Note that nearer object have a larger change in the angle than distant ones do.
The formula for the angle is expressing the angle in terms of what are called radians. (a full circle is 2 pi= 6.2832 radians) If one wants to express the angle in degrees, multiply the angle in radians by approximately 57
Note that the angular size of a body (like the moon say) is the size of the object divided by the distance. Thus the angular size of the moon is just the same as the parallax of you to an observer on the moon who moves across the moon.
After three saros periods (19756 days or about 54 years and a month), the eclipse will be approximately 360 degrees or a full circle and will thus be a very similar eclipse to the first on, in the same region of the earth. If we are talking about lunar eclipse, you will get another lunar eclipse. If solar that new one will be about 10-15 degrees different in lattitude, and will be a partial eclipse if the original was a total eclipse.
Of course in that 18 years of a saros cycle, there will be many other eclipses (about 2 a year somewhere on earth). They will not be part of the saros cycle of the original eclipse, but each will have their own saros cycle.
A Saros cycle of eclipses last about 1000 years, before the location of the eclipses falls off either the north of south poles of the earth. Looking at the central eclipses (which occur somewhere on the earth) roughly 60% of the eclipses are Annular and about 35-40% are total .
Today the Saros cycle can be explained in terms of the motions of the sun and moon, it is a fluke that three aspects of the lunar orbit which have just a fortuitous relation to each other give these long stretches about 1000 yrs of periodicity in eclipses. We will come back to this when we look in more detail at what the Greek astronomers discovered about the motion of the sun and especially the moon.
It was the Babylonians, who kept detailed records of eclipses for astrological puproses (ie, divination of earthly, and especially royal events), who noticed this cycle, and it formed a key part of their ability to predict eclipses. Depite their extensive catalogs, they seem to have not ever tried to understand why anything happened in terms of physical, rather than human fate, terms. Those records however turned out to be incredibly useful to those Greeks who were interested precisely in understanding the heavens in physical terms.
Note that another civilisation in the same area who kept written records (other than for bureaucratic reasons) were the Jews, but at least the records that have been preserved are almost exclisively concerned with their relation to God, and preserve almost no astronomical records. IIn the New Testament about the only astronomical item is the "star in the east" seen by the wise men who took it as an omen of the birth of a King. These wise men are usually taken to be Chaldeans, who were the Babylonian "priests".
Solar Eclipses are cataloged into three types, two called central, and one type called partial. In a partial eclipse the moon only trespasses part way across the sun. It never covers the sun completely (Total eclipse), nor has the remaning sun form a complete ring around the sun (an annular eclipse). These latter two types are called central eclipses.
A central eclipse is visible only along a very narrow band of the earth, a maximum of about 200km wide. (but over 10,000 km long). This is surrounded by regions of the earth where a parial eclipse can be seen, which is about 3-4000km wide. Of course the Babylonians and the Greeks had no idea of this since they did not receive reports from places 2000 km away. Even 100Km reports were fairly rare.
Lunar eclipses are categorized into total (when the whole full moon goes much darker. It does not get black, but has a copper-rust colour. The illusion is also that it is much closer to the observer than a normal full moon.), partial, when that darkness covers only a part of the moon at maximum coverage, or Penumbral, when there is a slight darkening of the full moon, but no definite edge to the darkening.