Cardano, Girolamo, De subtilitate, 1663

Table of figures

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              <s id="s.010916">
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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. </s>
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            <p type="main">
              <s id="s.010917">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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              </s>
              <s id="s.010918">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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            <p type="margin">
              <s id="s.010919">
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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.</s>
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            <p type="margin">
              <s id="s.010920">
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              Centrum &
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              verſa in hy­
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              perbolis tria
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              priuilegia.</s>
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            <p type="main">
              <s id="s.010921">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. </s>
              <s id="s.010922">Et habes exemplum in quarta
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              figura. </s>
              <s id="s.010923">Nam A vocatur verſa, & L centrum
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              hiperbolis. </s>
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              <s id="s.010924">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. </s>
              <s id="s.010925">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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              </s>
              <s id="s.010926">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. </s>
              <s id="s.010927">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. </s>
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            <p type="margin">
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              Hyperbolis
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              tria priuile­
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              gia.</s>
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            <p type="margin">
              <s id="s.010929">
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              Duarum li­
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              nearum, quæ
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              ſemper
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                proximantur</expan>
              ,
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              & nunquam
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              coëunt &
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              demonſtra­
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              tio.</s>
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            <p type="main">
              <s id="s.010930">Sit igitur conus ABCD: nunc triangu­
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              lum nullum ſecantem intelligo. </s>
              <s id="s.010931">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. </s>
              <s id="s.010932">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. </s>
              <s id="s.010933">Dico primò, hanc
                <expan abbr="ſuperficiẽ">ſuperficiem</expan>
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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. </s>
              <s id="s.010934">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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              </s>
              <s id="s.010935">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. </s>
              <s id="s.010936">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. </s>
              <s id="s.010937">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, </s>
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