Doesn't the g'mara on the top of the Daf seem to indicate that we use a solar calendar? Of what use is the g'matria being equal to 364?
You are correct -- the Jewish year (with regard to the festivals) follows the lunar calendar, and if so, what significance can be ascribed to the Gematriya of "ha'Satan" being 364?
The NETZIV (in Meromei Sadeh) asks your question, and answers simply that it is a "Remez b'Alma." That is, the fact that the Gematriya of "ha'Satan" is one less than the number of days in the solar year is significant enough to allude the fact that he has no reign on one day out of the year.
(We may add that every 19 solar 365 day years equal 19 lunar years, because of the lunar leap years -- which is Chazal used a 19 year lunar cycle. If so, Satan indeed gets, on average 364 out of every 365 days, if his record is reviewed after 18 years. In fact, this method of calculating has a decisive advantage over listing the number of days he gets in a lunar year, since the lunar years vary in length (depending on whether they are Malei/Chaser/k'Sidran, and whether they are leap years), and therefore it is preferable to count how many days he has in the unchanging solar year. - M. Kornfeld)
On a deeper level, there is a link between the reign of the Satan and the solar year. The Midrash (Bereishis Rabah 6:3) says that Esav (and the forces in the world which he represents) follows the solar year, while Yakov (and the forces in the world which he represents) follows the lunar year. Esav represents the forces that pull a person away from spirituality, just like ha'Satan. (For more on this topic, I recommend Nosson Slifkin's book, "Seasons of Life.")
I believe you mean to say that the number of lunar and solar days are the same after x19x years and not x18x. Also, it is just an xapproximationx, not xexactlyx ...
Average month in our calendar = 29 days 12 hours 793 chalakim = 29 + 0.5 + 793/1080 days = 29.530594 days #of months in 19 years in our calendar = 235 (7 years of 13 months and 12 years of 12 months): 12x12+7x13 = 144+91=235.
So average # of days in 19 years = 6939.68962 days (approximately)
Average # of days in 1 solar year (Julian calendar) = 365.25 Average # of days in 19 solar years (Julian calendar) = 365.25 x19 = 6939.75 days
The Gregorian calendar is missing 3 leap years every 400 years (every 4 years is a leap year except when year is divisible by 100 except when year is divisible by 400) Average # of days in 1 solar year (Gregorian calendar) = ((365.25x400)-3) / (365.25x400) = 365.2425
Average # of days in 19 solar years (Gregorian calendar) = 365.2425 x 19 = 6939.6075 days
Thank you for correcting that, I indeed meant 19, and not 18, years, as our calendar follows a 19 year lunar cycle, and "meets" the solar year after 19 years. As you pointed out, the lunisolar calendar is itself an approximation (the Kollel discussed this in its Insights to Eruvin 56:1), but for the purposes of answering the question at hand, I think what we wrote will still suffice.