BOOK I OF THE CONICS.

pertain [to the elementary introduction]. The first contains the origins of the three sections and of the opposite ones, and their principal properties, exposed more broadly and universally than what others have written about them. The second contains what diameters and axes of sections and asymptotes have as their own, and other things that provide general and necessary use for determinations. You will discover from this book which diameters and which axes I name. The third, however, contains many and marvelous theorems useful both for the composition of solid loci and for determinations, most of which, and the most beautiful ones, are new. By these findings, I recognized that the locus for three and four lines was not composed by Euclid, but only a chance part of it, and that not optimally; for it could not be that the composition was perfected without the propositions added by us. The fourth contains in how many ways the sections of cones meet both among themselves and with the circumference of a circle, and besides that, certain other things, neither of which has been treated by our predecessors, [such as] in how many points a section of a cone or the circumference of a circle meet [with the opposite sections]. The remaining books, however, progress further. For the first of them treats of minima and maxima at greater length, the second of equal and similar conic sections, the third of theorems pertaining to determination, [and] the fourth has determinate conic problems. But truly, when all have been published, it is permitted to those who read them to judge them according to the will of each. Farewell.