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Si quotlibet lineis ordinatim applicatis ad diametrum circuli, aliæ numero
et quantitate æquales similiter applicentur ad similes partes lineæ maioris vel
minoris de data diametro circuli: termini illarum linearum sunt in
[Translation: If any number of ordinate lines are dropped to the diameter of a circle, that number of other lines, equal in quantity and similarly applied to similar parts of lines greater or less than the given diameter of a circe, then the ends of those lines are on an ]
Sit diameter circuli . lineæ ordinatim applicatæ et . sit etiam
linea maior quam cui applicentur ad angulos rectos et æquales
lineis et Et sint partes lineæ videlicet [???] quotlibet
[???] sed ita fiat ud ad ita ad et ad . Quod etiam fit
si utraque lineæ et similter dividantur et ad utraque æquales similes et
similiter sitas æquales lineæ ordinatim applicentur.
Dico quod puncta et termini linearum et sunt in ellipsi.
Quoniam ex hypothesi
[…] Ergo: per 21, prop: primi Apollonij, puncta et
sunt in ellipsi. quod demonstrare
[Translation: Let the diameter of the circle be , and the ordinate lines and ; also let the line greater than be , to which are applied at right angles and euqal to the lines and in the circle. But thus, as is to so is to and to . Which also happens if the two lines et are similarly divided and to both parts, similar and similarly situated, equal ordinate lines are applied.
I say that the points and , the ends of the lines and , are on an ellipse.
Because from the hypothesis:
Therefore, by Proposition 21 of the first Book of Apollonius, the points et are on an ellipse; which was to be demonstrated.
[Commentary:
This page refers to Proposition 21 of 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 ]
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 ]
De
[Translation: On the ]
[Translation: On the ]
Si quotlibet lineis ordinatim applicatis ad diametrum circuli, aliæ numero
et quantitate æquales similiter applicentur ad similes partes lineæ maioris vel
minoris de data diametro circuli: termini illarum linearum sunt in
[Translation: If any number of ordinate lines are dropped to the diameter of a circle, that number of other lines, equal in quantity and similarly applied to similar parts of lines greater or less than the given diameter of a circe, then the ends of those lines are on an ]
Sit diameter circuli . lineæ ordinatim applicatæ et . sit etiam
linea maior quam cui applicentur ad angulos rectos et æquales
lineis et Et sint partes lineæ videlicet [???] quotlibet
[???] sed ita fiat ud ad ita ad et ad . Quod etiam fit
si utraque lineæ et similter dividantur et ad utraque æquales similes et
similiter sitas æquales lineæ ordinatim applicentur.
Dico quod puncta et termini linearum et sunt in ellipsi.
Quoniam ex hypothesi
[…] Ergo: per 21, prop: primi Apollonij, puncta et
sunt in ellipsi. quod demonstrare
[Translation: Let the diameter of the circle be , and the ordinate lines and ; also let the line greater than be , to which are applied at right angles and euqal to the lines and in the circle. But thus, as is to so is to and to . Which also happens if the two lines et are similarly divided and to both parts, similar and similarly situated, equal ordinate lines are applied.
I say that the points and , the ends of the lines and , are on an ellipse.
Because from the hypothesis:
Therefore, by Proposition 21 of the first Book of Apollonius, the points et are on an ellipse; which was to be demonstrated.