Cardano, Girolamo
,
De subtilitate
,
1663
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ſint directæ, & ſint trigoni per axem in eo
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dem plano ABC, & ADE, & puncta præ
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ter verticem ſignata in vno G, in altero H,
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& planum K, per ambo puncta ad perpendi
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culum ſuper ambos trigonos ductum, &
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clarum eſt quòd facit duas hyperboles, quia
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axes figurarum occurrunt extra trigonum
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lateri oppoſito: quia tales ſunt in ambobus
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planis, videlicet duorum triangulorum & K,
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igitur illæ figuræ erunt ambæ hyperbo
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les, & vocantur ab Apollonio contra
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poſitæ. </
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">Ex his patet igitur, quòd omnes hæ fi
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guræ conueniunt in hoc, quòd generantur
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ex ſectione coni, aut conorum bifariam: per
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planum ad perpendiculum erectum ſuper
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ſuperficiem triangulorum, quòd non tran
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ſeat per verticem coni: & quod latera ha
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rum ſuperficierum ſunt lineæ obliquæ: &
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quòd non poſſunt eſſe plures his quinque.
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">Omnibus igitur his quinque figuris com
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mune eſt, vt cùm duæ quæ illas contangant
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rectè in vnum coierint, ducta recta linea ex
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concurſus loco vſque ad aduerſam figuræ
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partem, vel in contrapoſitis, vſque ad rectam
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lineam, quæ per puncta contactus ducitur,
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proportionem totius lineæ ad partem, quæ
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eſt extra obliquas, eſſe velut partium in
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tra obliquas ad lineam quæ contactus pun
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cta iungit terminatarum.
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Omnium co
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ni quinque
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figurarum
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commune
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priuilegium.</
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Centrum &
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verſa in hy
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perbolis tria
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priuilegia.</
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<
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">Cùm igitur (vt dictum eſt) tertium la
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tus trigoni diuidentis conum per axem coni
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neceſſariò occurrat, deductum axi hyperbo
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lis extra conum, pars axis hyperbolis inter
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verticem hyperbolis, & punctum concur
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ſus, cum latere oppoſito trianguli vocatur
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verſa, & punctus in medio verſæ centrum
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hyperbolis. </
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<
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id
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">Et habes exemplum in quarta
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figura. </
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<
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">Nam A vocatur verſa, & L centrum
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hiperbolis. </
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<
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id
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">Sunt autem hyperboli tria maximè præ
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cipua, quorum primum eſt, quòd in quaque
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illius parte circumferentiæ duo puncta ſu
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mantur, à quibus binæ, & binæ ad non tan
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gentes rectæ lineæ deducantur mutuò inter
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ſe æquidiſtantes, rectangula, contecta ab his
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lineis, quæ ab imo eueniunt, atque ab his
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quæ ab aliis punctis inuicem æqualia erunt.
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Secundum eſt, quòd inuenire contingit duas
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lineas in eodem plano, quarum altera erit
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recta, reliqua latus hyperboles, quæ ſemper
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ſibi inuicem magis approximabuntur, &
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nunquam ſe tangent. </
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id
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">Tertium ex ſecundo
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pendet, quòd erit inuentu facilè, duas li
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neas, quæ ſemper magis in eodem plano
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approximabuntur, & quanquam etiam in
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infinitum protraherentur, nunquam erunt
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proximiores mille ſtadiis, gratia exempli.
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<
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id
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">Demonſtrato enim ſecundo, ſi ſumatur li
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nea æquidiſtans rectæ ex aduerſa parte mil
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le ſtadiis, patebit quod dictum eſt. </
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<
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id
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">Igitur
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demonſtremus ſecundum, quod licet ab
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Apollonio demonſtretur, volo tamen vti
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demonſtratione Rabbi Moyſis Narbonen
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ſis exponentis dictum Rabbi Moyſis Ægy
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ptij, in libro cui titulus eſt, Directio dubi
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tantium, quod erat: Quædam intelligi poſſe,
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quæ imaginari nequeunt: vnde concludit,
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quòd intellectus ab imaginatione differat,
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non ſolùm ob nouitatem, ſed ob facilitatem
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& pulchritudinem. </
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Hyperbolis
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tria priuile
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gia.</
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Duarum li
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nearum, quæ
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ſemper
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ap-proximãtur
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proximantur</
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,
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& nunquam
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coëunt &
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demonſtra
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tio.</
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<
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">Sit igitur conus ABCD: nunc triangu
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lum nullum ſecantem intelligo. </
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<
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">Sed per
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A B D, intelligo connexam coni ſuperfi
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ciem, in qua protraho AC, à vertice, vſque
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ad baſim. </
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<
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">Et ſit K plana ſuperficies contan
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gens conum in recta linea AC: quæ ſuperfi
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cies intelligatur in infinitum cum coni ſu
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perficie extendi. </
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<
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">Dico primò, hanc
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ſuperficiẽ
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planam non poſſe tangere coni ſuperficiem
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alibi, quàm in linea AC: quòd ſi poteſt, tan
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gat in G, & duco circulum æquidiſtantem
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per G baſi BCD: cùm igitur circulus ſit in
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vna ſuperficie, erunt puncta contactus plani
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K, & periferiæ circuli illius in vna recta linea
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ex demonſtratis in vndecimo elementorum
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Euclidis. </
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<
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">Quamobrem cùm illa linea iam
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tangat circuli periferiam in linea AC, ca
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det ex demonſtratis ab Euclide in tertio ele
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mentorum extra circumferentiam circuli
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VXG, igitur non tanget illum in puncto G.
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<
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id
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">Aſſumo igitur E F rectam æquidiſtantiam
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AC, in ſuperficie K, & adeò propinquam
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rectæ AC, vt ſuperficies H ducta ad per
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pendiculum ſuper ſuperficiem K, ſecet co
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num, & illius ſuperficiem in punctis, puta S
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& G, & palam eſt ex dictis, partem ſuper
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ficiei ex H, cono incluſam eſſe hyperbolem,
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& lineam G S, quæ eſt in coni ſuperficie,
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eſſe latus hyperbolis. </
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<
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id
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">Conſtat igitur iam
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latus hyperbolis GS eſſe in ſuperficie eadem
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cum linea EF, ſcilicet in ſuperficie H: &
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quòd iſtæ duæ lineæ cùm ſint in eodem pla
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no H, nunquam ſe tangent: ſi enim ſe tan
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gent, vel in linea AC, & ita AC, & EF
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æquidiſtantes concurrent, quòd includit
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contradictionem: vel extra lineam A C,
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& ita cùm G S ſemper ſit in ſuperficie coni
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& EF ſemper in ſuperficie K, igitur K, tange
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ret conum extra lineam AC, cuius iam op
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poſitum demonſtrauimus. </
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<
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id
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">Dico modò, quòd
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cùm EF recta, & GS latus hyperbolis ſint
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in eadem ſuperficie H, & protractæ in infi
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nitum nunquam conueniunt, quòd ſem
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per vt magis à vertice coni elongantur, </
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