4. The ascendent at any given moment is that degree of the zodiac which is then seen upon the Eastern horizon. Chaucer says that astrologers reckoned in also 5 degrees _of the zodiac_ above, and 25 below; the object being to extend the planet's influence over a whole 'house,' which is a space of the same length as a _sign_, viz. 30°. See § 36 below. 5. This merely amounts to taking the mean between two results. 6. This depends upon the refraction of light by the atmosphere, owing to which light from the sun reaches us whilst he is still 18° below the horizon. The nadir of the sun being 18° high on the W. side, the sun itself is 18° below the Eastern horizon, giving the time of dawn; and if the nadir be 18° high on the E. side, we get the time of the end of the evening twilight. Thus, at the vernal equinox, the sun is 18° high soon after 8 A.M. (roughly speaking), and hence the evening twilight ends soon after 8 P.M., 12 hours later, sunset being at 6 P.M. 7. Ex. The sun being in the first point of Cancer on the longest day, its rising will be shewn by the point in fig. 5 where the _horizon obliquus_ and _Tropicus Cancri_ intersect; this corresponds to a point between P and Q in fig. 2, or to about a quarter to 4 A.M. So too the sunset is at about a quarter past 8, and the length of the day 16½ hours; hence also, the length of the night is about 7½ hours, neglecting twilight. 8. On the same day, the number of degrees in the whole day is about 247½, that being the number through which the _Rete_ is turned in the example to § 7. Divide by 15, and we have 16½ equal hours. 9. The 'day vulgar' is the length of the 'artificial day,' with the length of the twilight, both at morn and at eve, added to it. 10. If, as in § 7, the day be 16½ hours long, the length of each 'hour inequal' is 1 h. 22½ m.; and the length of each 'hour inequal' of the night is the 12th part of 7½ hours, or 37½ m.; and 1 h. 22½ m., added to 37½ m., will of course make up 2 hours, or 30°. 11. This merely repeats that 15° of the border answer to an hour of the clock. The '4 partie of this tretis' was never written. 12. This 'hour of the planet' is a mere astrological supposition, involving no point of astronomy. Each hour is an 'hour inequal,' or the 12th part of the artificial day or night. The assumptions are so made that _first_ hour of every day may resemble the _name of the day_; the first hour of Sunday is the hour of the _Sun_, and so on. These hours may be easily found by the following method. Let 1 represent both Sunday and the Sun; 2, Monday and the Moon; 3, Tuesday and Mars; 4, Wednesday and Mercury; 5, Thursday and Jupiter; 6, Friday and Venus; 7, Saturday and Saturn. Next, write down the following succession of figures, which will shew the hours at once. 1642753|16427531642753164275316.