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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THEOREM. ARIT.
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            <div xml:id="echoid-div71" type="math:theorem" level="3" n="32">
              <p>
                <s xml:id="echoid-s299" xml:space="preserve">
                  <pb o="21" rhead="THEOREM. ARIT." n="33" file="0033" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0033"/>
                retur .20. ſcilicet et .4. certè .24. perſingulas partes diuiſo, daretur vnum proue-
                  <lb/>
                niens ſex integra, & alterum vnum & quinta pars, quorum ſumma eſſet ſeptem in-
                  <lb/>
                tegra cum quinta parte, tum altera parte per alteram diuiſa, daretur vnum proue-
                  <lb/>
                niens quinque integrorum & alterum vnius quinti tantum, quorum ſumma eſſet
                  <lb/>
                quinque integra, & vna quinta pars, minor prima reliquorum duorum prouenien-
                  <lb/>
                tium per binarium.</s>
              </p>
              <p>
                <s xml:id="echoid-s300" xml:space="preserve">Cuius conſiderationis cauſa, propoſitus numerus linea
                  <var>.q.p.</var>
                ſignificetur, eius duę
                  <lb/>
                partes lineis
                  <var>.q.x.</var>
                et
                  <var>.x.p.</var>
                  <reg norm="tum" type="context">tũ</reg>
                  <var>.q.f.</var>
                ſit proueniens ex diuiſione totius
                  <var>.q.p.</var>
                per
                  <var>.x.p.</var>
                et
                  <var>.
                    <lb/>
                  q.i.</var>
                ſit proueniens ex diuiſione eiuſdem
                  <var>.q.p.</var>
                per
                  <var>.q.x.</var>
                adhæc
                  <var>.h.m.</var>
                ſit proueniens,
                  <lb/>
                ex diuiſione
                  <var>.q.x.</var>
                per
                  <var>x.p.</var>
                et
                  <var>.h.k.</var>
                proue-
                  <lb/>
                niensex diuiſione
                  <var>.p.x.</var>
                per
                  <var>.q.x.</var>
                patet igi-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0033-01a" xlink:href="fig-0033-01"/>
                tur ex .22. theoremate huiuslibri proue-
                  <lb/>
                niés.h.m. minus eſſe proueniente
                  <var>.q.f.</var>
                per
                  <lb/>
                vnitaté, & proueniens
                  <var>.h.k.</var>
                minus proue-
                  <lb/>
                niente
                  <var>.q.i.</var>
                per alteram vnitatem. </s>
                <s xml:id="echoid-s301" xml:space="preserve">Itaque
                  <var>.
                    <lb/>
                  f.q.i.</var>
                maior erit
                  <var>.m.h.k.</var>
                per numerum binarium, quoderat propoſitum.</s>
              </p>
              <div xml:id="echoid-div71" type="float" level="4" n="1">
                <figure xlink:label="fig-0033-01" xlink:href="fig-0033-01a">
                  <image file="0033-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0033-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div73" type="math:theorem" level="3" n="33">
              <head xml:id="echoid-head49" xml:space="preserve">THEOREMA.
                <num value="33">XXXIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s302" xml:space="preserve">
                  <emph style="sc">QVilibet</emph>
                numerus, medius eſt
                  <lb/>
                proportionalis inter numerum
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0033-02a" xlink:href="fig-0033-02"/>
                ſui quadrati & vnitatem.</s>
              </p>
              <div xml:id="echoid-div73" type="float" level="4" n="1">
                <figure xlink:label="fig-0033-02" xlink:href="fig-0033-02a">
                  <image file="0033-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0033-02"/>
                </figure>
              </div>
              <p>
                <s xml:id="echoid-s303" xml:space="preserve">Detur enim numerus propoſitus,
                  <lb/>
                qui linea
                  <var>.a.u.</var>
                ſignificetur, cuiusqua-
                  <lb/>
                dratum ſit
                  <var>.u.n.</var>
                vnitas linearis ſit
                  <var>.i.a.</var>
                  <lb/>
                et ſuperficialis
                  <var>.o.</var>
                patebit ex .18. ſexti
                  <lb/>
                aut 11. octaui proportionem
                  <var>.u.n.</var>
                ad
                  <var>.
                    <lb/>
                  o.</var>
                futuram duplam proportioni
                  <var>.u.a.</var>
                  <lb/>
                ad
                  <var>.i.a.</var>
                ſed
                  <var>.i.a.</var>
                e
                  <unsure/>
                t.o. eadem (ſpecie)
                  <lb/>
                res
                  <reg norm="sunt" type="context">sũt</reg>
                , tanta ſcilicet
                  <var>.a.i.</var>
                quanta
                  <var>.o.</var>
                vni
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0033-03a" xlink:href="fig-0033-03"/>
                tas eſt, Itaque proportio numeri
                  <var>.u.n.</var>
                  <lb/>
                ad
                  <var>.u.a.</var>
                æqualis erit proportioni
                  <var>.u.a.</var>
                  <lb/>
                ad
                  <var>.i.a</var>
                . </s>
                <s xml:id="echoid-s304" xml:space="preserve">Quare numerus
                  <var>.u.a.</var>
                inter nu-
                  <lb/>
                merum
                  <var>.u.n.</var>
                & vnitatem, medius erit
                  <lb/>
                proportionalis.</s>
              </p>
              <div xml:id="echoid-div74" type="float" level="4" n="2">
                <figure xlink:label="fig-0033-03" xlink:href="fig-0033-03a">
                  <image file="0033-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0033-03"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div76" type="math:theorem" level="3" n="34">
              <head xml:id="echoid-head50" xml:space="preserve">THEOREMA
                <num value="34">XXXIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s305" xml:space="preserve">
                  <emph style="sc">HOc</emph>
                ipſum quod diximus & alia ratione ſpeculari licebit.</s>
              </p>
              <p>
                <s xml:id="echoid-s306" xml:space="preserve">Propoſitus numerus, nunc etiam per
                  <var>.a.u.</var>
                ſignificetur, eius quadratum per
                  <var>.
                    <lb/>
                  u.n.</var>
                vnitas linearis per
                  <var>.a.i.</var>
                  <reg norm="productumque" type="simple">productumq́;</reg>
                  <var>.a.u.</var>
                in
                  <var>.a.i.</var>
                terminetur,
                  <reg norm="ſitque" type="simple">ſitq́;</reg>
                  <var>.n.i</var>
                . </s>
                <s xml:id="echoid-s307" xml:space="preserve">quare
                  <lb/>
                  <var>n.i.</var>
                conſtabit numero íuperficiali æquali numero lineari
                  <var>.a.u.</var>
                & ex prima fexti aut .
                  <lb/>
                18. vel .19. ſeptimi, eadem erit proportio
                  <var>.u.n.</var>
                ad
                  <var>.i.n.</var>
                quæ eſt
                  <var>.a.u.</var>
                ad
                  <var>.a.i.</var>
                ſed nu-
                  <lb/>
                merus
                  <var>.a.u.</var>
                cum numero
                  <var>.n.i.</var>
                idem ſpecie eſt. </s>
                <s xml:id="echoid-s308" xml:space="preserve">Itaque medius eſt proportiona-
                  <lb/>
                lis inter
                  <var>.u.n.</var>
                & vnitatem.</s>
              </p>
            </div>
          </div>
        </div>
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