“_The units of the time-reckoning_ are given by the motions of the heavenly bodies (expressed according to the Ptolemaic system), and the more intimately these enter into the life of man, the more important do they become. For this reason only those units which depend upon the sun have asserted themselves in our calendar, those depending upon the moon having been dropped, except for the movable paschal term, which has been kept on religious grounds. The units are the year, the month, and the day. Other units more convenient for time-reckoning play no part in the arrangement of the calendar since they are without importance for practical life. _The day_ (= 24 hours, νυχθήμερον) is determined from the apparent motion of the heavenly bodies about the earth, which is caused by the rotation of the earth on its axis; but since the sun also, on account of the annual revolution of the earth about it, runs through the zodiac in an opposite direction to its daily movement and completes the circle of the ecliptic in a year, a day will be a little longer than a complete rotation of the earth. Or to put it otherwise:--The time between two successive upper culminations of a star, i. e. between the moments at which the star passes through the meridian-line of one and the same place (= attains the zenith), represents an axial rotation: that is a _stellar day_. The time between two successive culminations of the sun is, on account of the annual motion of the sun (really that of the earth), 3 min. 56.=5= secs. longer than a stellar day: that is a _solar day_. The number of stellar days in a year is greater by one day than the number of solar days. The stellar day does not follow the variations of light and darkness and therefore does not enter into the calendar. The difference between the actual solar day, which is of slightly varying length, and the mean solar day abstracted from it for the purposes of our clock-regulated time-reckoning has no significance for antiquity. The second unit determined by the sun is the _year_, the period of a revolution of the earth about the sun. In relation to the apparent motion of the sun it may be defined as the time which the sun takes to come back again to the same fixed star. This is a _stellar_ or _sidereal year_, the length of which amounts to 365 days 6 hrs. 9 min. 9.=34= secs. The _tropic year_ is the time which the sun takes to come back to the crossing point of the equator, viz. the vernal equinox. This is the natural year. Its length varies a little; it is about 20 minutes shorter than the stellar year. The _lunar_ or _moon-month_ is determined from the visible phases of the moon. This term will be used only when it is necessary to make an express distinction between the lunar and our Roman month; the latter is a conventional subdivision of the year which has nothing to do with the moon, and has the name ‘month’ only because it historically arose from the lunar month and in its duration comes fairly near the latter.