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[Commentary:
This page refers to Propositions 48 and 49 of Book III of Apollonius, as edited by Commandino Conicorum libri quattuor
(Apollonius .
III.48 With the same things being so, it must be shown that the straight lines drawn from the point of contact to the points produced by the application make equal angles with the
]
III.48 With the same things being so, it must be shown that the straight lines drawn from the point of contact to the points produced by the application make equal angles with the
]
Sit ellipsis :
cuius axis
centroides puncta , .
diametroides, recta
centrum, .
circulus circa axim, .
circulus circa diametroides, .
recta contingens ellipsin in
puncto , fit .
perpendicularis a centroide
ad illam fit
per 49.3 conicorum est angulus
rectus
ergo punctum in
[Translation: Let there be an ellipse with axis , centroids at points and , diametroid the line , centre . The circle about the axis is ; the circle about the diametroid is ; the line touching the ellipse at the point is . Perpendicular to it from the centroid , construct . By Proposition III.49 of the Conics, is a right angle. Therefore, the point is on the ]
cuius axis
centroides puncta , .
diametroides, recta
centrum, .
circulus circa axim, .
circulus circa diametroides, .
recta contingens ellipsin in
puncto , fit .
perpendicularis a centroide
ad illam fit
per 49.3 conicorum est angulus
rectus
ergo punctum in
[Translation: Let there be an ellipse with axis , centroids at points and , diametroid the line , centre . The circle about the axis is ; the circle about the diametroid is ; the line touching the ellipse at the point is . Perpendicular to it from the centroid , construct . By Proposition III.49 of the Conics, is a right angle. Therefore, the point is on the ]
hinc sequitur
Si producatur ad periferium in
et ducatur parallela ad
continget etiam
[Translation: If is produced to the perpiphery at , and is taken paralle to , it will also touch the ellipse.
Si producatur ad periferium in
et ducatur parallela ad
continget etiam
[Translation: If is produced to the perpiphery at , and is taken paralle to , it will also touch the ellipse.
Si puncta et in periferia
connectantur
linea transibit per alterum
centroides
[Translation: If the points and in the periphery are joined, the line will pass through the other centroid, .
connectantur
linea transibit per alterum
centroides
[Translation: If the points and in the periphery are joined, the line will pass through the other centroid, .
Hinc. conclusio
Si circa axim ellipseos describatur circulus
et in circulo inscribatur parallelogrammum
ita ut duo latera transeant per centroides:
reliqua duo contingent ellipsin.
et si duo latera contingent ellipsin; reliqua
duo transibunt per centroides.
Ita etam:
Si circa axim Hyperboles &
[Translation: Hence, the conclusion.
If around the axis of an ellipse there is described a circle, and in the circle there is inscribed a parallelogram so that two sides pass thorugh the centroids, the other two are tangents to the ellipse. And if two sides are tangents to the ellipse, the other two will pass throug the centroids.
Thus also: if around the axis of a hyperbola, ]
Si circa axim ellipseos describatur circulus
et in circulo inscribatur parallelogrammum
ita ut duo latera transeant per centroides:
reliqua duo contingent ellipsin.
et si duo latera contingent ellipsin; reliqua
duo transibunt per centroides.
Ita etam:
Si circa axim Hyperboles &
[Translation: Hence, the conclusion.
If around the axis of an ellipse there is described a circle, and in the circle there is inscribed a parallelogram so that two sides pass thorugh the centroids, the other two are tangents to the ellipse. And if two sides are tangents to the ellipse, the other two will pass throug the centroids.
Thus also: if around the axis of a hyperbola, ]
Alia conclusiones
iisdem positis.
per 48.3. conicorum. et
faciunt æquale angulos ad
contingentem.
Si et agantur
parallelæ ad contingentes:
puncta et sunt in peri-
feria cuius diameter
[Translation: Other conclusions form the same assumptions.
By Proposition III.48 of the Cinics, and make equal angles to the tangent.
If and are taken parallel to the tangents, the points and are on the circumference whose diamter is .
iisdem positis.
per 48.3. conicorum. et
faciunt æquale angulos ad
contingentem.
Si et agantur
parallelæ ad contingentes:
puncta et sunt in peri-
feria cuius diameter
[Translation: Other conclusions form the same assumptions.
By Proposition III.48 of the Cinics, and make equal angles to the tangent.
If and are taken parallel to the tangents, the points and are on the circumference whose diamter is .
Conveniat cum in .
et cum in .
Dico quod:
nam:
[Translation: Let meet with at , and with at . I say that:
for the ]
et cum in .
Dico quod:
nam:
[Translation: Let meet with at , and with at . I say that:
for the ]
Dico
[Translation: I also ]
[Translation: I also ]
Dico
[Translation: I also ]
[Translation: I also ]