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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Therefore AT x AO is greater than o AO
Or AT greater than Ao.
Corollary II. If the three given points be I, A, E; and O falls between
A and I, ſo as to make AO x PE: IOE: : AL: LI, I ſay then O will fall
beyond L.
For let us ſuppoſe that O and L coincide; then by hypotbeſis AL: LI: :
AL x PE: IL x LE
And by the next following Lemma IV. AL x IL: IL x PE: : AL: LE
i. e. AL: PE: : AL: LE
Hence PE is equal to LE, a part to the whole, which is manifeſtly abſurd.
LEMMA IV.
If it be as a line to a line ſo a rectangle to a rectangle; then I ſay it will be
as the flrſt line into the breadth of the ſecond rectangle to the ſecond line into
the breadth of the firſt rectangle, ſo the length of the firſt rectangle to the
length of the ſecond.
Suppoſition. AE: IO: : UYN: SRL.
Concluſion. AE x RL: IO x YN: : UY: SR.
Dem. AE: IO: : AE x YN: IO x YN: : UYN: SRL
And by Permutation AE x YN: UYN: : AE: UY: : IO x YN: SRL
But SR: AE: : SRL: : AE x RL
Therefore ex æquo perturbatè SR: UY: : IO x YN: AE x RL
Q. E. D.

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