On the significance of KEPLER'S discovery
The conic sections play a significant role in the history of projective geometry. The Brouillon project d'une atteinte aux événemens des rencontres d'un cône avec une plan by GIRARD DESARGUES from 1639 is considered to be the cornerstone of this area. DESARGUES looks at the conic sections in a projective manner and thereby rewrites the teaching of hyperbola, parabola, ellipse and circle on a new basis.
The projective consideration of the conic sections led him to points at infinity. But these infinitely distant points do not appear there for the first time! They can already be found in the little discourse on conic sections that JOHANNES KEPLER wrote at the beginning of that century. That was the time of KEPLER'S stay in Prague, when he advanced on the planetary laws of the Astronomia nova. His discussion of the conic sections differs significantly from the mathematics of the astronomical work, because here, in a geometrical digression of his optics, KEPLER undertook a purely qualitative investigation, starting from the free observation of the curves.
This method led him to remarkable discoveries. Above all, KEPLER'S references to the importance of analogies as a guide to mathematical research became famous: »Analogia monstravit et Geometria comprobat«: »the analogy has shown it and the geometry provides the confirmation«, so he explains his procedure. His analogies are based on the transition from one conic section to another, as ERNST MACH states: »With these classic words, KEPLER emphasizes not only the value of analogy, but also rightly the principle of continuity, which alone could guide him to the degree of abstraction that allowed him the comprehension of such deep-seated analogies.«
A decade later, KEPLER came back to the conic sections. This time in the Messekunst Archimedis he describes them in his mother tongue. The Duden or fixed grammatical rules did not yet exist at that time: It was written as it was spoken. And KEPLER spoke a lively, robust German! In this text, some mathematical terms appear for the first time in German. In particular the term "Kegelschnitt" was introduced into German by Kepler.
These two treatises have been edited here together, the new translation of the first text is juxtaposed with the Latin original. The introduction deals with the importance of geometry for KEPLER'S work.