Adhika-Māsa: The Intercalary Month

Why the Hindu calendar adds a whole month every few years, and once in a long while loses one

Twelve lunar months fall about eleven days short of the solar year, so every two or three years the Hindu calendar adds a whole month. Learn Bhāskara's rule for finding it (a lunar month with no saṅkrānti), the rare lost month, why the extra month is kept for Viṣṇu as Puruṣottama, how the Hebrew calendar solves the same problem with a fixed count, and what the Kumbh Mela's computed timing has in common with it.

The Month That Would Vanish

In the year 1150, a thirty-six-year-old astronomer named Bhāskara was finishing the book that would make his name, the Siddhānta Śiromaṇi, "the crest-jewel of the siddhāntas". Picture him at night: an oil lamp, a pot of ink, pages of his own sums piled at his elbow, each number checked twice because a wrong one would travel into every almanac copied from his.

In one short section he did something bold. He wrote a rule for a month that would disappear from the calendar. Then he gave the years. One such month had fallen in Śaka 974, about a century before him. The next, he wrote, would come in Śaka 1115, around 1193 CE, more than forty years away. He would be nearly eighty by then, if he lived at all.

How could a man at a desk know, decades ahead, that a whole month would go missing? And why would a calendar ever need to lose a month, or add one? The answer begins with a small gap of eleven days.

Eleven Days a Year

Twelve lunar months, new moon to new moon, add up to about 354 days. The Sun takes about 365 days to bring the seasons round again. So every year the Moon's calendar falls about eleven days short of the Sun's.

That sounds small, but it adds up. After three years the gap is a whole month. After sixteen or seventeen years it is half a year. A spring festival would be falling in the autumn rains. The Egyptian calendar of the last lesson drifted by a quarter-day a year. A pure lunar year drifts more than forty times faster.

India's oldest text already knew the cure: add a month. A hymn of the Ṛgveda, sung to Varuṇa, the keeper of cosmic order, says he knows the twelve months, and also the one that is born in addition.