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-div269" type="math:theorem" level="3" n="141">
              <p>
                <s xml:id="echoid-s1236" xml:space="preserve">
                  <pb o="95" rhead="THEOREM. ARITH." n="107" file="0107" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0107"/>
                  <var>e.o.</var>
                eſſe duas primas vrnas vini miſti hoc eſt primæ miſtionis, vnde cum eadem pro
                  <lb/>
                portio ſit
                  <var>.a.i.</var>
                ad
                  <var>.i.u.</var>
                vt
                  <var>.e.i.</var>
                ad
                  <var>.i.o.</var>
                ita erit (ex .19. quinti).
                  <var>a.e.</var>
                ad
                  <var>.o.u.</var>
                ut
                  <var>.a.i.</var>
                ad
                  <var>.i.u.</var>
                &
                  <lb/>
                  <reg norm="componendo" type="context">componẽdo</reg>
                ita erit
                  <var>.a.e.</var>
                cum
                  <var>.o.u.</var>
                hoc eſt
                  <var>.i.o.u.</var>
                (proptereà quòd
                  <var>.i.o.</var>
                æqualis eſt
                  <var>.a.e.</var>
                  <lb/>
                vt reſidua totorum æqualium) ad
                  <var>.o.u.</var>
                quemadmodum
                  <var>.a.i.u.</var>
                ad
                  <var>.i.u</var>
                . </s>
                <s xml:id="echoid-s1237" xml:space="preserve">Quare
                  <var>.i.u.</var>
                erit
                  <lb/>
                media proportionalis inter
                  <var>.a.u.</var>
                et
                  <var>.o.u.</var>
                vnde proportio
                  <var>.a.u.</var>
                ad
                  <var>.o.u.</var>
                dupla erit pro
                  <lb/>
                portioni
                  <var>.i.u.</var>
                ad
                  <var>.o.u</var>
                . </s>
                <s xml:id="echoid-s1238" xml:space="preserve">Nunc autem cum extracta fuerit quantitas
                  <var>.e.o.</var>
                ex primo mi-
                  <lb/>
                ſto, & poſteà infuſa aqua vſque ad plenitudinem dolij, proportio ingredientium
                  <lb/>
                huius ſecundi miſti erit ea, quæ eſt inter
                  <var>.o.u.</var>
                et
                  <var>.o.a.</var>
                eo quòd in prima miſtione pro-
                  <lb/>
                proportio ingredientium erat ea, quæ eſt inter
                  <var>.o.u.</var>
                et
                  <var>.a.e.</var>
                vel inter
                  <var>.a.e.</var>
                et
                  <var>.o.u.</var>
                  <lb/>
                vt demonſtrauimus. </s>
                <s xml:id="echoid-s1239" xml:space="preserve">Accipiamus ergo
                  <var>.t.m.</var>
                huiuſmodi ſecundi mifti, magnitudi-
                  <lb/>
                nis
                  <var>.a.i.</var>
                vel
                  <var>.e.o.</var>
                ſignificantis duas vrnas, & permutemus eum in tantam aquam,
                  <lb/>
                  <reg norm="ſitque" type="simple">ſitq́;</reg>
                punctum
                  <var>.o.</var>
                quod nobis diuidat
                  <var>t.m.</var>
                in
                  <var>.o.m.</var>
                et,
                  <var>o.t.</var>
                partes ſimplices, tali propor
                  <lb/>
                tione inuicem relatas, vt ſunt
                  <var>.o.u.</var>
                et
                  <var>.o.a.</var>
                vnde habebimus ex ſupradictis rationibus
                  <lb/>
                eandem proportionem ipſius
                  <var>.a.t.</var>
                ad
                  <var>.m.u.</var>
                vt
                  <var>.a.o.</var>
                ad
                  <var>.o.u.</var>
                & componendo
                  <var>.a.t.</var>
                cum
                  <var>.m.
                    <lb/>
                  u.</var>
                hoc eſt
                  <var>.i.m.u.</var>
                (eo quod cum
                  <var>.t.m.</var>
                æqualis ſit
                  <var>.a.i.</var>
                per conſequens
                  <var>.i.m.</var>
                æqualis erit
                  <var>.
                    <lb/>
                  a.t.</var>
                ) ad
                  <var>.m.u.</var>
                vt
                  <var>.a.o.u.</var>
                ad
                  <var>.o.u.</var>
                ſed proportio
                  <var>.a.o.u.</var>
                ad
                  <var>.o.u.</var>
                dupla erat proportioni
                  <var>.i.o.
                    <lb/>
                  u.</var>
                ad
                  <var>.o.u.</var>
                quemadmodum ſupra diximus. </s>
                <s xml:id="echoid-s1240" xml:space="preserve">Ergo proportio
                  <var>.i.m.u.</var>
                ad
                  <var>.m.u.</var>
                erit dupla
                  <lb/>
                ſimiliter proportioni
                  <var>.i.o.u.</var>
                ad
                  <var>.o.
                    <lb/>
                  u.</var>
                quapropter
                  <var>.o.u.</var>
                erit media pro­
                  <lb/>
                  <figure xlink:label="fig-0107-01" xlink:href="fig-0107-01a" number="147">
                    <image file="0107-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0107-01"/>
                  </figure>
                portionalis inter
                  <var>.i.u.</var>
                et
                  <var>.m.u</var>
                . </s>
                <s xml:id="echoid-s1241" xml:space="preserve">Ec-
                  <lb/>
                ce igitur quomodo eadem eſt pro
                  <lb/>
                portio
                  <var>.a.u.</var>
                ad
                  <var>.i.u.</var>
                quæ
                  <var>.i.u.</var>
                ad
                  <var>.o.u.</var>
                & quæ
                  <var>.o.u.</var>
                ad
                  <var>.m.u.</var>
                qui quidem modus neceſſarius
                  <lb/>
                eſt vt intellectus acquieſcat, id quod experientia non facit.</s>
              </p>
            </div>
            <div xml:id="echoid-div271" type="math:theorem" level="3" n="142">
              <head xml:id="echoid-head161" xml:space="preserve">THEOREMA
                <num value="142">CXLII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1242" xml:space="preserve">PRæcedens Tartaleæ quæſitum elegans quidem eſt, ſed pulchrum etiam vide-
                  <lb/>
                tur quærere proportionem ingredientium in ultima miſtione, cum cognita fue
                  <lb/>
                rit nobis proportio continentiæ dolij ad capacitatis vrnæ ſimul
                  <reg norm="cum" type="context">cũ</reg>
                numero vitium
                  <lb/>
                extractionum & impletionum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1243" xml:space="preserve">Exempli gratia, ſi proportio
                  <var>.a.u.</var>
                ad
                  <var>.a.i.</var>
                cognita nobis fuerit, cognoſcemus etiam
                  <lb/>
                  <var>e.i.</var>
                ex regula de tribus & per conſequens etiam
                  <var>.i.o.</var>
                reſiduum ex
                  <var>.e.o.</var>
                & ſimiliter ag-
                  <lb/>
                gregatum
                  <var>.a.i.</var>
                cum
                  <var>.i.o.</var>
                & ſic
                  <var>.o.u.</var>
                reſiduum totius, et
                  <var>.o.t.</var>
                ſimiliter, eo quòd
                  <var>.a.u.</var>
                ad
                  <var>.a.
                    <lb/>
                  o.</var>
                eſt ut
                  <var>.t.m.</var>
                ad
                  <var>.o.t.</var>
                vnde cognoſcemus etiam
                  <var>.o.m.</var>
                vt reſiduum
                  <var>.t.m.</var>
                & ſimiliter ag-
                  <lb/>
                gregatum
                  <var>.a.o.</var>
                cum
                  <var>.o.m.</var>
                hoc eſt
                  <var>.a.m.</var>
                & etiam
                  <var>.m.u.</var>
                reſiduum totius.</s>
              </p>
              <p>
                <s xml:id="echoid-s1244" xml:space="preserve">Cognoſcere autem proportionem totius dolij ad vrnam, vel ècontrà, cum cogni
                  <lb/>
                ta nobis fuerit proportio ingredientium in vltima miſtione ſimul cum numero vi-
                  <lb/>
                tium extractionum, & repletionum, quod ſcribit Tartalea, hoc etiam modo
                  <lb/>
                poſſumus.</s>
              </p>
              <p>
                <s xml:id="echoid-s1245" xml:space="preserve">Exempli gratia, ſi proportio
                  <var>.m.u.</var>
                ad
                  <var>.m.a.</var>
                cognita nobis fuerit, illicò ſcie-
                  <lb/>
                mus proportionem
                  <var>.a.u.</var>
                ad
                  <var>.m.u.</var>
                & cum ſciuerimus numerum vitium extractionum,
                  <lb/>
                & impletionum illicò cognoſci-
                  <lb/>
                mus multiplicitatem proportio-
                  <lb/>
                nis
                  <var>.a.u.</var>
                ad
                  <var>.m.u.</var>
                ad proportionem
                  <var>.
                    <lb/>
                    <figure xlink:label="fig-0107-02" xlink:href="fig-0107-02a" number="148">
                      <image file="0107-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0107-02"/>
                    </figure>
                  o.u.</var>
                ad
                  <var>.m.u.</var>
                quapropter propor-
                  <lb/>
                tio
                  <var>.o.u.</var>
                ad
                  <var>.m.u.</var>
                nobis cognita erit
                  <lb/>
                hoc eſt
                  <var>.a.u.</var>
                ad
                  <var>.i.u.</var>
                & ſimiliter ea, quæ eſt
                  <var>.a.u.</var>
                ad
                  <var>.a.i.</var>
                & è conuerſo ſimiliter.</s>
              </p>
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