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

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          <p>
            <s xml:id="echoid-s4096" xml:space="preserve">
              <pb o="122" file="0146" n="146" rhead=""/>
            cadat inter I, & </s>
            <s xml:id="echoid-s4097" xml:space="preserve">B; </s>
            <s xml:id="echoid-s4098" xml:space="preserve">tunc enim m
              <unsure/>
            portione
              <unsure/>
            IBF, per punctum E, Ellipſis alteri
              <lb/>
            GEHD ſimilis inſcribi nũquam poterit, qualis ſemper inſcribi poteſt m
              <unsure/>
            triã-
              <lb/>
            gulis IBF, MBL, &</s>
            <s xml:id="echoid-s4099" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4100" xml:space="preserve">primæ figuræ; </s>
            <s xml:id="echoid-s4101" xml:space="preserve">quod omne, vel leuiter intuenti ſatis
              <lb/>
            patebit) illę omnino his minores erunt, cum verticibus ſint propiores; </s>
            <s xml:id="echoid-s4102" xml:space="preserve">& </s>
            <s xml:id="echoid-s4103" xml:space="preserve">ob
              <lb/>
            id, quæ ad oppoſitas partes ibi inſcribuntur, erunt quidem _MAXIMAE_
              <lb/>
            quæſitæ.</s>
            <s xml:id="echoid-s4104" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div398" type="section" level="1" n="168">
          <head xml:id="echoid-head173" xml:space="preserve">THEOR. XXXVII. PROP. LXXVII.</head>
          <p>
            <s xml:id="echoid-s4105" xml:space="preserve">MAXIMI circuli m
              <unsure/>
            Parabolæ inſcripti, & </s>
            <s xml:id="echoid-s4106" xml:space="preserve">à vertice ſucceſſiuè ſe
              <lb/>
            mutuò contingentes, ſunt inter ſe in ratione quadratorũ, diſparium
              <lb/>
            numerorum ab vnitate incipientium.</s>
            <s xml:id="echoid-s4107" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4108" xml:space="preserve">SIt Parabole ABC, cuius axis BH, vertex B; </s>
            <s xml:id="echoid-s4109" xml:space="preserve">& </s>
            <s xml:id="echoid-s4110" xml:space="preserve">_MAXIMI_ circuli m
              <unsure/>
            e@ in-
              <lb/>
            ſcripti, & </s>
            <s xml:id="echoid-s4111" xml:space="preserve">à vertice ſucceſſiuè ſe mutuò contingentes ſint, quorum dia-
              <lb/>
            metri BE, EF, FG, GH, &</s>
            <s xml:id="echoid-s4112" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4113" xml:space="preserve">contactus veròſint, primi vertex B, ſecundi
              <lb/>
            punctum L, ter@ij O, quarti R, &</s>
            <s xml:id="echoid-s4114" xml:space="preserve">c. </s>
            <s xml:id="echoid-s4115" xml:space="preserve">dico huiuſmodi circulos, eſſe inter ſe, vt
              <lb/>
            quadrata numerorum diſparium ab vnitate incipientium, nempe 1. </s>
            <s xml:id="echoid-s4116" xml:space="preserve">9. </s>
            <s xml:id="echoid-s4117" xml:space="preserve">25.
              <lb/>
            </s>
            <s xml:id="echoid-s4118" xml:space="preserve">49. </s>
            <s xml:id="echoid-s4119" xml:space="preserve">&</s>
            <s xml:id="echoid-s4120" xml:space="preserve">c.</s>
            <s xml:id="echoid-s4121" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4122" xml:space="preserve">Ducantur, tum ex diametrorum terminis E, F, G; </s>
            <s xml:id="echoid-s4123" xml:space="preserve">tum ex contactibus L,
              <lb/>
            O, R ordinatæ EI, FN, GQ, LM, OP, RS.</s>
            <s xml:id="echoid-s4124" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s4125" xml:space="preserve">Iam cum circulus BE ſit _MAXIMVS_ in-
              <lb/>
              <figure xlink:label="fig-0146-01" xlink:href="fig-0146-01a" number="112">
                <image file="0146-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0146-01"/>
              </figure>
            ſcriptibilium per verticẽ B, erit BE
              <note symbol="a" position="left" xlink:label="note-0146-01" xlink:href="note-0146-01a" xml:space="preserve">1. Co-
                <lb/>
              roll. 20. h.</note>
            lis recto lateri Parabolę, ſed EI eſt
              <note symbol="b" position="left" xlink:label="note-0146-02" xlink:href="note-0146-02a" xml:space="preserve">Coroll.
                <lb/>
              I. h.</note>
            proportionalis inter EB, & </s>
            <s xml:id="echoid-s4126" xml:space="preserve">rectum latus,
              <lb/>
            hoc eſt inter æquales lineas, quare EI æ-
              <lb/>
            qualis erit ipſi EB, quæ concipiatur, vt
              <lb/>
            vnum; </s>
            <s xml:id="echoid-s4127" xml:space="preserve">eſtque EI æqualis EM, ergo
              <note symbol="c" position="left" xlink:label="note-0146-03" xlink:href="note-0146-03a" xml:space="preserve">2. Co-
                <lb/>
              roll. 75. h.</note>
            eſt vt 1, & </s>
            <s xml:id="echoid-s4128" xml:space="preserve">tota BM, vt 2; </s>
            <s xml:id="echoid-s4129" xml:space="preserve">ſed eſt vt BE
              <note symbol="d" position="left" xlink:label="note-0146-04" xlink:href="note-0146-04a" xml:space="preserve">1. Co-
                <lb/>
              roll. 13. h.</note>
            BM, ita BM ad BF, vel vt 1 ad 2, ita 2, ad
              <lb/>
            4; </s>
            <s xml:id="echoid-s4130" xml:space="preserve">erit ergo BF, 4: </s>
            <s xml:id="echoid-s4131" xml:space="preserve">eſtque BM, 2; </s>
            <s xml:id="echoid-s4132" xml:space="preserve">quare
              <lb/>
              <note symbol="e" position="left" xlink:label="note-0146-05" xlink:href="note-0146-05a" xml:space="preserve">2. Co-
                <lb/>
              roll. 75 h.</note>
            FM, ſiue FN, ſiue FP erit pariter 2; </s>
            <s xml:id="echoid-s4133" xml:space="preserve">
              <note symbol="f" position="left" xlink:label="note-0146-06" xlink:href="note-0146-06a" xml:space="preserve">ibidem.</note>
            detota BP erit 6; </s>
            <s xml:id="echoid-s4134" xml:space="preserve">eſtque BF ad BP, vel vt
              <lb/>
              <note symbol="g" position="left" xlink:label="note-0146-07" xlink:href="note-0146-07a" xml:space="preserve">1. Co-
                <lb/>
              roll. 13. h.</note>
            4 ad 6, ita BP ad BG, & </s>
            <s xml:id="echoid-s4135" xml:space="preserve">vt 4 ad 6, ita 6 ad 9, vnde BG erit 9, ſed eſt BP, 6, ergo GP,
              <lb/>
            ſiue GQ, vel GS erit 3; </s>
            <s xml:id="echoid-s4136" xml:space="preserve">quare tota BS, erit
              <lb/>
            12, ſed vt BG ad BS, vel vt 9 ad 12, ita BS
              <lb/>
            ad BH, & </s>
            <s xml:id="echoid-s4137" xml:space="preserve">vt 9 ad 12, ita 12 ad 16, quare
              <lb/>
            BH, erit 16. </s>
            <s xml:id="echoid-s4138" xml:space="preserve">Siergo dum BE eſt, vt I, BF
              <lb/>
            eſt 4, BG, 9, & </s>
            <s xml:id="echoid-s4139" xml:space="preserve">BH, 16; </s>
            <s xml:id="echoid-s4140" xml:space="preserve">ipſa BE cum ea-
              <lb/>
            rum differentijs EF, FG, GH, erunt, vt
              <lb/>
            ſunt numeri 1, 3, 5, 7, qui ſunt numeri im-
              <lb/>
            pares ab vnitate incipientes, ſed circuli
              <lb/>
            ſunt, vt quadrata ſuorum diametrorum,
              <lb/>
            ipſæque BF, EF, FG, GH ſunt inſcripto-
              <lb/>
            rum circulorum diametri, quare hi </s>
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