Huygens, Christiaan, Christiani Hugenii opera varia; Bd. 2: Opera geometrica. Opera astronomica. Varia de optica

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            <s xml:id="echoid-s1541" xml:space="preserve">
              <pb o="372" file="0082" n="87" rhead="CHRISTIANI HUGENII"/>
            ad A G, ita B C ad C F, propter triangulos ſimiles D A G,
              <lb/>
            B C F. </s>
            <s xml:id="echoid-s1542" xml:space="preserve">Erit proinde ut E D ad C B, ita quoque C B ad C F.
              <lb/>
            </s>
            <s xml:id="echoid-s1543" xml:space="preserve">Quod erat demonſtrandum.</s>
            <s xml:id="echoid-s1544" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div83" type="section" level="1" n="39">
          <head xml:id="echoid-head62" xml:space="preserve">
            <emph style="sc">Lemma</emph>
          .</head>
          <p>
            <s xml:id="echoid-s1545" xml:space="preserve">Eſto linea B C diviſa æqualiter in R; </s>
            <s xml:id="echoid-s1546" xml:space="preserve">& </s>
            <s xml:id="echoid-s1547" xml:space="preserve">inæqualiter in F,
              <lb/>
              <note position="left" xlink:label="note-0082-01" xlink:href="note-0082-01a" xml:space="preserve">TAB. XXXIX.
                <lb/>
              Fig. 5.</note>
            ſitque ſegmentum majus F C; </s>
            <s xml:id="echoid-s1548" xml:space="preserve">& </s>
            <s xml:id="echoid-s1549" xml:space="preserve">fiat B O æqualis utrique
              <lb/>
            ſimul B C, C F; </s>
            <s xml:id="echoid-s1550" xml:space="preserve">B M vero utrique B C, C R. </s>
            <s xml:id="echoid-s1551" xml:space="preserve">Dico ma-
              <lb/>
            jorem eſſe rationem R B ad B F, quam triplicatam ejus,
              <lb/>
            quam habet O B ad B M. </s>
            <s xml:id="echoid-s1552" xml:space="preserve">Sumatur enim ipſi O M æqualis
              <lb/>
            utraque harum M L, L P. </s>
            <s xml:id="echoid-s1553" xml:space="preserve">Quoniam igitur M O ipſi R F
              <lb/>
            æqualis eſt, (nam hoc ex conſtructione intelligitur) erit P O
              <lb/>
            tripla ipſius F R. </s>
            <s xml:id="echoid-s1554" xml:space="preserve">Sed & </s>
            <s xml:id="echoid-s1555" xml:space="preserve">B M tripla eſt B R. </s>
            <s xml:id="echoid-s1556" xml:space="preserve">Ergo ut B R
              <lb/>
            ad B M, ita F R ad P O. </s>
            <s xml:id="echoid-s1557" xml:space="preserve">Et permutando ut B R ad F R,
              <lb/>
            ſic B M ad P O. </s>
            <s xml:id="echoid-s1558" xml:space="preserve">Major autem eſt B O quam B M. </s>
            <s xml:id="echoid-s1559" xml:space="preserve">Ergo
              <lb/>
            major erit ratio B O ad O P, quam B R ad R F: </s>
            <s xml:id="echoid-s1560" xml:space="preserve">& </s>
            <s xml:id="echoid-s1561" xml:space="preserve">per
              <lb/>
            converſionem rationis minor O B ad B P, quam R B ad
              <lb/>
            B F. </s>
            <s xml:id="echoid-s1562" xml:space="preserve">Porro quoniam æquales ſunt O M, M L, major erit
              <lb/>
            ratio B O ad O M, quam B M ad M L: </s>
            <s xml:id="echoid-s1563" xml:space="preserve">& </s>
            <s xml:id="echoid-s1564" xml:space="preserve">per converſio-
              <lb/>
            nem rationis minor O B ad B M, quam M B ad B L. </s>
            <s xml:id="echoid-s1565" xml:space="preserve">Eo-
              <lb/>
            dem modo minor adhuc oſtendetur ratio M B ad B L, quam
              <lb/>
            L B ad B P. </s>
            <s xml:id="echoid-s1566" xml:space="preserve">Itaque omnino ratio triplicata ejus quam ha-
              <lb/>
            bet O B ad B M minor erit quam compoſita ex rationibus
              <lb/>
            O B ad B M, B M ad B L, & </s>
            <s xml:id="echoid-s1567" xml:space="preserve">B L ad B P, hoc eſt,
              <lb/>
            quam ratio O B ad B P. </s>
            <s xml:id="echoid-s1568" xml:space="preserve">Major autem erat R B ad B F,
              <lb/>
            quam O B ad B P. </s>
            <s xml:id="echoid-s1569" xml:space="preserve">Ergo omnino major erit ratio R B ad
              <lb/>
            B F, quam triplicata rationis O B ad B M. </s>
            <s xml:id="echoid-s1570" xml:space="preserve">Quod erat pro-
              <lb/>
            poſitum.</s>
            <s xml:id="echoid-s1571" xml:space="preserve"/>
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        <div xml:id="echoid-div85" type="section" level="1" n="40">
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            <emph style="sc">Theor</emph>
          . XI.
            <emph style="sc">Prop</emph>
          . XIV.</head>
          <p style="it">
            <s xml:id="echoid-s1572" xml:space="preserve">
              <emph style="bf">O</emph>
            Mnis circuli circumferentia minor eſt minore
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            duarum mediarum proportionalium inter peri-
              <lb/>
            metros polygonorum ſimilium, quorum alterum or-
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            dinate circulo inſcriptum ſit, alterum </s>
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