# Full text: Pergaeus, Apollonius: The two books of Apollonius Pergaeus, concerning tangencies, as they have been restored by Franciscus Vieta and Marinus Ghetaldus

at I, the perpendicular IG: by Eu. VI. 13. 17. the ſquare on IG is equal
to the rectangle contained by IB and IC; and the ſquare on HO is equal
to the rectangle contained by IB and UC. Now IC is by ſuppoſition
greater than UC, and therefore the rectangle IB, IC is greater than the
rectangle IB, UC: conſequently the ſquare on IG is greater than the
ſquare on HO, and IG than HO; whence O muſt fall between I and B,
much more between I and A. And in the ſame manner it may be proved
that the point o falls between U and E.

Case IV. In which the order of the given points is U, E, A, I; it is
conſtructed exactly in the ſame manner as Caſe III, and is exhibited by
Fig. 49.

Epitagma II. There are here only four Caſes, becauſe, as in Epi-
tagma I. it is indifferent whether the given ratio be of a leſs to a greater,
or of a greater to a leſs; and the two laſt of thoſe, viz. where the order
of the given points is E, A, U, I; or E, U, A, I, being reducible to the
two former by reading every where I for A, E for U, and the contrary,
I ſhall omit ſaying any thing of their conſtructions, except that they are
exhibited by Fig. 52 and 53.

Case I. The order of the given points, being A,E,I,U, make B to
fall between A and I, C between E and U, and draw DH through the
center of the circle on BC, as is done in Fig. 50; and O will fall as re-
quired for reaſons ſimilar to thoſe urged in Caſe I. of the firſt Epitagma
of this Problem.

Case II. If the order of the given points be A, I, E, U, the conſtruc-
tion will be as in Fig. 51, where B and C are made to fall, and DH is
drawn as in Caſe I.

Epitagma III. Here there are eight Caſes, viz. four where in the
order of the given points is A, U, E, I; A, U, I, E; U, A, E, I; and
U, A, I, E, and the given ratio of a greater to a leſs, when O will fall
between the two given points, which bound the conſequent rectangle; and four others@ wherein the order of the given points is the ſame as
here, but the given ratio of a leſs to a greater, and in which the point
O will fall between the points that bound the antecedent rectangle; but
as theſe laſt are reducible to the former by the ſame means which have been
uſed on former ſimilar occaſions, I ſhall not ſtop to ſpecify them.

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