Clavius, Christoph, Geometria practica

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        <div xml:id="echoid-div958" type="section" level="1" n="339">
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            <s xml:id="echoid-s15662" xml:space="preserve">
              <pb o="335" file="365" n="365" rhead="LIBER OCTAVVS."/>
            ex coroll. </s>
            <s xml:id="echoid-s15663" xml:space="preserve">propoſ. </s>
            <s xml:id="echoid-s15664" xml:space="preserve">2. </s>
            <s xml:id="echoid-s15665" xml:space="preserve">cap. </s>
            <s xml:id="echoid-s15666" xml:space="preserve">5. </s>
            <s xml:id="echoid-s15667" xml:space="preserve">Num. </s>
            <s xml:id="echoid-s15668" xml:space="preserve">1. </s>
            <s xml:id="echoid-s15669" xml:space="preserve">lib. </s>
            <s xml:id="echoid-s15670" xml:space="preserve">5. </s>
            <s xml:id="echoid-s15671" xml:space="preserve">Igitur erit rectangulum ſub diametro
              <lb/>
            AC, & </s>
            <s xml:id="echoid-s15672" xml:space="preserve">circumferentia ABCD, contentum, ad rectangulum ſub diametro E G,
              <lb/>
            & </s>
            <s xml:id="echoid-s15673" xml:space="preserve">circumferentia EFGH, comprehenſum, vt quadratum ex A C, ad quadra-
              <lb/>
            tum ex E G; </s>
            <s xml:id="echoid-s15674" xml:space="preserve">Et permutan do erit rectangulum ſub diametro A C, & </s>
            <s xml:id="echoid-s15675" xml:space="preserve">circumfe-
              <lb/>
            rentia ABCD, ad quadratũ ex AC, vt rectangulum ſub diametro EG, & </s>
            <s xml:id="echoid-s15676" xml:space="preserve">circum-
              <lb/>
            ferentia EF GH, ad quadratũ ex E G. </s>
            <s xml:id="echoid-s15677" xml:space="preserve"> Eſt autem rectangulum ſub A C, & </s>
            <s xml:id="echoid-s15678" xml:space="preserve">
              <note symbol="a" position="right" xlink:label="note-365-01" xlink:href="note-365-01a" xml:space="preserve">1. ſexti.</note>
            cta, quæ circumferentiæ ABCD, ſit æqualis, ad quadratum ex AC, vt recta cir-
              <lb/>
            cumferentiæ æqualis ad A C: </s>
            <s xml:id="echoid-s15679" xml:space="preserve">propterea quod rectangulum, & </s>
            <s xml:id="echoid-s15680" xml:space="preserve">quadratum
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            eandem habent altitu dinem A C. </s>
            <s xml:id="echoid-s15681" xml:space="preserve">Eodemque modo eſt rectangulum ſub E G,
              <lb/>
            & </s>
            <s xml:id="echoid-s15682" xml:space="preserve">recta, quæ circum ferentiæ EFGH, ſit æqualis, ad quadratum ex EG, vt recta
              <lb/>
            circumferentiæ æqualis ad EG. </s>
            <s xml:id="echoid-s15683" xml:space="preserve">Igitur erit, vt circumferentia A B C D, ad dia-
              <lb/>
            metrum A C, ita circumferentia EFGH, ad diametrum EG: </s>
            <s xml:id="echoid-s15684" xml:space="preserve">Et permutando cir-
              <lb/>
            cumferentia ad circumferentiam, vt diameter ad diametrum, quod demon-
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            ſtrandum erat.</s>
            <s xml:id="echoid-s15685" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div960" type="section" level="1" n="340">
          <head xml:id="echoid-head367" xml:space="preserve">SCHOLIVM.</head>
          <p>
            <s xml:id="echoid-s15686" xml:space="preserve">
              <emph style="sc">Svnt</emph>
            qui putent, fruſtrà à Pappo hoc theorema demonſtrari, cum videatur
              <lb/>
            eſſe per ſe notum, ita eſſe circumferentiam cuiuſuis circuli ad ſuam diametrum,
              <lb/>
            vt eſt circumferentia alterius circuli ad ſuam diametrum. </s>
            <s xml:id="echoid-s15687" xml:space="preserve">ac proinde permutan-
              <lb/>
            do eſſe circumferentiam ad circumferentiam, vt eſt diameter ad diametrum.
              <lb/>
            </s>
            <s xml:id="echoid-s15688" xml:space="preserve">Qua in re mirum in modum decipiuntur. </s>
            <s xml:id="echoid-s15689" xml:space="preserve">Cum enim à Ptolomæo (quod & </s>
            <s xml:id="echoid-s15690" xml:space="preserve">à
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            nobis propoſ. </s>
            <s xml:id="echoid-s15691" xml:space="preserve">10. </s>
            <s xml:id="echoid-s15692" xml:space="preserve">Sinuum factum eſt) demonſtretur, maiorem eſſe proportio-
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            nem maioris arcus ad minorem eiuſdem circuli, quam chordæ ad chordam,
              <lb/>
            (quod etiam de arcubus, & </s>
            <s xml:id="echoid-s15693" xml:space="preserve">chordis in circulis inæqualibus verum eſt, niſi ar-
              <lb/>
            cus illi ſimiles ſint, vt in ſequenti Theoremate oſtendemus) quis ſine demon-
              <lb/>
            ſtratione concederet, eandem eſſe proportionem circumferentiæ ad circumfe-
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            rentiã, quæ eſt diametri ad diametrum? </s>
            <s xml:id="echoid-s15694" xml:space="preserve">Quod ſi demonſtratum eſſet, ita eſſe ar-
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            cum cuiuſuis circuli ad ſimilem arcum alterius circuli, vt eſt corda ad chordam,
              <lb/>
            tum demum conſtaret, ita eſſe circumferentiam ad circumferentiam, ac
              <note symbol="b" position="right" xlink:label="note-365-02" xlink:href="note-365-02a" xml:space="preserve">15. quinti.</note>
            de & </s>
            <s xml:id="echoid-s15695" xml:space="preserve">ſemir cumferentiam ad ſemicircumferentiam, vt eſt diameter ad diame-
              <lb/>
            trum: </s>
            <s xml:id="echoid-s15696" xml:space="preserve">propterea quod arcus ſemicirculorum ſimiles ſunt, quorum chordæ ſunt
              <lb/>
            diametri. </s>
            <s xml:id="echoid-s15697" xml:space="preserve">Verum hoc demonſtrari non poteſt, niſi prius demonſtretur, ita eſ-
              <lb/>
            ſe circumferentiam ad circumferentiam, vt eſt diameter ad diametrum, vt in
              <lb/>
            Theoremate ſequenti conſtabit. </s>
            <s xml:id="echoid-s15698" xml:space="preserve">Meritò ergo, & </s>
            <s xml:id="echoid-s15699" xml:space="preserve">non ſine cauſa, theorema
              <lb/>
            præcedens à Pappo fuit demonſtratum.</s>
            <s xml:id="echoid-s15700" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div962" type="section" level="1" n="341">
          <head xml:id="echoid-head368" xml:space="preserve">THEOR. 3. PROPOS. 3.</head>
          <p>
            <s xml:id="echoid-s15701" xml:space="preserve">ARCVS cuiuſuis circuli ad arcum ſimilem alterius circuli eandem
              <lb/>
            habet proportionem, quam chorda ad chordam. </s>
            <s xml:id="echoid-s15702" xml:space="preserve">Et contra arcus
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            candem habentes proportionem, quam chordæ, ſimiles ſunt.</s>
            <s xml:id="echoid-s15703" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s15704" xml:space="preserve">
              <emph style="sc">In</emph>
            figura præ cedentis propoſ. </s>
            <s xml:id="echoid-s15705" xml:space="preserve">ducantur ad diametros perpen diculares P B,
              <lb/>
            QF, ex centris P, Q, diuidentes ſemicirculos in binos quadrantes: </s>
            <s xml:id="echoid-s15706" xml:space="preserve">ſintque ar-
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            cus BI, BK, æquales, quibus ſimiles capiantur FL, FM; </s>
            <s xml:id="echoid-s15707" xml:space="preserve">adeò vt toti arcus </s>
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