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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I O. BAPT. BENED.
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            <div xml:id="echoid-div67" type="math:theorem" level="3" n="30">
              <p>
                <s xml:id="echoid-s285" xml:space="preserve">
                  <pb o="20" rhead="I O. BAPT. BENED." n="32" file="0032" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0032"/>
                biturum, ſicut
                  <var>.u.x.</var>
                ad
                  <var>.n.x.</var>
                ex prima ſexti aut .18. vel .19. ſeptimi, </s>
                <s xml:id="echoid-s286" xml:space="preserve">quare ex 11.
                  <lb/>
                quinti ita ſe habebit
                  <var>.o.x.</var>
                ad
                  <var>.e.x.</var>
                ſicut
                  <var>.s.x.</var>
                ad vnitatem; </s>
                <s xml:id="echoid-s287" xml:space="preserve">ſed ſicut ſe habet
                  <var>.s.x.</var>
                ad.
                  <lb/>
                vnitatem, ita ſe habet pariter
                  <var>.o.x.</var>
                ad
                  <var>.m</var>
                . </s>
                <s xml:id="echoid-s288" xml:space="preserve">vnde ex .11. prædicta ita ſe habebit
                  <var>.o.
                    <lb/>
                  x.</var>
                ad
                  <var>.m.</var>
                ſicut idipſum
                  <var>.o.x.</var>
                ad
                  <var>.e.x.</var>
                itaq́ue ex .9. prædicti quinti
                  <var>.m.</var>
                æqualis erit
                  <var>.o.x</var>
                .</s>
              </p>
              <div xml:id="echoid-div67" type="float" level="4" n="1">
                <figure xlink:label="fig-0031-02" xlink:href="fig-0031-02a">
                  <image file="0031-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0031-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div69" type="math:theorem" level="3" n="31">
              <head xml:id="echoid-head47" xml:space="preserve">THEOREMA
                <num value="31">XXXI</num>
              .</head>
              <p>
                <s xml:id="echoid-s289" xml:space="preserve">CVR propoſito aliquo numero in duas partes inæquales diuiſo, ſi rurſus per
                  <lb/>
                quamlibet ipſarum diuidatur, prouenientia tantumdem coniuncta quantum
                  <lb/>
                multiplicata efficiant.</s>
              </p>
              <p>
                <s xml:id="echoid-s290" xml:space="preserve">Exempli gratia, ſit denarius prop oſitus numerus, per binarium & octonarium
                  <lb/>
                diuiſus, prouenientia erunt quinque & vnum cum quarta parte, quæ coniuncta
                  <lb/>
                crunt .6. cum quarta parte lineari, quæ ſi mul multiplicata, pariter erunt .6. cum
                  <lb/>
                quarta parte ſuperficiali.</s>
              </p>
              <p>
                <s xml:id="echoid-s291" xml:space="preserve">Cuius ſpeculationis cauſa, totalis numerns, linea
                  <var>.q.p.</var>
                ſignificetur, eius duæ
                  <lb/>
                partes, per
                  <var>.k.</var>
                maiorem et
                  <var>.u.</var>
                minorem, ipſa vnitas per .t: proueniens ex diuiſio-
                  <lb/>
                ne
                  <var>.q.p.</var>
                per
                  <var>.k.</var>
                ſit
                  <var>.q.i.</var>
                proueniens autem ipſius
                  <var>.q.p.</var>
                per
                  <var>.u.</var>
                ſit
                  <var>.q.f.</var>
                </s>
                <s xml:id="echoid-s292" xml:space="preserve">quare ex defini-
                  <lb/>
                tione diuiſionis ita ſe habebit
                  <var>.q.p.</var>
                ad
                  <var>.q.i.</var>
                ſicut
                  <var>.k.</var>
                ad
                  <var>.t.</var>
                et
                  <var>.q.p.</var>
                ad
                  <var>.q.f.</var>
                ſicut
                  <var>.u.</var>
                ad
                  <var>.t.</var>
                  <lb/>
                hoc eſt
                  <var>.q.f.</var>
                ad
                  <var>.q.p.</var>
                ſicut
                  <var>.t.</var>
                ad
                  <var>.u.</var>
                vnde ex æqualitate
                  <reg norm="proportionum" type="context">proportionũ</reg>
                ſic ſe habebit
                  <var>.q.f.</var>
                  <lb/>
                ad
                  <var>.q.i.</var>
                ſicut
                  <var>.k.</var>
                ad
                  <var>.u.</var>
                et conuerſim. </s>
                <s xml:id="echoid-s293" xml:space="preserve">Ad hæc in linea
                  <var>.q.p.</var>
                vnitas, per lineam
                  <var>.q.o.</var>
                ſigni-
                  <lb/>
                ficetur, quo facto, dicamus, ſi
                  <var>.q.p.</var>
                ad
                  <var>.q.i.</var>
                ſic ſe habet vt
                  <var>.k.</var>
                ad
                  <var>.q.o.</var>
                itaque permu-
                  <lb/>
                tando, ſic ſe habebit
                  <var>.q.p.</var>
                ad
                  <var>.k.</var>
                ſicut
                  <var>.q.i.</var>
                ad
                  <var>.q.o.</var>
                hoc eſt
                  <var>.k.u.</var>
                ad
                  <var>.k.</var>
                ſicut
                  <var>.i.q.f.</var>
                ad
                  <var>.
                    <lb/>
                  q.f.</var>
                (nam
                  <var>.k.u.</var>
                partes ſunt integrales totius
                  <var>.q.p.</var>
                et
                  <var>.k.u.</var>
                ad
                  <var>.k.</var>
                eſt ſicut
                  <var>.i.q.f.</var>
                ad
                  <var>.q.f.</var>
                  <lb/>
                ex .18. quinti) </s>
                <s xml:id="echoid-s294" xml:space="preserve">Quare ita erit
                  <var>.i.q.f.</var>
                ad
                  <var>.q.f.</var>
                ſicut
                  <var>.q.i.</var>
                ad vnitatem
                  <var>.q.o.</var>
                ex .11. quinti
                  <lb/>
                Addatur deinde
                  <var>.q.i.</var>
                ad
                  <var>.q.f.</var>
                et
                  <var>.q.i.</var>
                per
                  <var>.
                    <lb/>
                  q.f.</var>
                multiplicetur, cuius multiplicatio-
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0032-01a" xlink:href="fig-0032-01"/>
                nis productum, ſit
                  <var>.x.f.</var>
                quod probabo
                  <lb/>
                æquale eſſe ſummæ
                  <var>.f.q.</var>
                cum
                  <var>.q.i</var>
                . </s>
                <s xml:id="echoid-s295" xml:space="preserve">Sece-
                  <lb/>
                tur enim linea
                  <var>.q.x.</var>
                in puncto
                  <var>.s.</var>
                ita. vt
                  <var>.
                    <lb/>
                  q.s.</var>
                æqualis ſit
                  <var>.q.o.</var>
                ſigneturq́ue pro-
                  <lb/>
                ductum
                  <var>.s.f.</var>
                </s>
                <s xml:id="echoid-s296" xml:space="preserve">quare
                  <reg norm="eadem" type="context">eadẽ</reg>
                erit propor-
                  <lb/>
                tio quantitatis
                  <var>.x.f.</var>
                ad
                  <var>.s.f.</var>
                quæ eſt
                  <var>.q.x.</var>
                  <lb/>
                ad
                  <var>.q.s.</var>
                ex prima ſexti, aut .18. vel 19.
                  <lb/>
                ſeptimi, hoc eſt, ſicut
                  <var>.q.i.</var>
                ad
                  <var>.q.o.</var>
                et
                  <lb/>
                ex .11. quinti (vt dictum eſt) ſicut
                  <var>.i.q.
                    <lb/>
                  f.</var>
                ad
                  <var>.q.f.</var>
                ſed numerus
                  <var>.s.f.</var>
                fuperficia-
                  <lb/>
                lis tantus eſt, quantus linearis
                  <var>.q.f</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s297" xml:space="preserve">quare ex .9. quinti tantus erit (ſu-
                  <lb/>
                perficialiter) numerus
                  <var>.x.f.</var>
                quantus
                  <lb/>
                (lineariter).
                  <var>f.q.i.</var>
                quod erat pro-
                  <lb/>
                poſitum.</s>
              </p>
              <div xml:id="echoid-div69" type="float" level="4" n="1">
                <figure xlink:label="fig-0032-01" xlink:href="fig-0032-01a">
                  <image file="0032-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0032-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div71" type="math:theorem" level="3" n="32">
              <head xml:id="echoid-head48" xml:space="preserve">THEOREMA.
                <num value="32">XXXII</num>
              .</head>
              <p>
                <s xml:id="echoid-s298" xml:space="preserve">CVR numero aliquo in duas partes inæquales diuiſo, ſi rurſus diuidatur per
                  <lb/>
                ſingulas partes, ſumma duorum prouenientium per binarium, ſemper ma-
                  <lb/>
                ior ſit ſumma prouenientium ex diuiſione vnius partis per alteram.</s>
              </p>
              <p>
                <s xml:id="echoid-s299" xml:space="preserve">
                  <reg norm="Exempli" type="context">Exẽpli</reg>
                gratia, ſi proponeretur numerus .24. qui in duas partes inæquales diuide­ </s>
              </p>
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