Chinese syzygy calculation established in the 13th century

In the 13th century, a new type of Chinese astronomical calculation was put into use, and it was almost entirely included in the calendar of Shoushi (formulated by Guo shou-jing and his collaborators, epoch AD 1281, adopted from AD 1281-1644). Afterwards, this calendar has been thought of as one of the most accurate in the history of China. The perfectly systematic theoretical models at that time had been evolved since thousands of years. Nearly every part has its correspondences in modern astronomical year books, e.g., the syzygy calculation. By research on the origin of the Shoushi Calendar, we express the detailed and integral syzygy computing methods with several mathematical formulas, to simplify the complex reading and understanding of this calendar. The corrections of solar and lunar motions of this calendar for computing syzygys were compared with Newcomb's solar motion theory and the lunar ones of Chapront-Touze and Chapront. The results show that they have good correspondence. This calendar is of high precision, and the mean error summed of the syzygy calculation is less than 21 minutes in its publishing periods.