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

Table of contents

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[221.] THEOR. I. PROP. III.
[222.] LEMMA III. PROP. IV.
[223.] THEOR. II. PROP. V.
[224.] THEOR. III. PROP. VI.
[225.] LEMMA IV. PROP. VII.
[226.] THEOR. IV. PROP. VIII.
[227.] THEOR. V. PROP. IX.
[228.] SCHOLIVM.
[229.] THEOR. VI. PROP. X.
[230.] THEOR. VII. PROP. XI.
[231.] THEOR. VIII. PROP. XII.
[232.] THEOR. IX. PROP. XIII.
[233.] THEOR. X. PROP. XIV.
[234.] THEOR. XI. PROP. XV.
[235.] LEMMA V. PROP. XVI.
[236.] COROLL.
[237.] THEOR. XII. PROP. XVII.
[238.] THEOR. XIII. PROP. XVIII.
[239.] THEOR. XIV. PROP. XIX.
[240.] PROBL. I. PROP. XX.
[241.] MONITVM.
[242.] THEOR. XV. PROP. XXI.
[243.] PROBL. II. PROP. XXII.
[244.] PROBL. III. PROP. XXIII.
[245.] MONITVM.
[246.] THEOR. XVI. PROP. XXIV.
[247.] THEOR. XVII. PROP. XXV.
[248.] COROLL.
[249.] THEOR. XIIX. PROP. XXVI.
[250.] COROLL. I.
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            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
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            puncta B, E ſint earundem portionum vertices; </s>
            <s xml:id="echoid-s6581" xml:space="preserve">quare vt baſis H G ad ba-
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            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-
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            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-
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            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-
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            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"/>
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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>
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            <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>
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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>
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          </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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