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

THE CONICS 131
diorismi. Nicoteles indeed, on account of his controversy
with Conon, will not have it that any use can be made of the
discoveries of Conon for the purpose of diorismi; he is,
however, mistaken in this opinion, for, even if it is possible,
without using them at all, to arrive at results in regard to
limits of possibility, yet they at all events afford a readier
means of observing some things, e.g. that several or so many
solutions are possible, or again that no solution is possible
and such foreknowledge secures a satisfactory basis for investigations,
while the theorems in question are again useful
for the analyses of diorismi. And, even apart from such
usefulness, they will be found worthy of acceptance for the
sake of the demonstrations themselves, just as we accept
many other things in mathematics for this reason and for
no other.
The prefaces to Books V-VII now to be given are reproduced
for Book V from the translation of L. Nix and for
Books VI, VII from that of Halley.
Preface to Book V.
Apollonius to Attalus, greeting.
In this fifth book I have laid down propositions relating to
maximum and minimum straight lines. You must know
that my predecessors and contemporaries have only superficially
touched upon the investigation of the shortest lines,
and have only proved what straight lines touch the sections
and. conversely, what properties they have in virtue of which
they are tangents. For my part, 1 have proved these properties
in the first book (without however making any use, in
the proofs, of the doctrine of the shortest lines), inasmuch as
I wished to place them in close connexion with that part
of the subject in which I treat of the production of the three
conic sections, in order to show at the same time that in each
of the three sections countless properties and necessary results
appear, as they do with reference to the original (transverse)
diameter. The propositions in which I discuss the shortest
lines I have separated into classes, and I have dealt with each
individual case by careful demonstration ; I have also connected
the investigation of them with the investigation of
the greatest lines above mentioned, because I considered that
those who cultivate this science need them for obtaining
a knowledge of the analysis, and determination of limits of
possibility, of problems as well as for their synthesis : in
addition to which, the subject is one of those which seem
worthy of study for their own sake. Farewell.
k2
132 APOLLONIUS OF PERGA
Preface to Book VI.
Apollonius to Attalus, greeting.
I send you the sixth book of the conies, which embraces
propositions about conic sections and segments of conies equal
and unequal, similar and dissimilar, besides some other matters
left out by those who have preceded me. In particular, you
will find in this book how, in a given right cone, a section can
be cut which is equal to a given section, and how a right cone
can be described similar to a given cone but such as to contain
a given conic section. And these matters in truth I have
treated somewhat more fully and clearly than those who wrote
before my time on these subjects. Farewell.
Preface to Book VII.
Apollonius to Attalus, greeting.
I send to you with this letter the seventh book on conic
sections. In it are contained a large number of new propositions
concerning diameters of sections and the figures described
upon them ;
and all these propositions have their uses in many
kinds of problems, especially in the determination of the
limits of their possibility. Several examples of these occur
in the determinate conic problems solved and demonstrated
by me in the eighth book, which is by way of an appendix,
and which I will make a point of sending to you as soon
as possible. Farewell.
Extent of claim to originality.
We gather from these prefaces a very good idea of the
plan followed by Apollonius in the arrangement of the subject
and of the extent to which he claims originality. The
first four Books form, as he says, an elementary introduction,
by which he means an exposition of the elements of conies,
that is, the definitions and the fundamental propositions
which are of the most general use and application ; the term
'
elements ' is in fact used with reference to conies in exactly
the same sense as Euclid uses it to describe his great work.
The remaining Books beginning with Book V are devoted to
more specialized investigation of particular parts of the subject.
It is only for a very small portion of the content of the
treatise that Apollonius claims originality ; in the first three
Books the claim is confined to certain propositions bearing on
the ' locus with respect to three or four lines '
; and in the
fourth Book (on the number of points at which two conies