Viviani, Vincenzo - De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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144120 ptarum, ſed eſſe MAXIMAS quoque earum, quæ ad partes verticum in-
ſcribuntur; id ſequenti Theoremate, in angulo, & qualibet coni-ſectione,
vel circulo conſequetur, ſimulque dabitur Methodus ipſis inſcribendi ſimiles
Ellipſes, quæ ſucceſsiuè ſe mutuò, & anguli, vel ſectionum latera contin-
gant.

THEOR. XXXVI. PROP. LXXVI.

Ellipſes inſcriptæ eidem angulo, vel Parabolæ, vel Hyperbolę,
aut portioni Ellipticæ, vel circulari, quæ non excedat Ellipſis, vel
circuli dimidium, ſe mutuò, & anguli latera, vel ſectionem, vel
circulum contingentes, & quarum diagonales menſalium, quibus
inſcribuntur, inter ſe æquidiſtent, ſunt ſimiles, & quæ propior eſt
vertici, minor eſt remotiori.

SIt ABC, vel angulusrectilineus, vt in prima figura, vel Parabolæ, vel
Hyperbolæ, aut portio non maior ſemi-circuli, vel ſemi-Ellipſis dimi-
dio, vt in ſecunda, cuius vertex B, diameter BD, & circa ipſius ſegmentum
DE, inter applicatas AC, IF, ducta diagonali AF, ſecan@ diametrum ED
in K, & applicata per K recta GKH, per extrema G, E, H, D, 11Coroll.
57. h. Ellipſis GEHD, quæ per Scholium 62. huius, menſali AIFC, hoc eſt dato
angulo, vel ſectioni crit inſcripta. Et per I ducta IL parallela diagonali AF,
diametrum ſecan@ in O, conſimili conſtructione, ac ſupra, deſcribatur in
menſali INLF Ellipſis PMQE. Dico primùm has Ellipſes inter ſe ſimiles
eſſe.

Nam, in prima figura, proportio rectanguli GKH, ad rectangulum AKF,
componitur ex ratione GK ad KA, ſiue (per triangulorum ſimilitudinem)
PO ad OI, & ex ratione HK ad KF, ſiue QO ad OL, ſed etiam proportio re-
ctanguli POQ, IOL, ex ijſdem rationibus componitur, quare in triangulo,
rectangulum GKH ad AKF, eſt vt rectangulum POQ ad IOL.

Iam, in ſecunda figura, eadem ratione, vt in 64. huius, oſtendetur huiuſ-
modi Ellipſium centra cadere infra K, O, nempe in R, S, per quæ ſi appli-
centur RT, SV, ipſæ, diagonales ſecabunt in T, V; cum ſit DR ęqua-
lis RE, erit AT æqualis TF, ob parallelas; item IV æqualis VL; ſuntque
AF, IL æquidiſtanter ductæ in ſectione, vel circulo, quare iuncta TV erit
earundem æquidiſtantium diameter, quæ producta ad aliud punctum, præ-
ter B, ſectioni occurret, vt in Z, eritque ſectionis, vel circuli diameter: 2228. ſec.
conic. ergo ex verticibus B, Z, agantur BX, ZX ordinatim applicatis GH, AF æ-
quidiſtantes, hæ ſectionem contingent, & ſimul conuenient in X; 3317. pri-
mi conic.4459. h. rectangulum GKH ad rectangulum AKF, vt quadratum BX ad quadratum
ZX; item erit rectangulum POQ ad IOL, vt idem quadratum BX ad 5517. tertij
conic. ZX, quapropter rectangulum GKH ad AKF, erit vt rectangulum POQ ad
IOL, quod etiam ſuperius in prima figura demonſtratum fuit. Itaque, cum
ſit in vtraque, rectangulum GKH ad AKF, vt rectangulum POQ ad IOL, &

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