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

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          <pb o="2" file="0022" n="22" rhead=""/>
          <p>
            <s xml:id="echoid-s351" xml:space="preserve">Itaque, quoniam rectangulum BKC ad quadratum AK eſt vt LF ad FH
              <lb/>
            per conſtrutionem, vel vt XN ad NH, & </s>
            <s xml:id="echoid-s352" xml:space="preserve">quadratum AK ad rectangulum
              <lb/>
            AKC eſt vt AK ad KC, vel HG ad GC, vel HN ad NS, ergo ex æqualire-
              <lb/>
            ctangulum BKC ad rectangulum AKC, ſiue recta BK ad KA, ſiue BG ad
              <lb/>
            GF, vel RN ad NF, eſt vt XN ad NS, ac propterea rectangulum ſub extre-
              <lb/>
            mis RN, NS, hoc eſt quadratum MN æquale rectangulo ſub medijs XN, NF:
              <lb/>
            </s>
            <s xml:id="echoid-s353" xml:space="preserve">_linea igitur MN poteſt ſpatium XF, & </s>
            <s xml:id="echoid-s354" xml:space="preserve">c._ </s>
            <s xml:id="echoid-s355" xml:space="preserve">vt ibi vſque ad finem.</s>
            <s xml:id="echoid-s356" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s357" xml:space="preserve">Quo tandem ad 13. </s>
            <s xml:id="echoid-s358" xml:space="preserve">primi poſt ea verba _ergo rectangulum PMR æquale eſt_
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            _LM quadrato_ legatur ſic.</s>
            <s xml:id="echoid-s359" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s360" xml:space="preserve">Cumque ſit rectangulum BKC ad quadratum AK ita HE ad ED ex con-
              <lb/>
            ſtrutione, vel XM ad MD, & </s>
            <s xml:id="echoid-s361" xml:space="preserve">vt quadratum AK ad rectangulum AKC ita
              <lb/>
            AK ad KC, vel DG ad GC, vel vt DM ad MR, erit ex æquo rectangulum
              <lb/>
            BKC ad rectangulum AKC, vel BK ad KA, ſiue BG ad GE, vel PM ad ME
              <lb/>
            vt XM ad MR, quare rectangulum ſub extremis PM, MR, vel quadratum
              <lb/>
            ML æquatur rectangulo XME ſub medijs. </s>
            <s xml:id="echoid-s362" xml:space="preserve">_Liuea igitur LM poteſt ſpatinm_
              <lb/>
            _MO &</s>
            <s xml:id="echoid-s363" xml:space="preserve">c._ </s>
            <s xml:id="echoid-s364" xml:space="preserve">vſque ad finem.</s>
            <s xml:id="echoid-s365" xml:space="preserve"/>
          </p>
          <p style="it">
            <s xml:id="echoid-s366" xml:space="preserve">Sed iam ad propoſitas Apollonij propoſitiones accedamus, quas ſimul ſequenti
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            Theoremate amplectemur, itemque ſine compoſita proportione demonſtrabimus.</s>
            <s xml:id="echoid-s367" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div12" type="section" level="1" n="11">
          <head xml:id="echoid-head15" xml:space="preserve">THEOR. I. PROP. I.</head>
          <p>
            <s xml:id="echoid-s368" xml:space="preserve">Si conus plano per axem fecetur, fecetur autem & </s>
            <s xml:id="echoid-s369" xml:space="preserve">altero plano
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            baſi coni non æquidiſtante, quorum communis ſectio conueniat,
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              <note position="left" xlink:label="note-0022-01" xlink:href="note-0022-01a" xml:space="preserve">Prop. 11.
                <lb/>
              12. 13.
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              primi co-
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              nic.</note>
            vel cum vnotantum, vel cum vtroque latere trianguli per axem vl-
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            tra, vel infra ſui ipſius verticem, planum verò, in quo eſt baſis co-
              <lb/>
            ni, & </s>
            <s xml:id="echoid-s370" xml:space="preserve">ſecans planum, conueniant ſecundum rectam lineam, quæ ſit
              <lb/>
            perpendicularis, vel ad baſim trianguli per axem, vel ad eam, quæ
              <lb/>
            indirectum ipſi conſtituitur, & </s>
            <s xml:id="echoid-s371" xml:space="preserve">fiat, vt rectangulum ſegmentorum
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            diametri ſectionis inter latera, & </s>
            <s xml:id="echoid-s372" xml:space="preserve">baſim trianguli per axem interce-
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            ptorum, ad rectangulum ſegmentorum baſis, ita ſectionis diameter
              <lb/>
            ad aliam: </s>
            <s xml:id="echoid-s373" xml:space="preserve">recta linea, quę à ſectione coni ducitur æquidiſtans com-
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            muni ſectioni plani ſecantis, & </s>
            <s xml:id="echoid-s374" xml:space="preserve">baſis coni vſque ad ſectionis diame-
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            trum, poterit rectangulum adiacens lineæ quarto loco inuentæ, la-
              <lb/>
            titudinem habens lineam, quæ ex diametro abſcinditur inter ipſam,
              <lb/>
            & </s>
            <s xml:id="echoid-s375" xml:space="preserve">verticem ſectionis interiectam (ſi tamen ſectionis diameter ęqui-
              <lb/>
            diſtet alterutri laterum triãguli per axem) ſed ipſum excedet (ſi cum
              <lb/>
            vtroque latere vltra verticẽ conueniat) vel ab eo deficiet, (ſi ijſdem
              <lb/>
            lateribus infra verticem occurrat) rectangulo ſimili ſimiliterque po-
              <lb/>
            ſito ei, quod continetur prædicto diametri ſegmento, & </s>
            <s xml:id="echoid-s376" xml:space="preserve">quarta in-
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            uenta, iuxta quam poſſunt, quæ ad diametrum applicantur.</s>
            <s xml:id="echoid-s377" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s378" xml:space="preserve">SIt conus, cuius vertex A, baſis circulus BC, & </s>
            <s xml:id="echoid-s379" xml:space="preserve">ſecetur plano per axem,
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            quod ſectionem faciat triangulum B A C, ſecetur autem & </s>
            <s xml:id="echoid-s380" xml:space="preserve"/>
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