Investigating Chinese mathematical techniques to calculate sunrise and sunset times in Datongli

We investigate the calculation method of sunrise and sunset times in the Datongli (Great Concordance Calendar) of the Ming dynasty focusing on Chinese mathematical techniques. Referring to the Mingshi (History of the Ming Dynasty), we found that latitude and obliquity were determined from observations of the transit altitude of the Sun on the winter and summer solstices using the Guibiao (Gnomon) and the Jianyi (Simplified Armillary Sphere) as astronomical instruments. According to the Mingshi, 12 values for the equation of center of the Sun were also determined from observations. However, these values may have been rectified, at least to utilize a cubic equation for interpolating the equation of center. With regard to Chinese mathematical techniques, we found that the Zhaocha (solving the coefficients of a cubic equation), Hushigeyuan (finding approximate solutions of a quartic equation), and Chongcha (proportional property of similar right-angled triangles) were employed to calculate the sunrise and sunset times together with the Gougu (i.e., the Pythagorean theorem) and Kaifang (i.e., square root extraction) techniques. We reproduce the sunrise and sunset times by the Datongli, and verify the numerical values of the tables in the Mingshi associated with the calculations of these times. We found that the values recorded in the Mingshi agree satisfactorily with those reproduced in this study, except for several cases considered typographical errors. We think that this study will help understand the Datongli, particularly the Chinese mathematical techniques used in it.