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

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          <p>
            <s xml:id="echoid-s6580" xml:space="preserve">
              <pb o="53" file="0237" n="237" rhead=""/>
            titudo trianguli C B F ad altitudinem trianguli H E G, ſed horum triangu-
              <lb/>
            lorum altitudines eædem ſunt, ac altitudines portionum A B C, H E I, cum
              <lb/>
            puncta B, E ſint earundem portionum vertices; </s>
            <s xml:id="echoid-s6581" xml:space="preserve">quare vt baſis H G ad ba-
              <lb/>
            ſim C F, vel ſumptis duplis, vt H I baſis portionis H E I, ad A C baſim
              <lb/>
            portionis A B C, ita reciprocè altitudo portionis A B C ad altitudinem por-
              <lb/>
            tionis H E I, ſuntque huiuſmodi portiones Acuminata regularia, & </s>
            <s xml:id="echoid-s6582" xml:space="preserve">
              <note symbol="a" position="right" xlink:label="note-0237-01" xlink:href="note-0237-01a" xml:space="preserve">36. h.</note>
            portionalia, & </s>
            <s xml:id="echoid-s6583" xml:space="preserve">eorum baſes altitudinibus reciprocantur, quare ipſa Acumi-
              <lb/>
            nata, ſeu portiones H E I, A B C inter ſe ſunt æquales. </s>
            <s xml:id="echoid-s6584" xml:space="preserve">Quod
              <note symbol="b" position="right" xlink:label="note-0237-02" xlink:href="note-0237-02a" xml:space="preserve">37. h.</note>
            propoſitum fuit, quodque de ſola Parabola demonſtrauit Geometrarum
              <lb/>
            Princeps in 4. </s>
            <s xml:id="echoid-s6585" xml:space="preserve">Prop. </s>
            <s xml:id="echoid-s6586" xml:space="preserve">de Conoid. </s>
            <s xml:id="echoid-s6587" xml:space="preserve">ac Sphæroid. </s>
            <s xml:id="echoid-s6588" xml:space="preserve">ſuppoſita tamen eiuſdem Pa-
              <lb/>
            rabolę quadratura.</s>
            <s xml:id="echoid-s6589" xml:space="preserve"/>
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        <div xml:id="echoid-div685" type="section" level="1" n="272">
          <head xml:id="echoid-head281" xml:space="preserve">COROLL. I.</head>
          <p>
            <s xml:id="echoid-s6590" xml:space="preserve">HInc eſt, quod applicatæ ex terminis æqualium diametrorum in Para-
              <lb/>
            bola, vel ex punctis, in reliquis ſectionibus, proportionaliter diuidẽ-
              <lb/>
            tibus ſemi-diametros ad angulum conſtitutas, omnino ſe mutuò ſecant; </s>
            <s xml:id="echoid-s6591" xml:space="preserve">& </s>
            <s xml:id="echoid-s6592" xml:space="preserve">
              <lb/>
            quod rectæ lineę, tùm harum applicatarum puncta media, tùm extrema iun-
              <lb/>
            gentes, rectæ ſemi-diametrorum terminos iungenti æquidiſtant. </s>
            <s xml:id="echoid-s6593" xml:space="preserve">Demon-
              <lb/>
            ſtratum eſt enim H I, A C ſecare ſe mutuò in M, & </s>
            <s xml:id="echoid-s6594" xml:space="preserve">iunctas H C, G F, A I
              <lb/>
            ipſi E B eſſe parallelas.</s>
            <s xml:id="echoid-s6595" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div686" type="section" level="1" n="273">
          <head xml:id="echoid-head282" xml:space="preserve">COROLL. II.</head>
          <p>
            <s xml:id="echoid-s6596" xml:space="preserve">PAtet quoq; </s>
            <s xml:id="echoid-s6597" xml:space="preserve">in quarta, quinta, ſeptima, & </s>
            <s xml:id="echoid-s6598" xml:space="preserve">octaua figura, portiones eiuſ-
              <lb/>
            dem Ellipſis, vel circuli, quarum baſes tranſeant per puncta earum ſe-
              <lb/>
            mi-diametros proportionaliter ſecantia, etiam ſi ipſæ ſemi-diametri ſint in
              <lb/>
            directum poſitæ, hoc eſt applicatæ inter ſe æquidiſtent, eſſe quoque inter ſe
              <lb/>
            æquales. </s>
            <s xml:id="echoid-s6599" xml:space="preserve">Vtra enim talium portionum æqualis demonſtratur, (vt in ſupe-
              <lb/>
            riori propoſitione) ei portioni, cuius baſis ſit applicata per punctum propor-
              <lb/>
            tionaliter ſecans aliam ſemi-diametrum, quæ cum prædictis angulum con-
              <lb/>
            ſtituat.</s>
            <s xml:id="echoid-s6600" xml:space="preserve"/>
          </p>
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        <div xml:id="echoid-div687" type="section" level="1" n="274">
          <head xml:id="echoid-head283" xml:space="preserve">COROLL. III.</head>
          <p>
            <s xml:id="echoid-s6601" xml:space="preserve">EX ijſdem conſtat, quod ſi quotcunque applicatæ in eadem Ellipſi, vel
              <lb/>
            circulo integras diametros proportionaliter ſecent, abſciſſæ portiones
              <lb/>
            viciſſim æquales erunt, hoc eſt minor minori, & </s>
            <s xml:id="echoid-s6602" xml:space="preserve">maior maiori.</s>
            <s xml:id="echoid-s6603" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6604" xml:space="preserve">Si enim in prædictis figuris ſint duæ diametri B R E L, ita ſectæ in F, G;
              <lb/>
            </s>
            <s xml:id="echoid-s6605" xml:space="preserve">vt R F ad F B ſit vt L G ad G E, erit componendo, & </s>
            <s xml:id="echoid-s6606" xml:space="preserve">ſumptis antece-
              <lb/>
            dentium ſubduplis D B ad B F, vt D E ad E G; </s>
            <s xml:id="echoid-s6607" xml:space="preserve">applicatis ergo A F C,
              <lb/>
            H G I erunt portiones A B C, H E I inter ſe æquales, & </s>
            <s xml:id="echoid-s6608" xml:space="preserve">reliqua portio
              <lb/>
            A R C reliquæ portioni H R I æqualis erit.</s>
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