The Conics Generated by the Method of Application of Areas

TL;DR

This paper explores how Euclid's method of application of areas can generate conic sections as loci of points satisfying quadratic equations, suggesting conics were likely discovered through geometric constructions before being studied as conic sections.

Contribution

It provides a geometric reconstruction showing conics can be derived from Euclidean methods, offering historical insight into their discovery process.

Findings

Conics can be generated via Euclid's application of areas.

Conic equations arise naturally from geometric constructions.

Supports the idea that conics predate Apollonius' study.

Abstract

The method of application of areas as presented in Euclid's Elements, is employed to generate the three conics as the loci of points with Cartesian coordinates satisfying quadratic equations with coefficients defined by the initial settings of the geometric constructions produced by the applications method. This conceptual reconstruction supports the view that the conics were most probably discovered as plane curves by fusing the method of the application of areas with the concept of locus, long before Apollonius studied them as conic sections.