Square Fields and Millet: Everyday Problems in a Han Reckoning Book

A problem is short: 'a field fifteen paces wide and sixteen paces long: what is the field?' — 'Answer: one mu' (今有田廣十五步，從十六步。問：為田幾何？荅曰：一畝). One question, one answer, and then the procedure. The everyday life of a Han reckoning book begins with a piece of ground like this.

One field problem: fifteen paces by sixteen

The first two problems, as written: 'a field fifteen paces wide and sixteen paces long: what is the field? Answer: one mu.' 'Another field twelve paces wide and fourteen long: what is the field? Answer: one hundred and sixty-eight paces' (答曰：一百六十八步). The second falls short of a mu, so the answer is given in paces — the unit changes with the question. The procedure for these problems is not under them; it stands at the head of the chapter for all of them.

The chapter's procedure, as written: 'the square-field procedure: multiply the numbers of paces of width and length to obtain the pace-area; divide by the mu-divisor of two hundred and forty paces, and you have the number of mu; a hundred mu make a qing' (方田術曰：廣從步數相乘得積步。以畝法二百四十步除之，即畝數。百畝為一頃). Three actions: multiply to get the area, divide by two hundred and forty, get the mu; a hundred mu to a qing. This article does not rewrite the procedure as a modern formula, nor reduce 'multiply' to modern signs. Nor is the answer 'one mu' checked by a modern calculation: the book gives that answer, and the answer is copied.

The chapter changes its unit and its procedure together. A third problem: 'a field one li wide and one li long: what is the field? Answer: three qing seventy-five mu' (答曰：三頃七十五畝). A fourth: 'two li by three li: answer, twenty-two qing fifty mu.' And the procedure for these: 'the li-field procedure: multiply the numbers of li of width and length to obtain the li-area; multiply by three hundred and seventy-five, and you have the number of mu' (里田術曰：廣從里數相乘得積里。以三百七十五乘之，即畝數). Same chapter, same idea — multiply the sides, turn the result into mu — but the multiplier changes because the unit does. The figures are copied; none of them is checked.

Another chapter: the millet rates

The millet chapter opens with a table of rates: 'grain 50, husked rice 30; polished rice 27, bleached rice 24; imperial rice 21, small bran 13½; large bran 54, husked-rice cooked 75…' — down to 'malt 175' (櫱一百七十五), and in between 'rice 60, bean paste 63' and 'cooked grain 90, cooked beans 103½'. A dozen and more entries, two to a line, some carrying fractions, all kept as written. Below the table stands the general rule: 'multiply the given number by the rate sought for the dividend, take the given rate as the divisor, and divide' (今有術曰：以所有數乘所求率為實，以所有率為法，實如法而一). Multiply, divide, result.

The problems follow those three moves. The first: 'a dou of grain, to be made into husked rice: how much? Answer: six sheng of husked rice. Procedure: to get husked rice from grain, three it and divide by five' (以粟求糲米，三之，五而一). The second: 'two dou one sheng of grain, to be made into polished rice… Answer: one dou one sheng and seventeen fiftieths of a sheng. Procedure: to get polished rice from grain, twenty-seven it and divide by fifty' (二十七之，五十而一). The fraction in the answer (seventeen fiftieths) is copied as written, neither decimalised nor reduced. A third: 'four dou five sheng of grain, to be made into bleached rice: answer, two dou one sheng and three fifths of a sheng; procedure, twelve it and divide by twenty-five' (十二之，二十五而一).

Problems and procedures: two arrangements

The two chapters are arranged differently. The field chapter puts one procedure at the head and every problem uses it; the millet chapter gives each problem its own — 'three it, divide by five', 'twenty-seven it, divide by fifty', 'twelve it, divide by twenty-five', one line each. That is a difference of arrangement, not of mathematics: each says what to multiply by and what to divide by, then gives the result. The li-field procedure is a third arrangement again, inserted after its own problems rather than at the head of the chapter. This article states the order of the moves and does not turn them into modern formulas, nor claim that the steps were ahead of anything.

Whose book this is

The Han imperial catalogue lists a different set of reckoners' books: 'a sundial book, thirty-four scrolls; Xu Shang's arithmetic, twenty-six scrolls; Du Zhong's arithmetic, sixteen scrolls' (日晷書三十四卷，許商算術二十六卷，杜忠算術十六卷) — in the section 'calendrical tables', which it totals as 'eighteen schools, six hundred and six scrolls' (右曆譜十八家，六百六卷), with a preface saying that calendrical tables 'order the places of the four seasons, rectify the nodes of solstice and equinox, and bring together the stations of sun, moon and five planets'. Note what the catalogue does and does not have: two books of 'arithmetic', and no title called Nine Chapters. Above the calendrical tables the same catalogue has 'astronomy: twenty-one schools, four hundred and forty-five scrolls' — the reckoners' books are filed as tables, not under a class of their own. The catalogue is one layer, the reckoning book another.

So the making of the book must be stated plainly: the Nine Chapters is a Han compilation of a reckoning tradition (this is generally held), not direct evidence for 'the level of Han mathematics'. Its problems and procedures accumulated over a long time; which belong to the Han and which are earlier, neither page separates, and the Hanshu catalogue does not date it. The book itself is divided into nine chapters, named on its headings — square fields, millet, and so on — but the two pages give neither an editor nor a date. This article copies problems, answers and procedures, and uses them to prove nothing about a period.

What these problems do not say

What neither page gives: who used these problems, how they were taught and learned, the physical counting rods and how they were laid out, how the procedures were applied in administration, or where the problems came from. The gaps are listed and not filled. Nor does this article run modern checks or point out that 'the ancients got it wrong' — unless the book itself records a variant; there is none on this page, so there is nothing to mark. Nor do the pages say how any answer was checked: each problem has one answer and no working.

Back to the two problems at the top: one multiplies and divides by two hundred and forty; one threes and divides by five. The book copies out the steps and does not explain why; all a reader can do is work through them — one mu, six sheng, the two numbers are there. How the whole thing was built up, both pages leave at the surface of the problem.

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