But when in relation to antiquity--apart from Rome and Egypt--we speak of months, lunar months are as a rule to be understood. The moon revolves around the earth twelve times a year and a little more: consequently it moves backwards in the zodiac much more rapidly than the sun. The interval between two successive moments at which the moon culminates at the same spot at the same time as one and the same star is a _sidereal month_ (cp. the sidereal year); its length is 27 days 7 hrs. 43 min. 11.=42= secs., but it does not follow the phases of the moon and is therefore of no consequence for the calendar. The phases of the moon are dependent upon the position of the moon in relation to the sun and the earth. When the three bodies are in a straight line (or rather in a plane perpendicular to the plane of the ecliptic) in such a way that the earth is in the middle, the side of the moon turned towards the earth is completely illuminated and we have full moon: when the moon is in the middle, the side turned towards the earth is completely overshadowed, and that is new moon. In between lie the separate phases of the waxing and waning moon. The _synodic month_ is the interval between two new moons and comprises on an average 29 days 12 hrs. 44 min. 2.=98= secs. This is the true lunar month: other varieties of month are of no importance for us. "_The risings and settings of the stars._ It has already been remarked that the sun in the course of a year runs through the zodiac backwards, so that one particular star culminates 3 min. 56 secs. earlier every day. Hence it is evident that if we indicate the exact interval of time between the culmination of the sun and that of one particular star, or name the star with which the sun precisely culminates, we can determine the day of the solar year. This is the principle of one method of computing time which was very common among ancient and primitive peoples, but has entirely dropped out of use in modern times owing to our paper calendar. The stars are so to speak the stationary ciphers on the clock-face and the sun is the hand. In practice we naturally have to do not with the invisible culmination of the stars but with the position of the sun and certain neighbouring stars on the edge of the horizon, whereby the matter becomes more complicated on the astronomical side. For this observation the so-called circumpolar stars are singled out, that is to say the stars situated so near the pole that they do not set (e. g. the Great Bear). If the star rises or sets simultaneously with the rising of the sun, this is called the _true cosmic rising_ or _setting_. If the star rises or sets simultaneously with the setting of the sun, this is termed the _true acronychal rising_ or _setting_. These risings and settings of the star are not visible, since the sun hides them by its light: the rising and setting are perceptible only when the star stands at some distance from the sun, i. e.