How did Archimedes and Apollonius influence mathematicians during the Renaissance and the 17th century?

During the Renaissance and the 17th century, the works of Archimedes and Apollonius of Perga served as essential foundations for the rapid advancement of mathematics, providing both a model of rigor and a source of inspiration for new discoveries.12

Influence of Archimedes

Archimedes’ influence was paramount in the development of infinitesimal research and geometric mechanics. His method of exhaustion was revitalized by mathematicians like Bonaventura Cavalieri, who developed the method of indivisibles to calculate volumes and areas. Figures such as Galileo Galilei, Johannes Kepler, and Isaac Newton drew direct inspiration from his work, using his geometric style as a standard for mathematical precision.132

Influence of Apollonius

Apollonius’s Conics was instrumental in the birth of analytic geometry. By recasting Apollonius’s geometric arguments into algebraic terms, René Descartes and Pierre de Fermat founded a new field that allowed for the study of curves through equations. Furthermore, the application of conic sections to astronomy—most notably in Kepler’s laws of planetary motion—made the study of these shapes a practical necessity for 17th-century science.45