The Jewish calendar and seasonal drift

Am I right in thinking that Hillel’s calendar introduced in 4119 was the first Jewish mathematical calendar? In which case, did they use the Babylonian calendar in Babylon, and was Hillel’s calendar related to that introduced by Cyrus? And after the return to the holy land? Did they revert to the observational calendar, or if it was another mathematical calendar, was it ever corrected for drift? And if Hillel’s calendar is gaining half a second per month (as this article argues), are there any measures to reset it, or does it just keep drifting? http://yourjerusalem.org/2010/02/jewish-calendar-is-slowly-drifting-off-track/ I’m wondering about any drift before Hillel, and if it was corrected. Thanks and Shalom, Danny

1. Rav Sa’adya Gaon and Rabenu Chananel, among others, believe that the mathematical calculated Jewish calendar was always used, and that the witnesses came to validate that which was expected. 2. I’m not familiar with the issue of the Babylonian calendar, and I suggest you contact Rabbi Dr. Nachum Rabinowitz, the rosh yeshiva of Ma’aleh Adumim, who specializes in this field (he speaks English, as well). But, as I mentioned above, after the return to Israel in the time of Cyrus, and during the entire span of the 2nd Temple, many believe that a mathematical calculated Jewish calendar was used, in addition to the witnesses who saw the New Moon.