The philosophy of mathematics

This work maps the structure and purpose of mathematical science, defining its object, methods, and two main branches: concrete mathematics, which seeks equations of phenomena linking geometry and mechanics, and abstract mathematics, concerned with analysis and the calculus. It surveys algebraic and transcendental techniques, contrasts methods of infinitesimals, fluxions, and analytical derivation, and develops differential and integral calculus, the calculus of variations, and applications to tangents, quadratures, and extrema. Alongside technical exposition it proposes a systematic classification of functions, equations, and methods, and offers pedagogical guidance for students to study parts in relation to the whole.