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

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          <p>
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              <pb o="21" file="0041" n="41" rhead=""/>
            plicetur NMO fectionem, ac diametrum ſecans in N, O.</s>
            <s xml:id="echoid-s796" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s797" xml:space="preserve">Quoniam igitur eodem pænitus argumento, quo ſuperius demonſtratum
              <lb/>
            eſt rectangulum AHB ad quadratum HI, eſſe vt quadratum CB ad BD, eſt
              <lb/>
            quoque rectangulum AOB ad quadratum ON, vt idem quadratum C B ad
              <lb/>
            BD, vel vt quadratum BO ad OM, erit permutando, rectangulum AOB ad
              <lb/>
            quadratum BO, vt quadratum NO ad OM, ſed rectangulum AOB ſuperat
              <lb/>
            quadratum BO, (exceſſus enim eſt rectangulum ABO) ergo & </s>
            <s xml:id="echoid-s798" xml:space="preserve">quadratum
              <lb/>
            NO, maius eſt quadrato MO; </s>
            <s xml:id="echoid-s799" xml:space="preserve">ſed punctum N eſt in ipſa ſectione, quare pun-
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            ctum M cadit intra: </s>
            <s xml:id="echoid-s800" xml:space="preserve">ideoque iuncta CM ſectionem prius ſecat. </s>
            <s xml:id="echoid-s801" xml:space="preserve">Non eſt ergo
              <lb/>
            altera aſymptotos, quæ diuidat angulum ab aſymptotis factum. </s>
            <s xml:id="echoid-s802" xml:space="preserve">Quod erat
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            ſecundò demonſtrandum.</s>
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        <div xml:id="echoid-div59" type="section" level="1" n="35">
          <head xml:id="echoid-head40" xml:space="preserve">MONITVM.</head>
          <p style="it">
            <s xml:id="echoid-s804" xml:space="preserve">HIs itaque præoſtenſis, ipſarum ope, ac tertiæ ſecundi conico-
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            rum demonſtremus aliter decimam quartam eiuſdem, abſq;
              <lb/>
            </s>
            <s xml:id="echoid-s805" xml:space="preserve">auxilio præcedentium 5. </s>
            <s xml:id="echoid-s806" xml:space="preserve">10. </s>
            <s xml:id="echoid-s807" xml:space="preserve">12. </s>
            <s xml:id="echoid-s808" xml:space="preserve">ac 13. </s>
            <s xml:id="echoid-s809" xml:space="preserve">quibus ipſa 14. </s>
            <s xml:id="echoid-s810" xml:space="preserve">in-
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            diget, præmiſſo tantum ſequenti Lemmate.</s>
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        <div xml:id="echoid-div60" type="section" level="1" n="36">
          <head xml:id="echoid-head41" xml:space="preserve">LEMMA II. PROP. IX.</head>
          <p>
            <s xml:id="echoid-s812" xml:space="preserve">Sit rectangulum ABD æquale quadrato BC. </s>
            <s xml:id="echoid-s813" xml:space="preserve">Dico addita qua-
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            cunque BE, rectangulum AED maius eſſe quadrato EC.</s>
            <s xml:id="echoid-s814" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s815" xml:space="preserve">CVm enim rectangulum ABD æquale ſit quadrato mediæ BC, erit AB
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            ad BC, vt BC ad BD, & </s>
            <s xml:id="echoid-s816" xml:space="preserve">diuidendo, & </s>
            <s xml:id="echoid-s817" xml:space="preserve">permutando AC ad CD, vt
              <lb/>
              <figure xlink:label="fig-0041-01" xlink:href="fig-0041-01a" number="17">
                <image file="0041-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0041-01"/>
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            CB ad BD. </s>
            <s xml:id="echoid-s818" xml:space="preserve">Et cum ſit DB minor
              <lb/>
            DE, habebit CD ad DB maiorem
              <lb/>
            rationem quam ad DE, & </s>
            <s xml:id="echoid-s819" xml:space="preserve">compo-
              <lb/>
            nendo CB ad BD, hoc eſt AC ad CD maiorem habebit rationem
              <note symbol="a" position="right" xlink:label="note-0041-01" xlink:href="note-0041-01a" xml:space="preserve">28. quin-
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              ti elem.</note>
            CE ad ED, & </s>
            <s xml:id="echoid-s820" xml:space="preserve">permutando AC ad CE maiorem rationem quam CD
              <note symbol="b" position="right" xlink:label="note-0041-02" xlink:href="note-0041-02a" xml:space="preserve">27. quin-
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              ti elem.</note>
            DE, & </s>
            <s xml:id="echoid-s821" xml:space="preserve">componendo AE ad EC maiorem quam EC ad ED. </s>
            <s xml:id="echoid-s822" xml:space="preserve">Si fiat ergo vt AE ad EC, ita EC ad EF, habebit quoque EC ad EF maiorem rationem
              <lb/>
              <note symbol="c" position="right" xlink:label="note-0041-03" xlink:href="note-0041-03a" xml:space="preserve">28. quin-
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              ti elem.</note>
            quam EC ad ED, vnde EF erit minor ED, ſed (cum factum ſit AE ad EC,
              <lb/>
            vt EC ad EF) rectangulum AEF æquale eſt quadrato EC, quare rectangu-
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            lum AED maius erit quadrato EC. </s>
            <s xml:id="echoid-s823" xml:space="preserve">Quod erat &</s>
            <s xml:id="echoid-s824" xml:space="preserve">c.</s>
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        <div xml:id="echoid-div62" type="section" level="1" n="37">
          <head xml:id="echoid-head42" xml:space="preserve">THEOR. III. PROP. X.</head>
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            <s xml:id="echoid-s826" xml:space="preserve">Aſymptoti, & </s>
            <s xml:id="echoid-s827" xml:space="preserve">ſectio in infinitum productæ ad ſe propius acce-
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              <note position="right" xlink:label="note-0041-04" xlink:href="note-0041-04a" xml:space="preserve">Prop. 14.
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              ſec. con.</note>
            dunt, & </s>
            <s xml:id="echoid-s828" xml:space="preserve">ad interuallum perueniunt minus quolibet dato interuallo.</s>
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          <p>
            <s xml:id="echoid-s830" xml:space="preserve">SIt Hyperbole, cuius aſymptoti CD, CE, & </s>
            <s xml:id="echoid-s831" xml:space="preserve">datum interuallum ſit M.
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            </s>
            <s xml:id="echoid-s832" xml:space="preserve">Dico aſymptotos CD, CE, & </s>
            <s xml:id="echoid-s833" xml:space="preserve">ſectionem productas, ad ſe ſe propius
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            accedere, & </s>
            <s xml:id="echoid-s834" xml:space="preserve">ad interuallum peruenire minus dato interuallo M.</s>
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