715358v
[Translation: ]
[Commentary:
This page refers to Propositions II.1, II.2, and I.21 from Conicorum libri quattuor
(Apollonius .
II.1 If a straight line touch an hyperbola at its vertex, and from it on both sides of the diameter a straight line is cut off equal in square to the fourth of the figure, then the straight lines drawn from the centre of the section to the ends thus taken on the tangent will not meet the
II.2 With the same things it is to be shown that a straight line cutting the angle contained by the straight lines and is not another asymptote.
I.21 If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter, the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of the transverse side of the figure, as the upright side of the figure is to the transverse, and to each other as the areas contained by the straight lines cut off, as we have ]
II.1 If a straight line touch an hyperbola at its vertex, and from it on both sides of the diameter a straight line is cut off equal in square to the fourth of the figure, then the straight lines drawn from the centre of the section to the ends thus taken on the tangent will not meet the
II.2 With the same things it is to be shown that a straight line cutting the angle contained by the straight lines and is not another asymptote.
I.21 If in a hyperbola or ellipse or circumference of a circle straight lines are dropped as ordinates to the diameter, the square on them will be to the areas contained by the straight lines cut off by them beginning from the ends of the transverse side of the figure, as the upright side of the figure is to the transverse, and to each other as the areas contained by the straight lines cut off, as we have ]
prop.
[Translation: Proposition ]
[Translation: Proposition ]
lib. 2. prop:
[Translation: Book 2, Proposition ]
[Translation: Book 2, Proposition ]
per.
[Translation: by Proposition ]
[Translation: ]
[Translation: by Proposition ]
[Translation: ]
[Translation: ]
prop.
[Translation: Proposition ]
[Translation: Proposition ]
Ergo
[Translation: Therefore ]
[Translation: Therefore ]