From later commentary, it appears that the binomial coefficients and the additive formula for generating them, formula_10, were known to Pingala in or before the 2nd century BC. While Pingala's work only survives in fragments, the commentator Varāhamihira, around 505, gave a clear description of the additive formula, and a more detailed explanation of the same rule was given by Halayudha, around 975. Halayudha also explained obscure references to "Meru-prastaara", the Staircase of Mount Meru, giving the first surviving description of the arrangement of these numbers into a triangle. In approximately 850, the Jain mathematician Mahāvīra gave a different formula for the binomial coefficients, using multiplication, equivalent to the modern formula formula_11. In 1068, four columns of the first sixteen rows were given by the mathematician Bhattotpala, who was the first recorded mathematician to equate the additive and multiplicative formulas for these numbers.