How did Brahmagupta divide mathematics into pati-ganita and bija-ganita?

In his seminal work, the Brahma-sphuta-siddhanta, Brahmagupta established a foundational distinction between two major fields of Indian mathematics: pati-ganita and bija-ganita. This classification, which evolved during the second half of the 1st millennium, categorized mathematical knowledge based on the nature of the quantities involved.12

• Pati-ganita (literally "board-computations") refers to arithmetic and practical mathematics, often called "manifest" calculation. It encompassed eight fundamental operations—including addition, subtraction, multiplication, division, and root extraction—alongside techniques for proportions, mixtures, series, and geometry. Brahmagupta emphasized these as essential skills for a mathematician (ganaka).21
• Bija-ganita (literally "seed-computations") refers to algebra, or "unmanifest" calculation. It focused on the manipulation of equations involving unknown quantities (the "seeds"). This field covered the handling of signs, coefficients, surds, and the resolution of equations up to the second order, including indeterminate equations.21

While pati-ganita dealt with known, tangible values, bija-ganita provided the framework for solving problems where the values were initially unknown.2