Apollonius <Pergaeus>; Lawson, John, The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus : with a supplement to which is now added, a second supplement, being Mons. Fermat's Treatise on spherical tangencies

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For by conſtruction IU = AE, and UN = PE. therefore IN = AP. But by
Lemma I.
oN = AO. therefore (o falling beyond I by hypotbeſis) O will fall
beyond
P;
but by hypotbeſis it falls ſhort of E; therefore O falls between
P
and E.
And by Diviſion AT: AI: : AE: AP
Hence AT x AP = IAE or o AO

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