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 |< < (19) of 347 > >|
    <echo version="1.0RC">
      <text xml:lang="la" type="free">
        <div xml:id="echoid-div579" type="section" level="1" n="234">
          <p>
            <s xml:id="echoid-s5606" xml:space="preserve">
              <pb o="19" file="0201" n="201" rhead=""/>
            caret rectam G I, vt in L, tunc G L æquaretur ipſi H B, ideoque G
              <note symbol="a" position="right" xlink:label="note-0201-01" xlink:href="note-0201-01a" xml:space="preserve">16. ſec.
                <lb/>
              conic.</note>
            G L inter ſe æquales eſſent, totum, & </s>
            <s xml:id="echoid-s5607" xml:space="preserve">pars, quod eſt abſurdum.</s>
            <s xml:id="echoid-s5608" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5609" xml:space="preserve">Cum ergo puncta I, A cadant in oppoſitas ſectiones, iunctaque ſit I
              <lb/>
            A ſecans rectas C O, C D continentes angulum O C D, qui deinceps eſt
              <lb/>
            angulo D C E ſectionem A B continenti erunt ex ipſa abſciſſæ lineæ
              <note symbol="b" position="right" xlink:label="note-0201-02" xlink:href="note-0201-02a" xml:space="preserve">ibidem.</note>
            I, N A inter aſymptotos, & </s>
            <s xml:id="echoid-s5610" xml:space="preserve">ſectiones interiectæ inter ſe æquales. </s>
            <s xml:id="echoid-s5611" xml:space="preserve">Pro-
              <lb/>
            ducantur F A, F B vſque ad aſymptotos in D, E, agaturque ex I recta
              <lb/>
            I O æquidiſtans ad C D. </s>
            <s xml:id="echoid-s5612" xml:space="preserve">Cumque triangulorum I O M, N D A, baſes I
              <lb/>
            M, N A ſint in directum conſtitutæ ſintque latera I O, N D; </s>
            <s xml:id="echoid-s5613" xml:space="preserve">M O, A D
              <lb/>
            inter ſe parallela, ſingula ſingulis, erunt quoque anguli ad I, & </s>
            <s xml:id="echoid-s5614" xml:space="preserve">N; </s>
            <s xml:id="echoid-s5615" xml:space="preserve">vti
              <lb/>
            etiam ad M, & </s>
            <s xml:id="echoid-s5616" xml:space="preserve">A inter ſe æquales; </s>
            <s xml:id="echoid-s5617" xml:space="preserve">ſed & </s>
            <s xml:id="echoid-s5618" xml:space="preserve">baſes I M, N A inter ſe ſunt
              <lb/>
            æquales, vt ſuperiùs demonſtratum fuit, quare, & </s>
            <s xml:id="echoid-s5619" xml:space="preserve">reliqua latera M O,
              <lb/>
            A D æqualia erunt.</s>
            <s xml:id="echoid-s5620" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5621" xml:space="preserve">Præterea cum ſit linea B G æqualis H I, erunt quoque E G, C O inter
              <lb/>
            ſe æquales (ob æ quidiſtantiam linearum I O, H C, B E,) quibus addita
              <lb/>
            communi G C in prima, ſecunda, & </s>
            <s xml:id="echoid-s5622" xml:space="preserve">tertia figura, vel dempta in quin-
              <lb/>
            ta, proueniet E C, æqualis ipſi G O, ſed F D, E C ſunt æquales (nam
              <lb/>
            ſunt latera oppoſita in parallelogrammo C F,) quare F D ipſi G O ęqua-
              <lb/>
            lis erit; </s>
            <s xml:id="echoid-s5623" xml:space="preserve">ſi ergo ex his demantur æquales M O, A D; </s>
            <s xml:id="echoid-s5624" xml:space="preserve">reliquæ G M, F A
              <lb/>
            æquales erunt, at ſunt quoque parallelæ, vnde G F, I A inter ſe ęquidi
              <lb/>
            ſtabunt. </s>
            <s xml:id="echoid-s5625" xml:space="preserve">Quod demonſtrare oportebat.</s>
            <s xml:id="echoid-s5626" xml:space="preserve"/>
          </p>
        </div>
        <div xml:id="echoid-div582" type="section" level="1" n="235">
          <head xml:id="echoid-head243" xml:space="preserve">LEMMA V. PROP. XVI.</head>
          <p>
            <s xml:id="echoid-s5627" xml:space="preserve">Sint duæ rationes, A B nempe ad B C, & </s>
            <s xml:id="echoid-s5628" xml:space="preserve">D E ad F maioris
              <lb/>
            inæqualitatis, & </s>
            <s xml:id="echoid-s5629" xml:space="preserve">ſit ratio A B ad B C, minor ratione D E ad F.
              <lb/>
            </s>
            <s xml:id="echoid-s5630" xml:space="preserve">Oportet B C, ita ſecare in G, ita vt A G ad G C ſit vt D E
              <lb/>
            ad F.</s>
            <s xml:id="echoid-s5631" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5632" xml:space="preserve">FIat E H æqualis F, & </s>
            <s xml:id="echoid-s5633" xml:space="preserve">vt D H ad H E, ita A C ad C G, & </s>
            <s xml:id="echoid-s5634" xml:space="preserve">punctum
              <lb/>
            G erit quæſitum. </s>
            <s xml:id="echoid-s5635" xml:space="preserve">Quoniam cum A C ad C G ſit vt D H ad H E,
              <lb/>
            erit componendo A G ad G C,
              <lb/>
              <figure xlink:label="fig-0201-01" xlink:href="fig-0201-01a" number="161">
                <image file="0201-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/QN4GHYBF/figures/0201-01"/>
              </figure>
            vt D E ad E H, velad F. </s>
            <s xml:id="echoid-s5636" xml:space="preserve">Quod
              <lb/>
            faciendum erat.</s>
            <s xml:id="echoid-s5637" xml:space="preserve"/>
          </p>
          <p>
            <s xml:id="echoid-s5638" xml:space="preserve">Quod autem punctum G ca-
              <lb/>
            dat infra B, patet. </s>
            <s xml:id="echoid-s5639" xml:space="preserve">Nam ex hy-
              <lb/>
            poteſi, A B ad B C habet mi-
              <lb/>
            norem rationem quàm D E ad
              <lb/>
            F, vel ad E H, quare diuiden-
              <lb/>
            do A C ad C B habebit mino-
              <lb/>
            rem rationem, quàm D H ad H
              <lb/>
            E, vel quàm eadem A C ad C
              <lb/>
            G; </s>
            <s xml:id="echoid-s5640" xml:space="preserve">ergo C B eſt maior C G; </s>
            <s xml:id="echoid-s5641" xml:space="preserve">ſiue
              <lb/>
            punctum G cadit infra B. </s>
            <s xml:id="echoid-s5642" xml:space="preserve">Quod demonſtrandum erat.</s>
            <s xml:id="echoid-s5643" xml:space="preserve"/>
          </p>
        </div>
      </text>
    </echo>