Viviani, Vincenzo
,
De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei
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<
s
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<
s
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xml:space
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">cum ſit EH æqualis HT, eſſet etiam EH æqualis ST, vnde
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eius fegmentum EB mins eſſet diſtantia ST. </
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<
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xml:space
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">Si ſecundùm; </
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<
s
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echoid-s2846
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xml:space
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">cum ſit HT æ-
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qualis HE omnino ST maior eſſet eadem HE, & </
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<
s
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xml:space
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">eò maior ipſius ſegmento
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BE. </
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<
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xml:space
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">Si tertiùm; </
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<
s
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">vt in hac ipſa figura, in qua centrum H interioris cadit inter
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S, & </
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<
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">G; </
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<
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">cum ſit HE æqualis HT, & </
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<
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xml:space
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">ablata HB maior ablata HS (nam eſt to-
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ta SB ſecta bifariam in G) erit reliqua BE maior reliqua ST. </
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<
s
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">Quapropter in
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hoc caſu, in quo centrum H interioris cadit vltra centrum G exterioris, vbi-
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cunq; </
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<
s
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xml:space
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">ſit eius incidentia, demonſtratum eſt ſemper diſtantiam verticum B, E,
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minorem eſſe ipſa ST diſtantia inter ſuperiora extrema tranſuerſorum late-
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rum ET, BS. </
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<
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">Quod memento.</
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<
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</
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<
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<
s
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xml:space
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">Ampliùs ſint harum ſectionum recta latera BV, EX, & </
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">regulæ TV, TX.
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</
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<
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">Patet ob ſectionum ſimilitudinem, vt SB ad BV, ita eſſe TE ad EX, ſed an-
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guli ad B, E, ſunt æquales (cum ſectiones ſint ſimul adſcriptæ, &</
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<
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">c.) </
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<
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">quare
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in triangulis SBV, TEX, anguli ad S, T, æquales erunt, ac ideò regulæ SV,
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TX inter ſe æquidiſtabunt. </
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<
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">Cumque ſit ST maior BE, ſi dematur SK ipſi BE
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æqualis, ducaturque SY parallela ad EX, & </
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<
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">abſcindatur EL æqualis SY, ac
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iungantur KY, BL: </
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<
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">erunt in triangulis KSY, BEL, in quibus latera circum
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æquales angulos S, E, ſunt æqualia, vtrunque vtrique, anguli quoq; </
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<
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">SKY,
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EBL æquales; </
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<
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">ſuntque alterni, quare KY, & </
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">BL inter ſe ęquidiſtant, ſed KY
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ſecat TX, vnde & </
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<
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xml:space
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">BL producta ſecabit TX, vt in N: </
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<
s
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xml:space
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">Iam per N ordinatim
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ductis æquidiſtans applicetur NQDP, regulam SV, ſecans in Z, communem
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diametrum in Q, exteriorem ſectionem CBA in P, & </
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<
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">interiorem in D: </
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<
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">Cum
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in triangulo BQN ſit EL ipſi QN parallela, erit BQ ad QN, vt BE ad EL,
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& </
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<
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">permutando QB ad BE, vt QN ad EL, ſiue ad SY, vel ZN, & </
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<
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uerſionem rationis BQ ad QE, vt NQ ad QZ, vnde rectangulum BQZ
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">Coroll.
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1. huius.</
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quadratum applicatæ PQ æquale eſt rectangulo EQN, ſiue quadrato appli-
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catæ DQ ex quo puncta P, D in vnum conueniunt, hoc eſt interior Hyper-
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bole FED exteriori ABC occurrit in D; </
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<
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">eademque ratione oſtendetur ipſas
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ſimul occurrere in F, altero extremo eiuſdem applicatæ DQF, quare in ipſis
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occurſibus ſe mutuò ſecant: </
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<
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">quoniam ſi exempli gratia, huiuſmodi ſectiones
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non ſe ſecarent, ſed contigerent in D, contingerent ſe quoque in F, vt fa-
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cillimum eſt demonſtrare, ſed Hyperbole ED ſecat omnino rectam GI extra
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ſectionem BA, vti ſuperius oſtendimus, quare hæc inter ſectio alio in loco
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cadet quàm in D, pariterque ad alteram partem ſectio EF ſecabit BC in alio
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puncto, præter in F: </
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<
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">Quapropter coni-ſectio coni-ſectionem contingeret in
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duobus punctis D, F, & </
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<
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">in alijs duobus punctis ſibi ipſis occurrerent, quod
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eſt impoſſibile: </
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<
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">vnde in ipſis occurſibus D, F ſe mutuò ſecant; </
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<
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conic.</
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abundanti oſtendere propoſuimus.</
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<
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">Si verò centrum H interioris idem fuerit cum G centro exterioris, etiam
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aſymptotos GI eadem erit cum aſymptoto HM, cum angulus IGB æqualis,
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vel idem ſit cum angulo MHE; </
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<
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40. huius.</
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larum aſymptoti communes ſunt. </
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<
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<
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</
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<
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</
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<
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">huius, vel quàm breuiſſimè ex propoſ. </
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<
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<
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<
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<
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<
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">Si autem centrum H interioris DEF cadat infra G centrum exterioris
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ABC, vt in ſecunda figura, per verticem E contingenter applicata CEA;
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</
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<
s
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">cum HM ſit intra angulum IGO ab aſymptotis factum, ac ipſi GI </
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