Full text: Viviani, Vincenzio: De maximis et minimis, geometrica divinatio

& ſectioni occurrentibus in L,I. Conſtat Hyperbolen ex F ad partes H
omnino incedere intra angulum L F I, & cum ipſa in infinitum extendi
poſſit, cumque in ſecunda figura ſpatium F I B ſit occluſum ad I, & ad
rectam L B nunquam poſſit prouenire, eò quod ipſa L B ponatur Hyper-
bole G F H aſymptotos: in tertia verò cum ſpatium F I N ſit vndique oc-
cluſum, neceſſariò, in vtraque figura, deſcripta Hyperbole G F H in ali-
quo puncto datam ſectionem ſecabit. Sit ergo harum mutua interſectio
punctum M, per quod ductis, vt factum fuit in prima figura, rectis lineis
quæ aſymptotis E D, E C æquidiſtent, ijſdem penitus argumentis, ac in
primo caſu, demonſtrabitur ipſam Hyperbolen in nullo alio puncto quàm
M cum data ſectione A B conuenire. Quare ſi per punctum in angulo, & c. Quod erat demonſtrandum.

232. THEOR. IX. PROP. XIII.

Si in Hyperbola, ſumpta fuerint duo quælibet puncta, à qui-
bus ductæ ſint aſymptotis æquidiſtantes, eiſque occurrentes: re-
cta linea iungens occurſus; lineæ, data puncta iungenti, æqui-
diſtabit.

ESto Hyperbole A B, cuius aſymptoti C D, C E, ſumptaque ſint in
ſectione duo quælibet puncta A, B, à quibus ductæ ſint A F, B G,
aſymptotis æquidiſtantes. Dico iunctas A B, F G, eſſe inter ſe paralle-
las.

Nam vtrinque producta A B vſque-
ad aſymptotos in D, & E. Erit in pri- ma figura, B D æqualis A E: in ſecun-
da verò, cum ſit A D æqualis B E, ad-
dita communi A B, erit item D B æ-
qualis ipſi A E. Sed in triangulis D B
G, E A F, anguli ad D, B, æquantur
angulis ad A, & E, vterque vtrique,
ob paralellas D G, A F, & B G, E F; quare triangula D B G, A E F ſunt ſimi-
lia inter ſe, ac propterea vt D B ad B
G, ita A E ad E F, ſed antecedentes
D B, A E ſunt ęquales, vt modò oſten-
dimus, ergo, & conſequentes B G, E F,
æquales erunt, at ſunt quoque inter ſe
parallelæ, quare, & F G ipſi A B ęqui-
diſtabit. Quod, & c.

232.1.

0198-01
8. ſec.
conic.

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