [the] lower divisor retreateth twice.
  • 再： twice; or again
復置上商八、次前商。 Put again [for the] upper quotient eight, next [after the] former quotient.
副置八於方法之下、下法之上、名為隅法。 Subsidiarily put eight below [the] upright divisor, [and] above [the] lower divisor, [its] name being [the] moral divisor.
  • 隅： moral; or corner
方廉隅各命上商八、以除實。 Each of [the] upright, [the] incorrupt, [and the] moral nameth [the] upper quotient's eight, to remove from [the] dividend.
  • Version C erroneously has 上前 for 上商八.
除訖、倍隅法、從方法。 [The] removal finished, double [the] moral divisor, [which] followeth [the] upright divisor.
上商得六百四十八、下法得一千二百九十六、不盡九十六。 [The] upper quotient resulteth in six hundred [and] forty-eight, [and the] lower divisors result in one thousand two hundred [and] ninety-six, remainder ninety-six.
  • Version A is missing 得 after 上商, and erroneously has 一千二百九十七 for 一千二百九十六 both here and in the line to follow.
是為方六百四十八步一千二百九十六分步之九十六。 This be [a] circumference of six hundred [and] forty-eight paces [and] ninety-six one thousand two hundred [and] ninety-sixths of [a] pace.
  • Thus the algorithm gives
    \sqrt{420000 \unit{paces}^2} \approx \colg{648} \tfrac{\colb{96}}{\colv{1296}} \unit{paces},
    which has relative error 6.5 \times 10^{-9}.

Cite this page

Conway (2023). "Sun Tzŭ's Computational Classic: Volume II §20". <https://yawnoc.github.io/sun-tzu/ii/20> Accessed yyyy-mm-dd.