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

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11288

THEOR. XXVII. PROP. XXXXIX.

Si binæ Parabolæ, aut binæ concentricæ Hyperbolæ fuerint per
diuerſos vertices ſimul adſcriptæ, ipſæ, vel ad neutram partem ſe
vnquam ſecabunt, vel ſi ad alteram partem occurrant, occurrent
quoque ad aliam, punctaque occurſuum erunt extrema eiuſdem
communis applicatæ, ac in ijſdem occurſibus ſe mutuò ſecabunt.

SInt duæ Parabolæ ABC, DEF, vel duæ concentricæ Hyperbolæ per di-
uerſos vertices B, E ſimul adſcriptæ. Dico primùm, ſi huiuſmodi ſectio-
nes ad alteram partium, vt ad A nunquam conueniunt, ad aliam quoque C
nunquam conuenire.

Nam ſumpto in ſectione DEF, ad

77[Figure 77] partes C, quocunque puncto F, per ip-
sũ ordinatim applicetur recta CFGDA,
quæ vtrinque producta, vtrique ſectio-
ni occurret (cum ipſæ ob Hypoteſim,
ſint ſectiones in infinitam diſtantiam
abeuntes ad inferiores partes) cumque
in ſectione ABC ſit ſemi-applicata AG,
æqualis GC, & in ſectione DEF, ſemi-
applicata DG, æqualis GF, ſitque ante-
cedens AG maior antecedente DG (cum ad partes A nunquam cõueniant)
erit etiam conſequens GC, maior conſequenti GF, quare punctum F, ſe-
ctionis DEF cadit intra ABC, & ſic de reliquis. Quod primò, & c.

Si verò ſectiones ad alteras partes, veluti ad A, & D conueniant vt in H.
Dico ipſas ad alias quoque partes ſimul occurrere ad extrema puncta eiuſ-
de m applicatæ.

Nam, ducta per H communi applicata HI, ipſa producatur ſecans ſectio-
nem BC in L; & EF in M. Erit in ſectione ABC ſemi- applicata HI æqualis
IL, & in ſectione DEF eadem HI æqualis IM; ergo IL, & IM æquales, ideo-
que ſectionum puncta L, M in vnum conueniunt; Quare cum ſectiones
ABC, DEF non per vertices ſimul adſcriptæ ad alteram partem occurrunt,
occurrent quoque ad aliam, punctaque occurſuum erunt extrema eiuſdem
communis applicatæ.

Quod autem in occurſibus H, & L ſe mutuò ſecent, ſic demonſtratur. Nam
ſi huiuſmodi ſectiones tangerent ſe mutuò in occurſu H, ita vt ſectionis, ver-
bi gratia, ED partes HO, HN cadant totæ intra ſectionem ABC: ſumpto in
altera ipſarum partium, vt puta OH, quolibet puncto D, & per ipſum ducta
communi applicata ADGFG vtranque ſectionem ſecane, erunt in ſectione
ABC rectæ AG, GC æquales, & in ſectione DEF rectæ DG, GF itẽ ę; qua-
les, ſed eſt AG maior GD cum ponatur peripheria OH cadere intra BH, vn-
de & CG maior erit ipſa FG, hoc eſt punctum F cadet intra. Idemque de-
monſtrabitur de quolibet alio extremo puncto cuiuſcunque applicatæ inter
O, & N, tùm ſupra, tùm infra occurſum L: quare ſectio DEF

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