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

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        <div xml:id="echoid-div227" type="section" level="1" n="104">
          <head xml:id="echoid-head109" xml:space="preserve">THEOR. XXI. PROP. XXXXI.</head>
          <p>
            <s xml:id="echoid-s2409" xml:space="preserve">Similes Hyperbolæ per eundem verticem ſimul adſcriptæ, ſunt
              <lb/>
            inter ſe nunquam coeuntes, & </s>
            <s xml:id="echoid-s2410" xml:space="preserve">ſemper magis recedentes, ſed ad in-
              <lb/>
            teruallum nunquam perueniunt æquale cuidam dato interuallo.</s>
            <s xml:id="echoid-s2411" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2412" xml:space="preserve">SInt duæ ſimiles Hyperbolæ ABC, DBE per eundem verticem B ſimul
              <lb/>
            adſcriptæ, & </s>
            <s xml:id="echoid-s2413" xml:space="preserve">Hyperbolæ ABC maiora ſint latera, tranſuerſum nempe
              <lb/>
            FB, rectũ autem BG; </s>
            <s xml:id="echoid-s2414" xml:space="preserve">& </s>
            <s xml:id="echoid-s2415" xml:space="preserve">DBE minora ſint, tranſuerſum HB, rectum verò BI.
              <lb/>
            </s>
            <s xml:id="echoid-s2416" xml:space="preserve">Patet primùm has ſectiones eſſe inter ſe nunquam coeuntes;</s>
            <s xml:id="echoid-s2417" xml:space="preserve">cum enim DBE
              <lb/>
            alteri ABC ſit inſcripta, ipſæ, licet in infinitum producantur,
              <note symbol="a" position="left" xlink:label="note-0094-01" xlink:href="note-0094-01a" xml:space="preserve">5. Co-
                <lb/>
              roll. 19. h.</note>
            conuenient.</s>
            <s xml:id="echoid-s2418" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2419" xml:space="preserve">Dico ampliùs, eaſdem in-
              <lb/>
              <figure xlink:label="fig-0094-01" xlink:href="fig-0094-01a" number="64">
                <image file="0094-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0094-01"/>
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            ter ſe longiùs ſemper recede-
              <lb/>
            re. </s>
            <s xml:id="echoid-s2420" xml:space="preserve">Applicatis enim in alte-
              <lb/>
            ra ſectionum, nempe in in-
              <lb/>
            ſcripta, duabus vbicunque
              <lb/>
            rectis MN, LD, fiat vt BH ad
              <lb/>
            BF, ita BM ad BQ, & </s>
            <s xml:id="echoid-s2421" xml:space="preserve">BL ad
              <lb/>
            BR, & </s>
            <s xml:id="echoid-s2422" xml:space="preserve">per Q, R, applicentur
              <lb/>
            in circumſcripta Hyperbola
              <lb/>
            rectæ QS, RA; </s>
            <s xml:id="echoid-s2423" xml:space="preserve">& </s>
            <s xml:id="echoid-s2424" xml:space="preserve">cum dia-
              <lb/>
            metrorum ſegmẽta BM, BQ,
              <lb/>
            BL, BR ſint tranſuerſis BH,
              <lb/>
            B F proportionalia,
              <note symbol="b" position="left" xlink:label="note-0094-02" xlink:href="note-0094-02a" xml:space="preserve">38. h.</note>
            quoque applicatæ MN, QS;
              <lb/>
            </s>
            <s xml:id="echoid-s2425" xml:space="preserve">LD, RA ijſdẽ lateribus pro-
              <lb/>
            portionales, quare M N ad
              <lb/>
            QS erit vt LD ad RA; </s>
            <s xml:id="echoid-s2426" xml:space="preserve">cum-
              <lb/>
            que ſit vt tota BL ad totam BR, ita pars BM ad partem BQ, erit & </s>
            <s xml:id="echoid-s2427" xml:space="preserve">reliqua
              <lb/>
            ML ad reliquam QR, vt tota BL ad totam BR, vel vt BH ad BF, vel vt MN
              <lb/>
            ad QS, & </s>
            <s xml:id="echoid-s2428" xml:space="preserve">vt LD ad RA, vt nuper oſtendimus; </s>
            <s xml:id="echoid-s2429" xml:space="preserve">quare iunctis rectis DN, AS
              <lb/>
            in menſalibus DM, AQ, erunt ipsæ DN, AS inter ſe parallelæ. </s>
            <s xml:id="echoid-s2430" xml:space="preserve">Iam
              <note symbol="c" position="left" xlink:label="note-0094-03" xlink:href="note-0094-03a" xml:space="preserve">39. h.</note>
            ductis MN, LD vſque ad circumſcriptam ſectionem ABC, in P, & </s>
            <s xml:id="echoid-s2431" xml:space="preserve">O, ſi iun-
              <lb/>
            gatur PO, hæc omninò ſecabit iunctam AS, vel intra ipſam ſectionem; </s>
            <s xml:id="echoid-s2432" xml:space="preserve">(ſi
              <lb/>
            nempe vnius iunctarum ſectioni occurſus, alterius occurſibus contineatur)
              <lb/>
            vel extra (ſi nullius occurſus, alterius occurſibus amplectatur) ſed AS
              <note symbol="d" position="left" xlink:label="note-0094-04" xlink:href="note-0094-04a" xml:space="preserve">25. ſe-
                <lb/>
              cundi co-
                <lb/>
              nic</note>
            ducta ad partes verticis tota cadit extra ſectionem in SK, & </s>
            <s xml:id="echoid-s2433" xml:space="preserve">punctum P eſt in ipſa ſectione, quare punctum P eſt inter parallelas lineas ASK, DN, ſed
              <lb/>
              <note symbol="e" position="left" xlink:label="note-0094-05" xlink:href="note-0094-05a" xml:space="preserve">10. pri-
                <lb/>
              mi. conic.</note>
            producta PO conuenit cum altera parallelarum AS, vt modò monitum fuit;
              <lb/>
            </s>
            <s xml:id="echoid-s2434" xml:space="preserve">taliſq; </s>
            <s xml:id="echoid-s2435" xml:space="preserve">occurſus eſt omnino ad partes O infra applicatam PN, cum punctum
              <lb/>
            S cadat infra P (nam ex ipſa conſtructione applicata QS eſt infra applicatam
              <lb/>
            MP) quare eadem OP producta conueniet quoque cum altera æquidiſtan-
              <lb/>
            tium DN, ad oppoſitas tamen partes, vtputa ſupra ipſam applicatam PN,
              <lb/>
            vnde intercepta applicata OD, maior erit intercepta PN vertici B </s>
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