Viviani, Vincenzo, De maximis et minimis, geometrica divinatio : in qvintvm Conicorvm Apollonii Pergaei

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        <div xml:id="echoid-div788" type="section" level="1" n="310">
          <head xml:id="echoid-head319" xml:space="preserve">LEMMA XIII. PROP. LXVII.</head>
          <p>
            <s xml:id="echoid-s7658" xml:space="preserve">Si in angulo A B C applicatæ ſint duæ rectæ lineæ D E, A
              <lb/>
            C, quæ ab eadem recta B G per verticem B ducta proportio-
              <lb/>
            naliter ſecentur, ita vt ſit A G ad G C, homologè, vt D F ad
              <lb/>
            F E. </s>
            <s xml:id="echoid-s7659" xml:space="preserve">Dico ipſas A C, D F inter ſe æquidiſtare.</s>
            <s xml:id="echoid-s7660" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7661" xml:space="preserve">SI enim A C non eſt ipſi D E parallela, ſit alia
              <lb/>
              <figure xlink:label="fig-0274-01" xlink:href="fig-0274-01a" number="225">
                <image file="0274-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0274-01"/>
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            applicata A H, ſecans B G in I: </s>
            <s xml:id="echoid-s7662" xml:space="preserve">erit ergo
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            (ob parallelas) A I ad I H, vt D F ad F E; </s>
            <s xml:id="echoid-s7663" xml:space="preserve">vel
              <lb/>
            ob hypotheſim, vt A G ad G C, ergo in trian-
              <lb/>
            gulo A C H erit I G parallela ad H C, ſed ipſæ
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            conueniunt in B. </s>
            <s xml:id="echoid-s7664" xml:space="preserve">Quare non erit alia ex A ipſi
              <lb/>
            D E parallela, quàm A C. </s>
            <s xml:id="echoid-s7665" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s7666" xml:space="preserve">c.</s>
            <s xml:id="echoid-s7667" xml:space="preserve"/>
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        <div xml:id="echoid-div790" type="section" level="1" n="311">
          <head xml:id="echoid-head320" xml:space="preserve">THEOR. XLII. PROP. LXVIII.</head>
          <p>
            <s xml:id="echoid-s7668" xml:space="preserve">Baſes æqualium portionum, ex eodem angulo, ſiue ex eadem
              <lb/>
              <note position="left" xlink:label="note-0274-01" xlink:href="note-0274-01a" xml:space="preserve">Conuer-
                <lb/>
              ſum Pro-
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              p. 45. h.</note>
            quacunque coni- ſectione, vel circulo abſciſſarum, eandem in-
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            ſcriptam eiuſdem nominis ſectionem ſimilem, & </s>
            <s xml:id="echoid-s7669" xml:space="preserve">concentricam
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            ad puncta media contingunt.</s>
            <s xml:id="echoid-s7670" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7671" xml:space="preserve">SInt de angulo rectilineo, vt in prima figura, vel de qualibet alia coni- ſe-
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            ctione, vel circulo, vt in ſecunda, abſciſſæ duæ æquales portiones A
              <lb/>
            B C, D E F, quarum baſes A C, D F ſint bifariam ſectæ in G, H, & </s>
            <s xml:id="echoid-s7672" xml:space="preserve">per G
              <lb/>
            inſcribatur eiuſdem nominis ſectio ſimilis, & </s>
            <s xml:id="echoid-s7673" xml:space="preserve">concentrica exteriori A B
              <note symbol="a" position="left" xlink:label="note-0274-02" xlink:href="note-0274-02a" xml:space="preserve">4. ſec.
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              conic &
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              5 6.7. p. h.</note>
            quæ ſit I G H. </s>
            <s xml:id="echoid-s7674" xml:space="preserve">Dico baſim A C ſectionem I G H contingere in G, & </s>
            <s xml:id="echoid-s7675" xml:space="preserve">ba-
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            ſim quoque D F eandem ſectionem contingere in H.</s>
            <s xml:id="echoid-s7676" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s7677" xml:space="preserve">Iungatur, in prima, B G, & </s>
            <s xml:id="echoid-s7678" xml:space="preserve">producatur, nam ipſa erit diameter Hyper-
              <lb/>
            bolæ I G H (cum ſit B eius centrum) bifariam ſecans omnes in ea applica-
              <lb/>
            tas, quæ ſi vſque ad aſymptotos producantur, erunt, & </s>
            <s xml:id="echoid-s7679" xml:space="preserve">ipſarum ſegmenta
              <lb/>
            inter aſymptotos, & </s>
            <s xml:id="echoid-s7680" xml:space="preserve">ſectionem æqualia inter ſe, quare ſi ipſa
              <note symbol="b" position="left" xlink:label="note-0274-03" xlink:href="note-0274-03a" xml:space="preserve">8. ſecũd.
                <lb/>
              conic.</note>
            concipiantur addita æqualibus ſemi- applicatis in ſectione eis in directum
              <lb/>
            poſitis, prouenient totæ applicatæ in angulo A B E biſariam ſectæ à dia-
              <lb/>
            metro B G producta, ſed ponitur quoque applicata A C bifariam ſecta in
              <lb/>
            G, quare A C ipſis applicatis in ſectione æquidiſtabit, ac ideò
              <note symbol="c" position="left" xlink:label="note-0274-04" xlink:href="note-0274-04a" xml:space="preserve">67. h.</note>
            I G H continget in G.</s>
            <s xml:id="echoid-s7681" xml:space="preserve"/>
          </p>
          <note symbol="d" position="left" xml:space="preserve">32. pri-
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          mi conic.</note>
          <p>
            <s xml:id="echoid-s7682" xml:space="preserve">In ſecunda autem figura quaſcunque coni- ſectiones exhibente ducatur
              <lb/>
            ex G diameter G B, quæ vtriuſque ſectionis A B E, I G H erit communis
              <lb/>
            diameter (cumipſæ ponantur ſectiones concentricæ, &</s>
            <s xml:id="echoid-s7683" xml:space="preserve">c.) </s>
            <s xml:id="echoid-s7684" xml:space="preserve">ad applicatas </s>
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