130 
APOLLONIUS OF PERGA 
minima and maxima, another with equal and similar sections 
of cones, another with theorems of the nature of determina 
tions 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. F are well. 
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 Attains, greeting. 
Some time ago I expounded and sent to Eudemus of Per- 
gamum the first three books of my conics 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 hyper 
bolas 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, with 
out, however, showing proper mastery of the proofs, and on 
this ground Nicoteles of Gyrene, 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