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

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          <p>
            <s xml:id="echoid-s2600" xml:space="preserve">Ampliùs dico ipſam DEF quò longiùs aberit à vertice E infra EA, eò
              <lb/>
            magis appropinquare ſectioni B A M. </s>
            <s xml:id="echoid-s2601" xml:space="preserve">quoniam ducta D M parallela ad
              <lb/>
            OEB, & </s>
            <s xml:id="echoid-s2602" xml:space="preserve">MN ad DO, fiet parallelogrammum DN, cuius oppoſita latera
              <lb/>
            MN, DO æqualia erunt: </s>
            <s xml:id="echoid-s2603" xml:space="preserve">Itaqueregulæ IG occurrat producta MN in Q, & </s>
            <s xml:id="echoid-s2604" xml:space="preserve">
              <lb/>
            regulæ LH producta DO in R: </s>
            <s xml:id="echoid-s2605" xml:space="preserve">cum ſit oſtenſa MN æqualis DO, erit qua-
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            dratum MN ſiue rectangulum BNQ, æquale quadrato DO ſiue
              <note symbol="a" position="right" xlink:label="note-0101-01" xlink:href="note-0101-01a" xml:space="preserve">Coroll.
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              1. huius.</note>
              <note symbol="b" position="right" xlink:label="note-0101-02" xlink:href="note-0101-02a" xml:space="preserve">ibidem.</note>
            lo EOR: </s>
            <s xml:id="echoid-s2606" xml:space="preserve">ſed in triangulis IBG, LEH ſunt latera IB, LE, & </s>
            <s xml:id="echoid-s2607" xml:space="preserve">BG, EH inter ſe
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            æqualia, alterum alteri, quapropter æqualium rectangulorum BNQ, EOR
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            latera BN, & </s>
            <s xml:id="echoid-s2608" xml:space="preserve">EO æqualia erunt: </s>
            <s xml:id="echoid-s2609" xml:space="preserve">quare cum diametri ſegmenta BN,
              <note symbol="c" position="right" xlink:label="note-0101-03" xlink:href="note-0101-03a" xml:space="preserve">43. h.</note>
            ſint æqualia, facta proſtaphereſi, proueniet BE æqualis NO, ſed eſt quoque
              <lb/>
            MD æqualis NO in parallelogrammo DN, igitur rectæ BE, & </s>
            <s xml:id="echoid-s2610" xml:space="preserve">MD inter ſe
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            ſunt æquales, at ſunt etiam parallelæ, ergo coniuncta BM iunctæ ED æqui-
              <lb/>
            diſtat, ſed BM ſecat NM, quare producta ſecabit quoque OD, ſed extra ſe-
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            ctionem BMA (cum BM ſit intra ſectionem, producta verò tota cadat extra)
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            ſitque occurſus in P, & </s>
            <s xml:id="echoid-s2611" xml:space="preserve">OD occurrat ſectioni BMA in S, PM verò contin-
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            gentem EA ſecet in T; </s>
            <s xml:id="echoid-s2612" xml:space="preserve">& </s>
            <s xml:id="echoid-s2613" xml:space="preserve">in ſecunda figura, in qua punctum A cadit inter
              <lb/>
            puncta S, & </s>
            <s xml:id="echoid-s2614" xml:space="preserve">M, iungatur SM, quæ cum tota cadat intra ſectionem, neceſſa-
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            riò ſecabit applicatam AE: </s>
            <s xml:id="echoid-s2615" xml:space="preserve">veluti in V.</s>
            <s xml:id="echoid-s2616" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2617" xml:space="preserve">Iam, in prima figura, cum in parallelogrammo P E oppoſita latera ET,
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            DP ſint æqualia, ſitque EA maius ET, erit EA quoque maius ipſo DP, ſed
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            eſt DP maius intercepto applicatæ ſegmento DS, erit ergo AE, eò maius
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            ipſo DS. </s>
            <s xml:id="echoid-s2618" xml:space="preserve">In ſecunda autem figura cum pariter ET, DP ſint æquales, ſitque
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            dempta TV minor dempta PS, erit reliqua EV maior reliqua DS, & </s>
            <s xml:id="echoid-s2619" xml:space="preserve">eò ma-
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            gis EA maior eadem DS. </s>
            <s xml:id="echoid-s2620" xml:space="preserve">Eodè penitùs modo oſtendetur, quamlibet aliam
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            interceptam ZY infra SD minorem eſſe ipſa SD: </s>
            <s xml:id="echoid-s2621" xml:space="preserve">nam ducta YZ æquidiſtan-
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            ter ad EB, demonſtrabitur item YZ æqualem eſſe eidem BE, ac ideo YZ, & </s>
            <s xml:id="echoid-s2622" xml:space="preserve">
              <lb/>
            DM eſſe inter ſe ęquales, & </s>
            <s xml:id="echoid-s2623" xml:space="preserve">parallelas: </s>
            <s xml:id="echoid-s2624" xml:space="preserve">ex quo ſi iungantur MZ, & </s>
            <s xml:id="echoid-s2625" xml:space="preserve">DY, ipſę
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            æquales erunt, & </s>
            <s xml:id="echoid-s2626" xml:space="preserve">parallelæ; </s>
            <s xml:id="echoid-s2627" xml:space="preserve">completa igitur conſimili conſtructione, ac ſu-
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            pra, idem omnino inſequetur, hoc eſt interceptam YX minorem adhuc eſſe
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            DS: </s>
            <s xml:id="echoid-s2628" xml:space="preserve">tales ergo interceptæ quò magis à tangente EA remouentur continuè
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            decreſcunt. </s>
            <s xml:id="echoid-s2629" xml:space="preserve">Quare ſectiones ABC, DEF ſunt ſemper ſimul accedentes.
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            </s>
            <s xml:id="echoid-s2630" xml:space="preserve">Quod ſecundò, &</s>
            <s xml:id="echoid-s2631" xml:space="preserve">c.</s>
            <s xml:id="echoid-s2632" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s2633" xml:space="preserve">Præterea, ſi ad euitandam in hiſce figuris linearum implicationem, con-
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            cipiatur circumſcriptæ Hyperbolæ ABC centrum eſſe I, aſymptoton IG, & </s>
            <s xml:id="echoid-s2634" xml:space="preserve">
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            contingens ex vertice BG; </s>
            <s xml:id="echoid-s2635" xml:space="preserve">at inſcriptæ DEF centrum L, aſymptoton LH,
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            contingens autem ex vertice ſit EH: </s>
            <s xml:id="echoid-s2636" xml:space="preserve">cum harum ſectionum latera ſint data
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            æqualia, erunt quoque ipſorum rectangula inter ſe æqualia, ideoque, & </s>
            <s xml:id="echoid-s2637" xml:space="preserve">eo-
              <lb/>
            rum ſubquadrupla hoc eſt quadrata contingentium BG, EH, vnde ipſæ
              <note symbol="d" position="right" xlink:label="note-0101-04" xlink:href="note-0101-04a" xml:space="preserve">1. ſecú-
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              diconic.</note>
            neæ BG, EH æquales erunt, ſed eſt etiam BI æqualis EL (nam vtra eſt dimi-
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            dium æqualium verſorum laterum) quare in triangulis IBG, LEH, cum ſint
              <lb/>
            latera IB, BG, lateribus LE, EH æqualia, & </s>
            <s xml:id="echoid-s2638" xml:space="preserve">anguli ad B, E æquales, etiam
              <lb/>
            anguli ad baſes I, L æquales erunt, vnde aſymptoti IG, LG inter ſe æqui-
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            diſtant; </s>
            <s xml:id="echoid-s2639" xml:space="preserve">& </s>
            <s xml:id="echoid-s2640" xml:space="preserve">cum ſit à puncto L, quod eſt intra angulum ab aſymptotis cir-
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            cumſcriptæ ſectionis factũ, ducta LH alteri aſymptoto IK æquidiſtans, pro-
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            ducta ſecabit omnino Hyperbolen ABC: </s>
            <s xml:id="echoid-s2641" xml:space="preserve">quare LH aſymptotos
              <note symbol="e" position="right" xlink:label="note-0101-05" xlink:href="note-0101-05a" xml:space="preserve">11. h.</note>
            ſecat Hyperbolen circumſcriptam; </s>
            <s xml:id="echoid-s2642" xml:space="preserve">ſecet ergo in 1, per quod applicetur
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            2 1 3: </s>
            <s xml:id="echoid-s2643" xml:space="preserve">Dico harum ſectionum interuallum infra applicatam 2 1 3 per </s>
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