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

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          <p>
            <s xml:id="echoid-s3496" xml:space="preserve">
              <pb o="105" file="0129" n="129" rhead=""/>
            recta AE vtranque ſectionem contingat. </s>
            <s xml:id="echoid-s3497" xml:space="preserve">Dico ſectionem ADC, cuius ver-
              <lb/>
            tex D eſt infra alterius verticem B, totam cadere intra ſectionem ABC, hoc
              <lb/>
            eſt ei eſſe inſcriptam, & </s>
            <s xml:id="echoid-s3498" xml:space="preserve">in extremis A, C, ſe mutuò contingere.</s>
            <s xml:id="echoid-s3499" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3500" xml:space="preserve">Nam producta AE vſque ad occurſum cum diametro in E (ſi tamen ap-
              <lb/>
            plicata AC non fuerit diameter circuli, vel Ellipſis, vt ſecunda figura, quo
              <lb/>
              <note symbol="a" position="right" xlink:label="note-0129-01" xlink:href="note-0129-01a" xml:space="preserve">27. ſec.
                <lb/>
              conic. &
                <lb/>
              6. eiuſd.</note>
            in caſu contingentes AE, CE ſibi ipſis, & </s>
            <s xml:id="echoid-s3501" xml:space="preserve">coniugatæ diametro BT æquidi- ſtabunt) iungatur ECO, quæ item vtranque ſectionem continget in C: </s>
            <s xml:id="echoid-s3502" xml:space="preserve">&</s>
            <s xml:id="echoid-s3503" xml:space="preserve">
              <note symbol="b" position="right" xlink:label="note-0129-02" xlink:href="note-0129-02a" xml:space="preserve">59. h.</note>
            applicetur quæcunque L O. </s>
            <s xml:id="echoid-s3504" xml:space="preserve">_lo_ eaſdem ſectiones ſecans in I, N, G, M. </s>
            <s xml:id="echoid-s3505" xml:space="preserve">_i, n,_
              <lb/>
            _g, m_, contingentes verò in L, O. </s>
            <s xml:id="echoid-s3506" xml:space="preserve">_l, o_; </s>
            <s xml:id="echoid-s3507" xml:space="preserve">ducanturque ex verticibus tangentes
              <lb/>
            BQ, DP, quę ordinatim ductis æquidiſtabunt, & </s>
            <s xml:id="echoid-s3508" xml:space="preserve">iungatur AB, ſecans DP
              <lb/>
            in S.</s>
            <s xml:id="echoid-s3509" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3510" xml:space="preserve">Iam cum ſit AH æqualis H C, erit LF. </s>
            <s xml:id="echoid-s3511" xml:space="preserve">_l f_ æqualis F O. </s>
            <s xml:id="echoid-s3512" xml:space="preserve">_f o_, eſtque IF. </s>
            <s xml:id="echoid-s3513" xml:space="preserve">_if_
              <lb/>
            æqualis FN. </s>
            <s xml:id="echoid-s3514" xml:space="preserve">_fn_, & </s>
            <s xml:id="echoid-s3515" xml:space="preserve">GF. </s>
            <s xml:id="echoid-s3516" xml:space="preserve">_gf_, ipſi FM. </s>
            <s xml:id="echoid-s3517" xml:space="preserve">_fm_ (ſunt enim ſectionum ſemi-appli-
              <lb/>
            catæ) quare reliquæ LI. </s>
            <s xml:id="echoid-s3518" xml:space="preserve">_li_, ON. </s>
            <s xml:id="echoid-s3519" xml:space="preserve">_on_, æquales erunt, itemque LG. </s>
            <s xml:id="echoid-s3520" xml:space="preserve">_lg_, OM.
              <lb/>
            </s>
            <s xml:id="echoid-s3521" xml:space="preserve">_o m_ inter ſe æquales, ideoque rectangulum OIL. </s>
            <s xml:id="echoid-s3522" xml:space="preserve">_oil_ æquabitur rectangulo
              <lb/>
            NLI. </s>
            <s xml:id="echoid-s3523" xml:space="preserve">_nli_, & </s>
            <s xml:id="echoid-s3524" xml:space="preserve">rectangulum OGL. </s>
            <s xml:id="echoid-s3525" xml:space="preserve">_ogl_ rectangulo MLG. </s>
            <s xml:id="echoid-s3526" xml:space="preserve">_mlg_. </s>
            <s xml:id="echoid-s3527" xml:space="preserve">Et cum in ſe-
              <lb/>
              <note symbol="c" position="right" xlink:label="note-0129-03" xlink:href="note-0129-03a" xml:space="preserve">16. tertij
                <lb/>
              conic.</note>
            ctione ABC ſit quadratum BQ ad quadratum QA, hoc eſt quadratum SP ad PA, vt rectangulum NLI. </s>
            <s xml:id="echoid-s3528" xml:space="preserve">_nli_ ad quadratum L A. </s>
            <s xml:id="echoid-s3529" xml:space="preserve">_l_A, & </s>
            <s xml:id="echoid-s3530" xml:space="preserve">in ſectione
              <lb/>
              <note symbol="d" position="right" xlink:label="note-0129-04" xlink:href="note-0129-04a" xml:space="preserve">ibidem.</note>
            ADC quadratum DP ad idem PA ſit vt rectangulum MLG. </s>
            <s xml:id="echoid-s3531" xml:space="preserve">_mlg_ ad idem quadratum L A. </s>
            <s xml:id="echoid-s3532" xml:space="preserve">_l_A, habeatque quadratum SP ad PA minorem rationem
              <lb/>
            quàm DP quadratum, ad idem quadratum PA, habebit quoque rectangu-
              <lb/>
            lum NLI. </s>
            <s xml:id="echoid-s3533" xml:space="preserve">_nli_ ad quadratum LA. </s>
            <s xml:id="echoid-s3534" xml:space="preserve">_l_A minorem rationem quàm rectangulum
              <lb/>
            MLG. </s>
            <s xml:id="echoid-s3535" xml:space="preserve">_mlg_ ad idem quadratum LA. </s>
            <s xml:id="echoid-s3536" xml:space="preserve">_l_A; </s>
            <s xml:id="echoid-s3537" xml:space="preserve">quare rectangulum NLI. </s>
            <s xml:id="echoid-s3538" xml:space="preserve">_nli_, hoc
              <lb/>
            eſt OIL. </s>
            <s xml:id="echoid-s3539" xml:space="preserve">_oil_, minus eſt rectangulo MLG. </s>
            <s xml:id="echoid-s3540" xml:space="preserve">_mlg_, ſiue rectangulo OGL. </s>
            <s xml:id="echoid-s3541" xml:space="preserve">_ogl_;
              <lb/>
            </s>
            <s xml:id="echoid-s3542" xml:space="preserve">
              <note symbol="e" position="right" xlink:label="note-0129-05" xlink:href="note-0129-05a" xml:space="preserve">conuerſ.
                <lb/>
              60. h.</note>
            vnde punctum I remotius eſt ab ipſo F quàm pũctum G. </s>
            <s xml:id="echoid-s3543" xml:space="preserve">_g_, ſed I. </s>
            <s xml:id="echoid-s3544" xml:space="preserve">_i_ eſt in ipſa ſectione ABC; </s>
            <s xml:id="echoid-s3545" xml:space="preserve">quare punctum G. </s>
            <s xml:id="echoid-s3546" xml:space="preserve">_g_ ſectionis ADC cadet intra ABC, & </s>
            <s xml:id="echoid-s3547" xml:space="preserve">ſic
              <lb/>
            de quolibet alio puncto ſectionis SADCT, præter A, C: </s>
            <s xml:id="echoid-s3548" xml:space="preserve">vnde ipſa ADC in-
              <lb/>
            ſcripta erit ſectioni ABC, & </s>
            <s xml:id="echoid-s3549" xml:space="preserve">in punctis tantùm A, C extremis eiuſdem ap-
              <lb/>
            plicatæ ſe mutuò contingent. </s>
            <s xml:id="echoid-s3550" xml:space="preserve">Quod erat dei
              <unsure/>
            nonſtrandum.</s>
            <s xml:id="echoid-s3551" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div339" type="section" level="1" n="146">
          <head xml:id="echoid-head151" xml:space="preserve">THEOR. XXXIII. PROP. LXII.</head>
          <p>
            <s xml:id="echoid-s3552" xml:space="preserve">Siextrema inæqualium baſium menſalis, cuiuſcunque coni- ſe-
              <lb/>
            ctionis, vel circuli, ad vtranque diametri partem rectis lineis iun-
              <lb/>
            gantur, ipſæ ſimul, & </s>
            <s xml:id="echoid-s3553" xml:space="preserve">in eodem diametri puncto conuenient, à
              <lb/>
            quo, ſi ad terminos ordinatim ductæ per interſectionem diagona-
              <lb/>
            lis cum diametro, ducantur aliæ rectæ lineæ, hæ omnino ſectionem
              <lb/>
            contingent.</s>
            <s xml:id="echoid-s3554" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3555" xml:space="preserve">SIt menſalis coni-ſectionis, vel circuli ABCD, cuius baſis, AD maior,
              <lb/>
            BC minor, diameter E F. </s>
            <s xml:id="echoid-s3556" xml:space="preserve">Dico ſi iungantur AB, DC, ipſas cum dia-
              <lb/>
            metro, & </s>
            <s xml:id="echoid-s3557" xml:space="preserve">in eodem puncto conuenire, ac ducta diagonali AC ſecant dia-
              <lb/>
            metrum in G, & </s>
            <s xml:id="echoid-s3558" xml:space="preserve">applicata LGM, ſi per extrema puncta L, M, ad prędictum
              <lb/>
            occurſum ducantur rectæ, ipſas ſectionem contingere.</s>
            <s xml:id="echoid-s3559" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s3560" xml:space="preserve">Cum ſit enim AF maior BE, & </s>
            <s xml:id="echoid-s3561" xml:space="preserve">ipſi parallela, occurret AB cum FE ad par-
              <lb/>
              <note symbol="a" position="right" xlink:label="note-0129-06" xlink:href="note-0129-06a" xml:space="preserve">22. pri-
                <lb/>
              mi conic.</note>
            tes B, E, vt in H; </s>
            <s xml:id="echoid-s3562" xml:space="preserve">itemq; </s>
            <s xml:id="echoid-s3563" xml:space="preserve">DC cum eadem FE, vt in I, vtraque verò extra </s>
          </p>
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