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-div154" type="math:theorem" level="3" n="78">
              <pb o="52" rhead="IO. BAPT. BENED." n="64" file="0064" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0064"/>
              <p>
                <s xml:id="echoid-s688" xml:space="preserve">Sint exempli gratia .4. quantitates
                  <var>.a.b</var>
                :
                  <var>c.d</var>
                :
                  <var>e.f</var>
                : et
                  <var>.g.h</var>
                : inuicem proportionales in
                  <lb/>
                proportionalitate arithmetica. </s>
                <s xml:id="echoid-s689" xml:space="preserve">Hoc eſt vt quæ proportio (licet impropriè dicta)
                  <lb/>
                eſt ipſius
                  <var>.a.b.</var>
                ad
                  <var>.c.d.</var>
                  <reg norm="eadem" type="context">eadẽ</reg>
                ſit ipſius
                  <var>.e.f.</var>
                ad
                  <var>.g.h</var>
                . </s>
                <s xml:id="echoid-s690" xml:space="preserve">Tunc permutando dico eandem pro
                  <lb/>
                portionem fore ipſius
                  <var>.a.b.</var>
                ad
                  <var>.e.f.</var>
                quæ ipſius
                  <var>.c.d.</var>
                ad
                  <var>.g.h</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s691" xml:space="preserve">Nam, ex hypotheſi, differentia qua
                  <var>.a.b.</var>
                ſuperat
                  <var>.c.d.</var>
                (quæ ſit
                  <var>.m.b.</var>
                ) æqualis eſt
                  <lb/>
                differentiæ qua
                  <var>.e.f.</var>
                ſuperat
                  <var>.g.h.</var>
                (quæ ſit
                  <var>.i.f.</var>
                ) vnde
                  <var>.a.m.</var>
                reſiduum ex
                  <var>.a.b.</var>
                æquale erit
                  <lb/>
                  <var>c.d.</var>
                & reſiduum
                  <var>.e.i.</var>
                æquale
                  <var>.g.h</var>
                . </s>
                <s xml:id="echoid-s692" xml:space="preserve">Sit igitur exempli gratia
                  <var>.c.d.</var>
                maior
                  <var>.g.h.</var>
                per
                  <var>.c.n.</var>
                  <lb/>
                vnde
                  <var>.n.d.</var>
                æqualis erit
                  <var>.g.h.</var>
                </s>
                <s xml:id="echoid-s693" xml:space="preserve">quare
                  <var>.a.m.</var>
                maior erit
                  <var>.e.i.</var>
                per
                  <var>.a.K.</var>
                æqualem
                  <var>.c.n.</var>
                ex com-
                  <lb/>
                muni ſcientia. </s>
                <s xml:id="echoid-s694" xml:space="preserve">Vnde
                  <var>.K.m.</var>
                æqualis erit
                  <var>.n.d.</var>
                hoc eſt ipſi
                  <var>.g.h.</var>
                hoc eſt ipſi
                  <var>e.i</var>
                . </s>
                <s xml:id="echoid-s695" xml:space="preserve">Quare ex
                  <lb/>
                communi conceptu
                  <var>.b.K.</var>
                æqualis erit ipſi
                  <var>.f.e.</var>
                ſed
                  <var>.n.d.</var>
                æqualis eſt
                  <var>.g.h.</var>
                vt dictum eſt.
                  <lb/>
                </s>
                <s xml:id="echoid-s696" xml:space="preserve">Cum ergo
                  <var>.b.K.</var>
                æqualis ſit
                  <var>.e.f.</var>
                et
                  <var>.d.n.</var>
                ipſi
                  <var>.g.h.</var>
                et
                  <var>.a.b.</var>
                maior ſit ipſa
                  <var>.K.b.</var>
                per
                  <var>.a.K.</var>
                æqua-
                  <lb/>
                lem ipſi
                  <var>.c.n.</var>
                per quam
                  <var>c.n</var>
                :
                  <var>d.c.</var>
                maior eſt ipſa
                  <var>.d.n.</var>
                ſequitur verum eſſe
                  <reg norm="propoſitum" type="context">propoſitũ</reg>
                hoc
                  <lb/>
                eſt, quod eadem proportio ſit ipſius
                  <var>.a.b.</var>
                ad
                  <var>.e.f.</var>
                quæ
                  <var>.c.d.</var>
                ad
                  <var>.g.h.</var>
                arithmetice ſcilicet.</s>
              </p>
              <figure position="here" number="87">
                <image file="0064-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-01"/>
              </figure>
            </div>
            <div xml:id="echoid-div155" type="math:theorem" level="3" n="79">
              <head xml:id="echoid-head96" xml:space="preserve">THEOREMA
                <num value="79">LXXIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s697" xml:space="preserve">CVR prouenientia duorum numerorum diuidentium eiuſdem numeri diuiſi-
                  <lb/>
                bilis, geometricè
                  <reg norm="eandem" type="context">eandẽ</reg>
                inter ſe
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                ſeruant,
                  <reg norm="quam" type="context">quã</reg>
                ipſimet
                  <reg norm="diuidentes" type="context">diuidẽtes</reg>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s698" xml:space="preserve">Exempli gratia ſi per ſenarium & octonarium numerus vigintiquatuor diuida-
                  <lb/>
                tur, prouenientia erunt .4. et .3. eadem proportione, qua diuidentes.</s>
              </p>
              <p>
                <s xml:id="echoid-s699" xml:space="preserve">Cuius eſt ratio numerus diuiſibilis ſignificetur rectangulis
                  <var>.u.x.</var>
                et
                  <var>.n.e.</var>
                diuidentes
                  <lb/>
                autem ſint
                  <var>.u.o.</var>
                et
                  <var>.e.o.</var>
                </s>
                <s xml:id="echoid-s700" xml:space="preserve">quare ex ijs, quæ .10.
                  <lb/>
                  <figure xlink:label="fig-0064-02" xlink:href="fig-0064-02a" number="88">
                    <image file="0064-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-02"/>
                  </figure>
                theoremate dicta fuerunt
                  <var>.u.x.</var>
                per
                  <var>.u.o.</var>
                diui-
                  <lb/>
                ſo dabit
                  <var>.x.o.</var>
                & diuiſo
                  <var>.n.e.</var>
                per
                  <var>.e.o.</var>
                dabit
                  <var>.o.
                    <lb/>
                  n</var>
                . </s>
                <s xml:id="echoid-s701" xml:space="preserve">Dicimus itaque
                  <reg norm="eandem" type="context">eandẽ</reg>
                eſſe
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                  <lb/>
                  <var>o.x.</var>
                ad
                  <var>.o.n.</var>
                quæ
                  <var>.e.o.</var>
                ad
                  <var>.o.u.</var>
                quod patet ſub
                  <lb/>
                ſcriptam figuram conſiderantibus, in qua,
                  <lb/>
                ex .15. ſexti aut .20. ſeptimi, eadem propor-
                  <lb/>
                tio cernitur
                  <var>.o.x.</var>
                ad
                  <var>.o.n.</var>
                quæ
                  <var>.o.e.</var>
                ad
                  <var>.o.u</var>
                .</s>
              </p>
            </div>
            <div xml:id="echoid-div157" type="math:theorem" level="3" n="80">
              <head xml:id="echoid-head97" xml:space="preserve">THEOREMA
                <num value="80">LXXX</num>
              .</head>
              <p>
                <s xml:id="echoid-s702" xml:space="preserve">CVR quauis quantitate, tribus
                  <lb/>
                  <figure xlink:label="fig-0064-03" xlink:href="fig-0064-03a" number="89">
                    <image file="0064-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0064-03"/>
                  </figure>
                aut quatuor aut etiam pro libi-
                  <lb/>
                to pluribus diuidentibus numeris di-
                  <lb/>
                uifa, prouenientia eandem prorſus
                  <lb/>
                inter ſe proportionem ſeruabunt,
                  <lb/>
                quam ipſi diuidentes habere compe
                  <lb/>
                riuntur.</s>
              </p>
              <p>
                <s xml:id="echoid-s703" xml:space="preserve">Exempli gratia, proponitur nu-
                  <lb/>
                merus .60. quinque numeris diuiden
                  <lb/>
                dus, vtpotè .30. 20. 15. 12. 10. pro-
                  <lb/>
                uenientia erunt .2. 3. 4. 5. 6. eadem </s>
              </p>
            </div>
          </div>
        </div>
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