Cardano, Geronimo, Opvs novvm de proportionibvs nvmerorvm, motvvm, pondervm, sonorvm, aliarvmqv'e rervm mensurandarum, non solùm geometrico more stabilitum, sed etiam uarijs experimentis & observationibus rerum in natura, solerti demonstratione illustratum, ad multiplices usus accommodatum, & in V libros digestum. Praeterea Artis Magnae, sive de regvlis algebraicis, liber vnvs abstrvsissimvs & inexhaustus planetotius Ariothmeticae thesaurus ... Item De Aliza Regvla Liber, hoc est, algebraicae logisticae suae, numeros recondita numerandi subtilitate, secundum Geometricas quantitates inquirentis ...

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    <archimedes>
      <text>
        <body>
          <chap>
            <pb pagenum="226" xlink:href="015/01/245.jpg"/>
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              8.
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              ter­
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              28.
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              eiuſ
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              dem.
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              </s>
            </p>
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              er
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              C
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              or
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              m.
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              15.
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              quarti
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                <lb/>
              E
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              lem.
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              </s>
            </p>
            <p type="margin">
              <s id="id003814">
                <margin.target id="marg720"/>
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                <emph type="italics"/>
              er
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                <emph type="italics"/>
              primi
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                <lb/>
              E
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              lem.
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              </s>
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            <p type="margin">
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            </p>
            <p type="head">
              <s id="id003816">LEMMA SECVNDVM.</s>
            </p>
            <p type="main">
              <s id="id003817">Quadratum lateris trianguli æquilateri ſe habet ad illius ſuperfi
                <lb/>
              ciem, ut latus eius ad mediam lineam inter latus dodrantis, & qua­
                <lb/>
              drantis proportione duplicata.
                <lb/>
                <arrow.to.target n="marg722"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003818">
                <margin.target id="marg722"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id003819">Quadratum a b eſt æquale quadratis a f, fb, & quadruplum qua</s>
            </p>
            <p type="main">
              <s id="id003820">
                <arrow.to.target n="marg723"/>
                <lb/>
              drato b f, igitur quadratum a f eſt dodrans quadrati a b. </s>
              <s id="id003821">Quod ue­
                <lb/>
              rò fit ex a fin f b eſt medium proportione inter quadrata a f, f b, re­
                <lb/>
                <arrow.to.target n="marg724"/>
                <lb/>
              ctangulum igitur ex a fin fb, eſt ex lateribus dodrantis a f, & qua­
                <lb/>
                <arrow.to.target n="marg725"/>
                <lb/>
              drantis b f quadrati a b, quare cum mediæ inter a f & fb æquale fa­
                <lb/>
              ciat quadratum rectangulo a fin fb, erit proportio quadrati a b ad
                <lb/>
              quadratum mediæ inter a f, fb, ut lateris trianguli ad mediam inter
                <lb/>
                <arrow.to.target n="marg726"/>
                <lb/>
              latera dodrantis, & quadrantis quadrati lateris ipſius duplicata: re­
                <lb/>
                <arrow.to.target n="marg727"/>
                <lb/>
              ctangulum autem a fin fb eſt æquale triangulo a b c, igitur propor
                <lb/>
              tio quadrati a b ad triangulum a b c eſt uelut lateris a b ad mediam
                <lb/>
              inter latera dodrantis & quadrantis duplicata.</s>
            </p>
            <p type="margin">
              <s id="id003822">
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              P
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              er
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              27.
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              pri
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              E
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              lement.
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              </s>
            </p>
            <p type="margin">
              <s id="id003823">
                <margin.target id="marg724"/>
              P
                <emph type="italics"/>
              er
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              1.
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              ſexti
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                <lb/>
              E
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              lem.
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              </s>
            </p>
            <p type="margin">
              <s id="id003824">
                <margin.target id="marg725"/>
              P
                <emph type="italics"/>
              er eandem
                <lb/>
              &
                <emph.end type="italics"/>
              11.
                <emph type="italics"/>
              quin
                <lb/>
              ti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
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              </s>
            </p>
            <p type="margin">
              <s id="id003825">
                <margin.target id="marg726"/>
              P
                <emph type="italics"/>
              er
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              17.
                <emph type="italics"/>
              &
                <emph.end type="italics"/>
                <lb/>
              20.
                <emph type="italics"/>
              ſexti
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              l.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003826">
                <margin.target id="marg727"/>
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              pri­
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              E
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              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s id="id003827">LEMMA TERTIVM.</s>
            </p>
            <p type="main">
              <s id="id003828">Propoſitio quadrati cubi ſphæræ incluſi ad triangulum pyrami
                <lb/>
              dis eidem ſphæræ incluſæ, eſt uelut lateris pyramidis ſeu trianguli
                <lb/>
              eius ad cathetum ſuum.
                <lb/>
                <arrow.to.target n="marg728"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003829">
                <margin.target id="marg728"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id003830">Proponatur enim ſphæræ diameter g, & latus pyramidis b a, & </s>
            </p>
            <p type="main">
              <s id="id003831">
                <arrow.to.target n="marg729"/>
                <lb/>
              latus cubi b h, quæ corpora illi ſphæræ includuntur: igitur g erit
                <lb/>
              poteſtate ſexquialtera ad a b, & tripla ad b h, igitur b a eſt poteſtate
                <lb/>
                <arrow.to.target n="marg730"/>
                <lb/>
              dupla ad b h, quod igitur fit ex b a in dimidium ſuum, eſt æquale
                <lb/>
              quadrato b h, igitur b h eſt media inter b a & b f, b f enim eſt dimi­
                <lb/>
              dium b a, ut probatum eſt. </s>
              <s id="id003832">Quadratum igitur a b ſe habet ad trian­
                <lb/>
                <arrow.to.target n="marg731"/>
                <lb/>
              gulum a b c, ut a b ad mediam inter a f & fb duplicata: Quadratum
                <lb/>
              quoque a b ſe habet ad quadratum h b, ut a b ad mediam inter a b &
                <lb/>
              b f, duplicata igitur proportio quadrati b h ad triangulum a b c, eſt
                <lb/>
                <arrow.to.target n="marg732"/>
                <lb/>
              uelut lateris a b ad cathetum a f.</s>
            </p>
            <p type="margin">
              <s id="id003833">
                <margin.target id="marg729"/>
              P
                <emph type="italics"/>
              er
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              C
                <emph type="italics"/>
              or
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              ^{m}.
                <lb/>
              13.
                <emph type="italics"/>
              decimi­
                <lb/>
              tertij
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003834">
                <margin.target id="marg730"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.
                <lb/>
              15.
                <emph type="italics"/>
              decimi­
                <lb/>
              tertij
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003835">
                <margin.target id="marg731"/>
              P
                <emph type="italics"/>
              er
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              17.
                <emph type="italics"/>
              ſex
                <lb/>
              ti
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              E
                <emph type="italics"/>
              lem.
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                <lb/>
              L
                <emph type="italics"/>
              emmate
                <emph.end type="italics"/>
              1.</s>
            </p>
            <p type="margin">
              <s id="id003836">
                <margin.target id="marg732"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              67.</s>
            </p>
            <p type="head">
              <s id="id003837">LEMMA QVARTVM.</s>
            </p>
            <p type="main">
              <s id="id003838">Proportio lateris pyramidis ad axem illius eſt poteſtate ſex­
                <lb/>
              quialtera.
                <lb/>
                <arrow.to.target n="marg733"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003839">
                <margin.target id="marg733"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id003840">Intelligatur baſis pyramidis triangulus a b c, & conus pyrami­</s>
            </p>
            <p type="main">
              <s id="id003841">
                <arrow.to.target n="marg734"/>
                <lb/>
              dis k, & quæ per centrum ſphæræ tranſit ex cono k d, cumque k d a
                <lb/>
              angulus rectus ſit, erit quadratum k a æquale quadratis k d, d a, at
                <lb/>
              d a eſt dupla d f, ut probatum eſt, igitur poteſtate ſexquitertia f b,
                <lb/>
              k a uerò eſt quadrupla poteſtate fb, quia fb eſt dimidium k a, igitur
                <lb/>
              k a eſt tripla poteſtate a d, igitur k a poteſtate ſexquialtera k d, quod
                <lb/>
              erat demonſtrandum.
                <lb/>
                <arrow.to.target n="marg735"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003842">
                <margin.target id="marg734"/>
              P
                <emph type="italics"/>
              er
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              47.
                <emph type="italics"/>
              pri
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              mi
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              lem.
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                <lb/>
              L
                <emph type="italics"/>
              emmate
                <emph.end type="italics"/>
              1.</s>
            </p>
            <p type="margin">
              <s id="id003843">
                <margin.target id="marg735"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id003844">Ex hoc patet quod proportio axis pyramidis ad latus cubi ea­
                <lb/>
              dem ſphæra circumſcriptorum eſt poteſtate ſexquitertia.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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