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

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          <p>
            <s xml:id="echoid-s1368" xml:space="preserve">
              <pb o="39" file="0063" n="63" rhead=""/>
            rum ex eodem diametri puncto F: </s>
            <s xml:id="echoid-s1369" xml:space="preserve">idemque oſtendetur de omnibus alijs ex-
              <lb/>
            tremis punctis communium applicatarum ad vtraſque diametri partes: </s>
            <s xml:id="echoid-s1370" xml:space="preserve">qua-
              <lb/>
            re huiuſmodi ſectiones erunt in totum congruentes: </s>
            <s xml:id="echoid-s1371" xml:space="preserve">eruntque eiuſdem no-
              <lb/>
            minis; </s>
            <s xml:id="echoid-s1372" xml:space="preserve">quoniam cum regula Parabolæ æquidiſtet diametro; </s>
            <s xml:id="echoid-s1373" xml:space="preserve">Hyperbolæ au-
              <lb/>
            tem conueniat cum diametro extra ſectionem; </s>
            <s xml:id="echoid-s1374" xml:space="preserve">Ellipſis verò eidem diametro
              <lb/>
            intra ſectionem occurrat, hoc eſt ad extremum tranſuerſi lateris, cumque
              <lb/>
            harum ſectionum diametri ſimul congruant (nam ſectiones ſunt ſimul adſcri-
              <lb/>
            ptæ) ſi diuerſi nominis eſſent ipſarum regulæ nunquam congruerent, quod
              <lb/>
            eſt contra hypoteſim. </s>
            <s xml:id="echoid-s1375" xml:space="preserve">Sunt ergo tales ſectiones congruentes ſimul, & </s>
            <s xml:id="echoid-s1376" xml:space="preserve">eiuſ-
              <lb/>
            dem nominis. </s>
            <s xml:id="echoid-s1377" xml:space="preserve">Quod primò, &</s>
            <s xml:id="echoid-s1378" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1379" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1380" xml:space="preserve">Si verò regulæ GOI, HPL infra contingentem BGH nunquam conueniũt,
              <lb/>
            diſiunctim ſimul procedentes, vt in 26. </s>
            <s xml:id="echoid-s1381" xml:space="preserve">proximè ſubſequentibus figuris ap-
              <lb/>
            paret, in quarum primis quatuor, regulæ ſunt parallelæ, in reliquis autem à
              <lb/>
            contingente BGH ad partes ſectionum ſunt ſemper inter ſe recedentes, eſtq;
              <lb/>
            </s>
            <s xml:id="echoid-s1382" xml:space="preserve">regula GOI propinquior diametro quàm HPL; </s>
            <s xml:id="echoid-s1383" xml:space="preserve">facta eadem conſtructione,
              <lb/>
            vt ſupra; </s>
            <s xml:id="echoid-s1384" xml:space="preserve">quoniam latitudo FO minor eſt latitudine FP, & </s>
            <s xml:id="echoid-s1385" xml:space="preserve">altitudo BF eſt ea-
              <lb/>
            dem, erit rectangulum BFO ſiue quadratum applicatæ NF in
              <note symbol="a" position="right" xlink:label="note-0063-01" xlink:href="note-0063-01a" xml:space="preserve">Coroll.
                <lb/>
              prop. I. h.</note>
            DBE, maius rectangulo BFP ſiue quadrato applicatæ MF in ſectione
              <note symbol="b" position="right" xlink:label="note-0063-02" xlink:href="note-0063-02a" xml:space="preserve">Coroll.
                <lb/>
              prop. I. h.</note>
            C, hoc eſt applicata NF erit minor ipſa MF: </s>
            <s xml:id="echoid-s1386" xml:space="preserve">quare punctum m ſectionis AB
              <lb/>
            C cadit extra ſectionem DBE: </s>
            <s xml:id="echoid-s1387" xml:space="preserve">idemque de omnibus alijs punctis ſectionis
              <lb/>
            ABC ad vtranque diametri partem. </s>
            <s xml:id="echoid-s1388" xml:space="preserve">Vnde tota ſectio ABC cadit extra ſe-
              <lb/>
            ctionem DBE; </s>
            <s xml:id="echoid-s1389" xml:space="preserve">ideoq; </s>
            <s xml:id="echoid-s1390" xml:space="preserve">tales ſectiones ſunt in totum diſiunctæ (eò quod ſem-
              <lb/>
            per diſiunctim procedant ipſarum regulæ) & </s>
            <s xml:id="echoid-s1391" xml:space="preserve">in communi tantùm vertice B
              <lb/>
            ſe mutuò contingunt. </s>
            <s xml:id="echoid-s1392" xml:space="preserve">Quod ſecundò, &</s>
            <s xml:id="echoid-s1393" xml:space="preserve">c.</s>
            <s xml:id="echoid-s1394" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s1395" xml:space="preserve">Sitandem ſectionum regulę GOI, HPL infra contingentem BGH ad par-
              <lb/>
            tes ſectionum ſe mutuò ſecant in P, vt videre eſt in 9. </s>
            <s xml:id="echoid-s1396" xml:space="preserve">vltimis figuris; </s>
            <s xml:id="echoid-s1397" xml:space="preserve">duca-
              <lb/>
            tur ex P communis ſectionum applicata PFNM ſecans diametrum in F, ſe-
              <lb/>
            ctionem ABC in M, & </s>
            <s xml:id="echoid-s1398" xml:space="preserve">DBE in N. </s>
            <s xml:id="echoid-s1399" xml:space="preserve">Iam cum in ſectione ABC quadratum
              <lb/>
            applicatæ MF æquale ſit rectangulo BFP, & </s>
            <s xml:id="echoid-s1400" xml:space="preserve">quadratum applicatæ NF
              <note symbol="c" position="right" xlink:label="note-0063-03" xlink:href="note-0063-03a" xml:space="preserve">Coroll.
                <lb/>
              prop. I. h.</note>
            ſectione DBE æquale ſit eidem rectangulo BFP, erunt quadrata MF, NF in-
              <lb/>
            ter ſe æqualia, hoc eſt ipſæ applicatæ æquales; </s>
            <s xml:id="echoid-s1401" xml:space="preserve">quare huiuſmodi ſectiones
              <lb/>
            conueniunt ſimul in puncto M. </s>
            <s xml:id="echoid-s1402" xml:space="preserve">Eadem omnino ratione oſtendetur has ſe-
              <lb/>
            ctiones ad alteram quoque diametri partem ſimul conuenire in extremo pũ-
              <lb/>
            cto R reliquæ ad vnam ſectionum applicatæ ex eodem diametri puncto F:
              <lb/>
            </s>
            <s xml:id="echoid-s1403" xml:space="preserve">ergo in duobus punctis M & </s>
            <s xml:id="echoid-s1404" xml:space="preserve">R, præter in communi vertice B, ſimul conue-
              <lb/>
            niunt, in quibus patet has ſectiones ſe mutuò ſecare; </s>
            <s xml:id="echoid-s1405" xml:space="preserve">nam regulæ HL, GI
              <lb/>
            conueniunt ſimul in vnico puncto P, in quo ſe mutuò ſecantes, hinc inde di-
              <lb/>
            ſiunctim procedunt, cadens PH ſegmentum regulæ LPH remotius à diame-
              <lb/>
            tro BF, quàm PG ſegmentum regulæ GOI; </s>
            <s xml:id="echoid-s1406" xml:space="preserve">ideoque & </s>
            <s xml:id="echoid-s1407" xml:space="preserve">ſegmentum ſectionis
              <lb/>
            ABC ſupra applicatam MR totum cadet extra ſegmentum ſectionis DBE
              <lb/>
            ſupra eandem applicatam; </s>
            <s xml:id="echoid-s1408" xml:space="preserve">è contra verò reliquum portionis ABC infra ap-
              <lb/>
            plicatam MR cadet totum intra reliquum portionis DBE infra eandem ap-
              <lb/>
            plicatam, cum ſegmentũ PL propriæ regulæ HPL diſiunctum ſit, & </s>
            <s xml:id="echoid-s1409" xml:space="preserve">propius
              <lb/>
            diametro BF quàm ſegmentum PI propriæ regulæ GOI: </s>
            <s xml:id="echoid-s1410" xml:space="preserve">omneque id oſten-
              <lb/>
            ditur eadem penitus ratione, ac in ſecunda parte huius Theorematis demõ-
              <lb/>
            ſtrauimus: </s>
            <s xml:id="echoid-s1411" xml:space="preserve">quare huiuſmodi coni-ſectiones per vertices ſimul adſcriptæ, & </s>
            <s xml:id="echoid-s1412" xml:space="preserve">
              <lb/>
            quarũ regulæ ſe mutuò ſecant infra contingentem ex vertice, in ipſis </s>
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