John Pazmino 
 2005 October 19
    In early 2001, I explained how the ancestral astronomers of 
Babylonia sussed out the cycles of the planets. That article arose 
from a discussion among New York astronomers regarding the conjunction 
of Jupiter and Saturn. The Babylonians had nothing of a solar system 
or other model of the planetary motions in space. By merely noting the 
instances of such conjunctions they were able to foretell future ones. 
    The ensuing banter thru spring 2001 shifted to the ancient 
observations of the Moon. The Moon was -- and is still in some 
cultures -- crucial for timekeeping. The month was historicly founded 
on the cycle of lunar phases. Careful watch of the Moon's changing 
shape was an integral part of ancient astronomy. 
Some history
    Most of the peoples of ancestral importance to us in the western 
society lived in what we now call the Middle East. The center of 
culture in this region in the last couple millennia of the BC era was 
Babylon, near present day Baghdad, Iraq. The empire seated in Babylon, 
called Babylonia, underwent many mutations over the centuries, 
including being overrun by foreign peoples and being sacked many 
    Despite the repeated calamities, Babylon managed to adhaere to a 
conscientious scheme of astronomy, including careful logging of the 
Moon's motions. In the earlier period, before about 700 BC, the whole 
effort was for observing the Moon. By about 400 BC enough data were in 
hand and the civilization was matured enough to make some awfully 
shrewd predictions of the Moon's behavior. Similar enterprises were 
established for observing the planets and weather, but here I work 
with only the Moon. 
    Out of this legacy from Babylon the Greeks developed a true system 
of lunar motion. Modern cultures still today use this system (with 
suitable updates) for maintaining their calendars. In fact, very much 
of the skills and principles of contemporary stargazing are rooted in 
Babylonian astronomy. 
New Moons
    As the Moon circulates around the Earth she alters her shape due 
to our perspective on more or less of her sunlighted hemisphere. Of 
course, to the unaided eye, the Moon looks pretty much like a flat 
disc which by some magic grows and shrinks over the course of one 
circuit thru the zodiac. A complete circuit of the Moon from a one 
conjunction with the Sun to the next is one synodic cycle or one 
    When the Moon is against the Sun in the sky we can not see her at 
all, save for the rare instance of a solar eclipse. The moment of 
actual conjunction, when the Moon is directly alongside the Sun, 
either a bit to the north or to the south, is the instant of new Moon. 
This situation of the Moon is from the ground under our light 
scattering atmosphere utterly unobservable. 
    But the moment of new Moon was a cardinal event for Babylonians. 
For on the next day they could expect to catch sight of the merest 
thin crescent of Moon as she pulls away from the Sun into the evening 
sky at sunset. Seeing this Moon signals the start of a new month. 
    They could anticipate the new Moon by catching the last thin 
crescent west of the Sun in the morning twilight before sunrise. 
Within a day the actual new Moon would occur. The Babylonians then set 
out observers on the second day to see the first Moon in evening. 
    The idea sounds so simple. Indeed, today many authors of astronomy 
exercises implore you to go out and repeat this Babylonian task. The 
idea ends up being a most trying effort. The air near the horizon is 
almost always slicked with haze, mist, cloud. The crescent, so thin 
and obliquely lighted, is a low contrast target against the bright 
twilight sky. Observers with better eyesight may see the Moon sooner 
than others with more normal vision, throwing the start of the month 
one day off. 
    Similar considerations apply to the Moon in morning twilight. 
Full Moon
    When the Moon rounds to the halfway point in her march thru the 
zodiac and stands opposite in the sky from the Sun, she is fully 
lighted by the Sun. The moment of crossing the halfway mark, 180 
degrees downrange from the Sun, is the instant of full Moon. With the 
full Moon being in a dark night sky it should be trivial to observe 
it, right? 
    It is like really tough with bare eye to tell when the Moon is 
completely round, as against having some slight defect of shape along 
one edge. The Moon looks pretty full within a day on either side of 
the geometric full Moon moment. In fact, in the modern calculation of 
Easter, a schematic full Moon, not the real astronomy-based one, is 
employed in some cultures. It being easier to see the first crescent 
of the month and knowing the period of one lunation, some Easter 
computations clock off 14 days from that first Moon and declare that 
moment the full Moon. 
    The full Moon rises at sunset and sets at sunrise. This is 
approximately true and serves well for casual stargazing. Unless the 
Moon hits the opposition point right at sunset, she will not herself 
rise at sunset. Similarly for the event at sunrise. 
    The Moon will become full at some arbitrary instant within the 24 
hour span of a certain day and not exactly at sunrise or sunset. 
Hence, on the day of full Moon you may see the Moon rise a little 
before or a little after sunset. Or see it set before or after 
Other phases
    It is a curious fact of history that the other cardinal phases of 
the Moon, like half or a certain thickness of crescent, apparently 
were not chronicled by Babylonia. Nor were they observed by other 
peoples who followed them! It is only in very recent (by Babylonian 
standard) times that we as stargazers even note the half Moon phases 
in our almanacs. 
    I suppose a half Moon could be assayed by a ruler held against the
terminator but even strong-sighted folk tell me that it's hardly a 
simple feat. I tried this a couple times over the years. Apart from my 
weak eyes, the dark and light spots on the lunar disc distort the 
terminator into an crooked edge. Recall that Aristarchus, who used the 
half Moon to form the right triangle of Earth-Sun-Moon, had such 
trouble that he could only give the distance to the Sun as a lower 
    One intriguing suggestion for neglecting the half Moon is that it 
could be verified by the layman, thus potentially compromising the 
authority of the king's starwatchers. Only they had the special 
mission to look after the Moon's motion. I can see how a layman may 
challenge the declaration of exact half Moon. He could lay a ruler on 
the Moon in twilight (when the ruler is visible against the sky) and 
check up on the official sky watchers. 
    By picking intrinsicly difficult phaenomena, the officials can 
claim a special skill or training to determine new or full Moon. This 
is mission protection. 
Babylonian measurements
    As fascinating as Babylonian astronomy is -- and I do mean 
fascinating! -- I have to pass over a detailed account of their scheme 
of measurement. I'll use all contemporary measures. I also will not go 
into the Babylonian names of the various astronomy terms. I use only 
modern terminology here. 
Horizon events
    Normally the horizon is the worse place for a target to be 
studied. We would like to hold off serious study of the target until 
it is removed above the horizon as far as possible. Our views of Mars 
in the apparition of 2001 are complicated by the planet's low 
altitude, for one modern example. 
    The Babylonians had no choice. The horizon was the only obvious 
and secure coordinate circle on the sky. All other coordinate lines 
were mathematical constructs not directly observable. A passage of the 
Sun or Moon thru the horizon was a definite and verifiable event which 
can be (in theory) confidently timed. 
    But the air near the horizon is the thickest in the sky, full of 
varying densities and optical distortions. The grossest distortion is 
refraction. You would think that timings of the Sun and Moon on the
horizon would be killed by refraction, unknown and unknowable to the 
astronomers in Babylon. For a single timing in the absolute sense you 
are correct. Sunset in reality essentially never is seen to happen at 
the geometric moment cited in our almanacs. 
    The Babylonians used the interval between two horizon timings, not 
their actual values. This had the effect of netting out refraction! 
The cancellation of refraction was more perfect by timing the two 
events with an hourish span, so refraction was about the same in both 
instances. Refraction drifts in value due to changing temperature and 
other weather factors. So while the absolute clock readings of the two 
events may be shifted bodily by a couple minutes, the difference 
between them would be practicly unaltered. 
    And this is a good thing.
    The other problem on the horizon was the timing of sunset or 
sunrise. Geometricly we like to define these events as the passing of 
the Sun's center over the horizon, as is usual in the rest of 
astronomy. It is impossible, specially with a brilliant Sun, to tell 
when the disc is bisected. 
    The Babylonians used a far simpler and more precise definition of 
sunset and sunrise. The Sun rises when the first bit of his upper edge 
breaks thru the horizon. Sunset is when the last bit of the Sun's 
upper edge is covered by the horizon. Guess what? These definitions 
are the very same ones we still use today! This is why when you 
calculate sunset geometricly for the Sun's center, the answer is a 
minute or so earlier than the value given in almanacs. 
Horizontal parallax
    We think of parallax as a newer development in astronomy only 
seriously studied since the 1700s. Parallax was a concept in 
Ptolemaeus's works by which he assessed that the stars were 
indefinitely far away. But parallax for the Moon, amounting to a full 
degree of displacement between the horizon and zenith, was known and 
recognized in classical Greek days. It was thru this parallax that the 
Moon was fathomed to be quite 60 Earth radii away. 
    I haven't found that the Babylonians specificly knew about 
parallax for the Moon but they recognized that the Moon was ahead of 
her geometric position when rising and behind that position when 
setting. I omit the ancient treatment of parallax and apply here, 
given the mid northern latitude of Babylon and New York, an allowance 
for it. It is four minutes, the time needed to move the extra one 
degree of parallax-induced depression, times a latitude leverage of 
1.5. These six minutes are subtracted from the moonrise timings and 
six minutes are added to the moonset timings. That is, without 
parallax (Moon at infinity) she should have risen six minutes earlier 
but didn't because she was depressed by parallax below the horizon. 
Corresponding logic applies to moonset; parallax makes the Moon set 
too soon. 
    This is not exact, I know, but I do want to keep things simple, as if
they are otherwise so. Bear in mind that if this magilla with the Moon 
gives you a headache, consider that you're merely revisiting what iron and 
bronze age cultures doped out three thousand years ago. 
    The Moon does not run in the ecliptic; we would have eclipses 
every full and new Moon if she did. Our predecessor starwatchers knew 
this. The lunar orbit is inclined 5-1/2 degrees. As the Moon runs thru 
this inclined orbit she deviates north and then south of the ecliptic. 
She crosses the ecliptic twice, on the northbound arm and on the 
southbound arm, each month. At maximum excursion from the ecliptic, 
the Moon is 5-1/2 degrees away, north or south, from the ecliptic. 
    The north-south distance of the Moon from the ecliptic is her 
latitude. The downrange location along the ecliptic, counting from the 
vernal equinox, is her longitude. I consider here only the longitude 
    Moreover, the orbit is not fixed in the stars but swashes around 
the ecliptic once every 18ish years. Or, if it's any easier to 
visualize, the nodes of the orbit migrate thru the zodiac. The 
Babylonians figured all this out and laid out a cycle of lunar phases 
and positions, along with a theory of lunar and solar eclipses, based 
on this inclination and nodal migration. 
    The Moon's motion along her orbit is smooth but not steady. By the 
behavior of orbits, she runs faster round the perigee and slower round 
the apogee. Our Babylonian starwatcher knew nothing about this cause 
of the variable lunar speed but he did exploit this feature. He 
learned that within a month the Moon went thru a full cycle of speed 
from fastest to slowest. I'm going to make up Babylonian terms for the 
point in the Moon's path for the extrema of speed. The place where the 
Moon runs fastest is 'celerilune' and that where the Moon runs slowest 
is 'tardilune'. These are Latin-based words; I don't speak Babylonian. 
    The speed of the Moon was measured by the length of the arc run by 
the Moon in one day, or as we would say, the daily motion. The usual 
units are degree/day. The need for the daily motion was downright 
critical. With it the Babylonian was able to predict, to within 
minutes!, the moment of new or full Moon by taking a couple simple 
Timing the interval
    A cardinal reason for using the interval or difference between two 
events was the lack of good timekeeping devices. What our ancestors 
did was note the diurnal arc thru which the Sun or Moon travels during 
the span between the two events. This arc, by the known rate of 
diurnal rotation, was converted into time. 
    Cunning? Like a fox.
    In many inscriptions, this conversion was not explicitly done. The 
tables give the actual arc itself in [modern] degrees of angle. We may 
read that the Moon set four degrees after the Sun. We would transform 
this, at the rate of four minutes per degrees of diurnal motion, to 
sixteen minutes of time. By this trick the Babylonian stargazer could 
pin his timings to within a half degree of arc or two minutes of time. 
Boxing in new Moon 
    How did our ancestral stargazers determine the moment of new Moon? 
Or of full Moon?  By an amazing and deadly accurate program of timing 
certain risings and settings. In all there were six combinations of 
these events, four near full Moon and two near new Moon. In 
archaeology they are called the 'Lunar Six'. There are six only 
because Babylonian math wasn't up to understanding negative numbers. 
Today if we were to use these timings we would say there are four of 
    Consider the new Moon. As the Moon shrank her crescent toward the 
end of her lunation and edged toward the Sun in the predawn hours, 
timings were made of her rising. The Moon rose before the Sun, less 
and less so as new Moon approached. On the very last day before new 
Moon, when on the very next day there would be no Moon in sight, the 
final timing was made. And a timing was also made of the sunrise 
coming immediately after moonrise within a hour or so. 
    Now we have the interval between the moonrise and sunrise on the 
day of last visibility of the Moon immediately before new Moon. 
    We now wait one full day to let the Moon pass around the Sun and 
move into the evening sky. We inspect the sky near the sunset point 
and, lo!, we see the newly waxing Moon. The new month begins. We note 
when the Sun set and wait until moonset, coming within an hour or so. 
    We now got the difference between sunset and moonset on the day of 
first visibility of the Moon immediately after new Moon. 
    New Moon must have occurred sometime between these two timings. 
Boxing in full Moon 
    Despite the fullness of the disc, the Moon is revoltingly hard to 
see on the horizon in less than optimum viewing conditions. But 
gamefully our ancestors collected masses of full Moon observations. 
    Consider a Moon which is not quite full. She is less than 180 
degrees downrange from the Sun so at sunset she is already in the sky. 
We could have actually seen it rise some moments before sunset and 
indeed this is one of the two measurements the Babylonians made. So we 
have on the day before full Moon the interval from moonrise to sunset. 
    Much later on that same night, we see the fullish Moon in the west 
before sunrise. Of course it plodded along the zodiac for these many 
hours but let's say it still is not quite full. It will set a little 
before the Sun and we have an other timing. This is the span between 
moonset and sunrise just before full Moon. 
    When the Moon is a bit past full she is more than 180 degrees 
downrange and is not yet risen at sunset. We wait and then see the 
Moon rising. Our timing is now between sunset and moonrise on the day 
just after full Moon. Correspondingly near the next sunrise we still 
see the Moon in the west at sunrise; she sets shortly thereafter. The 
timing here is between sunrise and moonset just after full Moon. 
Lunar six
    Go back and verify that we accumulated six timings, set out here 
for reference. Scholars of Babylonian history call these te 'lunar 
six' but we'll see in a moment that modern astronomers would reduce 
these to a 'lunar four'. 
    just after new Moon ---- from sunset to moonset 
    just before full Moon -- from moonrise to sunset 
    just before full Moon -- from moonset to sunrise 
    just after full Moon --- from sunset to moonrise 
    just after full Moon --- from sunrise to moonset 
    just before new Moon --- from moonrise to sunrise 
    Looking over this table you notice that the events around full 
Moon are really not distinct events. The timings at sunset before full 
Moon are the same in substance as those after full Moon, but in 
reverse order. In the former case the Moon rises before sunset; 
latter, after. We may in modern context combine these into a single 
'full Moon at sunset' timing and define it as (moonrise) - (sunset). 
If the Moon rises first, this subtraction is negative and the minus 
sign is a flag for 'before full Moon'. The same subtraction taken 
after full Moon yields a positive result and the plus sign is the 
'after full Moon' flag. 
    A coalition of the timings at sunrise has the new one definition 
of (moonset) - (sunrise). A negative result signifies a Moon before 
full; positive, after. 
    As it turns out, historians already preempted the term 'lunar 
four' for the four timings around full Moon and have expressed no 
inclination to alter the ancient definitions. We will stay with the 
set of six here. Just be sure to subtract the EARLIER event from the 
LATER one, OK? 
Impediment against studies 
    Until literally the final couple decades of the 20th century, 
scholars of ancient astronomy suffered appallingly from the lack of 
cheap simple accurate means to reconstruct the skies in these early 
eras. Most historians knew little about astronomy and had almost no 
entree' to astronomy references. 
    In much of the older litterature on ancient astronomy we find 
crude graphical constructions of the skies and some conceptual 
mistakes. In fairness to these scholars, even astronomers had little 
to go by for far past aspects of the skies. They simply didn't need 
such information, not even for long ago solar eclipses. 
    Precession, which most astronomers ignore for the brief span of 
their own careers, becomes hugely important for ancient astronomy. 
Without good knowledge of the precession rate, attempts to build an 
ancient sky were chancy. In fact, as  late as 1960, astronomy 
litterature had annoyingly different precession rates! For shortterm 
applications, within a couple centuries, this was fine. For 
prolongation into the dawn of history you could slip the sky by many 
    Some historians tried using mechanical planetaria, like that at 
Hayden Planetarium in New York. As wonderful as these machines were, 
they suffered from being too, well, mechanical. They do not 
incorporate the subtle but crucial effects of interplanetary gravity. 
A planetarium's planet runs on a 'trolley track' made of rigid gears 
and cams and rods. 
    When the first home computers arrived, programs in BASIC were 
written for astronomy. Perfectly adequate for spans near to today (in 
the late 1970s) they fell apart for any longterm retrocycling. Part of 
the constraint was the low processing accuracy and small internal 
memory of the machines. The other was that nonastronomers didn't 
appreciate the nuances of solar system dynamics. They commonly wrote a 
'trolley track' orbital simulator with results that were as calamitous 
as a, erm, trolley car collision. 
Modern planetarium programs
    With the advent of large memory, accurate arithmetic processors, 
and a general maturation of software composing, computer programs for 
astronomy today, since the 1990s, are pretty much perfect for 
reproducing ancient skies. Many of these are free or cheap, obviating 
the nagging expense of collecting lunar ephemerides in books and 
tables. The computers themselfs are now so cheap, in the mid hundreds 
of dollars for an individual, or free as part of a college office's 
    While I could start now to simulate an ancient Babylonian sky, 
which a good computer planetarium can reliably do, there's no point to 
doing so for this article. In the stead we'll trace out how to find 
the moment of full Moon in July 2001, taking data from a planetarium 
as if they were extracted from sky watching. July in New York is 
routinely murky and hazy and humid. On many days the Moon is 
obliterated by shmutz several degrees above the horizon. 
Moonrise and moonset
    According as the operation of your planetarium you may develop a 
table of moonrise, sunset, &c. Or you may have to manually set the sky 
to each event and read the date/hour field on the screen. Either way 
is acceptable, altho the latter is more fiddly. 
    It is also more confident. In a table of Sun and Moon events, 
arranged in order of date like an ephemeris, there is a digusting 
ambiguity. Sunrise and sunset are firm. The Sun rises in the morning 
hours and sets in the afternoon hours. No problem. 
    Moonrise and moonset are tricky. You just have to know how your 
planetarium handles them. The Moon can rise or set at any hour of the 
day because our time system is based on the Sun and not the Moon. A 
table of events for a given day will show 'moonrise' and 'moonset' as 
occurring at certain hours. So far so good. 
    Usually the table has a fixed layout with moonrise and moonset in 
certain columns, regardless of the hour they occur at. You may at 
first assume that the Moon must rise first, cross the sky, and set. 
The event in the 'moonrise' column must precede the event in the 
'moonset' column. 
    It can turn out that what 'should' be earlier is in fact the later 
event. Or maybe not; it really is earlier. 
    Here's why. 
    For our example, elaborated below, on 4 July 2001 I find moonrise 
at 19:01 EST and moonset at 03:26. (To Hell with daylight time!) We 
must ask some questions. Is the 03:26 moonset taking place LATER in 
the night FOLLOWING the moonrise of 19:01? Which is to say, is the 
table STARTING its calculation at moonrise on the 4th and CONTINUING 
it to moonset on 5th? 
    Or is the planetarium starting AT 0h ON THE 4th and CONTINUING 
thru its OWN 24h? Which is to say, is the mooonset occurring first in 
time and some 16 hours LATER in the SAME day the Moon rises? 
    There is often no clue to what the program is doing; the 
instructions and explanatory material are silent. You may in the end 
have to figure this out by brute force. Set the program to moonset on 
and read out what the hour is. 
Full Moon of 5 July 2001 
    I deliberately picked this full Moon for a worked out example 
because there was a lunar eclipse with it. This means the Moon was 
close to the ecliptic. She was at her descending node so that assorted 
extra calculations which were part of the Babylonian lunar mechanism 
can be left out. 
    With a planetarium program I pretended I'm watching the Moon 
approach her full moment. I used SkyGlobe and manually rotated the sky 
to touch the Sun and Moon to the horizon. I read the times from the 
screen display. At sunset on July 4th, the last time the Moon is 
already in the sky at sunset, I capture the rising time of the Moon 
and the setting time of the Sun, all in EST from Brooklyn. (Hey, some 
people rank Babylon right up there with Brooklyn.) At sunrise on the 
morning of 5 July I see that the Moon set for the last time before 
sunrise. I capture the hour of moonset and of sunrise. I do thee 
measurements for the following two sunsets and sunrises and build this 
table below. 
    SkyGlobe probably can not give a true sunrise or moonrise against 
the upper limb. The symbols for the Sun and Moon are not of true 
angular size. I took these moments to be the instant the discs were 
bisected by the horizon. This does not alter the concept or the final 
results. And, yes, in the table, the rise and set times are entirely 
within the same 24h day; there is no rollover across midnight. 
            | 4 Jul | 5 Jul | 6 Jul | explanation
    moonset | 03:26 | 04:14 | 05:07 | bisection of Moon symbol
    paralx  | +0:06 | +0:06 | +0:06 | allowance for parallax
    corr MS | 03:32 | 04:20 | 05:13 | geocentric moonset
            |       |       |       |
    sunrise | 04:32 | 04:33 | 04:33 | bisection of Sun symbol
            |       |       |       |
    MS - SR | -1:00 | -0:13 | +0:40 | (moonset) minus (sunset) 
    arc     |  ---  | -5:53 |+18:07 | arc of 24h daily motion 
    moonrise| 19:01 | 19:52 : 20:37 | bisection of Moon symbol
    paralx  | -0:06 | -0:06 | -0:06 | allowance for parallax
    corr MR | 18:55 | 19:46 | 20:31 | geocentric moonrise
            |       |       |       | 
    sunset  | 19:24 | 19:24 | 19:23 | bisection of Sun symbol
            |       |       |       |
    MR - SS | -0:29 | +0:22 | +1:08 | (moonrise) minus (sunset)
    arc     |-13:39 |+10:21 |  ---  |  arc of 24h daily motion
    All this makes sense with a couple diagrams. The activity at 
sunrise has us looking toward the antisolar point on the western 
horizon; this is also the center line of the Earth's shadow for the 
lunar eclipse that happens at this full Moon. The Sun is at our back 
about to break the horizon. 
                           O  6 July, FM+ 
    western horizon            \ 
                                       O  5 July, FM-
    The hour is sunrise itself, essentially the same on the two days 
of 5 and 6 July, 04:33. At sunrise on the 5th, the Moon already set; 
she's now down. We observed it, in playing like we are Babylonians, to 
set at 03:32. (I'm including the parallax correction from now on.) 
    On the 6th, the very next sunrise, the Moon is still up and 
doesn't set until 05:13. Obviously sometime BETWEEN the two sunrises 
the Moon achieved her full phase. 
    If we recognize that the arcs computed in the table above are 
those from the points in this diagram labeled (FM- to FM) and (FM to 
FM+), we see immediately that the entire arc (FM- to FM+) is nothing 
more than the sum of the two pieces. That is, the entire arc is 
(+0:40) - (-0:13) = 53 minutes. In angular arc this amounts to 13.25 
degrees. Our Babylonian colleague would say that the Moon's speed at 
full Moon is 13.25 degree/day 
    Now in the ancient records not only are the partial arcs written 
down but also the sum to make the full daily arc. There is a vital 
reason for this. 
When was full Moon? 
    Without much error we can linearly interpolate between the 
endpoints of the daily arc by the portion of it cut by the two pieces. 
That is, the two pieces we measured on the 5th and 6th are in the 
ratio of the the moment of full Moon between the two sunrises. 
    We have 
    FM = ((FM- to FM) / (FM- to FM+)) * (24h) + (sunrise) 
       = ((0:13) / (0:53)) * (24:00) + (04:33)
       = (05:53) + (04:33) 
       = 10:25 
    A thoroly analogous diagram can be drawn for the events at sunset 
with the nearly full Moon rising. 
                                      O  4 July, FM-
    eastern horizon               /
                          0  5 July, FM+ 
    On the evening of the 4th of July the Moon is already up at sunset 
and we measured her rising at 18:55. Sunset came at 19:24. On the next 
evening, the 5th, the Moon is not yet up at sunset. We must wait until 
19:46 to see her rise. 
    From the table we see that the Moon rose on the 4th some 29 
minutes before sunset and on the 5th she rose 22 minutes after sunset. 
The entire arc between the two sunsets is the sum of these two 
timings, (+0:22) - (-0:29) = 51 minutes. In angular measure this is 
12.75 degree. The discrepancy between this and that derived from 
sunrise observations seems to be from the granulation of the timings, 
only to the nearest minute with a minute swing one way or an other. 
    The same math applies to get the moment of full Moon
    FM = ((FM- to FM) / (FM- to FM+)) * (24h) + (sunset) 
        = ((0:29) / (0:51)) * (24h) + (19.23)
        = (13:39) + (19:23)
        = 34:02
        -> 1 day + 10:02 
    The full Moon occurs at 10:02 on the day after the sunset, or on 
the 5th of July, not the 4th. 
    The two moments are a bit mismatched and we do a simple arithmetic 
    full Moon = ((sunrise full Moon) + (sunset full Moon)) / 2 
              = ((10:25) + (10:02)) / 2 
              = (20:27) / 2
              = 10:14
    The full Moon occurred by a regular ephemeris at 10:05 on 5 July 
2001. This method elaborated three millennia ago does work! 
The problem with new Moon
    For the moment of new Moon we would like to have a situation like 
in this diagram at sunrise 
                                            C  NM-
    eastern horizon                      / 
    -----------------------------------* NM ----------------------
                                 C NM+ 
    On the day before new Moon we see the vanishing crescent just 
before sunrise and can make a timing. We want to wait a day and see 
the Moon just after new as she rises following the Sun. We need the 
timing of the Moon at NM+ in the diagram. First the Sun rises and a 
bit later see the Moon come up. 
    This is an impossible observation! No way in Hell could Babylonian 
or us today see the NM+ alignment. Thus, totally unlike for the full 
Moon we can not directly get the arc of daily motion MN- to NM+. 
    By analogous fate we can not at sunset see the Moon setting before 
the Sun on the day before new Moon. Again we lose our grasp on the arc 
NM- to NM+. But we NEED this daily motion to perform our ratio of the 
timings to get the moment of new Moon! 
The extraordinary mental ingenuity
    An other diagram. 
                                      .     . 
                                 M                  . 
                              .                          . 
                            .                               . 
                         CL.    E                            .TL       
                            .                               .  
                              .                          . 
                                 .                  . 
                                      .     . 
    In this diagram we got the Moon's orbit (deliberately exaggerated 
in excentricity; we'll need this shape later) with the alignment for a 
new Moon. The Moon M stands between us on Earth E and Sun S. Imagine 
the Sun to be some 400 times farther off than the Moon. 
    Note that the Moon is in a part of her orbit where the daily 
motion is intermediate between the fastest and slowest. I labeled the 
fastest and slowest points as CL for celerilune and TL for tardilune. 
Please bear in mind that Babylonia never knew about the physical orbit 
of the Moon. They worked only with the path she traced in the sky. 
    The line of apsides, joining the perigee and apogee, rotate gently 
but not by much within one year. Hence the Moon can be seen with a 
certain daily motion in the same direction in the zodiac, closely, 
within a given year. Of course, the phase will continually shift as 
the Sun apparently rounds the Earth in annual motion. In particular, 
six months ago the Sun was off toward S'. The Moon is now near her 
full Moon phase. 
    Now pay attention. 
    When the Moon is near full we have the situation I elaborated 
earlier. We can actually obtain the arc traversed by the Moon in one 
day. This is the method of the two timings made on consecutive days 
bracketing the full Moon moment. This arc is characteristic of the 
part of the orbit where the Moon stands during that full Moon period. 
    But this part of the orbit is also where the new Moon is six 
months later when we can not by direct observation get the daily arc. 
So! The Babylonian astronomer figured out that if he used the daily 
motion of a full Moon six months BEFORE the new Moon he's now 
studying, he has nearly enough the correct daily motion of that very 
new Moon. It's the same as that of the full Moon! 
    With the surrogate arc and the one end of it actually timed, he 
can figure out when the new Moon took place! I do not go thru a worked 
computation because the maths are quite similar to those for the full 
Moon. The trick once again: Use as the daily motion the arc for a full 
Moon six months EARLIER than the instant new Moon. 
The saros cycle 
    Ancient astronomers, like those of Babylonia, used the saros cycle 
to predict eclipses. This cycle is 6585-1/3 days. Well, yes and no. To 
appreciate the eclipse features of the saros, you need eclipse records 
from all over the world. For starts, each eclipse of a given saros 
occurs one-third of way farther around the globe, so that, at best, 
only one in three of the eclipses take place in the longitude zone of 
    Of these one-third, Babylonia will miss some from a latitude too 
far north or south. On top of all this, there are many saros cycles 
operating at once, each with its own set of eclipses. Sorting all 
these out with scanty data was beyond hope for the Babylonians. 
    Now I guess if a computation showed that the new Moon occurred 
when it was down and an eclipse may be in the making, the Babylonian 
may have assumed that the eclipse happened in some other part of the 
world. I don't know if Babylonia had such a global picture of the 
Earth back then. More likely they treated the unseen eclipse as a dud 
in their theory. 
     Hence, while our ancestral stargazers knew of the saros they 
didn't make much use of it for eclipse predictions. They had a far far 
more important routine use of the saros. 
Round and round the Moon goes
    Moon phase (new for a solar eclipse; full, lunar) and the nodes 
must line up to make an eclipse. When the line of nodes rotates around 
to the same point on the ecliptic and a new Moon occurs there, we got 
the next solar eclipse of that saros. This is NOT the very next 
eclipse in time! There are several cycles running concurrently, each 
with its own eclipses. 
    Our colleagues of old found that also the line joining the 
celerilune and tardilune, what we now know as the line of apsides, 
also swings around the orbit in very nearly an integer number of laps 
along with the nodes and phases. So not only will an eclipse take 
place after one saros interval, the Moon will cross over the Sun at 
almost the same speed as for the previous saros's eclipse. 
    Look again at the orbit diagram above. This orientation of the 
orbit repeats after one saros so that when the Moon is new at M, her 
motion is the same as for the previous saros, 6585-1/3 days before. 
    With the extensive records of lunar behavior, the Babylonian 
assembled tables, called Goal-Year tables, with the lunar motion and 
position for the coming year, six months earlier in the previous year, 
and for one, or perhaps two, saros cycles earlier. 
    The six-month section we now can explain as being the data for 
getting the daily arc of a new Moon based on a full Moon six months 
earlier. In the latter six months of the instant year, the data from 
the front six months are used. 
Filling in the gaps 
    The Goal-Year table was used for anticipating the events of the 
instant year, as opposed to the diaries or logbooks which contained 
actual observations. But what happens, as it will, if bad weather 
intervenes against getting one or an other timing? In such a case, the 
ancient astronomer had to replicate the setup of the instant new or 
full Moon by using the alignment from one saros earlier! He knew that 
the daily arc of a full Moon 6585-1/3 days before would be the SAME as 
it is right now. If he loses one end of the arc from adversity of the 
weather, he can reconstruct it from a replicated value! 
    It is this regular use, potentially twice in every month!, of the 
saros that makes it one of humankind's most amazing intellectual 
achievements, alongside the Newton theory of gravity and Einstein 
theory of relativity. The bit in the usual astronomy books about 
eclipses is really an extra goodie. 
Summing up
    The lunar motion is far from simple, yet a people rising out of 
the bronze age living off of the soil of fickle rivers had the will 
and spirit to maintain a vigil on the Moon for nearly a thousand 
years. And they got her inconstancies sussed out pretty nearly right 
on the money. So well did they succede that much of the astronomy of 
these people is bodily carried over to today's home astronomy.