727364v
[Commentary:
This page refers to Propositions I.38 and I.40 from Conicorum libri quattuor
(Apollonius .
I.38 If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter, and if from the point of contact a straight line is dropped to that same [second] diameter parallel to the other diameter, then the straight line cut off from the centre of the section by the dropped straight line, together with the straight line cut off [on the second diameter] by the tangent from the centre of the section will contain an area equal to the square on the half of the second diameter, and, together with the straight line [on the second diameter] between the dropped straight line and the tangent, will contain an area having a ratio to the square on the dropped straight line which the upright side of the figure has to the transverse.
I.40. If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter, and if from the point of contact a straight line is dropped to the same diameter parallel to the other diameter, then whichever of the two straight lines is taken [along the second diameter], of which one is the straight line between the dropped straight line and the centre of the section, and the other is between the dropped straight line and the tangent, the dropped straight line will have to it the ratio compounded of the ratio of the transverse side to the upright and of the ratio of the other of the two straight lines to the dropped straight ]
I.38 If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter, and if from the point of contact a straight line is dropped to that same [second] diameter parallel to the other diameter, then the straight line cut off from the centre of the section by the dropped straight line, together with the straight line cut off [on the second diameter] by the tangent from the centre of the section will contain an area equal to the square on the half of the second diameter, and, together with the straight line [on the second diameter] between the dropped straight line and the tangent, will contain an area having a ratio to the square on the dropped straight line which the upright side of the figure has to the transverse.
I.40. If a straight line touching a hyperbola or ellipse or circumference of a circle meets the second diameter, and if from the point of contact a straight line is dropped to the same diameter parallel to the other diameter, then whichever of the two straight lines is taken [along the second diameter], of which one is the straight line between the dropped straight line and the centre of the section, and the other is between the dropped straight line and the tangent, the dropped straight line will have to it the ratio compounded of the ratio of the transverse side to the upright and of the ratio of the other of the two straight lines to the dropped straight ]
prop:
[Translation: proposition ]
per
[Translation: by proposition ]
[Translation: proposition ]
per
[Translation: by proposition ]
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