**abstract:**
It is repeatedly and appropriately stated that Apollonius' *Conics* played an important and indispensable role in the invention of analytic geometry in the seventeenth century.
True, nobody would ever have hit on the idea of expressing the straight and curved lines by means of coordinates, without the knowledge of Apollonius' *Conics*. Apollonius also provides us with the problem of three- and four-line locus, which may well have given a cue to Descartes for his invention of analytic geometry. However, if we spoke of Apollonius' *Conics* and the analytic geometry as something continuous, as if the latter had been a natural result of the development of the former, we would fail to duely evaluate the revolution of mathematics in the modern era. First of all, the analysis as used in the *Conics* and in other Greek mathematical works is different from, even alien to, the analysis in the modern sense, identified with the algebraic procedure. A more important and less known fact is the particular characteristics of the techniques in Apollonius' *Conics*. Apollonius uses various techniques to advance his investigation of conic sections, each of which is only "locally" valid, that is, particularly effective to some specific group of theorems and problems, while the analytic geometry provides us with a universally valid method of the presentation of figures by equations and their algebraic treatment. I will give a survey of Apollonius' major techniques that are effective but local, including those used to establish theorems intended for the solutions of the three- and four-line locus in my presentation. By showing the difference between the Apollonius' techniques and the modern analysis (= algebra), I hope to emphasize how revolutionary the invention of analytic geometry was.

Fri 16 Sep, 12:30 - 13:30, Aula Dini

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