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

Table of contents

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[41.] THEOR. IV. PROP. XI.
[42.] COROLL.
[43.] MONITVM.
[44.] LEMMA III. PROP. XII.
[45.] ALITER idem breuiùs.
[46.] ITER VM aliter breuiùs, ſed negatiuè.
[47.] COROLL.
[48.] THEOR. V. PROP. XIII.
[49.] COROLL. I.
[50.] COROLL. II.
[51.] COROLL. III.
[52.] THEOR. VI. PROP. XIV.
[53.] COROLLARIVM.
[54.] THEOR. VII. PROP. XV.
[55.] THEOR. VIII. PROP. XVI.
[56.] THEOR. IX. PROP. XVII.
[57.] MONITVM.
[58.] THEOR. X. PROP. XVIII.
[59.] Definitiones Secundæ. I.
[60.] II.
[61.] III.
[62.] IV.
[64.] VI.
[65.] VII.
[66.] VIII.
[67.] IX.
[68.] THEOR. XI. PROP. XIX.
[69.] COROLL. I.
[70.] COROLL. II.
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          <head xml:id="echoid-head51" xml:space="preserve">ITER VM aliter breuiùs, ſed negatiuè.</head>
          <p>
            <s xml:id="echoid-s987" xml:space="preserve">Sifuerit vt recta AD ad DC, ita quadratum AB ad BC, erunt
              <lb/>
            AD, DB, DC in continua ratione geometrica.</s>
            <s xml:id="echoid-s988" xml:space="preserve"/>
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          <p>
            <s xml:id="echoid-s989" xml:space="preserve">SIenim DB non eſt media proportionalis inter AD, DC, eſto ſi fieri po-
              <lb/>
            teſt media quæcunque DE; </s>
            <s xml:id="echoid-s990" xml:space="preserve">erit igitur, in prima figura, tota AD ad to-
              <lb/>
            tam DE, vt pars DE ad partem DC, ergo reliqua AE ad reliquam EC, erit
              <lb/>
              <note position="right" xlink:label="note-0047-01" xlink:href="note-0047-01a" xml:space="preserve">Vni-
                <lb/>
              uerſalius
                <lb/>
              quàm à
                <lb/>
              Caual. in
                <lb/>
              3. prop.
                <lb/>
              exerc. 6.</note>
            vt pars ED ad DC, vel vt tota AD ad totam DE: </s>
            <s xml:id="echoid-s991" xml:space="preserve">(ex cõ-
              <lb/>
            ſtructione) in ſecunda verò cum ſit AD ad DE vt DE ad
              <lb/>
              <figure xlink:label="fig-0047-01" xlink:href="fig-0047-01a" number="23">
                <image file="0047-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0047-01"/>
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            DC, erit componendo AE ad ED, vt EC ad CD, & </s>
            <s xml:id="echoid-s992" xml:space="preserve">per-
              <lb/>
            mutando AE ad EC, vt ED ad DC, vel vt AD ad DE
              <lb/>
            (ex conſtructione) cum ergo in vtraque figura ſit AE ad
              <lb/>
            EC, vt AD ad DE, erit quadratum AE ad EC, vt qua-
              <lb/>
            dratum AD ad DE, vel vt recta AD ad DC, vel vt qua-
              <lb/>
            dratum AB ad BC (ex ſuppoſitione) vel recta AE ad
              <lb/>
            EC, vt recta AB ad BC, & </s>
            <s xml:id="echoid-s993" xml:space="preserve">in prima figura componen-
              <lb/>
            do, at in ſecunda diuidendo, AC ad CE, vt AC ad CB,
              <lb/>
            quare CE, CB inter ſe ſunt &</s>
            <s xml:id="echoid-s994" xml:space="preserve">quales, totum, & </s>
            <s xml:id="echoid-s995" xml:space="preserve">pars,
              <lb/>
            quod eſt abſurdum: </s>
            <s xml:id="echoid-s996" xml:space="preserve">non eſt ergo media inter AD, & </s>
            <s xml:id="echoid-s997" xml:space="preserve">
              <lb/>
            DC, quæ ſit maior, vel minor DB; </s>
            <s xml:id="echoid-s998" xml:space="preserve">vnde ipſa DB erit
              <lb/>
            media proportionalis inter AD, & </s>
            <s xml:id="echoid-s999" xml:space="preserve">DC. </s>
            <s xml:id="echoid-s1000" xml:space="preserve">Quod demon-
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            ſtrare oportebat.</s>
            <s xml:id="echoid-s1001" xml:space="preserve"/>
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          <head xml:id="echoid-head52" xml:space="preserve">COROLL.</head>
          <p>
            <s xml:id="echoid-s1002" xml:space="preserve">HInc patet, quod, cum fuerint tres magnitudines continuè proportio-
              <lb/>
            nales, tùm exceſſus quibus differunt, tùm earum aggregata, erunt in
              <lb/>
            eadem ratione, in qua ſunt datæ magnitudines: </s>
            <s xml:id="echoid-s1003" xml:space="preserve">quando enim poſitum fuit
              <lb/>
            eſſe AD ad DE, vt DE ad DC oſtenſum quoque fuit AE ad EC eſſe vt AD
              <lb/>
            ad DE, ſed in prima figura AE, EC ſunt exceſſus datarum magnitudinum,
              <lb/>
            in ſecunda verò ſunt aggregata primæ cum ſecunda, & </s>
            <s xml:id="echoid-s1004" xml:space="preserve">ſecundæ cum tertia;
              <lb/>
            </s>
            <s xml:id="echoid-s1005" xml:space="preserve">quare patet, &</s>
            <s xml:id="echoid-s1006" xml:space="preserve">c.</s>
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          <head xml:id="echoid-head53" xml:space="preserve">THEOR. V. PROP. XIII.</head>
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            <s xml:id="echoid-s1008" xml:space="preserve">Si duæ Parabolæ ad eaſdem partes deſcriptæ ad idem punctum
              <lb/>
            ſimul occurrant, ſintque earum diametri inter ſe æquidiſtantes, & </s>
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            applicatæ ſint eædem, ac ipſarum vertices ſint in eadem recta, quæ
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            ducitur ex occurſu; </s>
            <s xml:id="echoid-s1010" xml:space="preserve">ipſæ in nullo alio puncto ſimul conuenient, & </s>
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              <lb/>
            omnes, quæ ex contactu in ipſis ducuntur in eadem ratione à ſe-
              <lb/>
            ctionibus diuidentur.</s>
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          <p>
            <s xml:id="echoid-s1013" xml:space="preserve">ESto Parabole ABC, cuius diameter BF, & </s>
            <s xml:id="echoid-s1014" xml:space="preserve">ducta BA, ſit quælibet DE
              <lb/>
            ipſi BF æquidiſtans, & </s>
            <s xml:id="echoid-s1015" xml:space="preserve">AC ordinatim applicata BF, & </s>
            <s xml:id="echoid-s1016" xml:space="preserve">per verticem </s>
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