Benedetti, Giovanni Battista de, Io. Baptistae Benedicti ... Diversarvm specvlationvm mathematicarum, et physicarum liber : quarum seriem sequens pagina indicabit ; [annotated and critiqued by Guidobaldo Del Monte]

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          <div xml:id="echoid-div441" type="chapter" level="2" n="5">
            <div xml:id="echoid-div454" type="section" level="3" n="2">
              <div xml:id="echoid-div455" type="section" level="4" n="2">
                <p>
                  <s xml:id="echoid-s2434" xml:space="preserve">
                    <var>
                      <pb o="201" rhead="IN QVINT. LIB. EVCLI." n="213" file="0213" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0213"/>
                    b.</var>
                  et
                    <var>.b.</var>
                  ad
                    <var>.g.</var>
                  & ſimiliter proportio
                    <var>.f.</var>
                  ad
                    <var>.h.</var>
                    <reg norm="componitur" type="context">cõponitur</reg>
                  ex eis quæſunt
                    <var>.f.</var>
                  ad
                    <var>.c.</var>
                  et
                    <var>.c.</var>
                  ad
                    <var>.d.</var>
                    <lb/>
                  et
                    <var>.d.</var>
                  ad
                    <var>.h</var>
                  .</s>
                </p>
              </div>
              <div xml:id="echoid-div457" type="section" level="4" n="3">
                <head xml:id="echoid-head343" xml:space="preserve">THEOR.V. ET VI.</head>
                <p>
                  <s xml:id="echoid-s2435" xml:space="preserve">
                    <emph style="sc">Circa</emph>
                  5. et .6. theorema nihil notandum occurrit.</s>
                </p>
              </div>
              <div xml:id="echoid-div458" type="section" level="4" n="4">
                <head xml:id="echoid-head344" xml:space="preserve">THEOR. VII. VIII. IX.X. XI. XII. XIII.</head>
                <p>
                  <s xml:id="echoid-s2436" xml:space="preserve">THeoremata à .6. in .13. cum ſint de obiectis intelligibilibus, ſine vllo medio,
                    <lb/>
                  ab intellectu cognitis, inter axiomata à me relata fuerunt .7. inquam quinti
                    <lb/>
                  Euclid. fecimus tertium Poſtulatum, .8. quintum, .9. quartum, .10. ſextum, .11. ſepti­
                    <lb/>
                  mum, .12. octauum, .13. nonum.</s>
                </p>
              </div>
              <div xml:id="echoid-div459" type="section" level="4" n="5">
                <head xml:id="echoid-head345" xml:space="preserve">THEOREM. XIIII.</head>
                <p>
                  <s xml:id="echoid-s2437" xml:space="preserve">QVartumdecimum Theorema ex Euclide demonſtrabitur, mutatis tantum
                    <lb/>
                  theorematibus ab interprete notatis, ita vt loco .7. 8. noni, & decimi citetur
                    <lb/>
                  tertium .5. 4. et .6. poſtulatum à me propoſitum.</s>
                </p>
              </div>
              <div xml:id="echoid-div460" type="section" level="4" n="6">
                <head xml:id="echoid-head346" xml:space="preserve">THEOR. XV.</head>
                <p>
                  <s xml:id="echoid-s2438" xml:space="preserve">QVintumdecimum Theorema ſic demonſtrabo; </s>
                  <s xml:id="echoid-s2439" xml:space="preserve">Sit, exempli gratia, a. termi-
                    <lb/>
                  nus antecedens. et
                    <var>.b.</var>
                  conſequens, qui-
                    <lb/>
                  bus duo multiplices ſumantur
                    <var>.c.</var>
                  et
                    <var>.d</var>
                  . </s>
                  <s xml:id="echoid-s2440" xml:space="preserve">Dico
                    <lb/>
                    <figure xlink:label="fig-0213-01" xlink:href="fig-0213-01a" number="263">
                      <image file="0213-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0213-01"/>
                    </figure>
                  eandem proportionem habiturum
                    <var>.c.</var>
                  ad
                    <var>.d.</var>
                    <lb/>
                  quam
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  habet. </s>
                  <s xml:id="echoid-s2441" xml:space="preserve">In primis enim manife-
                    <lb/>
                  ſtè patet quamlibet partem ipſius
                    <var>.c.</var>
                  habitu-
                    <lb/>
                  ram eandem proportionem cum qualibet par
                    <lb/>
                  te
                    <var>.d.</var>
                  quam habet
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  quare ex .7. et .8. po
                    <lb/>
                  ſtulato propoſitum eluceſcet.</s>
                </p>
              </div>
              <div xml:id="echoid-div462" type="section" level="4" n="7">
                <head xml:id="echoid-head347" xml:space="preserve">THEOREM. XVI.</head>
                <p>
                  <s xml:id="echoid-s2442" xml:space="preserve">SExtumdecimum theorema ſic demonſtrabitur. </s>
                  <s xml:id="echoid-s2443" xml:space="preserve">Sit, exempli cauſa, eadem pro
                    <lb/>
                  portio
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  quæ eſt
                    <var>.c.</var>
                  ad
                    <var>.d</var>
                  . </s>
                  <s xml:id="echoid-s2444" xml:space="preserve">Dico
                    <reg norm="quod" type="simple">ꝙ</reg>
                  ita ſe habebit
                    <var>.a.</var>
                  ad
                    <var>.c.</var>
                  ſicut
                    <var>.b.</var>
                  ad
                    <var>.d</var>
                  . </s>
                  <s xml:id="echoid-s2445" xml:space="preserve">Cogi-
                    <lb/>
                  temus itaque alterum iſtorum terminorum
                    <var>.c.</var>
                  aut
                    <var>.b.</var>
                  medium inter
                    <var>.a.</var>
                  et
                    <var>.d</var>
                  . </s>
                  <s xml:id="echoid-s2446" xml:space="preserve">quare
                    <lb/>
                  primum intelligamus
                    <var>.b.</var>
                  inter
                    <var>.a.</var>
                  et. d proportio ipſius
                    <var>.a.</var>
                  ad
                    <var>.d.</var>
                  componetur ex ea quę
                    <lb/>
                  eſt
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  & ea quæ eſt
                    <var>.b.</var>
                  ad
                    <var>.d.</var>
                  ex .12. poſtulato. </s>
                  <s xml:id="echoid-s2447" xml:space="preserve">Et ex eodem, illa ipſa proportio
                    <var>.
                      <lb/>
                    a.</var>
                  ad
                    <var>.d.</var>
                  pariter componetur ex ea quæ eſt
                    <var>.a.</var>
                  ad
                    <var>.c.</var>
                  & ea quæ eſt
                    <var>.c.</var>
                  ad
                    <var>.d.</var>
                  ſumpto
                    <var>.c.</var>
                  pro
                    <lb/>
                  medio termino. </s>
                  <s xml:id="echoid-s2448" xml:space="preserve">Ex quo ſequitur, aggregatum duarum proportionum, videlicet
                    <var>.a.</var>
                    <lb/>
                  ad
                    <var>.b.</var>
                  et
                    <var>.b.</var>
                  ad
                    <var>.d.</var>
                  æquale eſſe aggregato
                    <var>.a.</var>
                  ad
                    <var>.c.</var>
                  et
                    <var>.c.</var>
                  ad
                    <var>.d.</var>
                  ex quibus aggregatis æqua-
                    <lb/>
                  libus ſi duas proportiones æquales ſubtraxerimus, eam videlicet quæ eſt
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  & il
                    <lb/>
                  lam quæ eſt
                    <var>.c.</var>
                  ad
                    <var>.d.</var>
                  ſupererunt duæ proportiones
                    <lb/>
                  inter ſe æquales. </s>
                  <s xml:id="echoid-s2449" xml:space="preserve">erit enim proportio
                    <var>.a.</var>
                  ad
                    <var>.c.</var>
                  æqua
                    <lb/>
                    <figure xlink:label="fig-0213-02" xlink:href="fig-0213-02a" number="264">
                      <image file="0213-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0213-02"/>
                    </figure>
                  lis proportioni
                    <var>.b.</var>
                  ad
                    <var>.d.</var>
                  ex prima parte ſecundi po
                    <lb/>
                  ſtulati diuiſim.</s>
                </p>
                <p>
                  <s xml:id="echoid-s2450" xml:space="preserve">Alia etiam ratione idipſum
                    <reg norm="demonſtrari" type="context">demõſtrari</reg>
                  poteſt,
                    <lb/>
                  ſumpto
                    <var>.b.</var>
                  pro medio termino inter
                    <var>.a.</var>
                  et .c: et
                    <var>.c.</var>
                    <lb/>
                  pro termino medio inter
                    <var>.b.</var>
                  et
                    <var>.d</var>
                  . </s>
                  <s xml:id="echoid-s2451" xml:space="preserve">quare propor-
                    <lb/>
                  tio
                    <var>.a.</var>
                  ad
                    <var>.c.</var>
                  componetur ex
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  et
                    <var>.b.</var>
                  ad
                    <var>.c.</var>
                  illa
                    <lb/>
                  verò quæ eſt
                    <var>.b.</var>
                  ad
                    <var>.d.</var>
                  ex
                    <var>.b.</var>
                  ad
                    <var>.c.</var>
                  et
                    <var>.c.</var>
                  ad
                    <var>.d.</var>
                  ex .12. </s>
                </p>
              </div>
            </div>
          </div>
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