A history of Greek mathematics - Wilbourhall.org

130 APOLLONIUS OF PERGA
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.
The preface to Book II merely says that Apollonius is
sending the second Book to Eudemus by his son Apollonius,
and begs Eudemus to communicate it to earnest students of the
subject, and in particular to Philonides the geometer whom
Apollonius had introduced to Eudemus at Ephesus. There is
no preface to Book III as we have it, although the preface to
Book IV records that it also was sent to Eudemus.
Preface to Book IV.
Apollonius to Attalus, greeting.
Some time ago I expounded and sent to Eudemus of Pergamum
the first three books of my conies which I have
compiled in eight books, but, as he has passed away, I have
resolved to dedicate the remaining books to you because of
your earnest desire to possess my works. I am sending you
on this occasion the fourth book. It contains a discussion of
the question, in how many points at most it is possible for
sections of cones to meet one another and the circumference
of a circle, on the assumption that they do not coincide
throughout, and further in how many points at most a
section of a cone or the circumference of a circle can meet the
hyperbola with two branches, [or
two double-branch hyperbolas
can meet one another]; and, besides these questions,
the book considers a number of others of a similar kind.
Now the first question Conon expounded to Thrasydaeus, without,
however, showing proper mastery of the proofs, and on
this ground Nicoteles of Cyrene, not without reason, fell foul
of him. The second matter has merely been mentioned by
Nicoteles, in connexion with his controversy with Conon,
as one capable of demonstration ; but I have not found it
demonstrated either by Nicoteles himself or by any one else.
The third question and the others akin to it I have not found
so much as noticed by any one. All the matters referred to,
which I have not found anywhere, required for their solution
many and various novel theorems, most of which I have,
as a matter of fact, set out in the first three books, while the
rest are contained in the present book. These theorems are
of considerable use both for the syntheses of problems and for