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

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          <p>
            <s xml:id="echoid-s4393" xml:space="preserve">
              <pb o="131" file="0155" n="155" rhead=""/>
            ALCO eſt _MINIMA_ circumſcripta datæ Ellipſi ABCO, per terminos ap-
              <lb/>
            plicatæ AC, cum dato tranſuerſo DE: </s>
            <s xml:id="echoid-s4394" xml:space="preserve">immo ipſa ALCN vnica eſt, his con-
              <lb/>
            ditionibus circumſcriptibilis. </s>
            <s xml:id="echoid-s4395" xml:space="preserve">Quod faciendum, & </s>
            <s xml:id="echoid-s4396" xml:space="preserve">demonſtrandum erat.</s>
            <s xml:id="echoid-s4397" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div428" type="section" level="1" n="179">
          <head xml:id="echoid-head184" xml:space="preserve">SCHOLIVM.</head>
          <p>
            <s xml:id="echoid-s4398" xml:space="preserve">SIquæratur, qua nam ratione in prop. </s>
            <s xml:id="echoid-s4399" xml:space="preserve">82. </s>
            <s xml:id="echoid-s4400" xml:space="preserve">ad finem, dicatur _licet minor fue-_
              <lb/>
            _rit eadem ALCN_ in hac verò, _licet maior fuerit eadem ALCN_ (perinde ac
              <lb/>
            ſi, per terminos A, C, cum diametro æquali ipſi LN alia in ea deſcribi poſſit
              <lb/>
            Ellipſis minor ALCN, in hac verò alia maior ALCN) vtrunq; </s>
            <s xml:id="echoid-s4401" xml:space="preserve">noshaud te-
              <lb/>
            merè dixiſſe ex ſequéti Theoremate manifeſtum fiet, à quo habebitur quam-
              <lb/>
            libet aliam Ellipſim per A, C, adſcriptam, cum tranſuerſo ęquali ipſi LN, ſed
              <lb/>
            cuius ſegmenta ab applicata AC abſciſſa, ſint magis inæqualia quàm ſint ſe-
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            gmenta NF, FL, minorem eſſe ipſa ALCN; </s>
            <s xml:id="echoid-s4402" xml:space="preserve">& </s>
            <s xml:id="echoid-s4403" xml:space="preserve">è contra, eam quę cum ſegmentis
              <lb/>
            minus inæqualibus, quàm ſint NF, FL, eadem ALCN maiorem eſſe.</s>
            <s xml:id="echoid-s4404" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div429" type="section" level="1" n="180">
          <head xml:id="echoid-head185" xml:space="preserve">THEOR. XL. PROP. LXXXV.</head>
          <p>
            <s xml:id="echoid-s4405" xml:space="preserve">Ellipſium, perterminos communis applicatæ ſimul adſcripta-
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            rum, & </s>
            <s xml:id="echoid-s4406" xml:space="preserve">quarum tranſuerſa latera ſint æqualia, MINIMA eſt ea,
              <lb/>
            cuius communis ordinatim ducta ſit diameter coniugata: </s>
            <s xml:id="echoid-s4407" xml:space="preserve">aliarum
              <lb/>
            verò illa, cuius ſegmenta diametri ſunt minùs inæqualia, minor eſt
              <lb/>
            ea, cuius diametri ſegmenta ſunt magis inæqualia.</s>
            <s xml:id="echoid-s4408" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4409" xml:space="preserve">SInt duę Ellipſes ABCD, AECF, per terminos eiuſdem applicatæ AC
              <lb/>
            ſimul adſcriptæ, & </s>
            <s xml:id="echoid-s4410" xml:space="preserve">quarum tranſuerſa BD, EF ſint æqualia, ſitq; </s>
            <s xml:id="echoid-s4411" xml:space="preserve">AGC
              <lb/>
            coniugata diameter Ellipſis ABCD, ſiue G eius centrum. </s>
            <s xml:id="echoid-s4412" xml:space="preserve">Dico primùm
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            hanc minorem eſſe altera AECF, ſiue eſſe _MINIMAM_, &</s>
            <s xml:id="echoid-s4413" xml:space="preserve">c.</s>
            <s xml:id="echoid-s4414" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4415" xml:space="preserve">Etenim, cum ſit DB æqualis EF, & </s>
            <s xml:id="echoid-s4416" xml:space="preserve">DB
              <lb/>
              <figure xlink:label="fig-0155-01" xlink:href="fig-0155-01a" number="122">
                <image file="0155-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0155-01"/>
              </figure>
            bifariam ſecta in G, erit EF in pũcto Ginæ-
              <lb/>
            qualiter ſecta, vnde rectangulum BGD ma-
              <lb/>
            ius erit rectangulo EGF, cum ſit
              <note symbol="a" position="right" xlink:label="note-0155-01" xlink:href="note-0155-01a" xml:space="preserve">60. h.</note>
            _MVM_; </s>
            <s xml:id="echoid-s4417" xml:space="preserve">ideoque rectangulum BGD ad qua-
              <lb/>
            dratum AG, ſiue tranſuerſum BD ad
              <note symbol="b" position="right" xlink:label="note-0155-02" xlink:href="note-0155-02a" xml:space="preserve">21. pri-
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              mi conic.</note>
            ctum Ellipſis ABCD, maiorem habebit ra-
              <lb/>
            tionem quàm rectangulum EGF ad idem
              <lb/>
            quadratum AG, ſiue quàm
              <note symbol="c" position="right" xlink:label="note-0155-03" xlink:href="note-0155-03a" xml:space="preserve">ibidem.</note>
            EF ad rectum Ellipſis AECF: </s>
            <s xml:id="echoid-s4418" xml:space="preserve">ſed tranſuerſa
              <lb/>
            BD, EF ſunt æqualia, ergo rectũ Ellipſis AB
              <lb/>
            CD, minus erit recto AECF:</s>
            <s xml:id="echoid-s4419" xml:space="preserve">ſi igitur Ellipſis
              <lb/>
            huiuſmodi Ellipſes (cum ſint ęqualiter incli-
              <lb/>
            natæ) concipiantur eſſe per eundem verticem ſimul adſcriptæ, ita vt tranſ-
              <lb/>
            uerſæ diametri ſimul congruant, ipſa ABCD, cuius rectum minus eſt, inſcri-
              <lb/>
            pta erit, ſiue minor AECF, cuius rectum maius eſt, & </s>
            <s xml:id="echoid-s4420" xml:space="preserve">ſic minor
              <note symbol="d" position="right" xlink:label="note-0155-04" xlink:href="note-0155-04a" xml:space="preserve">2. Co-
                <lb/>
              roll. 19. h.</note>
            alia, cuius diametri ſegmenta ſint inæqualia: </s>
            <s xml:id="echoid-s4421" xml:space="preserve">quare ABCD erit _MINI-_
              <lb/>
            _MA_, &</s>
            <s xml:id="echoid-s4422" xml:space="preserve">c.</s>
            <s xml:id="echoid-s4423" xml:space="preserve"/>
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