The lunar orbit has a number of cycles.

Synodic month -- 29.53059 days. This is the time to go from new moon to new moon. Ie, the moon is in the same direction in the sky to the sun. The synodic month is longer than the siderial month because the sun is moving, and the moon needs more time (about 2 days) to catch up to the sun.
siderial month -- 27.32166 This is the time that the moon goes from being nearest some star in the sky, to its again beinging nearest that same star. Because the sun moves through the stars, this is different from the synodic month.
anomalistic month -- 27.55455 days The time the moon goes from being nearest the earth (perigee) to again being nearest the earth. This is greater than the siderial month because that point of closest approach also rotates around the earth (in about 9 years for a complete cycle) in the same direction as the rotation of the moon. We now know that this is caused by the gravitational effect of the other planets and especially the sun on the moon's orbit.
draconic month 27.21222 days moon . This is the time the moon goes from being on the ecliptic to the next time it is on the ecliptic and travelling in the same direction crossing the ecliptic (usually ascending node, which means that crossing of the ecliptic (sun's path) in which the moon goes from being south of the ecliptic to north of the ecliptic. This is shorter than the siderial month, because the whole orbit is rotating in the opposite direction as the moon orbits, and thus the moon catches up to the node more rapidly than one orbit.

Saros cycle 6585.3211 days. This is

241.99867 draconic months
238.992148 anomalistic months
222.9999841 synodic months

Note that this each of these are almost an exact number of months, to better than 1 percent. Ie, in one Saros period the moon returns to almost exactly the same place in its orbit with respect to the sun. The earth has rotated 6585 and 1/3 times with respect to the sun Thus the eclipse takes place 8 hours later in the day at any fixed location. (if it is a solar eclipse and 8 hours later is at night then of course you cannot see it). The next Saros cycle will another 8 hours later and the third will be 8 hours later again. Ie, it will be 24 hours after the first, which is a day. (it is actually slightly less than 8 hours later each time because .3211 is 7hr, 42 min later, not exactly 8 hours) Of course the Bablylonians and Greeks had no way of measuring time that exactly.

The Saros cycle is an period of eclipses. Ie, after a Saros cycle time, an eclipse of the Sun or Moon will again take place somewhere on the earth. because .3211 is approximately 1/3 of a day, the center of the next eclipse with be about 180 degrees to the westof the center of the last one, but after 3 cycles, it will occur at very close to the same place (lattitude) on the earth. It will however by about 5 degrees further south or north depending on whether the moon is at an ascending or descending node at that ecipse. (ie where the moon is going higher in the sky than the sun after the eclipse or lower. Eventually the eclipse path will go off the top of the earth or the bottom. The whole process takes a bit over a 1000 years, so given on eclipse one can predict the times of about 50 eclipses into the future and past. Since 3 Saros periods is 54 years and 32 days very closely, this means that after 54 years one will have an eclipse that occurs roughtly at the same time of day near where one sees the first eclipse. Because of the eclipse's change in lattitude, if the first was a central (total or annular) eclipse, the others in that cycle will all be partial eclipses.

Note that there are many (about 100 )Saros cycles going on at the same time. (Ie between any two Saros repetitions there are about 100 other eclipses somewhere on the earth, each being part of its own Saros cycle. ) Since there is a bit under 2 eclipses somewhere on earth in every year, in the 18 years Saros cycles, there are about 35 eclipses, each having its own saros cycle.

The lunar eclipses follow the same kind of cycle, but in the case of the moon, a bit over 1/2 the earth can see the eclipse, and thus in a set of 3 Saros cycles, any spot on the earth will be able to see at least parts of all three occurances of the triple Saros cycle in general.

This period was discovered by the Babylonian astologers, and used to predict eclipses. If you have the records of all eclipses over 18 years, then you can use those to predict the eclipses over the next 18 years, etc. They had no idea why this occured, nor did they care why they occured it seems. The were astrological engineers, not physicists:-)

Parallax

The change in position of foreground objects when the observer moves. The change in the angle at which one sees an object is (or small angles) the distance moved by the observer perpendicular to the line of sight, divided by the distance to the object. (This is measuring the angle in terms of radians. To get the angle in degrees, youhave to multiply the angle in radians by approximately 57.

(The definition of the angle is circular distance between the posions, divided by the distance from the source to the observer. The circular distance is the distance along the circle with center at the object between the two observation points. For small angles, this is just the same as the distance between the two, but for larger angles, the circular distance is larger than the direct distance.

Thus if you are looking at an object 100 meters away, and you move one meter, the obect seems to move (in theopposite direction to your motion) by 1/100 radians, or about .57 degrees ( which is about 35 minutes of arc, since 1 degree is 60 minutes of arc)

As an example, your two eyes are about 5cm apart. An object 3m away (300 cm) will have a parallax between the two eyes of 5/300 or 1/60 radians, which is about 1 degree. We know it is easy to "see 3D" for object 3 m away, but it gets hard if we go to 30 m. The difference between the two eyes becomes hard for the brain to distinguish.)

Measurement of angles.

The estimate of the best that the Greeks to do in measuring angles in the sky was about 10 monutes of arc (.166 degrees) which is about the angle of a grain of rice held at armslength away from the eye. This is about 1/3 of the moon or sun diameter. Clearly the eye can distinguish angles better than that. the Human eye can distinguish angles of about 1 minute of arc (or 1/30 of the lunur diameter). Ie if one has objects closer together than this, the eye cannot separate them into two objects. It took about 2000 years (Tycho Brahe) to reach this limit in being ablt to measure angles in the sky. This required huge instruments. Telescopes increase this resolution ability so that now one can measure angles to better that 1/1000 of a second of arc (or 600000 times better than the Greeks could.)

Measurement of time

Measuring time is very useful, as it can tell you where in the sky various objects are when you look at them. Unfortunately the Greeks did not have good clocks. It would be about another 1.5 thousand years before mechanical clocks were invented, and about 1.8 thousand years before they became accurate enough to measure to better than a minute.

The primary way of measuring time at any time of day was with clepsedra. These are water clocks and the theory is that if you have a little hole in a container, the water flows out at a constant rate. If you can measure how much water has flowed out, then you know (after calibrating your clepsedra) how much time has passed. Nice in theory, much more difficult in practice.

The first problem is that the water does not flow out at a constant rate. The water is forced out of the hole by the pressure of the water at the hole forcing it out. As the height of the water in the container decreases, the pressure decreases and the rate pf water coming out decreases. They triedto compensate by making the vessel narrower at the bottom so that the height of the water changed at a constant rate, even if the amount coming out in a certain time decreased.

In addition, there is friction between the flowing water and the walls of the tube out of which the water flows. If the tube is very short (eg the thickness of the walls of the vessel) then this effect is small, but then the water flows out so quickly you can only measure very short time intervals. If the tube is long and thin (eg so the water comes out drop by drop) then this friction is large. But the friction depends sensitively on temperature. Thus if the liquid is hot (middle of the day) the friction is smaller, and the water flows out more quickly, while at night it flows out more slowly. They had no theory about this were unable to compensate for this effect. Measuring over a long time, these clocks were only accurate to about an hour over 8 hours. They were pretty useless for accurate measurements.

Also if they measured long periods, they were not very useful for measuring short intervals (minutes) since the amount of water coming out would be difficult to measure how much had come out.