A history of Greek mathematics - Wilbourhall.org

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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.
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