Cardano, Girolamo, De subtilitate, 1663

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    <archimedes>
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        <body>
          <chap>
            <p type="main">
              <s id="s.011092">
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              minor ſit toto, erit angulus
                <expan abbr="contactuũ">contactuum</expan>
              quan­
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              tumcunque magnus minor angulo rectili­
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              neo quantumcunque paruo: quod erat de­
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              monſtrandum. </s>
              <s id="s.011093">Igitur primi argumenti diſ­
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              ſolutio eſſe videtur, quòd angulus ille non
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              æqualiter augetur eo motu linea CB, ſed vt
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              magis proximus fit ipſi A, eò maius fit ar­
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              gumentum, ideò ſtatim tranſit ab acuto in
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              obtuſum abſque recto. </s>
              <s id="s.011094">Demonſtratio huius
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              eſt, quòd in proceſſu primæ medietatis ſe­
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              micirculi à C vſque ad M. ſolùm acquiritur
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              angulus C M L, & in proceſſu alterius
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              medietatis ſemicirculi acquiritur angulus
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              CMA, ab M vſque ad A, ſed angulus CMA
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              eſt maior angulo C ML in angulo CM A
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              à rectis contento, qui eſt dimidium recti:
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              igitur multò magis augetur angulus recta &
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              circumferentia contentus in medietate ſemi­
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              circuli MA, quam CM. </s>
              <s id="s.011095">Eadem ratione de
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              partibus circumferentiæ M A inuicem col­
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              latis. </s>
              <s id="s.011096">Igitur incrementum anguli CEK, ſu­
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              per CKB maius eſt incremento CKB ſuper
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              CBL: & incrementum CAE ſuper CEK,
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              maius eſt incremento CEK ſuper CKB: igi­
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              tur etiam quòd linea tranſiret per omnia
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              media ex B in K, non tamen ex K in E, &
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              multò minùs ex E in A, ideóque nec ex A
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              in G Apparet igitur primi paralogiſmi diſ­
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              ſolutio. </s>
              <s id="s.011097">Sed ſecundus non eadem ratio­
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              ne diſſoluitur, verùm multi ſunt modi de­
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              monſtrationum, & longè plures aſſumptio­
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              num. </s>
              <s id="s.011098">In multis enim profuit ſcire, quòd ita
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              eſſet, vt in ſolidi cubi numeri generatio­
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              ne paulò pòſt demonſtrauimus. </s>
              <s id="s.011099">Et quod ma­
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              ximum productum ex parte cuiuſlibet quan­
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              titatis in reſidui quadratum eſt, cum tertia
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              pars quantitatis in reſidui quadratum duci­
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              tur: hoc enim in Geometricorum elemen­
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              torum duodecimo libro demonſtratur à no­
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              bis. </s>
              <s id="s.011100">Verùm id antiqui latêre voluerunt, vt
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              magis admiratione digni viderentur. </s>
              <s id="s.011101">Quæ
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              enim nobis non parùm profuere, etiam illis
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              præſidio fuerunt: quandoquidem nos in plu­
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              ribus principia cum tota arte inuenerimus.
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              </s>
              <s id="s.011102">Quod etiam feciſſe Apollonium, & Archi­
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              medem arbitror, non tamen Euclidem, ne­
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              que Philoſophorum quenquam aliorum:
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              nam demonſtratis, ab aliis adiuti ſunt. </s>
              <s id="s.011103">Ve­
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              luti penultimam primi elementorum refe­
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              runt inuentum eſſe Pythagoræ Samij, ob
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              cuius inuentionem lætatum adeò tradunt,
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              vt bouem immolauerit: quod tamen vix cre­
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              di poteſt, quandoquidem ac omni cæde ani­
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              malium abſtinuerit Pythagoras. </s>
              <s id="s.011104">Sed certum
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              eſt, ex demonſtratione Architæ Tarentini
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              illius diſcipuli de inuentione duarum linea­
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              rum inter alias duas continua proportione
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              iunctarum, ante tempora Euclidis Megaren­
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              ſis geometrica inuenta, & præſtantiſſima
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              floruiſſe. </s>
              <s id="s.011105">Neque tamen parum fuit, Eucli­
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              dem in eum ordinem adeò exquiſitum cun­
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              cta redegiſſe, & quæ defecerant adieciſſe. </s>
            </p>
            <p type="margin">
              <s id="s.011106">
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              Reſolutoriæ
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              methodi
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              exemplum.</s>
            </p>
            <p type="margin">
              <s id="s.011107">
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              Duæ quan­
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              titates
                <expan abbr="parũ">parum</expan>
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              magnitudi­
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              ne differen­
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              tes, quarum
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              maiore ſem­
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              per per me­
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              dium diuiſa,
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              & minore
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              ſemper du­
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              plicata, mi­
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              nor
                <expan abbr="nunquã">nunquam</expan>
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              maiorem ex­
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              cedere poteſt
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              aut æquare.</s>
            </p>
            <p type="main">
              <s id="s.011108">Sed vbi finis non adeò certus eſt, diffici­
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              lior profectò eſt inuenio. </s>
              <s id="s.011109">Quorundam qui­
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              dem difficillima prorſus, veluti, quæ maxi­
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              ma poſſit eſſe proportio dupli tertiæ quan­
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              titatis ad aggregatum primæ & quartæ con­
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              tinuæ proportionis. </s>
              <s id="s.011110">Nam ea conſtant in mi­
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              nore proportione ſeſquiquarta, & maiore
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              ſeſquiquinta. </s>
              <s id="s.011111">Si enim capiamus 64. & 80.
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              & 100. & 125. duplum tertiæ quantitatis eſt
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              200. aggregatum primæ, & quartæ 189. Pro­
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              portio autem 369/341 minor eſt 299/189. Eſt au­
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              tem 360 duplum tertiæ quantitatis, & 341.
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              eſt aggregatum primæ & quartæ in propor­
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              tione ſeſquiquinta, vt ſint quantitates ipſæ
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              125. 150. 180. 216. Hoc autem demonſtra­
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              ri poteſt, ducto 360. in 189. fit 68. M 40.
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              Eſt autem hoc minùs, quàm 68. M. 200.
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              In talibus igitur inuenire demonſtratione,
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              difficillimum eſt: quantò magis vbi duarum
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              quantitatum diuerſorum generum, quæ ad
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              æqualitatem nullam perueniunt, in genere
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              perfecto fit comparatio. </s>
              <s id="s.011112">Nam paraboles ad
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              trigonum interiorem exquiſita eſt ratio epi­
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              trita, vt ab Archimede demonſtratur, quod
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              in quinto priuilegio à nobis expoſito ſupe­
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              rius continetur. </s>
              <s id="s.011113">Hoc igitur principium fuit,
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              quo Archimedes proportionem, & menſu­
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              ram parabolis potuerit inuenire. </s>
              <s id="s.011114">Liquet ex
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              illius demonſtratione, quòd ſi proportio hæc
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              in abſurdam aliquam, & quæ nullo modo
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              numeris deſcribi poſſet quantitatem incidiſ­
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              ſet, Archimedem illam non potuiſſe demon­
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              ſtrare. </s>
              <s id="s.011115">Ita & ratio ſphæræ ad conum dupla
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              apud illum: & rurſus apud Euclidem cylin­
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              dri ad conum tripla exquiſita. </s>
              <s id="s.011116">Quibus inuen­
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              tis, facilè fuit etiam partium rationem inui­
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              cem declaraſſe. </s>
              <s id="s.011117">Nam quæ inuicem non iun­
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              guntur rationali proportione, medio dua­
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              rum proportionem innoteſcere
                <expan abbr="conſueuerũt">conſueuerunt</expan>
              . </s>
            </p>
            <p type="main">
              <s id="s.011118">Quamobrem quadratum circulo æquale
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              impoſſibile eſt inuenire: & qui conati ſunt,
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              non videntur demonſtrationes Archimedis,
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              aut Apollinij, aut Euclidis intellexiſſe: aut ſi
                <lb/>
              modò intellexerunt, non animaduerterunt.
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              </s>
              <s id="s.011119">Nam principium omne inuentionis à com­
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              poſitione fit, compoſitionem ſequitur reſo­
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              lutio. </s>
              <s id="s.011120">At in compoſitione notus neceſſariò
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              eſt finis, ob id igitur in genere diuerſarum
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              quantitatum finem, & rationem notam eſſe
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              illarum oportet. </s>
              <s id="s.011121">At in circuli magnitudine,
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              ſeu ſuperficies ac quadrati ſuperficiem refe­
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              ratur, ſeu periferia ad diametrum, nulla per
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              ſe nota proportio eſt: demonſtratum enim
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              eſt ab Archimede, rationem periferiæ ad
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              diametrum minorem eſſe, quàm 22. ad 7.
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              maiorem verò tripla, & 10/71. Atque id eſt di­
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              cere minorem tripla, & 19/70 maiorem tri­
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              pla, 10/71 ſeu inter proportionem 15/4 62/97 &
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              15/4 61/97. Sed neque in ſuperficiebus: nam po­
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              ſita diametro 7. erit quadratum interius
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              circuli 24. 1/22. Sed area circuli vt ab Archi­
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              mede, & nobis demonſtratum eſt, fit ex di­
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              midio diametri in dimidium periferiæ, qua­
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              re erit 308. 1/2. Igitur proportio qualis 77.
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              ad 49. quare vt 11. ad 7. Verùm vt dictum
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              eſt periferia eſt minor 22. quantitate non
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              ſenſili, nec rationali: quadratum autem inte­
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              rius non mutatur, igitur proportio circuli ad
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              quadratum interius inſcriptum eſt minor
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              aliquantò, quàm 11. ad 7. igitur abſurda, &
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              incognita. </s>
              <s id="s.011122">Conati autem ſunt antiquorum
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              plurimi, & alij noſtro tempore, quorum vix
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              eſt numerum, & nomina referre, verùm res
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              quæ poſſibilis non eſt, claritatem ingenij il­
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              lorum hebetiorem videri fecit. </s>
              <s id="s.011123">Sed ortum
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              hic conatus irritus habuit ex verbis Ariſto­
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              telis malè interpretantis. </s>
              <s id="s.011124">Exempli loco enim </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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