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="id002860">
                <pb pagenum="163" xlink:href="015/01/182.jpg"/>
              Si autem a d cadat extra a b ducatur d e: quæ ſi cadat ſupra b c uel
                <lb/>
              infra, cum totum ſit maius parte erit a d e, ut prius maior a b c quod
                <lb/>
                <arrow.to.target n="marg558"/>
                <lb/>
              eſt contra Euclidem. </s>
              <s id="id002861">Reliquum eſt ut d c cadat ſupra b c: hoc au­
                <lb/>
                <arrow.to.target n="marg559"/>
                <lb/>
              tem eſſe non poteſt, nam cum ſuppoſuerimus a b eſſe minorem a c
                <lb/>
              erit angulus a c b minor angulo a b c, quare a c b eſt minor recto, &
                <lb/>
                <arrow.to.target n="marg560"/>
                <lb/>
              ideò a c d maior recto, at a c d æqualis eſt a c d, alteri igitur a c d eſt
                <lb/>
                <arrow.to.target n="marg561"/>
                <lb/>
              maior recto a c b minor, erit ergo pars maior toto.</s>
            </p>
            <p type="margin">
              <s id="id002862">
                <margin.target id="marg555"/>
              C
                <emph type="italics"/>
              o
                <emph.end type="italics"/>
              ^{m}.</s>
            </p>
            <p type="margin">
              <s id="id002863">
                <margin.target id="marg556"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              23.
                <emph type="italics"/>
              pri
                <lb/>
              mi
                <emph.end type="italics"/>
              E
                <emph type="italics"/>
              lement.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id002864">
                <margin.target id="marg557"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              38.
                <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="id002865">
                <margin.target id="marg558"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              18.
                <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="id002866">
                <margin.target id="marg559"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              23.
                <emph type="italics"/>
              eiuſ
                <lb/>
              dem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id002867">
                <margin.target id="marg560"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              13.
                <emph type="italics"/>
              eiuſ
                <lb/>
              dem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="margin">
              <s id="id002868">
                <margin.target id="marg561"/>
              P
                <emph type="italics"/>
              er
                <emph.end type="italics"/>
              4.
                <emph type="italics"/>
              eiuſ­
                <lb/>
              dem.
                <emph.end type="italics"/>
              </s>
            </p>
            <p type="head">
              <s id="id002869">LEMMA.</s>
            </p>
            <p type="main">
              <s id="id002870">His demonſtratis quis dicere poſſet ex ſuperius expoſitis quod
                <lb/>
                <arrow.to.target n="marg562"/>
                <lb/>
              angulus rectilineus ſemper eſſet maior angulo contactus? </s>
              <s id="id002871">quia an­
                <lb/>
              gulus contactus non poteſt diuidi niſi obliqua linea, recti lineus
                <lb/>
              autem tam obliqua quam recta. </s>
              <s id="id002872">Propter hoc exponantur circuli
                <lb/>
                <figure id="id.015.01.182.1.jpg" xlink:href="015/01/182/1.jpg" number="193"/>
                <lb/>
              tres ſe tangentes a b, a c, a d hac rati­
                <lb/>
              one ut a b, b c, c d ſint æquales, erunt
                <lb/>
                <arrow.to.target n="marg563"/>
                <lb/>
              enim centra omnia in linea conta­
                <lb/>
              ctus, & ducatur a e f g recta quomo
                <lb/>
                <arrow.to.target n="marg564"/>
                <lb/>
              dolibet: & erunt ductis lineis b c,
                <lb/>
                <arrow.to.target n="marg565"/>
                <lb/>
              c f, d g anguli e f g recti, quare om­
                <lb/>
              nes trigoni a b e, a c f, a d g, ſimiles
                <lb/>
                <arrow.to.target n="marg566"/>
                <lb/>
              & ideo a e, e f, f g æquales, atque por­
                <lb/>
              tiones a g, a f, a e, iuxta proportio­
                <lb/>
              nem circulorum, quare a g, erit ſex­
                <lb/>
              quialtera a f & a f dupla a e, igitur
                <lb/>
                <arrow.to.target n="marg567"/>
                <lb/>
              per præcedentem maior erit angu­
                <lb/>
              lus e a f, quam f a g, & a d a ex recta
                <lb/>
                <arrow.to.target n="marg568"/>
                <lb/>
              & peripheria quam e a f, igitur augendo eadem ratione cum perue­
                <lb/>
              niamus ad angulum b a g qui fermè eſt recto æqualis cum deficiat
                <lb/>
              ſolo angulo contactus, liquet angulum e a g eſſe longè maiorem
                <lb/>
              multis rectilineis. </s>
              <s id="id002873">Iſtud poſſet etiam demonſtrari uia Archimedis
                <lb/>
              diuidendo arcus g a in h & f a in k bifariam ducendo que lineas re­
                <lb/>
              ctas g h & fk & ita diuidendo h a in 1, & k a in m bifariam, & ducen­
                <lb/>
              do rectas atque ita ſemper appropinquando puncto a. </s>
              <s id="id002874">Concludo er­
                <lb/>
              go quod angulus
                <expan abbr="cõtactus">contactus</expan>
              ex recta & peripheria eſt maior multis
                <lb/>
              rectilineis. </s>
              <s id="id002875">Cauſa autem erroris eſt quod multi exiſtimarunt corro­
                <lb/>
              larium illud eſſe Euclidis cum non ſit. </s>
              <s id="id002876">Nam Euclidi ſufficit hoc
                <lb/>
              quòd angulus contactus
                <expan abbr="nõ">non</expan>
              poſsit recta diuidi, nam eo utitur poſt
                <lb/>
                <expan abbr="modũ">modum</expan>
              in demonſtrationibus. </s>
              <s id="id002877">Eo uerò quod ſit minor omnibus re­
                <lb/>
              ctilineis angulis non utitur, ideò etiam ſi
                <expan abbr="uerũ">uerum</expan>
              fuiſſet
                <expan abbr="nõ">non</expan>
              addidiſſet:
                <lb/>
              quanto minus: cum uerum non ſit, ideò fuit
                <expan abbr="adiectũ">adiectum</expan>
              ab aliquo qui
                <lb/>
                <expan abbr="idẽ">idem</expan>
              fore credidit
                <expan abbr="nõ">non</expan>
              poſſe diuidi recta linea & eſſe minus quocunque
                <lb/>
              quod recta linea diuidi poſſet, quod apertè ut dixi falſum eſt.</s>
            </p>
          </chap>
        </body>
      </text>
    </archimedes>
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