linguistic:paraphrasing, structural:expansion, linguistic:voice_change, quality:correction
Methods of Apollonius

According to Heath, the "Methods of Apollonius" were not personal to him; whatever influence he had on later theorists was the influence of geometry, not of his own innovation of technique. Heath says,

As a preliminary to the consideration in detail of the methods employed in the Conics, it may be stated generally that they follow steadily the accepted principles of geometrical investigation which found their definitive expression in the Elements of Euclid.

When referring to golden age geometers, modern scholars use the term "method" to mean the visual, reconstructive way in which the geometer produces a result equivalent to that produced by algebra today. As a simple example, the algebraic method to compute the area of a square is to square its side length; the analogous geometric method is to construct a visual square. Geometric methods in the golden age could produce most of the results of elementary algebra.