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Leap Months and Kings

The Hebrew calendar is lunar, so that a leap-month has to be inserted every once in a while to keep the seasons and holidays at appropriate times. But when to insert the month, and what group should decide?
According to the Talmud (Sanhedrin 18b), the king was excluded from the group. Because he paid his soldiers on an annual basis, it was felt that he would have an incentive to insert extra months, since that lengthened the year and saved him money.
If the king only expected to be in power for a few years, or if he had a very high discount rate and didn’t care about the future, this explanation might make sense. But the calendar couldn’t get too far off because the king couldn’t keep inserting months year after year. Otherwise, for example, harvest holidays would come during planting times. One might even argue that the king had the opposite incentive: by avoiding excessive intercalations, he would demonstrate to the soldiers and the people his confidence in a long reign. (Hat tip: M.A.H. and S.C.H.)