CONICS I.

four [books] fall into an elementary introduction. The first contains the generations of the three sections i.e., the parabola, ellipse, and hyperbola and the opposite hyperbolic sections, and the initial properties therein, worked out more extensively and generally than those written by others. The second [contains] matters concerning the diameters and axes of the sections, and the asymptotes, and other things providing general and necessary utility for the determinations. You will learn from this book which lines I call diameters and which axes. The third [contains] many and paradoxical theorems useful for the compositions of solid loci and for the determinations, most of which and the most beautiful ones are new. Having grasped these, we recognized that the locus for three and four lines had not been composed by Euclid, but only a chance part of it, and even that not successfully; for it was not possible to complete the composition without the things we have discovered. The fourth [contains] how many ways the sections of cones meet each other and the circumference of a circle, and other things in addition, neither of which has been written by those before us, [namely] in how many points a section of a cone or a circumference of a circle meet. The remaining [books] are more advanced; for one is on minima and maxima at greater length, one on equal and similar conic sections, one on determining theorems, and one on determinate conic problems. Nevertheless, with all of them published, it is possible for those who encounter them to judge them as each one chooses. Farewell.