The Shoushi calendar and its evidence

The Yuan reform

In 1276, the Yuan court commissioned Xu Heng, Wang Xun and Guo Shoujing to prepare a new calendar. They worked with specialists from northern and southern China, comparing earlier methods with fresh observations of the Sun, Moon and stars. The calendar was completed at the winter solstice in 1280 and issued for use in 1281.

The reform addressed a continuing problem: the calendar had to connect lunar months with the seasonal year, although these cycles do not fit together exactly. It also had to predict particular events, including new moons and eclipses. Better observations supplied starting values; mathematical procedures carried those values forward into the dates of the coming year.

One important change was the abandonment of a remote “super-epoch,” an ideal ancient starting point at which several astronomical cycles were supposed to coincide. Shoushi instead used the winter solstice of 1280 as its epoch, or starting point for calculation. It recorded the intervals linking that moment with other relevant events, such as a preceding mean new moon. This removed the need to count enormous numbers of days from a hypothetical ancient beginning and allowed the different cycles to be tied to recent observations.

Calculating months and seasons

A mean lunar month supplied an average interval between new moons. Individual new moons could occur earlier or later than that average sequence because the Sun and Moon did not move at uniform apparent speeds. Shoushi applied corrections for these changing motions. The corrected new moon determined the beginning of a lunar month, so a change in the calculation could move the month beginning to a different day.

The reform refined interpolation: calculating intermediate values from a set of known values. This let a calculator obtain a correction for the particular stage of a celestial body's motion, rather than simply using the nearest tabulated entry. The system also accounted for precession, the gradual shift of seasonal reference points relative to the stars.

Solar terms continued to use pingqi, the mean-term method. The adopted seasonal year was divided into twenty-four equal time intervals, beginning from the winter solstice. Corrections for the Sun's uneven motion therefore coexisted with evenly spaced solar terms. Shoushi did not yet assign those terms by equal steps in the Sun's calculated position along its annual path.

Measuring the seasons at Gaocheng

At Gaocheng near Dengfeng in Henan, a horizontal bar cast its noontime shadow onto a long scale extending northward. Its height above the measuring surface was 40 chi, approximately 9.75 metres: five times the height of a standard eight-chi gnomon. A gnomon measures the Sun's changing height through its shadow. Raising the bar enlarged the changes in shadow position, while water-filled channels provided a level reference for the scale.

The larger instrument also produced a faint, blurred shadow. Guo's equipment addressed this with a small-aperture device that projected an image of the Sun crossed by the bar's shadow, making its position easier to identify.

Observers compared measurements on days before and after a solstice. Near the solstice itself, the noontime shadow changes very little, making the exact turning point difficult to locate from an apparently longest or shortest shadow. Repeated comparisons across a wider interval gave a firmer estimate of the solstice's time. Establishing that seasonal starting point was essential to calculating the year's solar terms.

Year length and the limits of prediction

Shoushi adopted a seasonal year of 365.2425 days and a mean lunar month of 29.530593 days. These numbers described astronomical cycles. The year value was not the length of every lunar calendar year, and the month value was not a prescription for fractional calendar days: individual lunar months contained 29 or 30 days, with leap months keeping the month sequence aligned with the seasons.

The year parameter governed the seasonal cycle, while the lunar-month parameter governed the average return of new moons. Accurate dates also depended on the epoch and the corrections for uneven motion. A close estimate of the year's length could not compensate for errors in those other parts of the calculation.

Eclipse prediction required still more information. The Moon's path is tilted relative to the Sun's apparent path, so at most new moons the Moon passes north or south of the Sun without covering it. A solar eclipse requires the new moon to occur near a crossing of those paths. Predicting its appearance and timing at a particular place also depends on the observer's location. The accuracy of the calendar cannot therefore be reduced to its famous year-length value.

The Ming inheritance

The Ming Datong calendar retained much of Shoushi's numerical and computational framework, with revisions to some starting values and rules. One difference concerned the length of the seasonal year. Shoushi included a long-term adjustment that made the adopted year shorter when calculating forward from its epoch and longer when calculating backward. Datong omitted this adjustment. The Yuan reform nevertheless continued to shape official calendar-making under a new dynasty through the procedures used to calculate months, seasons and celestial events.