La Géométrie

La Géométrie and two other appendices, also by Descartes, La Dioptrique (Optics) and Les Météores (Meteorology), were published with the Discourse to give examples of the kinds of successes he had achieved following his method[1] (as well as, perhaps, considering the contemporary European social climate of intellectual competitiveness, to show off a bit to a wider audience).

He introduces modern exponential notation for powers (except for squares, where he kept the older tradition of writing repeated letters, such as, aa).

He also breaks with the Greek tradition of associating powers with geometric referents, a2 with an area, a3 with a volume and so on, and treats them all as possible lengths of line segments.

These notational devices permit him to describe an association of numbers to lengths of line segments that could be constructed with straightedge and compass.

The third book, On the Construction of Solid and Supersolid Problems, is more properly algebraic than geometric and concerns the nature of equations and how they may be solved.

His exposition style was far from clear, the material was not arranged in a systematic manner and he generally only gave indications of proofs, leaving many of the details to the reader.

This enhancement of Descartes' work was primarily carried out by Frans van Schooten, a professor of mathematics at Leiden and his students.

The 1659−1661 edition was a two volume work more than twice the length of the original filled with explanations and examples provided by van Schooten and his students.