the comes 
131 
9 
ilar sections 
determina 
le problems, 
vili be open 
about them, 
11. 
pollonius is 
Apollonius, 
dents of the 
neter whom 
is. There is 
te preface to 
anus of Per- 
hich I have 
way, I have 
i because of 
sending you 
liscussion of 
possible for 
rcumference 
not coincide 
at most a 
îan meet the 
anch hyper- 
56 questions, 
similar kind, 
fdaeus, with- 
oofs, and on 
,son, fell foul 
lentioned by 
with Conon, 
not found it 
any one else, 
ve not found 
s referred to, 
sheir solution 
Inch I have, 
ks, while the 
theorems are 
>lems and for 
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 in 
vestigations, 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 repro 
duced 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 super 
ficially 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 pro 
perties 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 con 
nected 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. 
k 2