Cardano, Girolamo, De subtilitate, 1663

Table of figures

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb pagenum="642" xlink:href="016/01/289.jpg"/>
            <p type="margin">
              <s id="s.012907">
                <margin.target id="marg1765"/>
              Monſtrum
                <lb/>
              mirabile.</s>
            </p>
            <p type="margin">
              <s id="s.012908">
                <margin.target id="marg1766"/>
              Nauim quo­
                <lb/>
              modo quot
                <lb/>
              paſſuum M.
                <lb/>
              exegerit, vel
                <lb/>
              ploſtrum de­
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              prehenda­
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              mus.</s>
            </p>
            <p type="margin">
              <s id="s.012909">
                <margin.target id="marg1767"/>
              Meteoroſco­
                <lb/>
              pium.</s>
            </p>
            <p type="main">
              <s id="s.012910">Sit igitur meridiei circulus AEBF, fixus
                <lb/>
              ſuper pede AM. </s>
              <s id="s.012911">In eo poli fingantur KF, &
                <lb/>
              vertex tuus E. </s>
              <s id="s.012912">Alius circulus immobilis
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              æquinoctij ACBD, fixus ſuper pedem AM,
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              & ad rectos angulos ſecans priorem circu­
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                <figure id="id.016.01.289.1.jpg" xlink:href="016/01/289/1.jpg" number="152"/>
                <lb/>
              lum AKBF. </s>
              <s id="s.012913">Sit alius circulus FGHK per
                <lb/>
              polos, & in ipſis polis F & K, per paxillos
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              verſatilis C E D L. </s>
              <s id="s.012914">Sit igitur diſtantia EN,
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              nota rectáque, numerentur autem diuiſis
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              ſingulis circulis ex his in partes ter mille
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              ſexcentas, partes illæ in C E D per EN, &
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              conſtituatur C N D, ſuper viam rectam ex
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              ciuitate tua in N locum, & vbi punctus N,
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              cadit, ducatur CKHF, circulus mobilis me­
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              ridiei. </s>
              <s id="s.012915">Habebis igitur per arcum KN latitu­
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              dinem loci, ſeu poli eleuationem, & per GC
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              differentiam longitudinis loci N à tua vr­
                <lb/>
              be: cúmque longitudo vrbis tuæ iam nota
                <lb/>
              ſit, erit & longitudo N. </s>
              <s id="s.012916">Quòd ſi altitudo N
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              nota fuerit, & iter rectum EN, circumdu­
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              ctis circulis CED & GNH, donec occurrant
                <lb/>
              extrema arcuum EN diſtantiæ rectæ, & KN
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              notæ altitudinis loci N in vnum, fiet arcus
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              tunc GC notus, differentia ſcilicet longitu­
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              dinis loci N, à patria tua. </s>
            </p>
            <p type="main">
              <s id="s.012917">Manifeſtum eſt autem, quòd contraria
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              ratione habitis longitudinibus, & latitudi­
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              nibus locorum, diſtantia quoque cognita
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              erit. </s>
              <s id="s.012918">Quòd ſi velis, vt inſtrumentum vnicui­
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              que ſeruiat regioni, facies paxillos EL, mo­
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              biles in circulo meridiei, AKBF, vt ſub qua­
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              cunque altitudine collocari vertex tuus poſ­
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              ſit. </s>
              <s id="s.012919">Porrò diuiſiones in ſingulas denas con­
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              ſpicuè, & in quinas minus, inde in quinqua­
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              genas aureo colore, vt diſtinguantur dili­
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              genter, velut in ſtateris. </s>
              <s id="s.012920">Numerus autem ne­
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              ceſſarius non eſt, quia vbique principium
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              ſtatuere oportet. </s>
              <s id="s.012921">Aſſequimur etiam & iſta
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              demonſtratione, ſed difficiliori modo: atque
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              hæc ad amuſſim omnia. </s>
              <s id="s.012922">Quòd verò ſemper
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              licet melius aſſequi, eſt proportio periferiæ
                <lb/>
              circuli ad diametrum, ab Archimede mira­
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              bili ingenio inuenta: quæ cùm facillima ſit,
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              eam quatuor verbis ſubſcribere placuit. </s>
              <s id="s.012923">Tri­
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              bus indiget illa ſuppoſitis: primùm, quòd
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              circuli periferia maior eſt aggregato late­
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              rum inſcriptæ figuræ, & minor circumſcri­
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                <arrow.to.target n="marg1768"/>
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              ptæ. </s>
              <s id="s.012924">De inſcripta patet ex rectæ lineæ diffi­
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              nitione: de circumſcripta, licet apud aliquos
                <lb/>
              videatur per ſe manifeſtum, à nobis tamen
                <lb/>
              in libris Elementorum per antiparalogiſ­
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              mum demonſtratur. </s>
              <s id="s.012925">Secundum eſt, quòd no­
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              ta quacunque linea in circulo collocata, il­
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              lius arcus per medium diuiſi, linea recta ſub­
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              tenſa cognita erit. </s>
              <s id="s.012926">Hæc licet à Ptolemæo de­
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              monſtretur, vt tamen quiſque rationem
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              habeat inueniendæ propoſitæ proportio­
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              nis, duobus verbis rem cum operatione de­
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              clarabo. </s>
            </p>
            <p type="margin">
              <s id="s.012927">
                <margin.target id="marg1768"/>
              Quomodo
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              ſciamus pro­
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              portionem
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              periferiæ cir­
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              culi ad dia­
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              metrum.</s>
            </p>
            <p type="main">
              <s id="s.012928">Sit A B nota in proportione ad BC, &
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              diuidatur arcus AB, per æqualia in D, & du­
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              catur AD, dico eam eſſe notam. </s>
              <s id="s.012929">Nam ducta
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              DCK, erit ex demonſtratis ab Euclide EA,
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              media proportione inter KE & ED, & qua­
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              dratum DA, æquale quadratis A E & E D.
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              </s>
              <s id="s.012930">Per quintam igitur ſecundi Elementorum
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              Euclidis detraham quadratum A E notum,
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              quia AE eſt dimidium AB, ex quadrato CD
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                <figure id="id.016.01.289.2.jpg" xlink:href="016/01/289/2.jpg" number="153"/>
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              cognito, & relinquatur quadratum CE co­
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              gnitum: igitur CE. </s>
              <s id="s.012931">Quare detracta CE ex
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              CD, relinquetur E D cognita. </s>
              <s id="s.012932">Iungam igi­
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              tur quadrata AE & E D, & per penultimam
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              primi Elementorum habebo quadratum AD
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              cognitum. </s>
              <s id="s.012933">Tertium ſuppoſitum eſt, quòd co­
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              gnito latere figuræ inſcriptæ circulo, co­
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              gnoſcam & latus circumſcribentis. </s>
              <s id="s.012934">Hoc li­
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              cet colligatur ab Euclide in quarto Ele­
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              mentorum, vt tamen habeamus demonſtra­
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              tionem cum operatione, vno eam verbo hîc
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              ſubiungam. </s>
              <s id="s.012935">Sit latus AB inſcriptæ figuræ, &
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              FG circumſcriptæ, ſub eodem enim angulo
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              continentur centri, ducantúrque A F &
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              CBG. </s>
              <s id="s.012936">Quia igitur nota eſt AB, nota eſt AE,
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              ideóque K E & E D, vt demonſtratum eſt.
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              </s>
              <s id="s.012937">Sed vt CE ad CD, ita AB ad FG. </s>
              <s id="s.012938">Ducta igi­
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              tur AB in CD, & quod producitur diuiſo
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              per CE, exit FG cognita. </s>
              <s id="s.012939">Suppoſito igitur
                <lb/>
              AB latere hexagoni, quod per demonſtrata
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              ab Euclide eſt æquale dimidio diametri, per
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              ſecundum ſuppoſitum habebo latus figuræ
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              duodecim baſium, & per eandem figuræ 24.
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              baſium, inde 48. inde 96. inde 192. & 384. &
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              768. & poſſum procedere abſque errore, imò
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              abſque extractione radicum. </s>
            </p>
            <p type="main">
              <s id="s.012940">Sit igitur gratia exempli ſtatus in latere
                <lb/>
              figuræ 768. laterum, habebo igitur per ter­
                <lb/>
              tium ſuppoſitum latus figuræ 768. circum­
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              ſcriptæ: duces vtrunque per numerum late­
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              rum, id eſt, per 768. & habebis ambitum in­
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              terioris, & exterioris figuræ, & proportio­
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              nem illorum ad diametrum circuli. </s>
              <s id="s.012941">Sed peri­
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              feria circuli maior eſt ambitu inſcriptæ figu­
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              ræ, & minor circumſcriptæ, ex primo ſup­
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              poſito igitur habebo proportionem perife­
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              riæ circuli ad diametrum, inter quas pro­
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              portiones debent collocari, & tamen nun­
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              quam ad perfectam cognitionem, & metam
                <lb/>
              peruenire poteſt. </s>
              <s id="s.012942">Ex quo patet, Archimedem
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              non indiguiſſe inuentis à Ptolemæo, nec
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              tabulis ſinuum, exquiſitiúſque ac purius ſine
                <lb/>
              illis, quàm cum illis geometram ad hanc </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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