Notes about Chronology

Let’s first discuss the translation from Gregorian to Julian dates. The original reform involved a moving ahead (or skipping) 10 days. October 4^th, 1582 was followed immediately by October 15^th, 1582. This means that the ten dates Oct. 5, 6, 7, 8, 9, 10, 11, 12, 13, 14 were skipped. Meanwhile, the Julian date for the day following October 4^th would be October 5^th. Consequently, to change a Gregorian date to the Julian date we need only subtract 10. Thus, December 25^th, 1599 (Gregorian) corresponds to December 15^th, 1599 (Julian).
However, this only works until the Gregorian calendar differs from the Julian calendar. The first year with a difference is 1700 – there was a February 29^th, 1700 in the Julian calendar, but not one in the Gregorian. Thus, after February 28^th, we need to subtract 11 days. And so, February 22, 1732 (Gregorian) is really February 11, 1732 (Julian). (These are of course the differing dates for George Washington’s birthday.)
So, for March 7^th, 2008, we have the 10 days Gregory left out, plus three other days left out: Feb. 29^th, 1700, 1800, 1900. Now March 7^th = February 36^th (it is a leap year), and so on the Julian calendar, this would be February 23^th, because 36-13=23.
Joseph Justus Scaliger, French classicist and critic (1540-1609), formulated a dating system designed to correlate historical events dated in different calendars.
He brought together three important cycles:

The 28 year Julian Cycle, denoted by J
The 19 year Metonic Cycle, denoted by G (the Golden Number)
The 15 year Indiction Cycle, a tax cycle introduced by Constantine in 313 AD, denoted by I. Many dates in classical times included a reference to the indiction cycle, and so Scaliger found this a very useful cycle to include.
We denote place in the cycles by (J, G, I)
This is all strictly based on Julian years – not Gregorian. (Scaliger was a Protestant convert who stringently opposed the Gregorian reform, on religious/political grounds).

The complete cycle lasts (28)(19)(15)=7980 years.
We start with 1 BC = (9, 1, 3). We have seen before that J=Y+9(mod 28) and G=Y+1(mod 19), and since 1 BC = 0, we get the values for J and G. The value I =3 follows from the historical fact that I =1 for 313 AD. In general then, I=Y+3 (mod 15).
What’s the beginning of the cycle? We would need a number Z of years where (28, 19,15) = (0, 0, 0) + Z =(9, 1, 3). In other words, we have to solve,

Z=9 (mod 28)

Z=1(mod 19)

Z=3(mod 15).

If you do this, you get Z=4713. The Scaliger cycle begins at 4713 BC. But astronomers prefer to call 1BC year 0, and then 2BC = -1, etc. We will call 4713 BC = -4712 year number 1 in the Scaliger count. We denote the Scaliger count by S. We then have the following correspondence:

The advantage with the modern value Y is that J=Y+9 (mod 28), G=Y+1 (mod 19) and I=Y+3 (mod 15), for all values of Y (positive or negative). In particular, Y is a leap year exactly if it is divisible by 4, positive or negative.
Meanwhile, J=S (mod 28), G=S (mod 19), I=S (mod 15).
Given (J, G, I), we can then solve for S (or Y), by using the Chinese Remainder Theorem approach we encountered when studying the Mayan calendar – We won’t actually do such calculations in class.
Astronomers actually do a day count from Jan. 1, 4713 BC. This is day JD=0.
We assume the year begins on March 1^st – this way the leap day occurs at the end of the year.
Given a Scaliger year S, with month M and day D, we need to compute the days in S-1 years, plus the additional days in the partial year.
Let S-1 = 4q+r. Each four year cycle has 3(365)+366=1461 days. So the previous years give 1461q+365r days.
We have a month chart:

Then the day number is JD=59+1461q+365r+d(M)+D. The 59 comes from the days in the first January and February, with counting starting at 0 (it is a leap year).
For example, consider the date Sept. 17^th, 458AD (this is of course a Julian date). The Scaliger date is S=458+4713=5171. Then we have 5170=1292(4)+2.
So the day number for this date is JD=59+(1461)(1292)+365(2)+184+17=1888602. The 184 just comes from the table above for the month of September.
For another example, consider July 4, 1776. This is a Gregorian date, and so we must shift back 11 days, to get a Julian date. Now, July 4 = June 34 –11 days = June 23, 1776 Julian.
We now compute S=1776+4713=6489, and S-1 = 6488. Now 6488=(4)(1622)+0 (1776 is a leap year).
So the day number for this date is JD=59+(1461)(1622)+365(0)+92+23=2369916.
You can check these answers on the Julian Day calculator.
Remarkably, we can replace the month chart by another congruence, called Zeller’s congruence; it uses Z, which will measure the cumulative extra days in previous months, beyond a ‘month’ of length 28 (in the chart below these are labeled as Excess). It is just given by Z=(13M’+2)/5, where M’ is a count of the months, starting at March =0. We see this from the following chart:

Thus, we could use d(M)=28M’+(13M’+2)/5, as a non-table way to compute the values in the month table above.
For example, let’s re-do the calculation for July 4, 1776. All of the numbers above remain the same, except that this time we will compute the number 92. To do that, we observe that we are in June, which means that M’=3. Then we need to add on (28)(3)+(13(3)+2)/5=54+41/5=84+8=92.
Similarly, for Sept 17, 458. Then M = 9 and M’=6. Then (28)(6)+(13(6)+2)/5=168+16=184, as obtained from the table.
We can also do Julian day count calculations in reverse. For example, suppose that JD=520,000. We must first subtract off the first 59 days to get 519941. We then divide this by 1461 to get 599941=(1461)(355)+1286 – we have 355 four year cycles, and 1286 days left over. To account for these extra days, we divide by 365. Then 1286=3(365)+191. The 191 days belong to the current year. Looking on the chart, we see that September has 184 days prior. This makes the date September 7^th. To get the year, we have S-1=(355)(4)+3=1423, or S=1424. To get the Julian year, we subtract: Y=1424-4713=-3289. But this is in modern form; in traditional language, Y=3290BC. The day with Julian Day count 520000 is exactly September 7^th, 3290BC.