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

EX his patet, ſimilium Hyperbolarum per diuerſos vertices ſimul adſcri-
ptarum, & quarum eadem ſit regula, aſymptotos eſſe inter ſe paralle-
las, & aſymptoton inſcriptæ ſecare Hyperbolen circumſcriptam.

LEMMA VI. PROP. XXXXVI.

Si in quocunque triangulo ABC ducta ſit quæpiam linea DE
baſi BC parallela, rectangulum ABC ſuperabit ADE rectangu-
lo ſub DB, differentia altitudinum, & ſub aggregato baſium
BC, DE.

PRoducta enim BC, ac ſumpta CF æ-

73[Figure 73] quali ipſi DE, & completis in angulo
A B C parallelogrammis AE, AC, DF.
Conſtat parallelogrammum AC ſuperare
parallelogrammum AE gnomone DCG,
ſed gnomon DCG æquatur parallelogrã-
mis B E, GC, & GC æquatur DC, ſiue
EF, quare AC ſuperat AE parallelogram-
mo DF, hoc eſt rectangulum ABC ſupe-
rat rectangulum ADE, rectangulo DBF; ſed DB eſt differentia altitudinum,
& BF aggregatum baſium BC, DE. Quare patet propoſitum.

THEOR. XXV. PROP. XXXXVII.

Similes Hyperbolæ concentricæ per diuerſos vertices ſimul
adſcriptæ, ſunt inter ſe nunquam coeuntes, ac ſemper propiùs
accedentes, & ad interuallum perueniunt minus quocunque dato
interuallo.

SInt duæ ſimiles Hyperbolæ ABC, DEF per diuerſos vertices B, E ſimul
adſcriptæ, quarum commune centrum ſit G, ſitque Hyperbolæ ABC
tranſuerſum latus BH, rectum BI, Hyperbolæ autem DEF ſit tranſuerſum
EL, rectum EM. Dico primùm has, in infinitum productas, nunquam inter
ſe conuenire.

Producta enim contingente ME, donec vtrinque ſectioni ABC occurrat,
ipſa erit ordinata in ſectione ABC (cum ſint ſectiones ſimul adſcriptæ) ac
ſectio DEF cadet tota infra contingentem KEM; & ſumpto in DEC quoli-
bet puncto D, applicataque per D recta ADN, quæ iunctis regulis HI,

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