Apollonius to Eudemus, greeting.

If you are in good health and things are in other respects as you
wish, it is well; with me too things are moderately well. During
the time I spent with you at Pergamum I observed your eagerness
to become acquainted with my work in conics; I am therefore
sending the first book, which I have corrected, and I will
forward the remaining books when I have finished them to my
satisfaction. I dare say you have not forgotten my telling you
that I undertook the investigation of this subject at the request
of Naucrates the geometer, at the time when he came to Alexandria
and stayed with me, and, when I had worked it out in eight books,
I gave them to him at once, too hurriedly, because he was on the
point of sailing; they had therefore not been thoroughly revised,
indeed I had put down everything just as it occurred to me,
postponing revision till the end. Accordingly I now publish, as
opportunities serve from time to time, instalments of the work as
they are corrected. In the meantime it has happened that some
other persons also, among those whom I have met, have got the
first and second books before they were corrected; do not be
surprised therefore if you come across them in a different shape.

Now of the eight books the first four form an elementary
introduction. The first contains the modes of producing the three
sections and the opposite branches of the hyperbola, and the
fundamental properties subsisting in them, worked out more fully
and generally than in the writings of others. The second book
contains the properties of the diameters and axes of the sections
as well as the asymptotes, with other things needed for
determining limits of possibility; and what I mean by diameters
and axes you will learn from this book. The third book contains
many remarkable theorems useful for the construction of solid
loci and for limits of possibility; the largest number and most
beautiful of these theorems are new, and it was their discovery
which made me aware that Euclid did not work out the construction
of the locus with respect to three and four lines, but only a
chance portion of it and that not successfully; for it was not
possible for the said construction to be completed without the aid
of the additional theorems discovered by me. The fourth book shows
in how many ways the sections of cones can meet one another and
the circumference of a circle; it contains other things in
addition, none of which have been discussed by earlier writers,
namely the questions in how many points a section of a cone or a
circumference of a circle can meet a double-branch hyperbola, or
two double-branch hyperbolas can meet one another.

The rest of the books are more specialized: one of them deals
somewhat fully with minima and maxima, another with
equal and similar sections of cones, another with theorems of the
nature of determinations of limits, and the last with determinate
conic problems. But of course, when all of them are published, it
will be open to all who read them to form their own judgement
about them, according to their own individual tastes. Farewell.