Let us now consider the ascension of points on the ecliptic.
Applying Eq. (30) to a point on the celestial equator (*i.e.*, )
of right ascension , we obtain

(36) |

(37) |

(38) |

This expression specifies the ecliptic longitude, , of the point on the ecliptic circle which ascends simultaneously with a point on the celestial equator of right ascension . Note, incidentally, that points on the celestial equator ascend at a

The solution of Eq. (40) for observation sites lying above
the arctic circle is complicated by the fact that, at such sites, a section of the ecliptic never
sets, or *descends*, and a section never ascends. It is easily demonstrated that the
section which never descends lies between ecliptic longitudes
and
, whereas the section which
never ascends lies between longitudes
and
. Here,
.
Points on the ecliptic of longitude ,
,
, and
ascend simultaneously
with points on the celestial equator of right ascension
, ,
, and ,
respectively. Here,
.

Tables 5-17 list the ascensions of a series of equally spaced points on the ecliptic circle, as viewed from a set of observation sites in the earth's northern hemisphere with different terrestrial latitudes. The tables were calculated with the aid of formula (40). Let us now illustrate the use of these tables.

Consider a day on which the sun is at ecliptic longitude 14LE00 (*i.e.*,
into the sign of Leo). What is the length of the day (*i.e.*, the period between sunrise and sunset) at
an observation site on the earth's surface of latitude ? Consulting
Table 8, we find that the sun ascends simultaneously with a
point on the celestial equator of right ascension . Now,
the ecliptic is a great circle on the celestial sphere. Hence, exactly half of the ecliptic
is visible from any observation site on the earth's surface. This implies that
when a given point on the ecliptic circle is ascending, the point directly
opposite it on the circle is descending, and *vice versa*. Let us
term the directly opposite point the *complimentary point*.
By definition, the difference in ecliptic longitude between a given point on the
ecliptic circle and its complementary point is . Thus, the
complimentary point to is . It follows that
ascends at the same time that descends.
In other words, the sun sets when 14AQ00 ascends.
Consulting Table 8, we find that the sun sets at the
same time that a point on the celestial equator of right ascension
rises. Thus, in the time interval between the rising and
setting of the sun a
section
of the celestial equator ascends at the eastern horizon. However, points on the celestial
equator ascend at the uniform rate of an hour. Thus,
the length, in hours, of the period between the rising and setting of the sun is
In other words, the
length of the day in question is .

The above calculation is slightly inaccurate for a number of reasons. Firstly, it neglects the
fact that the sun is continuously *moving* on the ecliptic circle at the rate of about a day.
Secondly, it neglects the fact that the celestial equator ascends
at the rate of per *sidereal*, rather than
*solar*, hour. A sidereal hour is th of a sidereal day,
which is the time between successive upper transits of a fixed celestial object, such as a star. On the other hand, a solar hour is th of a
solar day, which is the mean time between successive upper transits
of the sun. A sidereal day is shorter than a solar day by 4 minutes.
Fortunately, it turns out that these first two inaccuracies largely
cancel one another out. Another source of inaccuracy is the fact that, due to
refraction of light by the atmosphere, the sun is actually below
the horizon when it appears to rise or set. The final source of inaccuracy
is the fact that the sun has a finite angular extent (of about half a degree),
and that, strictly speaking, dawn and dusk commence when the sun's upper limb rises and sets, respectively. Of course, our calculation
only deals with the rising and setting of the center of the sun.
All in all, the above mentioned inaccuracies can make the true length
of a day differ from that calculated from the ascension tables by up
to 15 minutes.

Tables 5-17 also effectively list the *descents* of a series of equally spaced points on the ecliptic
circle, as viewed from a set of observation sites in the earth's *southern*
hemisphere with different terrestrial latitudes (which are minus those specified in the various tables). For instance,
Table 6 gives the right ascensions of points on the celestial equator which *set* simultaneously
with points on the ecliptic, as seen from an observation site at latitude .

Consider a day on which the sun is at ecliptic longitude 08SC00. Let us calculate the length of the day at an observation site on the earth's surface of latitude ? Consulting Table 10, we find that the sun sets simultaneously with a point on the celestial equator of right ascension . Now, the complementary point on the ecliptic to 08SC00 is 08TA00. Consulting Table 10 again, we find that this point sets simultaneously with a point on the celestial equator of right ascension . It follows that the sun rises simultaneously with the latter point. Thus, the time interval between the rising and setting of the sun is time-degrees, or .

The *ascendent*, or *horoscope*, is defined as the point on the ecliptic which is ascending at the eastern horizon.
Suppose that we wish to find the ascendent hours after
sunrise, as seen from an observation site of latitude , on a day
on which the sun has ecliptic longitude 16SC00. Of course, knowledge
of the ascendent at the time of birth is key to drawing up a natal chart
in astrology. Hence, this type of calculation was of great importance to the ancients.
Consulting Table 11, we find that, on the day in question, the sun rises simultaneously with a point on the celestial equator of
right ascension . Now, hours corresponds to
. Thus, the ascendent rises simultaneously with a point
on the celestial equator of right ascension
. Consulting Table 11 again, we find that, to the
nearest degree, the ascendent at the time in question has ecliptic longitude 13SG00.

Suppose, next, that we wish to find the right ascension, , of the point on the celestial equator
which culminates simultaneously with a given point on the ecliptic of
ecliptic longitude .
From Eq. (33), we can see that if then
, and
, or

(41) |

Suppose, finally, that we wish to find the point on the ecliptic which culminates 7 hours after local noon on the aforementioned day. Since 7 hours corresponds to , the right ascension of the point on the celestial equator which culminates simultaneously with the point is question is . Consulting Table 5 again, we find that, to the nearest degree, the ecliptic longitude of the point in question is 21LE00.