Investigating calendrical methods of calculating sunrise and sunset times in the Shixian calendar

The Shixian calendar is the general name for Chinese calendars implemented in China's Qing dynasty (1636-1912) and Korea's Joseon dynasty (1392-1910). This calendar made under the influence of Western sciences was revised several times. Referring to the Yuzhi lixiang kaocheng, compiled by Mei Juecheng et al. in 1724, we analyze calendrical methods employed in the Shixian calendar for calculating sunrise and sunset times in terms of astronomy and mathematics. In astronomy, the Polaris was used to measure the latitude, and the Sun's diurnal parallax was considered to increase accuracy. In mathematics, Western mathematics, such as trigonometric functions and properties of similar spherical triangles, were used to calculate sunrise and sunset times. In addition, ready reckoner tables were used to obtain the trigonometric values. The reproduced sunrise and sunset times are compared the results recorded in Shixian annual almanacs published in China and Korea to obtain the rules regarding dates on which these recording were made in the almanacs. We find that sunrise and sunset times were changed before and after around 1726 and in 1728 in the Shixian almanacs of China and Korea, respectively. Furthermore, we verify the values of the latitude and obliquity of the ecliptic used for calculating the sunrise and sunset times in the almanacs in each dynasty. This study will help understand the Shixian calendar and its annual almanacs in China and Korea.