1. History of analysis

1.3. Foundations and the Rise of Calculus (Ancient to 1700s)

1. Greek and Pre-Modern Roots

  • Eudoxus developed the theory of proportions, avoiding irrational numbers by working with ratios.

  • Archimedes used the method of exhaustion to approximate areas and volumes—an early use of limits.

2. Birth of Calculus

  • In the 17th century, Newton and Leibniz independently invented calculus.

  • They relied on intuitive ideas like infinitesimals and fluxions—which lacked rigorous definitions.

  • Calculus allowed for major advances in physics and engineering, but its logical foundation was unclear.