Newton, Isaac - Philosophia naturalis principia mathematica

Si rectæ tres tangant quamcunque Coniſectionem, quarum duæ pa
rallelæ ſint ac dentur poſitione; dico quod Sectionis ſemidia
meter hiſce duabus parallela, ſit media proportionalis inter ha
rum ſegmenta, punctis contactuum & tangenti tertiæ inter
jecta.

Sunto AF, GBpa

58[Figure 58]
rallelæ duæ Coniſec
tionem ADBtan
gentes in A& B; EF
recta tertia Coniſec
tionem tangens in I,
& occurrens prioribus
tangentibus in F& G;
ſitque CDſemidiame
ter Figuræ tangenti
bus parallela: Dico
quod AF, CD, BG
ſunt continue proportionales.

Nam ſi diametri conjugatæ AB, DMtangenti FGoccurrant
in E& H,ſeque mutuo ſecent in C,& compleatur parallelogram
mum IKCL; erit, ex natura Sectionum Conicarum, ut ECad
CAita CAad CL,& ita diviſim EC-CAad CA-CL,ſeu
EAad AL,& compoſite EAad EA+ALſeu ELut ECad
EC+CAſeu EB; adeoque (ob ſimilitudinem triangulorum EAF,
ELI, ECH, EBG) AFad LIut CHad BG.Eſt itidem,
ex natura Sectionum Conicarum, LI(ſeu CK) ad CDut CDad
CH; atque, adeo ex æquo perturbate, AFad CDut CDad BG.
Q.E.D.

Corol.1. Hinc ſi tangentes duæ FG, PQtangentibus parallelis
AF, BGoccurrant in F& G, P& Q,ſeque mutuo ſecent in O;
erit (ex æquo perturbate) AFad BQut APad BG,& diviſim
ut FPad GQ,atque adeo ut FOad OG.

Corol.2. Unde etiam rectæ duæ PG, FQper puncta P& G,
F& Qductæ, concurrent ad rectam ACBper centrum Figuræ &
puncta contactuum A, Btranſeuntem.

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