A corrected planetary model produces a successful prediction

Descartes published La Géométrie as an appendix to his Discourse, linking algebraic equations with geometric curves.

The work helped establish methods later called Cartesian geometry, although modern coordinate notation developed gradually.

Publication in French rather than only Latin widened access while still excluding many people from formal scholarly networks.

Descartes built within traditions that included Greek geometry and algebra transmitted and extended through Arabic and European scholarship.

Jeremiah Horrocks predicted and observed Venus crossing the Sun; William Crabtree made an independent observation nearby.

Later astronomers used Venus transits and geometry to estimate the scale of distances between Earth, Venus and the Sun.

Horrocks and Crabtree worked outside major academies, showing that knowledge circulated through correspondence and local observation as well as institutions.

Henry Oldenburg launched a periodical that created a regular printed channel for reports in mathematics and natural philosophy.

Serial publication helped establish priority, criticism and cumulative discussion among dispersed correspondents.

Problems, observations, instruments and calculations could circulate more rapidly than through large books alone.