There seems to be little sound evidence in the Bible that the ancients, in Biblical times used a 360 day year. It is always assumed. The Bible always refers to time as a number of days, months, years, or "times" but never comes right out and says "a year is 360 days." The Biblical references of Genesis 7 and 8 suggesting that the month was 30 days is entirely coincidence. (The water was on the Earth for 150 days. 150/30=5! So what?) The fact that a 30 day month provides exactly 360 days when multiplied by 12, is at best, an attempt by Biblical scholars to "fit" what the Bible says about days, months, years, and "times" into a "year".
However, it seems entirely logical that the year WAS thought to be 360 days in Biblical times. The keen observer of old noticed that the number of days between the 12 sightings of the New Moon, led that same observer to insist that there must be some relationship between the number months, and the number of the days of the year and therefore the "month" must be 30 days. When you factor in the fact that the ancients did not comprehend or understand "fractions", you can immediately comprehend that they would try their level best do make things fit into integral days.
Now, consider you are a First Eon human and you are watching the cycle of the sun, exactly where on the horizon the sun rises each day. In what we call September today, the sun steadily rises further south, where on September 21 or 22nd, you notice the sun is due East. You know from your observations that the sun will once again appear due South, but not until it has gone way South, and then returned by progressing Northward each morning. So you wonder, when will the sun be back at this same spot, i.e., due East after having gone as far South as it gets.
You know from your previous observations that the sun continues rising further and further South on the horizon each morning, until one day (in what we call December today), the Sun stops going South and seems to literally stop, and then begins going North again! So you keep watching, and lo and behold, one day (in what we today call "March"), the sun is once again rising exactly due East. You look at how many days it's been since Sep 21 and what do you discover? The number of days elapsed was 180! Amazing! (to the First Eon human.) Therefore they concluded, 180 days must equal 1/2 year, and therefore the year must be 360 days long!
So you can see by pretending to be a First Eon human that you would conclude the year must be 360 days long, and you will quite naturally try to "force" the lunar month to be 30 days, because 30 "fits" into 360 exactly 12 times.
But as you said, if you try to create a calendar of 360 day years, the seasons get out of whack. The fact that the calendar got seriously out of whack is precisely why the Hebrew calendar adds a 13th month every 2nd or 3rd year! It is the very same reason that today, by the Gregorian calendar, we add one day every 4 years, but not if the year is divisible by 100 but do if the year is divisible by 400! Every calendar MUST have some complex sequence of leap days to account for the noninteger number of days of a tropical year.
Every society, regardless of the calendar they chose to observe, had to add days to their "Year" to bring the "calendar" back in line with the observed seasons. It really matters not exactly how many days we want to call the "year."
It is often argued that the citation in Esther 1:4 requires the year to be exactly 360 days! Why? Because they used a "180" day period. The reader assumes that 180 days had to be 1/2 of a year. Consider my argument above. "180" days was thought to be 1/2 of a year, simply because that matched observation of the rising sun. It is likely this reason that the reader is correct. But using that reasoning (i.e. taking a number of days quoted in the Bible and assuming it had something to do with a "year") wouldn't one also argue that the number "40" must have something to do with the "year"? After all, Moses was "40 days" on Mt Sinai. And it rained 40 days and 40 nights on Noach. Goliath came 40 days before being killed by David. So must there be a relationship between "40" and the number of days in a year? Well, coincidentally, there are "9" 40day periods in one 360 day year, but we see nothing special in the Bible about "9" or the relationship of "40" to a year.
What our article on the calendar attempted to convey was that the 7 day period of the week is what is truly important. It is so important that, today, we still measure the week in 7 days, and despite the enormous power of the ancient leaders to do so, they never broke the steady counting of the 7 days! Therefore we can be sure that when we honor the Sabbath on Saturday, the 7th day of the week, we are doing as did EVERYONE who is ever mentioned in the BIBLE, who obeyed Yahweh! We are truly honoring Him on the day He asked! That is simply one remarkable fact!
