Cardano, Girolamo, De subtilitate, 1663

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    <archimedes>
      <text>
        <body>
          <chap>
            <p type="main">
              <s id="s.010970">
                <pb pagenum="597" xlink:href="016/01/244.jpg"/>
              ſpatium ſpiralis ſuæ contentum intra ean­
                <lb/>
              dem cum recta, eſt velut quadrati ſemidia­
                <lb/>
              metri circuli ad rectangulum ex ſemidia­
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              metro circuli, in rectam præcedentis ſpira­
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              lis, cum tertia parte quadrati ſemidiametri
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              circuli ambientis primam ſpiram. </s>
              <s id="s.010971">Quintum
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              proportio ſectoris circuli circumſcribentis
                <lb/>
              ſpiralem primam aliquam portionem, ad
                <lb/>
              ipſam portionem ſpiralem terminatam in
                <lb/>
              centro, & angulum habentem eundem
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              cum ſectore, eſt veluti quadrati ſemidia­
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              metri eiuſdem circuli ad rectangulum ex
                <lb/>
              rectis ſpiralem ſectorem continentibus, ad­
                <lb/>
              dita tertia parte quadrati differentiæ earun­
                <lb/>
              dem linearum. </s>
              <s id="s.010972">Sextum, cùm ſectorem mi­
                <lb/>
              nore circulo abſcideris, conſtantem inter
                <lb/>
              duos circulos quorum ſemidiametri ex cir­
                <lb/>
              cumuolutione aliqua aucti ſint, ſpiralis quæ
                <lb/>
              à termino minoris ad maioris lineæ finem
                <lb/>
              procedit ſuperficiem diuidens, in duas par­
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              tes eam diuidit, quarum proportio exterio­
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              ris ad interiorem, eſt veluti ſemidiametri
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              minoris cum duplo tertiæ partis differentiæ
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              ſemidiametrorum ad ſemidiametrum mino­
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              ris cum tertia parte differentiæ ipſorum
                <lb/>
              etiam ſemidiametrorum.
                <lb/>
                <arrow.to.target n="marg1535"/>
              </s>
            </p>
            <p type="margin">
              <s id="s.010973">
                <margin.target id="marg1534"/>
              Spiralis li­
                <lb/>
              neæ priuile­
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              gia 6.</s>
            </p>
            <p type="margin">
              <s id="s.010974">
                <margin.target id="marg1535"/>
              Rectilinea­
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              rum om­
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              nium figura­
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              rum priuile­
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              gium.</s>
            </p>
            <p type="main">
              <s id="s.010975">Omnibus figuris rectilineis hoc vnum
                <lb/>
              eſt commune, quod protractis ſingulis la­
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              teribus exteriores omnes anguli pariter ac­
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              cepti, etiamſi mille fuerint, quatuor rectis
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              angulis ſunt æquales. </s>
              <s id="s.010976">Hoc autem ex hoc
                <lb/>
              pendet, quòd omnes qui intrà continen­
                <lb/>
              tur anguli, tot rectis æquantur, quotus eſt
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              numerus duplus laterum, ſeu angulorum,
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              quatuor demptis, quòd ex trigonorum ra­
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              tione pendet, in quos figura diuiditur.
                <arrow.to.target n="marg1536"/>
                <lb/>
              Nam cuiuſlibet trigoni tres anguli pariter
                <lb/>
              accepti duobus rectis ſunt æquales, exterior
                <lb/>
              verò angulus duobus interioribus ex aduer­
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              ſo poſitis pariter acceptis eſt æqualis. </s>
              <s id="s.010977">Area
                <lb/>
              etiam æqualis eſt producto ex dimidio ag­
                <lb/>
              gregati omnium laterum, in differentiam
                <lb/>
              cuiuſlibet lateris, ab eodem dimidio omnia
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              ſimul multiplicando, non iungendo, vt tres
                <lb/>
              fiant multiplicationes.
                <lb/>
                <arrow.to.target n="marg1537"/>
              </s>
            </p>
            <p type="margin">
              <s id="s.010978">
                <margin.target id="marg1536"/>
              Trigonorum
                <lb/>
              priuilegia.</s>
            </p>
            <p type="margin">
              <s id="s.010979">
                <margin.target id="marg1537"/>
              Quadrati
                <lb/>
              proprietas.</s>
            </p>
            <p type="main">
              <s id="s.010980">Quadrati verò proprium eſt, vt latus eius
                <lb/>
              inter aggregatum ex ipſo dimetiente, ac
                <lb/>
              inter differentiam eorundem proportione
                <arrow.to.target n="marg1538"/>
                <lb/>
              media conſiſtat. </s>
              <s id="s.010981">Hoc autem contingit, quia
                <lb/>
              dimetiens quadrati quadratum duplum effi­
                <lb/>
              cit ipſi quadrato, cuius erat dimetiens. </s>
              <s id="s.010982">Æ­
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              quilateri vero pentagoni, & æqui anguli
                <lb/>
              latus eſt maior pars lineæ diuiſæ, ſecundum
                <lb/>
              proportionem habentem medium, & duo
                <lb/>
              extrema, in comparatione ad lineam, quæ
                <lb/>
              duobus pentagoni eiuſdem lateribus ſubten­
                <lb/>
              ditur. </s>
              <s id="s.010983">Hexagoni verò latus, qui tamen ſit
                <lb/>
              ( vt dixi ) æquilaterus atque æquiangulus,
                <lb/>
              æquale eſt ſemidiametro circuli eundem he­
                <lb/>
                <arrow.to.target n="marg1539"/>
                <lb/>
              xagonum circumſcribentis. </s>
              <s id="s.010984">Heptagoni ve­
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              rò latus, & linea, quæ duobus eiuſdem la­
                <lb/>
              teribus, ac linea, quæ tribus pariter ſubten­
                <lb/>
              ditur conſtituunt trigonum, ſi fuerit ( vt di­
                <lb/>
              xi
                <emph type="italics"/>
              )
                <emph.end type="italics"/>
              æquilaterus ac æquiangulus, cuius pro­
                <lb/>
              portio aggregati ex latere & ſubtenſa tri­
                <lb/>
              bus ad ſubtenſam duobus; eſt vt ſubtenſæ
                <lb/>
              duobus ad latus eiuſdem, & rurſus late­
                <lb/>
              ris, & ſubtenſæ duobus ad ſubtenſam tri­
                <lb/>
              bus, eſt vt ſubtenſæ tribus ad lineam ſub­
                <lb/>
              tenſam duobus eiuſdem heptagoni late­
                <lb/>
              ribus. </s>
              <s id="s.010985">Hoc autem inferiùs demonſtrabitur. </s>
            </p>
            <p type="margin">
              <s id="s.010986">
                <margin.target id="marg1538"/>
              Pentagoni
                <lb/>
              æquilateri,
                <lb/>
              & æqui an­
                <lb/>
              guli proprie­
                <lb/>
              tas.</s>
            </p>
            <p type="margin">
              <s id="s.010987">
                <margin.target id="marg1539"/>
              Hexagonti,
                <lb/>
              & heptagoni
                <lb/>
              conſimilis
                <lb/>
              proprietas.</s>
            </p>
            <p type="main">
              <s id="s.010988">Habent & corpora, & ſuperficies planæ,
                <lb/>
                <arrow.to.target n="marg1540"/>
                <lb/>
              obliquæ proprietates: velut Ambiens ſphæ­
                <lb/>
              ram, quadrupla eſt illius maximo circulo,
                <lb/>
              ipſa verò ſphæra inter corpora omnium pro
                <lb/>
                <arrow.to.target n="marg1541"/>
                <lb/>
              ambitus ratione capaciſſima. </s>
              <s id="s.010989">Continet au­
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              tem, & contineri poteſt à quinque corpori­
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              bus, quæ ſola poſſunt æquas habere omnes
                <lb/>
              ſuperficies, æquoſque ſolidos angulos, ac la­
                <lb/>
              tera inuicem æqualia. </s>
              <s id="s.010990">Partium verò ſphæ­
                <lb/>
              ræ, quæ plano ſuper axem propendiculari
                <lb/>
              diuiduntur, tria ſunt priuilegia. </s>
              <s id="s.010991">Cuiuſcun­
                <lb/>
              que partis ſphæræ ſuperficies æqualis eſt
                <lb/>
              circulo, cuius ſemidiameter eſt linea à ver­
                <lb/>
              tice portionis ſphæræ ad terminum circuli,
                <lb/>
              qui eſt baſis eiuſdem portionis. </s>
              <s id="s.010992">Ex quo pa­
                <lb/>
              tet, quòd proportio ſuperficierum partium
                <lb/>
              ſphærę plano ſeparatarum, eſt veluti partium
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              diametri codem plano diuiſarum, cùm dime­
                <lb/>
              tiens tamen ſphæræ ſuper planum perpen­
                <lb/>
              dicularis fuerit. </s>
              <s id="s.010993">Proportio partium corpo­
                <lb/>
              rearum ſphæræ, quas planum vnum diſtin­
                <lb/>
              guit diametrum diuidens ei perpendicula­
                <lb/>
              rem, eſt veluti corporis producti ex quadra­
                <lb/>
              to maioris portionis axis, in lineam con­
                <lb/>
              ſtantem ex minore portione, & dimidio
                <lb/>
              axis, ad corpus, conſtans ex quadrato mino­
                <lb/>
              ris portionis in lineam conſtantem ex dimi­
                <lb/>
              dio, & maiore axis portione. </s>
            </p>
            <p type="margin">
              <s id="s.010994">
                <margin.target id="marg1540"/>
              Sphaera pri­
                <lb/>
              uilegia 2.</s>
            </p>
            <p type="margin">
              <s id="s.010995">
                <margin.target id="marg1541"/>
              Corporum
                <lb/>
              quinque
                <lb/>
              ſphæra con­
                <lb/>
              tentorum
                <lb/>
              proprietas.
                <lb/>
              </s>
              <s id="s.010996">Partium
                <lb/>
              ſphæræ pri­
                <lb/>
              uilegia 2.</s>
            </p>
            <p type="main">
              <s id="s.010997">Hoc autem ex iſto pendet, quòd onus ha­
                <lb/>
                <arrow.to.target n="marg1542"/>
                <lb/>
              bens baſim eandem cum portione ſphæræ,
                <lb/>
              ſi talem habeat illius altitudo, ad altitudi­
                <lb/>
              nem portionis portionem, qualis eſt aggre­
                <lb/>
              gati ex altitudine reſiduæ portionis, & dimi­
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              dio axis ad altitudinem eiuſdem reſidui por­
                <lb/>
              tionis, erit conus ille æqualis portionis. </s>
              <s id="s.010998">Ex
                <lb/>
              hoc patet, quòd quælibet ſphæra eſt qua­
                <lb/>
              drupla cono, cuius baſis eſt circulus maior,
                <lb/>
              altitudo verò medietas diametri ſphæræ.
                <lb/>
              </s>
              <s id="s.010999">Quilibet etiam conus æqualis eſt ſectori
                <lb/>
              ſphæræ ( Sectorem autem dico, corpus in
                <lb/>
              centrum ſphæræ terminatum, cuius baſis
                <lb/>
              portio eſt ſuperficiei ſphæræ: manifeſtum
                <lb/>
              eſt etiam illud conſtare portione ſphæræ, &
                <lb/>
              cono baſim habente ſuperficiem planam
                <lb/>
              eiuſdem portionis) cùm coni altitudo fuerit
                <lb/>
              ſemidiameter ſphæræ, & baſis æqualis ſuper­
                <lb/>
              ficiei ſectoris, igitur erunt baſes, & altitudi­
                <lb/>
              nes tunc æquales. </s>
            </p>
            <p type="margin">
              <s id="s.011000">
                <margin.target id="marg1542"/>
              Coni recti
                <lb/>
              priuilegia
                <lb/>
              tria.</s>
            </p>
            <p type="main">
              <s id="s.011001">Conoidali corpore rectangulo diuiſo per
                <lb/>
                <arrow.to.target n="marg1543"/>
                <lb/>
              planum, portio, quæ ad apicem terminatur,
                <lb/>
              ſeſquialtera eſt coni portioni eandem baſim
                <lb/>
              ac axem habenti: Conoidalium rurſus
                <expan abbr="rectã-gulorum">rectan­
                  <lb/>
                gulorum</expan>
              portiones inuicem proportionem
                <lb/>
              retinent, ſi plano diuiſæ fuerint, quàm axis
                <lb/>
              partium earundem ipſius quadrata. </s>
              <s id="s.011002">Cùm ve­
                <lb/>
                <arrow.to.target n="marg1544"/>
                <lb/>
              rò Conoidale obtuſiangulum ſecatur plano,
                <lb/>
              erit proportio partis ad verticem termina­
                <lb/>
              tæ, ad portionem coni eandem baſim, &
                <lb/>
              axis portionem habenti, qualis lineæ con­
                <lb/>
              ſtantis ex axis parte conoidalis portionis
                <expan abbr="cũ">cum</expan>
                <lb/>
              triplo lineæ, quæ ex centro hyperboles, ſeu
                <lb/>
              obliqui formæ lateris ad
                <expan abbr="eandẽ">eandem</expan>
              axis
                <expan abbr="portionẽ">portionem</expan>
                <lb/>
              cum duplo
                <expan abbr="eiuſdẽ">eiuſdem</expan>
              , quę ex centro hyperboles. </s>
            </p>
            <p type="margin">
              <s id="s.011003">
                <margin.target id="marg1543"/>
              Conoida­
                <lb/>
              lium rectan­
                <lb/>
              gulum pri­
                <lb/>
              uilegia 2.</s>
            </p>
            <p type="margin">
              <s id="s.011004">
                <margin.target id="marg1544"/>
              Conoida­
                <lb/>
              lium obtuſi­
                <lb/>
              ſi angulorum
                <lb/>
              priuilegium</s>
            </p>
            <p type="main">
              <s id="s.011005">Sed
                <expan abbr="ſphæroidaliũ">ſphæroidalium</expan>
              priuilegia quatuor ſunt
                <lb/>
                <arrow.to.target n="marg1545"/>
                <lb/>
              cùm enim plano diuiditur, per centrum, per
                <lb/>
              æqualia diuiditur, eritque quælibet portio
                <lb/>
              dupla cono baſim, & axem æquales portioni
                <lb/>
              ipſius ſphæroidis habenti. </s>
              <s id="s.011006">Si præter cen­
                <lb/>
              ſphæroides ſecetur, quomodolibet proportio </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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