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]

Page concordance

< >
Scan Original
241 229
242 230
243 231
244 232
245 233
246 234
247 235
248 236
249 237
250 238
251 239
252 240
253 241
254 242
255 243
256 244
257 245
258 246
259 247
260 248
261 249
262 250
263 251
264 252
265 253
266 254
267 255
268 256
269 259
270 258
< >
page |< < (202) of 445 > >|
    <echo version="1.0">
      <text type="book" xml:lang="la">
        <div xml:id="echoid-div7" type="body" level="1" n="1">
          <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>
              </div>
            </div>
          </div>
        </div>
      </text>
    </echo>
Impressum