238. The point _A_ is called _the Sun’s Apogee_, because when he is
there he is at his greatest distance from the Earth; the point _C_ his
_Perigee_, because when in it he is at his least distance from the
Earth: and a right line, as _AEC_, drawn through the Earth’s center,
from one of these points to the other, is called _the line of the
Apsides_.
[Sidenote: Mean Anomaly, what.]
239. The distance that the Sun has gone in any time from his Apogee (not
the distance he has to go to it though ever so little) is called _his
mean Anomaly_, and is reckoned in Signs and Degrees, allowing 30 Degrees
to a Sign. Thus, when the Sun has gone suppose 174 degrees from his
Apogee at _A_, he is said to be 5 Signs 24 Degrees from it, which is his
mean Anomaly: and when he is gone suppose 355 degrees from his Apogee,
he is said to be 11 Signs 25 Degrees from it, although he be but 5
Degrees short of _A_ in coming round to it again.
240. From what was said above it appears, that when the Sun’s Anomaly is
less than 6 Signs, that is, when he is any where between _A_ and _C_, in
the half _ABC_ of his orbit, the true noon precedes the fictitious; but
when his Anomaly is more than 6 Signs, that is, when he is any where
between _C_ and _A_, in the half _CDA_ of his Orbit, the fictitious noon
precedes the true. When his Anomaly is 0 Signs 0 Degrees, that is, when
he is in his Apogee at _A_; or 6 Signs 0 Degrees, which is when he is in
his Perigee at _C_; he comes to the Meridian at the moment that the
fictitious Sun does, and then it is noon by them both at the same
instant.
+----------------------------------------------------------+
| _Sun faster than the Clock if his Anomaly be_ |
+----+--------+-------+-------+-------+-------+-------+----+
| |0 Signs | 1 | 2 | 3 | 4 | 5 | |
| D. +--------+-------+-------+-------+-------+-------+ |
| | ʹ ʺ | ʹ ʺ | ʹ ʺ | ʹ ʺ | ʹ ʺ | ʹ ʺ | |
+----+--------+-------+-------+-------+-------+-------+----+
| 0 | 0 0 | 3 48 | 6 39 | 7 45 | 6 47 | 3 57 | 30 |
| 1 | 0 8 | 3 55 | 6 43 | 7 45 | 6 43 | 3 50 | 29 |
| 2 | 0 16 | 3 2 | 6 47 | 7 45 | 6 39 | 3 43 | 28 |
| 3 | 0 24 | 4 9 | 6 51 | 7 45 | 6 35 | 3 35 | 27 |
| 4 | 0 32 | 4 16 | 6 54 | 7 45 | 6 30 | 3 28 | 26 |
| 5 | 0 40 | 4 22 | 6 58 | 7 44 | 6 26 | 3 20 | 25 |
| 6 | 0 48 | 4 29 | 7 1 | 7 44 | 6 21 | 3 13 | 24 |
| 7 | 0 56 | 4 35 | 7 5 | 7 43 | 6 16 | 3 5 | 23 |
| 8 | 1 3 | 4 42 | 7 8 | 7 42 | 6 11 | 2 58 | 22 |
| 9 | 1 11 | 4 48 | 7 11 | 7 41 | 6 6 | 2 50 | 21 |
| 10 | 1 19 | 4 54 | 7 14 | 7 40 | 6 1 | 2 42 | 20 |
| 11 | 1 27 | 5 0 | 7 17 | 7 38 | 5 56 | 2 35 | 19 |
| 12 | 1 35 | 5 6 | 7 20 | 7 37 | 5 51 | 2 27 | 18 |
| 13 | 1 43 | 5 12 | 7 22 | 7 35 | 5 45 | 2 19 | 17 |
| 14 | 1 50 | 5 18 | 7 25 | 7 34 | 5 40 | 2 11 | 16 |
| 15 | 1 58 | 5 24 | 7 27 | 7 32 | 5 34 | 2 3 | 15 |
| 16 | 2 6 | 5 30 | 7 29 | 7 30 | 5 28 | 1 55 | 14 |
| 17 | 2 13 | 5 35 | 7 31 | 7 28 | 5 22 | 1 47 | 13 |
| 18 | 2 21 | 5 41 | 7 33 | 7 25 | 5 16 | 1 39 | 12 |
| 19 | 2 28 | 5 46 | 7 35 | 7 23 | 5 10 | 1 31 | 11 |
| 20 | 2 36 | 5 52 | 7 36 | 7 20 | 5 4 | 1 22 | 10 |
| 21 | 2 43 | 5 57 | 7 38 | 7 18 | 4 58 | 1 14 | 9 |
| 22 | 2 51 | 6 2 | 7 39 | 7 15 | 4 51 | 1 6 | 8 |
| 23 | 2 58 | 6 7 | 7 41 | 7 12 | 4 45 | 0 58 | 7 |
| 24 | 3 6 | 6 12 | 7 42 | 7 9 | 4 38 | 0 50 | 6 |
| 25 | 3 13 | 6 16 | 7 43 | 7 5 | 4 31 | 0 41 | 5 |
| 26 | 3 20 | 6 21 | 7 43 | 7 2 | 4 25 | 0 33 | 4 |
| 27 | 3 27 | 6 26 | 7 44 | 6 58 | 4 18 | 0 25 | 3 |
| 28 | 3 34 | 6 30 | 7 44 | 6 55 | 4 11 | 0 17 | 2 |
| 29 | 3 41 | 6 34 | 7 45 | 6 51 | 4 4 | 0 8 | 1 |
| 30 | 3 48 | 6 39 | 7 45 | 6 47 | 3 57 | 0 0 | 0 |
+----+--------+-------+-------+-------+-------+-------+----+
| |11 Signs| 10 | 9 | 8 | 7 | 6 | D. |
+----+--------+-------+-------+-------+-------+-------+----+
| _Sun slower than the Clock if his Anomaly be_ |
+----------------------------------------------------------+
[Sidenote: A Table of the Equation of Time, depending on the Sun’s
Anomaly.]
241. The annexed Table shews the Variation, or Equation of time
depending on the Sun’s Anomaly, and arising from his unequal motion in
the Ecliptic; as the former Table § 229 shews the Variation depending on
the Sun’s place, and resulting from the obliquity of the Ecliptic: this
is to be understood the same way as the other, namely, that when the
Signs are at the head of the Table, the Degrees are at the left hand;
but when the Signs are at the foot of the Table the respective Degrees
are at the right hand; and in both cases the Equation is in the Angle of
meeting. When both the above-mentioned Equations are either faster or
slower, their sum is the absolute Equation of Time; but when the one is
faster, and the other slower, it is their difference. Thus, suppose the
Equation depending on the Sun’s place, be 6 minutes 41 seconds too slow,
and the Equation depending on the Sun’s Anomaly, be 4 minutes 20 seconds
too slow, their Sun is 11 minutes 1 second too slow. But if the one had
been 6 minutes 41 seconds too fast, and the other 4 minutes 20 seconds
too slow, their difference had been 2 minutes 21 seconds too fast,
because the greater quantity is too fast.
242. The obliquity of the Ecliptic to the Equator, which is the first
mentioned cause of the Equation of Time, would make the Sun and Clocks
agree on four days of the year; which are, when the Sun enters Aries,
Cancer, Libra, and Capricorn: but the other cause, now explained, would
make the Sun and Clocks equal only twice in a year; that is, when the
Sun is in his Apogee and Perigee. Consequently, when these two points
fall in the beginnings of Cancer and Capricorn, or of Aries and Libra,
they concur in making the Sun and Clocks equal in these points. But the
Apogee at present is in the 9th degree of Cancer, and the Perigee in the
9th degree of Capricorn; and therefore the Sun and Clocks cannot be
equal about the beginning of these Signs, nor at any time of the year,
except when the swiftness or slowness of Equation resulting from one
cause just balances the slowness or swiftness arising from the other.
243. The last Table but one, at the end of this Chapter, shews the Sun’s
place in the Ecliptic at the noon of every day by the clock, for the
second year after leap-year; and also the Sun’s Anomaly to the nearest
degree, neglecting the odd minutes of a degree. Their use is only to
assist in shewing the method of making a general Equation Table from the
two fore-mentioned Tables of Equation depending on the Sun’s Place and
Anomaly § 229, 241; concerning which method we shall give a few examples
presently. The following Tables are such as might be made from these
two; and shew the absolute Equation of Time resulting from the
combination of both it’s causes; in which the minutes, as well as
degrees, both of the Sun’s Place and Anomaly are considered. The use of
these Tables is already explained, § 225; and they serve for every day
in leap-year, and the first, second, and third years after: For on most
of the same days of all these years the Equation differs, because of the
odd six hours more than the 365 days of which the year consists.
[Sidenote: Examples for making Equation Tables.]
EXAMPLE I. On the 15th of _April_ the Sun is in the 25th degree of ♈
Aries, and his Anomaly is 9 Signs 15 Degrees; the Equation resulting
from the former is 7 minutes 23 seconds of time too fast § 229; and from
the latter, 7 minutes 27 seconds too slow, § 241; the difference is 4
seconds that the Sun is too slow at the noon of that day; taking it in
gross for the degrees of the Sun’s Place and Anomaly, without making
proportionable allowance for the odd minutes. Hence, at noon the
swiftness of the one Equation balancing so nearly the slowness of the
other, makes the Sun and Clocks equal on some part of that day.
EXAMPLE II. On the 16th of _June_, the Sun is in the 25th degree of ♊
Gemini, and his Anomaly is 11 Signs 16 Degrees; the Equation arising
from the former is 1 minute 48 seconds too fast; and from the latter 1
minute 50 seconds too slow; which balancing one another at noon to 2
seconds, the Sun and Clocks are again equal on that day.
EXAMPLE III. On the 31st of _August_ the Sun’s place is 7 degrees 52
minutes of ♍ Virgo (which we shall call the 8th degree, as it is so
near) and his Anomaly is 2 Signs 0 Degrees; the Equation arising from
the former is 6 minutes 41 seconds too slow; and from the latter 6
minutes 39 seconds too fast; the difference being only 2 seconds too
slow at noon, and decreasing towards an equality will make the Sun and
Clocks equal in the afternoon of that day.
EXAMPLE. IV. On the 23d of _December_ the Sun’s place is 1 degree 41
minutes (call it 2 degrees) of ♑ Capricorn, and his Anomaly is 5 Signs
23 Degrees; the Equation for the former is 43 seconds too slow, and for
the latter 58 seconds too fast; the difference is 15 seconds too fast at
noon; which decreasing will come to an equality, and so make the Sun and
Clocks equal in the evening of that day.
And thus we find, that on some part of each of the above-mentioned four
days, the Sun and Clocks are equal; but if we work examples for all
other days of the year we shall find them different. And,
[Sidenote: Remark.]
244. On those days which are equidistant from any Equinox or Solstice,
we do not find that the Equation is as much too fast or too slow, on the
one side, as it is too slow or too fast on the other. The reason is,
that the line of the Apsides § 238, does not, at present, fall either
into the Equinoctial or Solsticial points § 242.
[Sidenote: The reason why Equation Tables are but temporary.]
245. If the line of the Apsides, together with the Equinoctial and
Solsticial points, were immoveable, a general Equation Table might be
made from the preceding Equation Tables, which would always keep true,
because these Tables themselves are permanent. But, with respect to the
fixed Stars, the line of the Apsides moves forwards 12 seconds of a
degree every year, and the above points 50 seconds backward. So that if
in any given year, the Equinoctial points, and line of the Apsides were
coincident, in 100 years afterward they would be separated 1 degree 43
minutes 20 seconds; and consequently in 5225.8 years they would be
separated 90 degrees, and could not meet again, so that the same
Equinoctial point should fall again into the Apogee in less than 20,903
years: and this is the shortest Period in which the Equation of Time can
be restored to the same state again, with respect to the same seasons of
the year.