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

Table of contents

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[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.
[251.] COROLL. II.
[252.] SCHOLIVM.
[253.] LEMMA VI. PROP. XXVII.
[254.] LEMMA VII. PROP. XXVIII.
[255.] LEMMA VIII. PROP. XXIX.
[256.] THEOR. XIX. PROP. XXX.
[257.] SCHOLIVM.
[258.] COROLL.
[259.] LEMMA IX. PROP. XXXI.
[260.] THEOR. XX. PROP. XXXII
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            <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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