Shushu Jiuzhang: Calculation in Calendrics, Surveying, and Administration

In 1247, Qin Jiushao of the Southern Song completed Shushu Jiuzhang. Its eighteen juan are divided into nine categories: Dayan, celestial phenomena, fields and land, surveying, taxation and labor service, money and grain, construction, military affairs, and miscellaneous matters. Each contains nine problems, for a total of eighty-one. It is not another name for The Nine Chapters on the Mathematical Art, but a separate work reorganizing Song-period problems and algorithms. Nor can all the activities described in its problems be dated to 1247.

Qin had experience in local office and practical administration, and his preface also mentions contact with calendrical mathematics in the capital. The book deals with calendrical cycles as well as land, labor obligations, grain transport, taxation, and trade. These subjects place calculation in a setting where officials needed to allocate resources, check accounts, and plan ahead. Carefully chosen numbers, however, may also have served to demonstrate algorithms; the problems are not necessarily unmodified archival records.

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The problem Guli Huiji combines years, months, and the sixty-day stem-and-branch cycle, seeking their conjunction under specified conditions. The difficulty is that the cycles have different lengths and the problem also gives offset positions within them. Testing day after day may be intuitive, but cannot efficiently handle very long cycles. The Dayan methods organize such conditions into systematic calculations with remainders.

In modern language, the task can be understood as finding an integer that simultaneously satisfies several remainder conditions. The original book has its own terms, including Dayan Zongshu Shu. Dayan Qiuyi Shu is the step used to find a crucial multiplier and should not be equated completely with the whole solution procedure. It is also distinct from the Tang dynasty’s similarly named Dayan calendar; similarity in names does not constitute evidence of direct inheritance.

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To combine different remainder conditions, one can first construct a number that gives the required remainder for one divisor while being divisible by the others, then combine such numbers. The challenge is finding the appropriate multiples. In a simplified example, thirty-five leaves a remainder of two when divided by three; multiplying it by two gives seventy, which leaves one. Two is therefore the multiplier that changes a remainder of two into one. More complex problems naturally cannot always be solved by mental trial.

Qin used organized division procedures to obtain such multipliers, making the handling of large numbers an executable process. The divisors are not always coprime, and the original problems require corresponding preparation and judgment. Applying only the simplest modern coprime case would overlook the original work’s processing of the conditions. The final result must be checked against every cycle or remainder requirement, rather than just one.

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The land category includes Sanxie Qiuji, a problem seeking a triangle’s area from its three sides. Without an available perpendicular height, the base cannot simply be multiplied by that height. The algorithm transforms relationships among the sides into a quantity obtainable by extracting a square root, then derives the area. It shows that land calculations could extend beyond regular rectangles, but the three input lengths still had to come from measurement: calculation could not repair incorrectly recorded sides.

The surveying problem Yaodu Yuancheng connects observation positions, distances, and the geometry of a circular city, ultimately reducing the task to a higher-degree equation. Its objective is not to solve an isolated equation in the abstract, but to infer a city’s diameter from accessible quantities when it cannot be measured directly. Complex relationships must be substituted and rearranged before they can become coefficient arrangements suitable for rod calculation.

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The book’s zhengfu kaifang procedures handle equations from lower to relatively high degrees. The calculator first organizes the coefficients, then determines successive place values of a root by trial, updating the numbers through repeated multiplication and addition so that the next digit is based on the residual relationship from the preceding step. Modern algebraic notation can display this structure, but the original work presents rod layouts, positions, and concrete operations.

The method’s value lies not only in obtaining an answer, but also in breaking a lengthy calculation into repeated, finite steps. It is not equivalent to a modern theory of all complex roots, and the later appearance of a similar algorithm does not by itself establish direct transmission. Historical comparison must examine results, operational sequences, representational media, and possible evidence of contact together.

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Problems concerning taxation and labor service, money and grain, and related subjects address the distribution of burdens, adjustment of taxes, transport costs, and grain volumes. Standardizing units and clarifying the basis of allocation before calculating are often as important as the operations themselves. For example, conflating distance, labor, and quantities to be borne into one measure could produce an inappropriate distribution even with faultless arithmetic. The problems therefore offer a window onto Song administrative ideals and quantitative management.

The work reached the present through later copying, editing, and printing, while modern critical explanations address textual errors, procedures, and terminology. Composition, transmission, and modern reinterpretation are distinct stages and cannot replace one another. Its significance in mathematical history derives from organizing the combination of remainders, numerical root finding, and diverse practical subjects into a systematic work, giving complex algorithms a written form that could be preserved, learned, and checked again.

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