Lecture I: Analogical reasoning has historically helped discover new ideas in mathematics by using familiar solutions to new problems. This talk investigates how analogy and meta-analogy are central to the development of mathematical knowledge, independent problem solving, and comparative reasoning. Tracing this process from ancient to modern times shows that diverse knowledge and insights have fundamentally shaped mathematics. This approach creates a more inclusive historical perspective, recognising diverse cultural and scientific heritages as essential to mathematical thought.

Lecture II: The rusty compass, or fixed-opening compass, is an instrument for polygon construction described by Abu al-Wafa al-Buzjani (940–998 CE), a Persian mathematician of the Islamic Golden Age, whose approximate constructions extended beyond Euclid’s compass and straightedge methods in the Elements. Centuries later, comparable techniques appear in Renaissance Europe in the work of Albrecht Dürer (1471–1528). Through historical tracing and practical comparison of polygon constructions, this study explores the circulation of the rusty compass as a case of ethnomathematics, examining whether its reappearance reflects cultural transmission, independent development, or shared practical needs in art and architecture.