723362v
[Translation: ]
per
[Translation: by proposition ]
per
[Translation: by proposition ]
per, 36:
[Translation: by proposition 36, ]
igitur:
[Translation: therefore, ]
[Commentary:
This page refers to Propositions 21, 36, and 37 from Book I of Apollonius, as edited by Commandino Conicorum libri quattuor
(Apollonius .
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 said.
I.36. If some straight line, meeting the transverse side of the figure touches an hyperbola or ellipse or circumference of a circle, and if a straight line is dropped from the point of contact as an ordinate to the diameter, then as the straight line cut off by the tangent from the end of the transverse side is to the straight line cut off by the tangent from the other end of that side, so will the straight line cut off by the ordinate from the end of the side be to the straight line cut off by the ordinate from the other end of the side in such a way that the corresponding straight lines are continuous; and another straight line will not fall into the space between the tangent and the section of the
I.37 If a straight line touching an hyperbola or ellipse or circumference of a circle meets the diameter, and from the point of contact to the diameter a straight line is dropped as ordinate, then the straight line cut off by the ordinate from the centre of the section with the straight line cut off by the tangent from the centre of the section will contain an area equal to the square on the radius of the section, and with the straight line between the ordinate and the tangent will contain an area having the ratio to the square on the ordinate which the transverse has to the upright.]
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 said.
I.36. If some straight line, meeting the transverse side of the figure touches an hyperbola or ellipse or circumference of a circle, and if a straight line is dropped from the point of contact as an ordinate to the diameter, then as the straight line cut off by the tangent from the end of the transverse side is to the straight line cut off by the tangent from the other end of that side, so will the straight line cut off by the ordinate from the end of the side be to the straight line cut off by the ordinate from the other end of the side in such a way that the corresponding straight lines are continuous; and another straight line will not fall into the space between the tangent and the section of the
I.37 If a straight line touching an hyperbola or ellipse or circumference of a circle meets the diameter, and from the point of contact to the diameter a straight line is dropped as ordinate, then the straight line cut off by the ordinate from the centre of the section with the straight line cut off by the tangent from the centre of the section will contain an area equal to the square on the radius of the section, and with the straight line between the ordinate and the tangent will contain an area having the ratio to the square on the ordinate which the transverse has to the upright.]
[Translation: ]
per
[Translation: by proposition ]
per
[Translation: by proposition ]
per, 36:
[Translation: by proposition 36, ]
igitur:
[Translation: therefore, ]
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