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

Table of contents

< >
[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.
< >
page |< < (39) of 347 > >|
    <echo version="1.0RC">
      <text xml:lang="la" type="free">
        <div xml:id="echoid-div642" type="section" level="1" n="256">
          <p>
            <s xml:id="echoid-s6186" xml:space="preserve">
              <pb o="39" file="0221" n="221" rhead=""/>
            quare angulus D A M, ſiue in ſimili triangulo D L I, angulus D L I erit
              <lb/>
            maior angulo A H M, ſiue angulo parallelarum externo I B L: </s>
            <s xml:id="echoid-s6187" xml:space="preserve">cum
              <note symbol="*" position="right" xlink:label="note-0221-01" xlink:href="note-0221-01a" xml:space="preserve">27. h.</note>
            tur in triangulo I B L ſit angulus I B L minor I L B, erit latus I L minus
              <lb/>
            latere I B.</s>
            <s xml:id="echoid-s6188" xml:space="preserve"/>
          </p>
          <figure number="183">
            <image file="0221-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0221-01"/>
          </figure>
          <p>
            <s xml:id="echoid-s6189" xml:space="preserve">Præterea, cum trian-
              <lb/>
            gula A P O, B P H
              <note symbol="a" position="right" xlink:label="note-0221-02" xlink:href="note-0221-02a" xml:space="preserve">I. tertij
                <lb/>
              conic.</note>
            æqualia, addito commu-
              <lb/>
            ni triangulo A P B, erunt
              <lb/>
            triangula A O B, A H B
              <lb/>
            ſuper eadem baſi A B in-
              <lb/>
            ter ſe ęqualia, quare O H
              <lb/>
            æquidiſtabit A B, ideo-
              <lb/>
            que vt D O ad O A, vel
              <lb/>
            D B ad B M, ita D H ad
              <lb/>
            H B, vel D A ad A I.
              <lb/>
            </s>
            <s xml:id="echoid-s6190" xml:space="preserve">Sunt ergo D M, D I pro-
              <lb/>
            portionaliter ſectæ in B,
              <lb/>
            A, quibus additæ ſunt D
              <lb/>
            E, D G, æquales ipſis D
              <lb/>
            B, D A, vtraq; </s>
            <s xml:id="echoid-s6191" xml:space="preserve">vtrique,
              <lb/>
            ſuntq; </s>
            <s xml:id="echoid-s6192" xml:space="preserve">rectangula triãgu-
              <lb/>
            la D M A, D I L ſimilia
              <lb/>
            inter ſe, quare recta ngu-
              <lb/>
            lum E M B ad quadratũ
              <lb/>
            M A, ſiue E B ad B
              <note symbol="b" position="right" xlink:label="note-0221-03" xlink:href="note-0221-03a" xml:space="preserve">21. primi
                <lb/>
              conic.</note>
            eſt vt rectãgulum G
              <note symbol="c" position="right" xlink:label="note-0221-04" xlink:href="note-0221-04a" xml:space="preserve">28. h.</note>
            ad quadratum I L, cumque ſit I L minor I B, erit quadratum I L minus
              <lb/>
            quadrato I B, ideoque rectangulum G I A ad quadratum I L, hoc eſt tranſ-
              <lb/>
            uerſum E B ad rectum B F, habebit maiorem rationem, quàm rectangu-
              <lb/>
            lum G I A ad quadratum I B, vel quàm tranſuerſum G A ad rectum A K;</s>
            <s xml:id="echoid-s6193" xml:space="preserve">
              <note symbol="d" position="right" xlink:label="note-0221-05" xlink:href="note-0221-05a" xml:space="preserve">21 pri-
                <lb/>
              mi conic.</note>
            ergo prima E B, ad ſecundam B F, maiorem habet rationem quàm tertia G
              <lb/>
            A ad quartam A K, ſed eſt prima E B minor tertia G A, ergo, & </s>
            <s xml:id="echoid-s6194" xml:space="preserve">
              <note symbol="e" position="right" xlink:label="note-0221-06" xlink:href="note-0221-06a" xml:space="preserve">24. h.</note>
            da B F erit minor quarta A K; </s>
            <s xml:id="echoid-s6195" xml:space="preserve">& </s>
            <s xml:id="echoid-s6196" xml:space="preserve">ſic de reliquis diametrorum rectis
              <note symbol="f" position="right" xlink:label="note-0221-07" xlink:href="note-0221-07a" xml:space="preserve">29. h.</note>
            teribus: </s>
            <s xml:id="echoid-s6197" xml:space="preserve">quare B F, rectum axis tranſuerſi, eſt _MINIMVM_, & </s>
            <s xml:id="echoid-s6198" xml:space="preserve">c.</s>
            <s xml:id="echoid-s6199" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6200" xml:space="preserve">Si autem axis E B æqualis fuerit eius recto B F; </s>
            <s xml:id="echoid-s6201" xml:space="preserve">cum demonſtratum ſit re-
              <lb/>
            ctangulum G I A ad quadratum I L eſſe vt tranſuerſus axis E B ad rectum
              <lb/>
            B F; </s>
            <s xml:id="echoid-s6202" xml:space="preserve">patet rectangulum quoque G I A æquari quadrato I L, ſed quando
              <lb/>
            E B æquatur B F, rectangulum etiam D M H æquatur quadrato M A, &</s>
            <s xml:id="echoid-s6203" xml:space="preserve">
              <note symbol="g" position="right" xlink:label="note-0221-08" xlink:href="note-0221-08a" xml:space="preserve">37. primi
                <lb/>
              conic.</note>
            tunc angulus D A M, ęqualis eſt angulo A H M, ergo etiam angulus D L
              <note symbol="h" position="right" xlink:label="note-0221-09" xlink:href="note-0221-09a" xml:space="preserve">27. h.</note>
            æquabitur angulo I B L, hoc eſt linea I B æqualis erit I L, ſed erat rectan-
              <lb/>
            gulum G I A æquale quadrato I L, ergo idem rectangulum G I A æqua-
              <lb/>
            bitur quadrato I B, ſiue tranſuerſa diameter A G, eius recto A K æqualis
              <lb/>
            erit, & </s>
            <s xml:id="echoid-s6204" xml:space="preserve">hoc ſemper, quæcunque ſit ducta tranſuerſa diameter præter axim.</s>
            <s xml:id="echoid-s6205" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s6206" xml:space="preserve">Cum ergo Hyperbole fuerit rectangula æquilatera, ad aliam quoque
              <lb/>
            diametri applicationem æquilatera erit, ſed axis eſt tranſuerſorum
              <note symbol="i" position="right" xlink:label="note-0221-10" xlink:href="note-0221-10a" xml:space="preserve">24. h.</note>
            _MIMVS_: </s>
            <s xml:id="echoid-s6207" xml:space="preserve">ergo in Hyperbola, cuius axis tranſuerſus eius rectum adæquet,
              <lb/>
            rectum axis aliorum rectorum eſt _MINIMVM_. </s>
            <s xml:id="echoid-s6208" xml:space="preserve">Quod erat, &</s>
            <s xml:id="echoid-s6209" xml:space="preserve">c.</s>
            <s xml:id="echoid-s6210" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>