Viviani, Vincenzo
,
De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei
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<
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">Sumpto enim in data recta A B quocunque alio puncto H, vel in ipſius
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parte producta vltra B, vt in prima figura, vel in ipſa A B, vt in ſecunda,
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& </
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">ex H ducta H I perpendiculari ad A B, ſecante diagonalem D B in I,
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ductaque A I ſecante circuli peripheriam in L, iunctiſque G L, G I: </
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<
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angulus A L G rectus, atque externus trianguli L I G; </
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<
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">quare internus L I
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G acutus erit, ac ideo recta I M, quæ ex I erigitur perpendicularis ad I A,
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hoc eſt, quæ ipſi L G æquidiſtat, ſecabit A B vltra punctum G, vt in M, ac
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ideo erit A G minor A M. </
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<
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">Et cum in triangulo rectangulo A I M, ſit vt A
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H ad H I, ita H I ad H M, ſitque H I æqualis H B, erit A H ad H B, vt
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H B ad H M, ergo A M eſt aggregatum extremarum proportionalium poſt
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partes A H, H B, ſed eſt A G minor A M, vt modò oſtendimus: </
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<
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xml:space
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">ergo ag-
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gregatum A G minus eſt aggregato A M: </
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<
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xml:space
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">& </
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<
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xml:space
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">hoc ſemper vbicunque aſſum-
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ptum fuerit punctum H extra C: </
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<
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xml:space
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">ergo aggregatum A G minus eſt aggrega-
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to A M: </
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<
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xml:space
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<
s
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xml:space
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">hoc ſemper vbicunque aſſumptum fuerit punctum H extra C:
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<
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xml:space
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">quare A G eſt _MINIMVM_ aggregatum quæſitum; </
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<
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">recta A B ſecta eſt in
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C, vt imperatum fuit. </
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">SI quæratur iuxta quam rationem repertum punctum C diuidat datam A
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B; </
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">id ex ipſa Theorematis conſtructione elicietur. </
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">Nam cum triangu-
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la D A B, B E F ſint ſimilia inter ſe, erit B D ad D A, ſiue diameter qua-
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drati ad latus, vt B F ad F E, vel ad F A, & </
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<
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ad C F, ſitque B C æqualis C E (cum & </
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<
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xml:space
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">B A æqualis ſit A D) erit etiam
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C E ſiue C B æqualis C F. </
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<
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">Quare ſi data recta B A diuidatur, ita vt pars
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B F ad reliquam partem F A, ſit vt diameter cuiuſdam quadrati ad eius la-
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tus, & </
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<
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">maior pars B F ſecetur bifariam in C, hoc ipſum punctum erit quæ-
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ſitum.</
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<
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tæ, erit A D, ſiue D E latus quadrati, & </
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diametri ſuper latus, ſed eſt A C ad C B, vt D E ad E B: </
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<
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">ergo quæſitum
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punctum C ſecat datam rectam A B, ita vt maior pars A C ad minorem C
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B, ſit vt latus cuiuſdam quadrati ad exceſſum diametri ſuper latus, quæ ra-
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tio, vt iam conſtat, cadit inter terminos incommenſurabiles.</
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