The Usefulness of Mathematical Learning Explained and Demonstrated: Being Mathematical Lectures Read in the Publick Schools at the University of Cambridge

The Usefulness Of Mathematical Learning Explained And Demonstrated: Being Mathematical Lectures Read In The Publick Schools At The University Of Cambridge by Isaac Barrow. This collection presents a rigorous program for understanding mathematics, defending Euclid’s Definition of Proportion, and treating proportion as a relation among homogeneous magnitudes rather than as a quantity. It distinguishes between geometrical (proportional) and arithmetical reasoning, emphasizes reductions to a common consequent, and offers a detailed taxonomy of expressible reasons (proportionals) and their inverses, alongside a critique of rival theories that blur the line between magnitude and relation. The work surveys the use of geometric constructions to illustrate universal proportional principles, defends the universality and demonstrability of proportion, and situates these ideas within a 17th-century Cambridge context that valorizes mathematical demonstration and the unity of arithmetic and geometry. It also anticipates future topics in analogy, hyperlogy/prology, and hypology as refined notions for equal, greater, and lesser reasons.