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[Commentary:
The reference to Apollonius is to pages 5 and 6 of Commandino's Apollonii Pergaei conicorum libri quattuor
(Apollonius . There are also references at the bottom of the page to Viète and Cardano.
The reference to Viète is to Apollonius Gallus, Appendix II, Problem 5 (Viete 1600a, Appendix II, Prob .
V. Dato triangulo, invenire punctum, a quo ad apices dati trianguli actæ tres lineæ rectæ imperatam teneant
Given a triangle, find a point from which there may be drawn three straight lines to the vertices of the given triangle, keeping a fixed ratio.
The reference to Cardano is to his Opus novum de proportionibus. The relevant Propositions are Prop (though mistakenly described in the Opus novum as 144) and Prop .
Propositio centesimaquadragesimaquarta
Sint lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes quam media inter totam & adiectam, ad adiectam erit punctus concursus a puncto extrema lineæ adiectæ distans per lineam mediam. Quod si ab extremo alicuius lineæ æqualis mediæ seu peripheria circuli cuius semidiameter sit media linea duæ lineæ ad prædicta puncta producantur, ipsæ erunt in proportione mediæ ad adiectam.
Hæc propositio est admirabilis:
Propositio centesimasexagesima
Proposita linea tribusque in ea signis punctum invenire, ex quo ductæ tres lineæ sint in proportionibus ]
The reference to Viète is to Apollonius Gallus, Appendix II, Problem 5 (Viete 1600a, Appendix II, Prob .
V. Dato triangulo, invenire punctum, a quo ad apices dati trianguli actæ tres lineæ rectæ imperatam teneant
Given a triangle, find a point from which there may be drawn three straight lines to the vertices of the given triangle, keeping a fixed ratio.
The reference to Cardano is to his Opus novum de proportionibus. The relevant Propositions are Prop (though mistakenly described in the Opus novum as 144) and Prop .
Propositio centesimaquadragesimaquarta
Sint lineæ datæ alia linea adiungatur, ab extremitatibus autem prioris lineæ duæ rectæ in unum punctum concurrant proportionem habentes quam media inter totam & adiectam, ad adiectam erit punctus concursus a puncto extrema lineæ adiectæ distans per lineam mediam. Quod si ab extremo alicuius lineæ æqualis mediæ seu peripheria circuli cuius semidiameter sit media linea duæ lineæ ad prædicta puncta producantur, ipsæ erunt in proportione mediæ ad adiectam.
Hæc propositio est admirabilis:
Propositio centesimasexagesima
Proposita linea tribusque in ea signis punctum invenire, ex quo ductæ tres lineæ sint in proportionibus ]
5. Appolonius. pag. 5.
[Translation: Apollonius, pages 5, ]
[Translation: Apollonius, pages 5, ]
Quæsitum:
ubicunque signatur in periferia punctum
erit; , : , : vel ,
[Translation: Sought:
Wherever a point is placed on the circumference, then or .
ubicunque signatur in periferia punctum
erit; , : , : vel ,
[Translation: Sought:
Wherever a point is placed on the circumference, then or .
Sint data puncta , ,
Data ratio. . .
producatur, , versus,
[Translation: Let the given points be , , the given ratio . Let be produced towards .
Data ratio. . .
producatur, , versus,
[Translation: Let the given points be , , the given ratio . Let be produced towards .
Dico
[Translation: I say ]
[Translation: I say ]
Inde: maior, quam
minor, quam
fiat
fiat periferia
sumatur quovis puncta
Ducantur: , ,
[Translation: Whence, is greater than , less than . Let , let be the circumference, taking any point . Let there be drawn , , .
minor, quam
fiat
fiat periferia
sumatur quovis puncta
Ducantur: , ,
[Translation: Whence, is greater than , less than . Let , let be the circumference, taking any point . Let there be drawn , , .
* Ducantur , parallela, .
ubicunque signatur in periferia punctum
erit; , : , : vel ,
[Translation: Taking parallel to , wherever the point is placed on the circumference, then or .
ubicunque signatur in periferia punctum
erit; , : , : vel ,
[Translation: Taking parallel to , wherever the point is placed on the circumference, then or .
Corollaria.
Hinc a tribus punctis sive sint in recta
vel non; possunt duci tres lineæ ad unum
punctum, ut s et erunt in data
[Translation: Corollary
Hence from three points, whether in a straight line or not, it is possible to draw three lines to a single point, and they will be in the given ratio.
Hinc a tribus punctis sive sint in recta
vel non; possunt duci tres lineæ ad unum
punctum, ut s et erunt in data
[Translation: Corollary
Hence from three points, whether in a straight line or not, it is possible to draw three lines to a single point, and they will be in the given ratio.
vide vertam
in Apolonio gallo
et card: de prop. pag. 145.
[Translation: see over, in Apollonius Gallus, and Cardano, De proportionibus, pages 145, 162.
in Apolonio gallo
et card: de prop. pag. 145.
[Translation: see over, in Apollonius Gallus, and Cardano, De proportionibus, pages 145, 162.