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

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5535

THEOR. IX. PROP. XVII.

Parabole ſeſquitertia eſt trianguli eandem ipſi baſim, & ean-
dem altitudinem habentis.

REpetito præcedenti diagrammate, dico Parabolen AB8 ſeſquitertiam
eſſe inſcripti trianguli AB8.

Nam ducta G9 parallela ad AC deſcribatur ſemi Parabole 9, 8, cuius dia-
meter ſit 9C, & ſemi-applicata ſit C8, æqualis baſi AC Parabolæ
AGC. Et cum ſit ſemi-Parabole ABC æqualis ſemi-Parabolæ CB8, & 11Coroll.
prop. 14. h. Parabole AGC æqualis ſemi-Parabolæ C98, ſitque C98 dimidium 22Coroll.
prop. 14. h. (nam eſt C9 dimidium CB & c.) erit Parabole AGC dimidium ſemi-Parabo-
3315. h. læ ABC, ſiue æqualis trilineo AHBCGA, ac etiam trilineo AEBH; 4416. h. totum triangulum AEC ſeſqui alterum erit ſemi-Parabolæ ABC, ſiuc erit
vt 6 ad 4, ſed ad triangulum ABC eſt vt 6 ad 3, cum ſit EC dupla CB, vnde
ſemi-Parabole ABC ad triangulum ABC, hoc eſt dupla ad duplum, nempe
Parabole AB8 ad inſcriptum triangulum AB8, erit vt 4 ad 3. Quod demon-
ſtrare oportebat.

MONITVM.

VT hoc loco, ex aduerſo indirectæ Antiquorum viæ per duplicem
poſitionem, luce clarius pateat quantum facilitatis, breuitatis,
atquæ euidentiæ naſciſcatur è noua, directaque methodo (rectè
tamen cautèque vſurpata) acutiſsimi Geometræ Caualerĳ,
per indiuiſibilium doctrinam, nobis amiciſsimam, ex hac alteram accipe
eiuſdem theorematis demonſtr ationem, conſimili arte cōp@catam, ac in præ-
cedenti.

THEOR. X. PROP. XVIII.

Parabole ſeſquitertia eſt trianguli eandem ipſi baſim, & ean-
dem altitudinem habentis.

SIt Parabole ABC, cuius diameter BD, baſis AC: dico ipſam ſeſquiter-
tiam eſſe inſcripti trianguli ABC.

Bifariam enim ſecta AD in G, per quod ducta GF parallela ad DB, & per
F, FH parallela ad AD, ac deſcriptis, vt in præcedenti figura Parabola
AED, & portione Parabolæ HCD, cuius diameter ſit HD, & ſemi-applica-
ta ſit DC ducatur in tota ABC quælibet applicata NI. diametrum ſecans in
M, eritque NM æqualis ML, & ſic de quibuslibet alijs applicatis ipſi AC æ-
quidiſtantibus, quare omnes ſimul in portione ABD, omnibus ſimul in por-
tione DBC æquales erunt, ſiue portio ABD æqualis DBC, nempè vtraque
erit ſemi-Parabole, & eadem ratione oſtendetur DHC ſemi-Parabolen eſſe.

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