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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PROBLEMS
CONCERNING
DETERMINATE SECTION.

PROBLEM I.

TO cut a given indefinite right line in one point, ſo that of the ſegments
intercepted between that point and two other points given in the inde-
finite right line, the ſquare of one of them may be to the rectangle under the
other and a given external right line, in a given ratio.

In the given indefinite right line let be aſſigned the points A and E, it is then
required to cut it in the point O, ſo that AO2 may be to OE into a given
line AU in the ratio of R to S; which ratio let be expreſſed by AI to AU,
ſetting off AI from A either way, either towards E or the contrary; and
then from A and I erect two perpendiculars AY equal to AE, and IR
equal to AI, and theſe on the ſame ſide of the given indefinite line, if AI
was ſet off towards E; but on oppoſite ſides, if AI was ſet off the other way.
The former conſtruction I will beg leave to call Homotactical, and the latter
Antitactical. Let now the extremities of theſe perpendiculars Y and R be
joined, and upon YR as a diameter let a circle be deſcribed, I ſay that the
interſection of this circle with the given indefinite line ſolves the Problem.
If it interſects the line in two places, the Problem admits of two Solutions;

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