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

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5131 curuis AB, DE; diametris BG, EH; & ſemi-applicatis AG, DH compre-
henſis, nempe trilineum ABG ad CBG, eſſe vt baſis AG ad GC, ſib æqua-
lem; ac propterea diametrum BG Parabolen ABC bifariam ſecare; & vnũ-
quodque trilineorum eſſe ſemi-Parabolen; & ſemi-Parabolen ABG ad ſe-
mi-Parabolen DEH æqualis altitudinis, eſſe vt baſis AG ad baſim DH, &
integram ABC ad dimidiam DEH eſſe vt baſis AC ad ſemi-baſim DH.

THEOR. VII. PROP. XV.

Parabolæ æqualium baſium ſunt inter ſe vt altitudines.

SInt primò duæ Parabolæ ABC, ABC ſuper eandem baſim AC, & circa
eandem diametrum BE. Dico has eſſe inter ſe vt earum altitudines FA,
GA; & quod de ſemi-Parabolis EBC, EDC demonſtrabitur, idem inſe-
quetur de duplis.

Si enim non eſt vt FA

28[Figure 28] ad AG, ita ſemi-Parabo-
le EBC ad EDC, erit al-
tera ipſarum minor quàm
ſit opus ad hoc vt ſint pro-
portionales altitudinibus
FA, AG, ſitque, ſi poſſi-
bile eſt, minor EBC de-
fectu R, & bifariam ſecta
EC in H, & iterum EH
bifariam in I, & c. circum-
ſcribatur, vt in pręceden-
ti, trilineo ſemi-Parabo-
læ ECB figura BLCE ex
parallelogrammis ęque
altis conſtans, cuius ex-
ceſſus ſupra ſemi-Parabolen ſit minor R, ita vt ipſa circumſcripta figura
BLCE ad ſemi-Parabolen EDC adhuc minorem habeat rationem quàm
altitudo FA ad AG; quo facto, ſemi-Parabolæ quoque EDC per æquidi-
ſtantium diametro interſectionem altera circumſcribatur figura DMNCE
ex totidem Parallelogrammis æque altis, & c. Et cum ſir ob Parabolas, re-
cta BE ad OI, vt rectangulum AEC ad AIC, vel vt DE ad PI, erit permu-
tando BE ad ED, vel parallelogrammum BI ad DI, vt OI ad IP, vel vt pa-
rallelogrammum OH, ad PH, & ſic de reliquis circumſcriptæ BL CE, ad re-
liqua circumſcriptæ DMCE, ſingula ſingulis, quare vniuerſa circumſcripta
ALCE ad vniuerſam DMCE, erit vt vnum parallelogrammum BI ad vnum
DI, vel vt baſis BE ad ED, vel vt FA ad AG, ſed FA ad AG habet maiorem
rationem quàm circumſcripta ALCE ad ſemi-Parabolen EDC; quare cir-
cumſcripta ALCE ad circumſcriptam DMCE, habebit maiorem rationem
quàm ad ſemi-Parabolen EDC, vnde circumſcripta DMCE minor erit in-
ſcripta ſemi-Parabola EDC; totum parte, quod eſt abſurdum. Non datur
ergo inter has ſemi-Parabolas minor quàm ſit opus, ad hoc vt ipſæ ſint

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Scan	Original
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