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

Table of contents

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[311.] THEOR. XLII. PROP. LXVIII.
Page: 274 (88)
[312.] COROLL. I.
Page: 276 (90)
[313.] COROLL. II.
Page: 276 (90)
[314.] MONITVM.
Page: 276 (90)
[315.] DEFINITIONES. I.
Page: 278 (92)
[316.] II.
Page: 278 (92)
[317.] III.
Page: 279 (93)
[318.] IIII.
Page: 279 (93)
[319.] PROBL. XIV. PROP. LXIX.
Page: 280 (94)
[320.] SCHOLIVM I.
Page: 282 (96)
[321.] COROLL. I.
Page: 282 (96)
[322.] SCHOLIVM II.
Page: 282 (96)
[323.] COROLL. II.
Page: 282 (96)
[324.] SCHOLIVM III.
Page: 283 (97)
[325.] COROLL. III.
Page: 283 (97)
[326.] THEOR. XLIII. PROP. LXX.
Page: 284 (98)
[327.] COROLL.
Page: 286 (100)
[328.] THEOR. XLIV. PROP. LXXI.
Page: 286 (100)
[329.] COROLL.
Page: 287 (101)
[330.] THEOR. XLV. PROP. LXXII.
Page: 288 (102)
[331.] SCHOLIVM.
Page: 288 (102)
[332.] THEOR. XLVI. PROP. LXXIII.
Page: 288 (102)
[333.] THEOR. XLVII. PROP. LXXIV.
Page: 288 (102)
[334.] MONITVM.
Page: 290 (104)
[335.] LEMMA XIV. PROP. LXXV.
Page: 290 (104)
[336.] SCHOLIVM.
Page: 291 (105)
[337.] LEMMA XV. PROP. LXXVI.
Page: 292 (106)
[338.] THEOR. XLVIII. PROP. LXXVII.
Page: 292 (106)
[339.] MONITVM.
Page: 293 (107)
[340.] THEOR. IL. PROP. LXXVIII.
Page: 294 (108)
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            de applicatis ad puncta arcus A I D, tum de ijs, quæ pertingunt ad puncta
              <lb/>
            reliqui arcus D B, hoc eſt prædicta rectangula hinc inde à puncto D, con-
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            tinuè decreſcere, quò magis diſtant à _MAXIMO_ rectangulo A E D.</s>
            <s xml:id="echoid-s8857" xml:space="preserve"/>
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          <p style="it">
            <s xml:id="echoid-s8858" xml:space="preserve">Hinc ſoluendum fit obuiam Problema huiuſmodi.</s>
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        <div xml:id="echoid-div919" type="section" level="1" n="368">
          <head xml:id="echoid-head377" xml:space="preserve">PROBL. XVIII. PROP. XCV.</head>
          <p>
            <s xml:id="echoid-s8860" xml:space="preserve">In dato ſemi - circulo, vel ſemi - Ellipſi, hinc inde à MA-
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            XIMO rectangulo nuper inuento, bina æqualia rectangula re-
              <lb/>
            perire.</s>
            <s xml:id="echoid-s8861" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s8862" xml:space="preserve">SIt datus ſemi- circulus, vel ſemi-Ellipſis, cuius diameter A B, centrum
              <lb/>
            C, & </s>
            <s xml:id="echoid-s8863" xml:space="preserve">punctum, ad quod peruenit _MAXIMVM_ rectangulum, ſit D,
              <lb/>
            (quod habebitur ſi diameter A B ſecetur in L, ita vt A L ſit tripla L
              <note symbol="a" position="left" xlink:label="note-0318-01" xlink:href="note-0318-01a" xml:space="preserve">Schol.
                <lb/>
              93. h. &
                <lb/>
              ex 94. h.</note>
            & </s>
            <s xml:id="echoid-s8864" xml:space="preserve">applicetur L D,) ſitque exempli gratia è quolibet puncto E arcus A E
              <lb/>
            D, applicata E F ad diametrum A B, & </s>
            <s xml:id="echoid-s8865" xml:space="preserve">oporteat in reliquo arcu D B pun-
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            ctum G reperire, ita vt ducta G H ipſi E F parallela, rectangula A F E, A
              <lb/>
            H G inter ſe ſint æqualia.</s>
            <s xml:id="echoid-s8866" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s8867" xml:space="preserve">Ducatur ex A ſectionem contingens A I, quę ipſis applicatis æquidiſta-
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            bit, atque in angulo aſymptotali I A B per punctum E deſcribatur
              <note symbol="b" position="left" xlink:label="note-0318-02" xlink:href="note-0318-02a" xml:space="preserve">4. ſec.
                <lb/>
              Conic.</note>
            perbole E G. </s>
            <s xml:id="echoid-s8868" xml:space="preserve">Dico hanc neceſſariò in aliquo puncto circuli arcum D B ſe-
              <lb/>
            care, vt in G, & </s>
            <s xml:id="echoid-s8869" xml:space="preserve">hoc eſſe quæſitum, atque vnicum.</s>
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          <p>
            <s xml:id="echoid-s8871" xml:space="preserve">Etenim demiſſa ordinata D L, cum hæc aſymptoto A I æquidiſtet, ipſa
              <lb/>
            neceſſariò Hyperbolen E G ſecabit, at
              <note symbol="c" position="left" xlink:label="note-0318-03" xlink:href="note-0318-03a" xml:space="preserve">Coroll.
                <lb/>
              11. primi
                <lb/>
              huius.</note>
            vno tantùm puncto, veluti in M, & </s>
            <s xml:id="echoid-s8872" xml:space="preserve">ob Hy-
              <lb/>
              <figure xlink:label="fig-0318-01" xlink:href="fig-0318-01a" number="254">
                <image file="0318-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0318-01"/>
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            perbolen, erit rectangulum A L M
              <note symbol="d" position="left" xlink:label="note-0318-04" xlink:href="note-0318-04a" xml:space="preserve">12. ſec.
                <lb/>
              Conic.</note>
            rectangulo A F E, ſed eſt rectangulùm A L
              <lb/>
            D maius eodem rectangulo A F E, cum ſit
              <lb/>
            _MAXIMVM_, ex hypotheſi, ergo idem rectan-
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            gulum A L D maius erit rectangulo A L M,
              <lb/>
            atq; </s>
            <s xml:id="echoid-s8873" xml:space="preserve">eſt A L communis eorum altitudo, qua-
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            re L D maior erit L M. </s>
            <s xml:id="echoid-s8874" xml:space="preserve">Hyperbole igitur E
              <lb/>
            G ſecat omnino D L inter D, & </s>
            <s xml:id="echoid-s8875" xml:space="preserve">L, vnde & </s>
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            producta neceſſariò ſecabit peripheriam arcus
              <lb/>
            D B, cum ſpatium L D B ſit vndique clau-
              <lb/>
            ſum, & </s>
            <s xml:id="echoid-s8877" xml:space="preserve">Hyperbole ſit infinitæ productionis:
              <lb/>
            </s>
            <s xml:id="echoid-s8878" xml:space="preserve">ſecet igitur in G. </s>
            <s xml:id="echoid-s8879" xml:space="preserve">Dico punctum G quæſitum ſoluere, vt ſatis patet, cùm
              <lb/>
            rectangulum G H A, ob Hyperbolen, ſit æquale rectangulo E F A.</s>
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          <note symbol="e" position="left" xml:space="preserve">ibidem.</note>
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            <s xml:id="echoid-s8881" xml:space="preserve">Quod autem in nullo alio puncto, præter in E, & </s>
            <s xml:id="echoid-s8882" xml:space="preserve">G, huiuſmodi Hyper-
              <lb/>
            bole arcui A D, vel arcui D B occurrat, manifeſtum eſt: </s>
            <s xml:id="echoid-s8883" xml:space="preserve">nam ſi alibi oc-
              <lb/>
            curreret, vt in N; </s>
            <s xml:id="echoid-s8884" xml:space="preserve">eſſet ob Hyperbolen, rectangulum pertingens ad N
              <lb/>
            æquale rectangulo A F E, quod eſt falſum, quoniam ob circulum, vel El-
              <lb/>
            lipſim, quando punctum N eſt inter E, & </s>
            <s xml:id="echoid-s8885" xml:space="preserve">D, rectangulum ad N maius eſt
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            quàm rectangulum ad E, & </s>
            <s xml:id="echoid-s8886" xml:space="preserve">ſi fuerit inter A, & </s>
            <s xml:id="echoid-s8887" xml:space="preserve">E, ipſo rectangulo ad </s>
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