A history of Greek mathematics Vol.II from Aristarchus to Diophantus by Heath, Thomas Little, Sir, 1921

THE CONIGS 129
well. During the time I spent with you at Pergamum
I observed your eagerness to become acquainted with my
work in conies; I am therefore sending you 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
the axes of the sections as well as the asymptotes, with other
things generally and necessarily used for determining limits
of possibility (Siopio-fioi) ;
and what I mean by diameters
and axes respectively you will learn from this book. The
third book contains many remarkable theorems useful for
the syntheses of solid loci and for diorismi ;
the most and
prettiest of these theorems are new, and it was their discovery
which made me aware that Euclid did not work out the
synthesis 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 synthesis 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 ;
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 by way of surplusage
(7r€piov(TLa(TTLK(OT€pa) : one of them deals somewhat fully with
1523.2 K
it
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
These theorems are
rest are contained in the present book.
of considerable use both for the syntheses of problems and for