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"The investigation of these theorems is of great service both for the synthesis of problems and the determinations of limits of possibility. On the other hand Nicoteles, on account of his controversy with Conon, will not have it that any use can be made of the discoveries of Conon for determinations of limits: in which opinion he is mistaken, for, even if it is possible, without using them at all, to arrive at results relating to such determinations, yet they at all events afford a more ready means of observing some things, e.g. that several solutions are possible or that they are so many in number, and again that no solution is possible; and such previous knowledge secures a satisfactory basis for investigations, while the theorems in question are further useful for the analyses of determinations of limits. Moreover, apart from such usefulness, they are worthy of acceptance for the sake of the demonstrations themselves, in the same way as we accept many other things in mathematics for this and for no other reason." - Apollonius of Perga
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