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-div462" type="section" level="4" n="7">
                <p>
                  <s xml:id="echoid-s2451" xml:space="preserve">
                    <pb o="202" rhead="IO. BAPT. BENED." n="214" file="0214" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0214"/>
                  poſtulato. </s>
                  <s xml:id="echoid-s2452" xml:space="preserve">Sed cum proportio
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  ęqualis ſit
                    <lb/>
                    <figure xlink:label="fig-0214-01" xlink:href="fig-0214-01a" number="265">
                      <image file="0214-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0214-01"/>
                    </figure>
                  proportioni
                    <var>.c.</var>
                  ad
                    <var>.d.</var>
                  communis autem
                    <var>.b.c</var>
                  : propor
                    <lb/>
                  tio. </s>
                  <s xml:id="echoid-s2453" xml:space="preserve">itaque
                    <var>.a.</var>
                  ad
                    <var>.c.</var>
                  æqualis erit
                    <var>.b.</var>
                  ad
                    <var>.d.</var>
                  ex ſecunda
                    <lb/>
                  parte .2. poſtulati compoſitè, & ſic habebimus pro
                    <lb/>
                  poſitum, ita quòd quotieſcunque
                    <reg norm="dabuntur" type="context">dabũtur</reg>
                  .4.
                    <reg norm="quam" type="context">quã</reg>
                    <lb/>
                  titates ex una parte proportionales, illæ ipſæ ex
                    <lb/>
                  altera proportionales erunt.</s>
                </p>
              </div>
              <div xml:id="echoid-div465" type="section" level="4" n="8">
                <head xml:id="echoid-head348" xml:space="preserve">THEOR. XVII.</head>
                <p>
                  <s xml:id="echoid-s2454" xml:space="preserve">DEcimiſeptimi theorematis hæc eſt demonſtratio. </s>
                  <s xml:id="echoid-s2455" xml:space="preserve">Ita ſe ha beat
                    <var>a.c.b.</var>
                  ad
                    <var>.c.
                      <lb/>
                    b.</var>
                  ſicut ſe habet
                    <var>.d.f.e.</var>
                  ad
                    <var>.f.e</var>
                  . </s>
                  <s xml:id="echoid-s2456" xml:space="preserve">Probo ita ſe habere
                    <var>.a.c.</var>
                  ad
                    <var>.c.b.</var>
                  ſicut ſe habet
                    <var>.d.
                      <lb/>
                    f.</var>
                  ad
                    <var>.f.e</var>
                  . </s>
                  <s xml:id="echoid-s2457" xml:space="preserve">Cogitemus itaque alterum terminum ſcilicet
                    <var>.n.f.</var>
                  qui ſic ſe habeat. ad
                    <var>.f.e.</var>
                    <lb/>
                  ſicut ſe habet
                    <var>.a.c.</var>
                  ad
                    <var>.c.b</var>
                  . </s>
                  <s xml:id="echoid-s2458" xml:space="preserve">Quare ex præcedenti theoremate ita ſe habebit
                    <var>.a.c.</var>
                  ad
                    <var>.n.
                      <lb/>
                    f.</var>
                  ſicut ſe habet
                    <var>.c.b.</var>
                  ad
                    <var>.f.e.</var>
                  & ex .8 poſtulato ita ſe habebit
                    <var>.a.c.b.</var>
                  ad
                    <var>.n.f.e.</var>
                  ſicut ſe ha-
                    <lb/>
                  bet
                    <var>.c.b.</var>
                  ad
                    <var>.f.e</var>
                  . </s>
                  <s xml:id="echoid-s2459" xml:space="preserve">Sed cum ex præſuppoſito ita ſe habeat
                    <var>.a.c.b.</var>
                  ad
                    <var>.c.b.</var>
                  ſicut ſe habet
                    <var>.
                      <lb/>
                    d.f.e.</var>
                  ad
                    <var>.f.e.</var>
                  ideo ex præcedenti theoremate ita ſe habebit
                    <var>.a.c.b.</var>
                  ad
                    <var>.d.f.e.</var>
                  ſicut ſe ha
                    <lb/>
                  bet
                    <var>.c.b.</var>
                  ad
                    <var>.f.e.</var>
                  demonſtratum autem eſt ita ſe habere
                    <var>.c.b.</var>
                  ad
                    <var>.f.e.</var>
                  ſicut ſe habet
                    <var>.a.c.b.</var>
                    <lb/>
                  ad
                    <var>.n.f.e</var>
                  . </s>
                  <s xml:id="echoid-s2460" xml:space="preserve">Quare ex .7. poſtulato proportio
                    <var>.a.c.b.</var>
                  ad
                    <var>.d.f.</var>
                  e, æqualis erit proportioni
                    <var>.a.
                      <lb/>
                    c.b.</var>
                  ad
                    <var>.n.f.e.</var>
                  & ex .4. poſtulato
                    <var>.d.f.e.</var>
                  æqualis erit
                    <var>.n.f.e</var>
                  . </s>
                  <s xml:id="echoid-s2461" xml:space="preserve">Itaque ex 3. poſtulato primi
                    <lb/>
                  Euclidis
                    <var>.f.d.</var>
                  æqualis erit
                    <var>.n.f</var>
                  . </s>
                  <s xml:id="echoid-s2462" xml:space="preserve">Quamob
                    <lb/>
                  rem proportio
                    <var>.a.c.</var>
                  ad
                    <var>.d.f.</var>
                  ęqualis erit
                    <lb/>
                    <figure xlink:label="fig-0214-02" xlink:href="fig-0214-02a" number="266">
                      <image file="0214-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0214-02"/>
                    </figure>
                  proportioni
                    <var>.a.c.</var>
                  ad
                    <var>.n.f.</var>
                  ex ſecunda par-
                    <lb/>
                  te tertij axiomatis præmiſſi. </s>
                  <s xml:id="echoid-s2463" xml:space="preserve">Igitur ita
                    <lb/>
                  ſe habebit
                    <var>.a.c.</var>
                  ad
                    <var>.d.f.</var>
                  ſicut
                    <var>.c.b.</var>
                  ad
                    <var>.f.e.</var>
                  ex
                    <lb/>
                  7. poſtulato. </s>
                  <s xml:id="echoid-s2464" xml:space="preserve">& ſic ex præcedenti theo-
                    <lb/>
                  remate ita ſe habebit
                    <var>.a.c.</var>
                  ad
                    <var>.c.b.</var>
                  ſicut
                    <var>.d.f.</var>
                  ad
                    <var>.f.e.</var>
                  quod erat propoſitum: </s>
                  <s xml:id="echoid-s2465" xml:space="preserve">Quotieſ-
                    <lb/>
                  cunque igitur dabuntur .4. quantitates coniunctim proportionales, diuiſim quoque
                    <lb/>
                  proportionales erunt.</s>
                </p>
              </div>
              <div xml:id="echoid-div467" type="section" level="4" n="9">
                <head xml:id="echoid-head349" xml:space="preserve">THEOREM. XVIII.</head>
                <p>
                  <s xml:id="echoid-s2466" xml:space="preserve">THeorema .18. hac ratione demonſtrari poteſt. </s>
                  <s xml:id="echoid-s2467" xml:space="preserve">Detur proportio
                    <var>.a.c.</var>
                  ad
                    <var>.c.b.</var>
                  ſi-
                    <lb/>
                  milis ei quæ eſt
                    <var>.d.f.</var>
                  ad
                    <var>.f.e.</var>
                  probo ita ſe habere
                    <var>.a.c.b.</var>
                  ad
                    <var>.c.b.</var>
                  ſicut ſe habet
                    <var>.d.f.
                      <lb/>
                    e.</var>
                  ad
                    <var>.f.e</var>
                  . </s>
                  <s xml:id="echoid-s2468" xml:space="preserve">In primis notum eſt ex .16. theoremate ita ſe habiturum,
                    <var>a.c.</var>
                  ad
                    <var>.d.f.</var>
                  ſi
                    <lb/>
                  cut
                    <var>.c.b.</var>
                  ad
                    <var>.f.e</var>
                  . </s>
                  <s xml:id="echoid-s2469" xml:space="preserve">Quare ex .8. poſtulato ita
                    <lb/>
                  ſe habebit
                    <var>.a.c.b.</var>
                  ad
                    <var>.d.f.e.</var>
                  ſicut
                    <var>.c.b.</var>
                  ad
                    <var>.f.e.</var>
                    <lb/>
                    <figure xlink:label="fig-0214-03" xlink:href="fig-0214-03a" number="267">
                      <image file="0214-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0214-03"/>
                    </figure>
                  </s>
                  <s xml:id="echoid-s2470" xml:space="preserve">Itaque ex .16. theoremate ita ſe habebit
                    <var>.
                      <lb/>
                    a.c.b.</var>
                  ad
                    <var>.c.b.</var>
                  ſicut
                    <var>.d.f.e.</var>
                  ad
                    <var>.f.e</var>
                  . </s>
                  <s xml:id="echoid-s2471" xml:space="preserve">Quod erat
                    <lb/>
                  propoſitum. </s>
                  <s xml:id="echoid-s2472" xml:space="preserve">Quotieſcunque igitur .4.
                    <lb/>
                  quantitates dabuntur vnius
                    <reg norm="eiuſdemque" type="simple">eiuſdemq́;</reg>
                  generis diſiunctim proportionales, coniun-
                    <lb/>
                  ctim quoque proportionales erunt.</s>
                </p>
              </div>
              <div xml:id="echoid-div469" type="section" level="4" n="10">
                <head xml:id="echoid-head350" xml:space="preserve">THEOREM. XIX.</head>
                <p>
                  <s xml:id="echoid-s2473" xml:space="preserve">THeorema .19. ſatis quidem apud Euclidem demonſtratur: </s>
                  <s xml:id="echoid-s2474" xml:space="preserve">eius tamentertia
                    <lb/>
                  pars commodius hac ratione demonſtrari poterit (nempe) quod cum ſit pro- </s>
                </p>
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