Lunar day around the Sun at the same time, the Moon must travel further to return to the same phase. On average, this synodic period lasts 29 days, 12 hours, 44 minutes, and 3 seconds. This is the mean figure since the speed of the Earth–Moon system around the Sun varies slightly during a year due to the eccentricity of its elliptical orbit, variances in orbital velocity, and a number of other periodic and evolving variations about its observed, relative, mean values, which are influenced by the gravitational perturbations of the Sun and other bodies in the Solar System. As a result,
History of mathematics the Roman Kingdom and included 356 days plus a leap year every other year. In contrast, the lunar calendar of the Republican era contained 355 days, roughly ten-and-one-fourth days shorter than the solar year, a discrepancy that was solved by adding an extra month into the calendar after the 23rd of February. This calendar was supplanted by the Julian calendar, a solar calendar organized by Julius Caesar (100–44 BC) and devised by Sosigenes of Alexandria to include a leap day every four years in a 365-day cycle. This calendar, which contained an error of 11 minutes and 14 seconds, was