The Egyptian calendar began in the year the summer solstice (modernly June 21) coincided with the helical rising (with the sun) of the Dog Star, Sirius, which the Egyptians called after the Goddess, Sopdet. But the star was also associated with Ast (Isis) which the Greeks called Sothis. This was called "the opening of the year," and the civil (administrative) calendar consisted of 3 seasons of 4 months each with each month being 30 days, for a total of 360 days. At the conclusion of the year there were 5 days which were not days counted in any year, but each day belonged to a God or Goddess. The opening of the year at the summer solstice was also the beginning of inundation of the Nile, and the 4 months were called Akhet, followed by the 4 months of emergence, called Proyet, and then 4 months of Shomu.
This put the civil calendar out of sync with the rising of Sirius by nearly one day every 4 years. While this was of little importance during the first years of Egypt, after the first generation (approximately 72 years) the rising of Sirius was 18 days later that the beginning of the Civil Year and had the Calendar been observed, the flooding of the Nile would have been a disaster. Sirius was the constant, and it was this star, and not the calendar that signaled the rising of the Nile. Thus began the first period of the Civil Calendar which fell behind the rising of Sirius by an increasing number of days each century. That is, Sirius' rising would advance one day every four years so that beginning with the seventy second year of the first year, Sirius would have rising (under the modern calendar) on July 10th instead of June 21st. Over the centuries Sirius would continue advancing until it again rose on the first day of the Civil Calendar.
This has led the brilliant minds of Egyptology to assign a period of 1460 years to the Sotic (original pronunciation of Sothic) Cycle, based on 365 days of the Egyptian year, times the 4 years in which Sirius rose a day later, to equal 1460. This calculation is of course, wrong, for several reasons. it assumes 365.25 instead of 365.2425 days per year, and that 10 minutes 48 seconds per year adds up over a thousand years. Also, the earth is not completely round, and it does not rotate in a perfect ball. Instead the earth wobbles in what is known as the Precession of the Equinox. This wobble take approximately 25,800 years to go from a beginning point and return to that point. But the earth doesn't really return to the exact point in space because of at least two other factors, the perihelion precession and the obliquity of the ecliptic. Simply put, The Sun appears (as viewed from the earth) to move backward through the stars at the rate of about 1 degree every 72 years.
This means that under our modern calendar, if Sirius rose with the sun on June 21 on year zero, the precession, or wobble of the earth, would have Sirius rise on June 22, 72 years later. That would happen, of course, only if there were no other factors to consider (which there are). In the 1460 years assigned to the Sothic Cycle by Egyptologists, Sirius would rise with the sun more than 20 days later, or on July 10th. So, if the brilliant Egyptologists ever went out and actually looked for the rising of Sirius on June 21, they would have a long wait. The Egyptians were not so brilliant. They actually went by what they could observe and made no such calculations. Empirical observations would have told the early Egyptians that Sirius was nowhere near the rising with the Civil Calendar 1460 years after the beginning of the first Cycle. If Sirius actually rose 20 days later (on July 10) it would take longer than 20 years for the Egyptians to discover this because their Civil Calendar was still out of sync with the Sotic (Sothic) Cycle by that one day every 4 years. In other words it would take an additional 80 years for the civil calendar to coincide with Sirius rising with the sun; and of course, during that additional 80 years, the precession would add over one additional day (4 years) to the watcher. That would have made the first Sothic Cycle 1540 years, and not the 1460 years assigned to it by Egyptologists. But there is another factor which makes 80 years inaccurate - after all that is a mathematical calculation, and not one that the Egyptians would have observed.
This would also put Sirius out of sync with the rising of the Nile. But it was actually Orion (Osiris) which heralded the approach of the Nile flooding, and Procyon, the Little Dog Star, or Barker Star would rise about a month ahead of Sirius. This gave the priests ample warning of the time of the new year. It should be noted that the Greeks who later ruled Egypt gave the name of Cyon to Sirius and Procyon (before Cyon) to the Little Dog Star. However, the Egyptians assigned this star to Thoth, the God of Wisdom, whose knowledge would redeem the dead.
The summer solstice (June 21) is the longest day of the year, June 22 is a shorter day, by less than a minute (June 24 would be a minute shorter) and July 10th would be even shorter by about 6 minutes. This means that for anyone who actually goes out and watches the rising of the sun, it is darker longer on the crepuscule morning of July 10 than it is on June 21; and the time the sun rises in New York is far different from when it rises in Karnack, Egypt. Thus, if the sun rose at 6:07 AM on June 21, the priests at Karnack, Egypt would have observed the sun rise at about 6:13 on July 10; and since it would be darker, they would be able to see Sirius rise (not be obscured by the light of the sun) earlier. Because of the crepuscule darkness, Sirius would rise with the sun a few days earlier. That is, Sirius would be visible anywhere between twenty and thirty minutes before the actual rising of the sun. But there are other factors, not the least of which is the speed of the rotation of the earth, its orbit around the sun, and the obliquity of the ecliptic, where the tilt of the earth has decreased over the millennia. It should also be noted that the years, and even centuries leading up to Sirius rising on the summer solstice, the days would have become longer. This would have kept the apparent rising of Sirius at about the same time (day of the modern calendar) for over a millennia and would have been unnoticed by ancient star watchers.
After taking all factors into consideration, it would have taken approximately 1525 years for the civil calendar to coincide with the Sothic Cycle, and not the 1460 years given by Egyptologists. Just exactly how long each Sothic Cycle was would be difficult, if not impossible, to calculate, but the periods given by the Priestess of our religion appears to be quite accurate.
We know that the Egyptian civil year and Sothic year coincided on July 20, 139 CE (Julian). That is nearly a whole month from the summer solstice. Modernly, Sirius rises with the sun in Egypt (if smog and weather do not interfere) on August 5 - 16 days later than it did in 139 CE. However
Most Egyptologists give the date of the founding the 19th Dynasty as 1292 BCE, even though, based on what Egyptologists accept as a 1460 year Menophres Era, the date would be 1321 BCE, which is probably within ±2 years, as the Greek rulers of Egypt changed the Egyptian calendar in 238 BCE.
It would appear a simple thing calculate backwards from 139 CE as the Egyptologists do and say that the Sothic previous cycles occurred in 1322, 2782, and 4242 BCE. That of course uses the 1460 year cycle, and the Sothic cycle actually began about 1371 (Gregorian) which is about 50 years longer. The next cycle would have taken more than 1507 Tropical years, because the sun would have risen earlier and obliterated the view of Sirius. (Remember, there are approximately 20 days more in the Sothic year because of the Precession of the Equinox, which equates to more than 80 years. The closer you get to the summer solstice, the more days it takes for the civil year to catch up with Sirius rising (called slipping).
The approximate dates (give or take 3 years) of the Sotic (Sothic) cycles would have been 1374 BCE, 2891 BCE and 4416 BCE, and the Egyptian calendar would actually have began approximately 4450 BCE.