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

Table of contents

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[261.] PROBL. IV. PROP. XXXIII.
[262.] PROBL. V. PROP. XXXIV.
[263.] DEFINITIONES. I.
[264.] II.
[265.] LEMMA X. PROP. XXXV.
[266.] THEOR. XXI. PROP. XXXVI.
[267.] THEOR. XXII. PROP. XXXVII.
[268.] SCHOLIVM.
[269.] LEMMA XI. PROP. XXXVIII.
[270.] LEMMA XII. PROP. XXXIX.
[271.] THEOR. XXIII. PROP. XXXX.
[272.] COROLL. I.
[273.] COROLL. II.
[274.] COROLL. III.
[275.] PROBL. VI. PROP. XXXXI.
[276.] PROBL. VII. PROP. XXXXII.
[277.] COROLL.
[278.] THEOR. XXIV. PROP. XXXXIII.
[279.] THEOR. XXV. PROP. XXXXIV.
[280.] SCHOLIVM.
[281.] THEOR. XXVI. PROP. XLV.
[282.] COROLL.
[283.] THEOR. XXVII. PROP. XLVI.
[284.] COROLL. I.
[285.] COROLL. II.
[286.] THEOR. XXVIII. PROP. XLVII.
[287.] THEOR. XXIX. PROP. XLVIII.
[288.] THEOR. XXX. PROP. XLIX.
[289.] THEOR. XXXI. PROP. L.
[290.] COROLL.
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          <p>
            <s xml:id="echoid-s8606" xml:space="preserve">
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            congruit cum maiori axe Sphæroidis) æquales ſecentur, atque ex interſe-
              <lb/>
            ctionum punctis plana ducantur portionum baſibus æquidiſtantia, abſcin-
              <lb/>
            dentur portiones ſolidæ æqualium axium, & </s>
            <s xml:id="echoid-s8607" xml:space="preserve">vnaquæque erit maior quali-
              <lb/>
            bet æqualium (totum enim ſua parte maius eſt) ac ideò maior ea portione,
              <lb/>
            cuius axi, vel cui portioni nihil additum fuit, quæ quidem eſt ea, cuius axis
              <lb/>
            congruit cum maiori axe Sphæroidis. </s>
            <s xml:id="echoid-s8608" xml:space="preserve">Itaque ſi omnes planæ portiones
              <lb/>
            æqualium axium ſunt hac portione maiores, erit è contra hæc ipſa portio,
              <lb/>
            cuius axis conuenit cum maiori axe Sphæroidis, _MINIMA_ earundem om-
              <lb/>
            nium portionum æqualium axium, in caſibus tamen poſſibilibus. </s>
            <s xml:id="echoid-s8609" xml:space="preserve">Quod vl-
              <lb/>
            timò demonſtrandum erat.</s>
            <s xml:id="echoid-s8610" xml:space="preserve"/>
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        <div xml:id="echoid-div899" type="section" level="1" n="358">
          <head xml:id="echoid-head367" xml:space="preserve">THEOR. LX. PROP. LXXXX.</head>
          <p>
            <s xml:id="echoid-s8611" xml:space="preserve">MAXIMA portionum de codem Cono recto, vel de quocun-
              <lb/>
            que Conoide, aut Sphæroide, & </s>
            <s xml:id="echoid-s8612" xml:space="preserve">quarum baſes ſint æquales, ea eſt,
              <lb/>
            cuius axis ſit ſegmentum maioris ſemi- axis genitricis ſectionis dati
              <lb/>
            ſolidi, reſpectiuè ad Sphæroides.</s>
            <s xml:id="echoid-s8613" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8614" xml:space="preserve">In Sphæroide autem, MINIMA, cuius axis ſit ſegmentum mi-
              <lb/>
            noris ſemi- axis Ellipſis, quæ ſolidum procreat.</s>
            <s xml:id="echoid-s8615" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8616" xml:space="preserve">QVando enim portiones eiuſdem Coni recti, vel cuiuslibet Conoidis,
              <lb/>
            aut Sphæroidis ſunt æquales, & </s>
            <s xml:id="echoid-s8617" xml:space="preserve">recti earum Canones ſunt
              <note symbol="a" position="left" xlink:label="note-0310-01" xlink:href="note-0310-01a" xml:space="preserve">84. h.</note>
            & </s>
            <s xml:id="echoid-s8618" xml:space="preserve">cum recti Canones, vel portiones de eodem angulo, vel de
              <lb/>
            eadem coni- ſectione, quæ ſolidum genuit æquales ſunt, inter ipſorum ba-
              <lb/>
              <note symbol="b" position="left" xlink:label="note-0310-02" xlink:href="note-0310-02a" xml:space="preserve">Schol.
                <lb/>
              poſt 5 1. h.
                <lb/>
              ad nu. 2.</note>
            ſes, _MINIMA_ eſt ea illius portionis, cuius diameter ſit ſegmentum maio- ris axis reſpectiuè ad Ellipſim, & </s>
            <s xml:id="echoid-s8619" xml:space="preserve">_MAXIMA_ eius, cuius diameter ſit ſegmen-
              <lb/>
            tum minoris, atque vt ſunt baſes æqualium planarum portionum de eodem
              <lb/>
            angulo, vel coni-ſectione, ita ſunt baſes ſolidarum portionum,
              <note symbol="c" position="left" xlink:label="note-0310-03" xlink:href="note-0310-03a" xml:space="preserve">2. Co-
                <lb/>
              roll. 78. h.</note>
            ipſæ planæ portiones ſint recti Canones, ergo & </s>
            <s xml:id="echoid-s8620" xml:space="preserve">inter baſes æqualium por-
              <lb/>
            tionum de eodem Cono recto, vel Conoide, aut Sphæroide quocunque,
              <lb/>
            _MINIMA_ erit ea illius portionis, cuius axis (qui idem eſt cum
              <note symbol="d" position="left" xlink:label="note-0310-04" xlink:href="note-0310-04a" xml:space="preserve">3. Schol.
                <lb/>
              69. h.</note>
            recti Canonis) congruat cum maiori axe genitricis ſectionis ſolidi, cuius
              <lb/>
            eſt portio, & </s>
            <s xml:id="echoid-s8621" xml:space="preserve">_MAXIMA_, in Sphæroide, erit baſis illius portionis, cuius axis
              <lb/>
            ſit ſegmentum minoris axis Ellipſis genitricis eiuſdem Sphæroidis; </s>
            <s xml:id="echoid-s8622" xml:space="preserve">quare ſi
              <lb/>
            primò intra has æquales portiones, dempta ea ſuper _MINIMA_ baſi, ducan-
              <lb/>
            tur plana baſibus æquidiſtantia, quorum vnumquodque efficiat in portione
              <lb/>
            ſectionem prædictæ _MINIMAE_ baſi æqualem (hoc autem ſieri poſſe, & </s>
            <s xml:id="echoid-s8623" xml:space="preserve">
              <lb/>
            quomodò infra docebimus) per huiuſmodi plana abſcindentur portiones
              <lb/>
            ſolidæ æqualium baſium, ſed harum quælibet minor erit quacunque æqua-
              <lb/>
            lium portionum (cum ſit pars minor ſuo toto) ideoque minor ea, à qua ni-
              <lb/>
            hil ablatum fuit, ſiue minor ea, cuius axis conuenit cum maiori axe dati ſo-
              <lb/>
            lidi. </s>
            <s xml:id="echoid-s8624" xml:space="preserve">Si ergo omnes aliæ portiones æqualium baſium hac portione ſunt mi-
              <lb/>
            nores, erit è contra hæc ipſa portio, cuius axis eſt ſegmentum maioris ſemi-
              <lb/>
            axis ſectionis genitricis dati ſolidi earundem portionum æqualium baſium,
              <lb/>
            ac de eodem ſolido _MAXIMA_, &</s>
            <s xml:id="echoid-s8625" xml:space="preserve">c.</s>
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