4929
COROLL. I.
HInc patet, quod ſi recta linea in Parabola vtcunque applicata ex
11Vni-
uerſalius
quàm in
4. prop.
exerc. 6.
Caual. vtraque parte ſectioni occurrens cum diametro, vel intra, vel extra
ſectionem conueniat, atque ex ipſius terminis cum ſectione, ad diametrum
ducantur ordinatæ, erunt ab his abſciſſa diametri ſegmenta ex vertice ſum-
pta, extremæ, & abſciſſum ab applicata, erit media trium continuè propor-
tionalium. Demonſtratum eſt enim in figuris Theorematis quando AH dia-
metrum ſecat in M, & ſectionem in A, H, quod ordinatim applicatis AF,
HI, eſt FB ad BM, vt BM ad BI.
11Vni-
uerſalius
quàm in
4. prop.
exerc. 6.
Caual. vtraque parte ſectioni occurrens cum diametro, vel intra, vel extra
ſectionem conueniat, atque ex ipſius terminis cum ſectione, ad diametrum
ducantur ordinatæ, erunt ab his abſciſſa diametri ſegmenta ex vertice ſum-
pta, extremæ, & abſciſſum ab applicata, erit media trium continuè propor-
tionalium. Demonſtratum eſt enim in figuris Theorematis quando AH dia-
metrum ſecat in M, & ſectionem in A, H, quod ordinatim applicatis AF,
HI, eſt FB ad BM, vt BM ad BI.
COROLL. II.
EX quo etiam elicitur, quod ſi in Parabola ABC ducta AH diametrum
ſecans in M producatur vſque ad occurſum cum contingente ex verti-
ce B in S, ſemper rectangulum ſub ſegmentis AS, & SH, inter ſectionem, &
contingentem interceptis æqua†@ quadrato ſegmenti SM inter contingẽtem,
ac diametrum intercepti. Nam cum ſit vt FB ad BM, ita BM ad BI erit quo-
que ob parallelas, AS ad SM, vt SM ad SH, quare rectangulum ASH æqua-
bitur quadrato SM.
ſecans in M producatur vſque ad occurſum cum contingente ex verti-
ce B in S, ſemper rectangulum ſub ſegmentis AS, & SH, inter ſectionem, &
contingentem interceptis æqua†@ quadrato ſegmenti SM inter contingẽtem,
ac diametrum intercepti. Nam cum ſit vt FB ad BM, ita BM ad BI erit quo-
que ob parallelas, AS ad SM, vt SM ad SH, quare rectangulum ASH æqua-
bitur quadrato SM.
COROLL. III.
HInc etiam eſt, quod, ſi ijſdem poſitis,
interior Parabole ADG habuerit ver-
26[Figure 26]
ticẽ in D puncto medio rectæ AB, ipſa quo-
que tranſibit per F medium punctum baſis
AC, & quæcunque educta ex contactu A,
qualis eſt AH, bifariam ſecabitur in O ab in-
terna ſectione; quare ſi ex O ducatur OLM
diametro BF æquidiſtans, ipſa erit diameter
portionis ALH, & AH vna applicatarum,
AO verò ſemi-applicata. Cumque ſit AP
contingens ABC in A, erit OL in trilineo
mixto ADFB, æqualis LM in trilineo mixto
ALBP, & ſic de omnibus vbicunque inter-
ceptis in ijſdem trilineis.
interior Parabole ADG habuerit ver-
que tranſibit per F medium punctum baſis
AC, & quæcunque educta ex contactu A,
qualis eſt AH, bifariam ſecabitur in O ab in-
terna ſectione; quare ſi ex O ducatur OLM
diametro BF æquidiſtans, ipſa erit diameter
portionis ALH, & AH vna applicatarum,
AO verò ſemi-applicata. Cumque ſit AP
contingens ABC in A, erit OL in trilineo
mixto ADFB, æqualis LM in trilineo mixto
ALBP, & ſic de omnibus vbicunque inter-
ceptis in ijſdem trilineis.
THEOR. VI. PROP. XIV.
Parabolæ æqualium altitudinum inter ſe ſunt vt baſes.
SInt duæ Parabolæ ABC, DEF æqualium altitudinum, hoc eſt concipian-
tur diſpoſitæ inter eaſdem parallelas BE, AF: dico eſſe vt baſis AC
vnius, ad baſim DF alterius, ita Parabole ABC ad Parabolen DEF.
tur diſpoſitæ inter eaſdem parallelas BE, AF: dico eſſe vt baſis AC
vnius, ad baſim DF alterius, ita Parabole ABC ad Parabolen DEF.
zoom in
zoom out
zoom area
full page
page width
set mark
remove mark
get reference
digilib