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>
            <p type="main">
              <s id="id003932">
                <pb pagenum="231" xlink:href="015/01/250.jpg"/>
              hoc leminate duo latera g d & g a deducta ad æquicrurium, erunt
                <lb/>
              maiora lateribus polygonię, & ſimiliter duo latera h d maiora late­
                <lb/>
              ribus polygoniæ incluſæ, ergo latera trapezij erunt maiora omni­
                <lb/>
              bus lateribus polygoniæ incluſæ.
                <lb/>
                <arrow.to.target n="marg765"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003933">
                <margin.target id="marg763"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="margin">
              <s id="id003934">
                <margin.target id="marg764"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              4.
                <emph type="italics"/>
              pri­
                <lb/>
              mi, &
                <emph.end type="italics"/>
              16.
                <lb/>
                <emph type="italics"/>
              tertij
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003935">
                <margin.target id="marg765"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="main">
              <s id="id003936">Ex hoc habetur demonſtratio propoſitionis: ſint duæ lineæ a b
                <lb/>
              & a c quæ comprehendant portionem cir­
                <lb/>
              culi b c, dico eas eſſe maiores b c portione,
                <lb/>
              ſi enim a b & a c ſunt æquales diuiſo arcu
                <lb/>
              b c per æqualia in f, ducam contingentem </s>
            </p>
            <p type="main">
              <s id="id003937">
                <arrow.to.target n="marg766"/>
                <lb/>
              h f k, ſi non faciant triangulum æquicruri­
                <lb/>
              um b c d ſuper b c, & cuius ambo latera pa
                <lb/>
              riter accepta ſint æqualia a b & a c. </s>
              <s id="id003938">Et du­
                <lb/>
              cam contingentem & habebo trapezium
                <lb/>
                <arrow.to.target n="marg767"/>
                <lb/>
              h b, c k. </s>
              <s id="id003939">Quare ſi peripheria circuli b c eſt
                <lb/>
                <figure id="id.015.01.250.1.jpg" xlink:href="015/01/250/1.jpg" number="248"/>
                <lb/>
              minor d b & d c pariter acceptis, habeo
                <expan abbr="intentũ">intentum</expan>
              , ſi non toties
                <expan abbr="diuidã">diuidam</expan>
                <lb/>
              peripheriam per æqualia ut fiat figura polygonia ſuper b c æquila­
                <lb/>
              tera & æquiangula, cuius differentia a peripheria ſit minor differen
                <lb/>
              tia d b & d c à trapezio b h, k c, id eſt, tribus eius lateribus, nam cum
                <lb/>
              d h & d k ſint maiores h k, conſtat quod d b & d e ſunt maiores h b,
                <lb/>
              & k c & h k igitur ſit differentia illa l, &
                <expan abbr="differẽtia">differentia</expan>
              peripherię à lineis
                <lb/>
              polygoniæ minori: igitur cum peripheria ſit æqualis aut maior
                <lb/>
              d b & d c, & differentia a lateribus polygoniæ minor quàm d b &
                <lb/>
              d c, a b, h b, h k, k c, erit minor proportio peripheriæ ad latera poly­
                <lb/>
                <arrow.to.target n="marg768"/>
                <lb/>
              goniæ quàm d b & d c ad tria latera trapezij, quare minor propor­
                <lb/>
                <arrow.to.target n="marg769"/>
                <lb/>
              tio peripheriæ ad d b & d c quàm laterum polygoniæ ad tria latera
                <lb/>
                <arrow.to.target n="marg770"/>
                <lb/>
              trapezij, ſed latera polygoniæ ſunt minora tribus lateribus. </s>
              <s id="id003940">trapezij,
                <lb/>
                <arrow.to.target n="marg771"/>
                <lb/>
              igitur peripheria b c eſt minor d b & d e, quod erat
                <expan abbr="demonſtrandũ">demonſtrandum</expan>
              .</s>
            </p>
            <p type="margin">
              <s id="id003941">
                <margin.target id="marg766"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              2.
                <emph type="italics"/>
              &
                <emph.end type="italics"/>
              1.
                <lb/>
                <emph type="italics"/>
              primi
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003942">
                <margin.target id="marg767"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              5.
                <emph type="italics"/>
              eiuſ­
                <lb/>
              dem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003943">
                <margin.target id="marg768"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              20.
                <emph type="italics"/>
              pri­
                <lb/>
              mi
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003944">
                <margin.target id="marg769"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              2
                <emph type="italics"/>
              lemma.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003945">
                <margin.target id="marg770"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              1
                <emph type="italics"/>
              lemma.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003946">
                <margin.target id="marg771"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}.
                <lb/>
              3
                <emph type="italics"/>
              lemmatis.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s id="id003947">SCHOLIVM.</s>
            </p>
            <p type="main">
              <s id="id003948">Hanc propoſitionem non ſcripſi quòd eſſet magni momenti, ſed
                <lb/>
              propter modum probandi, ſi enim reſpicis ex uno oppoſito ſcilicet
                <lb/>
              quod peripheria circuli ſit maior trianguli lateribus, oſtendo de­
                <lb/>
              monſtratione non ducente ad inconueniens, ſed ſimplici quod ipſa
                <lb/>
              peripheria eſt minor trianguli lateribus, & hoc nunquam fuit
                <expan abbr="factũ">factum</expan>
                <lb/>
              ab aliquo, imò uidetur plane impoſsibile. </s>
              <s id="id003949">Et eſt res admirabilior
                <lb/>
              quæ inuenta ſit ab orbe condito, ſcilicet oſtendere aliquid ex ſuo
                <lb/>
              oppoſito, demonſtratione non ducente ad impoſsibile & ita, ut
                <expan abbr="nõ">non</expan>
                <lb/>
              poſsit demonſtrari ea
                <expan abbr="demõ">demom</expan>
              ſtratione niſi per illud
                <expan abbr="ſuppoſitũ">ſuppoſitum</expan>
              quod
                <lb/>
              eſt contrarium concluſioni, uelut ſi quis demonſtraret quòd So­
                <lb/>
              crates eſt albus quia eſt niger, & non poſſet demonſtrare aliter, &
                <lb/>
              ideo eſt longè maius Chryſippeo Syllogiſmo.
                <lb/>
                <arrow.to.target n="marg772"/>
              </s>
            </p>
            <p type="margin">
              <s id="id003950">
                <margin.target id="marg772"/>
              C
                <emph type="italics"/>
              or
                <emph.end type="italics"/>
              ^{m}. 2.</s>
            </p>
            <p type="main">
              <s id="id003951">Ex hoc patet quod pars lineæ exterioris quæ tangit circulum </s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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