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-div7" type="chapter" level="2" n="1">
            <div xml:id="echoid-div52" type="math:theorem" level="3" n="23">
              <p>
                <s xml:id="echoid-s231" xml:space="preserve">
                  <pb o="16" rhead="IO. BAPT. BENED." n="28" file="0028" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0028"/>
                tas vero cui
                  <reg norm="differentiam" type="context">differentiã</reg>
                  <var>.n.c.</var>
                æquari dico, ſit
                  <var>.a.i</var>
                . </s>
                <s xml:id="echoid-s232" xml:space="preserve">Patet enim in primis, eandem propor
                  <lb/>
                tionem eſſe
                  <var>.a.e.</var>
                ad
                  <var>.a.c.</var>
                quæ eſt
                  <var>.u.e.</var>
                ad
                  <var>.a.i.</var>
                ex definitione diuiſionis, et eandem
                  <lb/>
                eſſe
                  <var>.a.u.</var>
                ad
                  <var>.a.n.</var>
                quæ eſt
                  <var>.u.e.</var>
                ad
                  <var>.a.i.</var>
                vnde ex .
                  <lb/>
                11. quinti ſic ſe habebit
                  <var>.a.e.</var>
                ad
                  <var>.a.c.</var>
                ſicut
                  <var>.a.
                    <lb/>
                    <figure xlink:label="fig-0028-01" xlink:href="fig-0028-01a" number="34">
                      <image file="0028-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-01"/>
                    </figure>
                  u.</var>
                ad
                  <var>.a.n.</var>
                et ex .19. eiuſdem ſic ſe habe-
                  <lb/>
                bit
                  <var>.u.e.</var>
                ad
                  <var>.n.c.</var>
                ſicut
                  <var>.a.e.</var>
                ad
                  <var>.a.c.</var>
                ſed. ſic ſe
                  <lb/>
                habebat
                  <var>.u.e.</var>
                ad
                  <var>.a.i</var>
                . </s>
                <s xml:id="echoid-s233" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                ex prædicta .11. quinti, ſic ſe habebit
                  <var>.u.e.</var>
                ad
                  <var>.n.c.</var>
                ſicut ad
                  <var>.a.
                    <lb/>
                  i</var>
                . </s>
                <s xml:id="echoid-s234" xml:space="preserve">Quare ex .9. eiuſdem
                  <var>.n.c.</var>
                æqualis erit
                  <var>.a.i.</var>
                etidcirco
                  <var>.n.c.</var>
                pariter vnitas erit.</s>
              </p>
            </div>
            <div xml:id="echoid-div55" type="math:theorem" level="3" n="24">
              <head xml:id="echoid-head40" xml:space="preserve">THEOREMA
                <num value="24">XXIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s235" xml:space="preserve">
                  <emph style="sc">CVr</emph>
                quibuslibet duobus numeris diuiſis adinuicem,
                  <reg norm="multiplicatisque" type="simple">multiplicatisq́</reg>
                prouenien
                  <lb/>
                tibus ſimul, productum, ſemper eſt vnitas ſuperficialis? </s>
                <s xml:id="echoid-s236" xml:space="preserve">Nempe ex .20. ſeptimi,
                  <lb/>
                quoniam vnitas linearis ſemper media proportionalis eſt inter bina prouenientia.
                  <lb/>
                </s>
                <s xml:id="echoid-s237" xml:space="preserve">Quodita ſpecularilicet.</s>
              </p>
              <p>
                <s xml:id="echoid-s238" xml:space="preserve">
                  <reg norm="Significentur" type="context">Significẽtur</reg>
                duo propoſiti numeri per
                  <var>.b.p.</var>
                et
                  <var>.b.d.</var>
                mutuo diuiſi, proueniens au-
                  <lb/>
                tem
                  <var>.b.p.</var>
                per
                  <var>.b.d.</var>
                diuiſum ſit
                  <var>.b.n.</var>
                tum proueniens
                  <var>.b.d.</var>
                diuiſum per
                  <var>.b.p.</var>
                ſit
                  <var>.b.a.</var>
                  <lb/>
                et
                  <var>.b.t.</var>
                ſit vnitas
                  <var>.b.p.</var>
                et
                  <var>.b.e.</var>
                vnitas
                  <var>.b.d.</var>
                ex quo
                  <var>.b.t.</var>
                æqualis erit
                  <var>.b.e</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s239" xml:space="preserve">Iam ex definitio ne diuiſionis, dabitur eadem proportio
                  <var>.b.p.</var>
                ad
                  <var>.b.n.</var>
                quæ eſt
                  <var>.b.d.</var>
                  <lb/>
                ad
                  <var>.b.e.</var>
                et proportio
                  <var>.b.d.</var>
                ad
                  <var>.b.a.</var>
                quæ eſt
                  <var>.b.p.</var>
                ad
                  <var>.b.t</var>
                . </s>
                <s xml:id="echoid-s240" xml:space="preserve">Sed cum ſic ſe habeat
                  <var>.b.
                    <lb/>
                  p.</var>
                ad
                  <var>.b.n.</var>
                ſicut
                  <var>.b.d.</var>
                ad
                  <var>.b.e.</var>
                permutando ſic ſe habebit
                  <var>.b.p.</var>
                ad
                  <var>.b.d.</var>
                ſicut
                  <var>.b.n.</var>
                ad
                  <var>.b.
                    <lb/>
                  e.</var>
                hoc eſt ad
                  <var>.b.t.</var>
                et cum ſic ſe habeat
                  <var>.b.d.</var>
                ad
                  <var>.b.a.</var>
                ſicut
                  <var>.b.p.</var>
                ad
                  <var>.b.t</var>
                : permutando ſic ſe
                  <lb/>
                habebit
                  <var>.b.d.</var>
                ad
                  <var>.b.p.</var>
                ſicut
                  <var>.b.a.</var>
                ad
                  <var>.b.t</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s241" xml:space="preserve">Quare euerſim ſic ſe habebit
                  <var>.b.p.</var>
                ad
                  <var>.
                    <lb/>
                    <figure xlink:label="fig-0028-02" xlink:href="fig-0028-02a" number="35">
                      <image file="0028-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-02"/>
                    </figure>
                    <lb/>
                  b.d.</var>
                ſicut
                  <var>.b.t.</var>
                ad
                  <var>.b.a.</var>
                ſed
                  <var>.b.n.</var>
                ad
                  <var>.b.t.</var>
                ſic
                  <lb/>
                ſe habebat vt
                  <var>.b.p.</var>
                ad
                  <var>.b.d</var>
                . </s>
                <s xml:id="echoid-s242" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                ex .11.
                  <lb/>
                quintiſic ſe habebit
                  <var>.b.n.</var>
                ad
                  <var>.b.t.</var>
                ſicut
                  <var>.b.
                    <lb/>
                    <figure xlink:label="fig-0028-03" xlink:href="fig-0028-03a" number="36">
                      <image file="0028-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-03"/>
                    </figure>
                  e.</var>
                ad
                  <var>.b.a</var>
                . </s>
                <s xml:id="echoid-s243" xml:space="preserve">Dictum autem eſt
                  <var>.b.e.</var>
                et
                  <var>.b.t.</var>
                idem omnino eſſe. </s>
                <s xml:id="echoid-s244" xml:space="preserve">Quare ex .20. ſeptimi pro-
                  <lb/>
                poſiti veritas innoteſcet.</s>
              </p>
            </div>
            <div xml:id="echoid-div57" type="math:theorem" level="3" n="25">
              <head xml:id="echoid-head41" xml:space="preserve">THEOREMA
                <num value="25">XXV</num>
              .</head>
              <p>
                <s xml:id="echoid-s245" xml:space="preserve">IDipſum & hac altera uia patebit.</s>
              </p>
              <p>
                <s xml:id="echoid-s246" xml:space="preserve">Duo illi numeri per
                  <var>.o.</var>
                et
                  <var>.u.</var>
                ſignificentur mutuo diuiſi, proueniens
                  <reg norm="autem" type="context">autẽ</reg>
                  <var>.o.</var>
                per
                  <var>.
                    <lb/>
                  u.</var>
                ſit
                  <var>.e.</var>
                et proueniens
                  <var>.u.</var>
                per
                  <var>.o.</var>
                ſit
                  <var>.x.</var>
                vnitas uerò per
                  <var>.i.</var>
                ſignificetur, quas tamen quanti-
                  <lb/>
                tates ſubſcripto modo ad inuicem diſponi-
                  <lb/>
                to. </s>
                <s xml:id="echoid-s247" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                ex definitione diuiſionis, eadem erit
                  <lb/>
                  <figure xlink:label="fig-0028-04" xlink:href="fig-0028-04a" number="37">
                    <image file="0028-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0028-04"/>
                  </figure>
                proportio
                  <var>.o.</var>
                ad
                  <var>.e.</var>
                quę eſt
                  <var>.u.</var>
                ad
                  <var>.i.</var>
                et
                  <var>.o.</var>
                ad
                  <var>.i.</var>
                quę
                  <lb/>
                eſt
                  <var>.u.</var>
                ad
                  <var>.x</var>
                . </s>
                <s xml:id="echoid-s248" xml:space="preserve">Quare ex æqualitate
                  <reg norm="proportionum" type="context">proportionũ</reg>
                  <var>.
                    <lb/>
                  c.</var>
                ad
                  <var>.i.</var>
                ſic ſe habebit ſicut
                  <var>.i.</var>
                ad
                  <var>.x.</var>
                erit enim
                  <var>.i.</var>
                  <lb/>
                media proportionalis inter
                  <var>.e.</var>
                et
                  <var>.x.</var>
                ex .20.
                  <reg norm="autem" type="context">autẽ</reg>
                  <lb/>
                ſeptimi propoſitum concludetur. </s>
                <s xml:id="echoid-s249" xml:space="preserve">Huiuſmodi rei cauſa etiam eſt, quod proueniens
                  <lb/>
                diuiſionis vnius eſt numerator æqualis denominatori diuiſionis alterius.</s>
              </p>
            </div>
            <div xml:id="echoid-div59" type="math:theorem" level="3" n="26">
              <head xml:id="echoid-head42" xml:space="preserve">THEOREMA
                <num value="26">XXVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s250" xml:space="preserve">
                  <emph style="sc">CVr</emph>
                duobus numeris mutuo diuiſis,
                  <reg norm="sumptis" type="context">sũptis</reg>
                deinde prouenientibus ſimul et adinui
                  <lb/>
                cem, & per hanc ſummam, diuiſa ſumma quadratorum dictorum
                  <reg norm="propoſitorum" type="context">propoſitorũ</reg>
                </s>
              </p>
            </div>
          </div>
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