Continuity/Roulean Calendar and Portal Years

In early 2003, Turbine released a timeline called The History of Auberean. We are presented with several events, and the year they occurred on both the Portal Year and Roulean calendars:
Portal Year	Roulean Year	Event
-1441 PY 	324 RC 	Jojiism founded.2003/03 The History of Auberean/Volume III: The Fall From Grace (-1,804 to -891)
-869 PY 	704 RC 	Viamont invades Aluvia. The reign of Pwyll II ends and the reign of Alfric begins.2003/03 The History of Auberean/Volume IV: Shifting Ways (-888 to -574)
-779 PY 	765 RC 	Reign of Alfrega begins. Harlune stays behind on Ispar during an expedition.
-758 PY 	779 RC 	Reign of Osric begins.
-540 PY 	924 RC 	Gharu'n armies seige the Roulean capital of Tirethas.2003/03 The History of Auberean/Volume V: New Arrivals (-540 to 13)
-358 PY 	1046 RC 	Emperor Kou unites the Sho under his rule.
With multiple matching pairs of dates, we can determine the equation to convert between calendars. All we have to do is treat the pairs of dates as coordinates of points, and find the equation for the line that intersects those points. We will determine the equation for each set of points:
Point A 	Point B 	Equation
324, -1441	704, -869 	PY = (143/95) * RC - (183227/95)
324, -1441	765, -779 	PY = (662/441) * RC - (94441/49)
324, -1441	779, -758 	PY = (683/455) * RC - (876947/455)
324, -1441	924, -540 	PY = (901/600) * RC - (96377/50)
324, -1441	1046, -358	PY = (3/2) * RC - (1927)
704, -869 	765, -779 	PY = (90/61) * RC - (116369/61)
704, -869 	779, -758 	PY = (37/25) * RC - (47773/25)
704, -869 	924, -540 	PY = (329/220) * RC - (9609/5)
704, -869 	1046, -358	PY = (511/342) * RC - (328471/171)
765, -779 	779, -758 	PY = (3/2) * RC - (3853/2)
765, -779 	924, -540 	PY = (239/159) * RC - (102232/53)
765, -779 	1046, -358	PY = (421/281) * RC - (540964/281)
779, -758 	924, -540 	PY = (218/145) * RC - (279732/145)
779, -758 	1046, -358	PY = (400/267) * RC - (513986/267)
924, -540 	1046, -358	PY = (91/61) * RC - (117024/61)
If we solve the division within the parentheses, we see these equations are all fairly similar. Below is a table with the equations, where the division has been solved to four decimal places:
Equation 	is similar to:
PY = (143/95) * RC - (183227/95) 	PY = (1.5053) * RC - (1928.7053)
PY = (662/441) * RC - (94441/49) 	PY = (1.5011) * RC - (1927.3673)
PY = (683/455) * RC - (876947/455)	PY = (1.5011) * RC - (1927.3560)
PY = (901/600) * RC - (96377/50) 	PY = (1.5017) * RC - (1927.54)
PY = (3/2) * RC - (1927) 	PY = (1.5) * RC - (1927)
PY = (90/61) * RC - (116369/61) 	PY = (1.4754) * RC - (1907.6885)
PY = (37/25) * RC - (47773/25) 	PY = (1.48) * RC - (1910.92)
PY = (329/220) * RC - (9609/5) 	PY = (1.4955) * RC - (1921.8)
PY = (511/342) * RC - (328471/171)	PY = (1.4942) * RC - (1920.8830)
PY = (3/2) * RC - (3853/2) 	PY = (1.5) * RC - (1926.5)
PY = (239/159) * RC - (102232/53) 	PY = (1.5031) * RC - (1928.9057)
PY = (421/281) * RC - (540964/281)	PY = (1.4982) * RC - (1925.1388)
PY = (218/145) * RC - (279732/145)	PY = (1.5034) * RC - (1929.1862)
PY = (400/267) * RC - (513986/267)	PY = (1.4981) * RC - (1925.0412)
PY = (91/61) * RC - (117024/61) 	PY = (1.4918) * RC - (1918.4262)
One thing is very clear, the slope of all of these equations is very close to 1.5, which we will express as (3/2). Its only the y-intercept that varies. We have determined that the y-intercept is somewhere between around -1,907 and -1,930. One way to further narrow this down is to find the equation that best works for each point, given that the slope is (3/2). If we do so, we get the following equations:
Input (RC)	Best Equation 	Expected Output (PY)	Actual Output (PY)
324 	PY = (3/2) * RC - 1927 	-1441 	-1,441
704 	PY = (3/2) * RC - 1925 	-869 	-869
765 	PY = (3/2) * RC - 1926.5	-779 	-779
779 	PY = (3/2) * RC - 1926.5	-758 	-758
924 	PY = (3/2) * RC - 1926 	-540 	-540
1046 	PY = (3/2) * RC - 1927 	-358 	-358
With this method, we can narrow the y-intercept to between -1925 and -1927. However, this is not perfect, because we are only using whole numbers. In actuality, an event could occur at any point in the year, not just the new year's day. So for example, the event that occurred in 324 RC could have occurred anywhere between 324 and 325 RC. And it's corresponding PY could be anywhere between -1440 and -1441 PY.
That expands the possible equations to the following:
Input (RC)	Equation 	Output (PY)
324 	PY = (3/2) * RC - 1926 	-1440
325 	PY = (3/2) * RC - 1928.5	-1441
704 	PY = (3/2) * RC - 1924 	-868
705 	PY = (3/2) * RC - 1926.5	-869
765 	PY = (3/2) * RC - 1925.5	-778
766 	PY = (3/2) * RC - 1928 	-779
779 	PY = (3/2) * RC - 1925.5	-757
780 	PY = (3/2) * RC - 1928 	-758
924 	PY = (3/2) * RC - 1925 	-539
925 	PY = (3/2) * RC - 1927.5	-540
1046 	PY = (3/2) * RC - 1926 	-357
1047 	PY = (3/2) * RC - 1928.5	-358
The only equations which will work will be somewhere between the following:
* PY = (3/2) * RC - 1926
* PY = (3/2) * RC - 1926.5
We can repeat this entire process with the points in reverse order (e.g. 324, -1441 becomes -1441, 324) to find the equation for converting PY to RC. When all is done, the only equations which will work will be somewhere between the following:
* RC = (2/3) * PY + 1284
* RC = (2/3) * PY + (1284 + (1/3))
== Conclusion ==
We don't have more information to help narrow down the range, and it would make equal sense for the y-intercept to be a whole number or a decimal. However, with noting else to go on, the simplest path seems like the best choice. Therefore, when calculating dates between the Roulean and Portal Year calendars, use the following equations:
* PY = (3/2) * RC - 1926
* RC = (2/3) * PY + 1284
== References ==