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-div19" type="math:theorem" level="3" n="8">
              <p>
                <s xml:id="echoid-s88" xml:space="preserve">
                  <pb o="6" rhead="IO. BAPT. BENED." n="18" file="0018" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0018"/>
                cato
                  <var>.a.e.</var>
                per
                  <var>.a.u.</var>
                dabitur productum
                  <var>.u.e.</var>
                  <reg norm="trigintatrium" type="context">trigintatriũ</reg>
                  <lb/>
                  <figure xlink:label="fig-0018-01" xlink:href="fig-0018-01a" number="10">
                    <image file="0018-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0018-01"/>
                  </figure>
                partium. </s>
                <s xml:id="echoid-s89" xml:space="preserve">ad hæc quadratum
                  <var>.u.i.</var>
                conſtabit ex duode-
                  <lb/>
                cim partibus eiuſdem rationis cum reliquis duobus
                  <lb/>
                productis, quod quadratum
                  <var>.u.i.</var>
                vnitas eſt ſuperficia-
                  <lb/>
                lis, & communis denominans duorum productorum.
                  <lb/>
                </s>
                <s xml:id="echoid-s90" xml:space="preserve">quod ſi in præſentiarum cogitabimus lineam
                  <var>.c.d.</var>
                tri-
                  <lb/>
                gintatrium partium æqualium, et
                  <var>.c.t.</var>
                duodecim ſimi-
                  <lb/>
                lium, et
                  <var>.c.f.</var>
                viginti
                  <var>.c.n.</var>
                duodecim, multiplicato
                  <var>.c.
                    <lb/>
                  d.</var>
                cum
                  <var>.c.f.</var>
                dabitur ſuperficies
                  <var>.f.d.</var>
                660. fractorum
                  <lb/>
                ſuperficialium, quorum vnitas integra ſuperficialis
                  <lb/>
                erit quadratum
                  <var>.n.t.</var>
                144. partium cuiuſmodi
                  <var>.f.d.</var>
                  <lb/>
                partes habet .660. diuiſo itaque
                  <var>.f.d.</var>
                per
                  <var>.n.t.</var>
                pro-
                  <lb/>
                poſitum conſequetur. </s>
                <s xml:id="echoid-s91" xml:space="preserve">eo quòd eadem proportio erit
                  <lb/>
                  <figure xlink:label="fig-0018-02" xlink:href="fig-0018-02a" number="11">
                    <image file="0018-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0018-02"/>
                  </figure>
                producti
                  <var>.f.d.</var>
                ad
                  <var>.n.t.</var>
                quæ producti eius quòd fit ex
                  <var>.
                    <lb/>
                  a.e.</var>
                in
                  <var>.a.o.</var>
                ad
                  <var>.u.i.</var>
                nam proportio
                  <var>.c.d.</var>
                ad
                  <var>.c.t.</var>
                ea-
                  <lb/>
                dem eſt quæ
                  <var>.a.e.</var>
                ad
                  <var>.a.i.</var>
                &
                  <var>c.f.</var>
                ad
                  <var>.c.n.</var>
                vt
                  <var>.a.o.</var>
                ad
                  <var>.a.
                    <lb/>
                  u.</var>
                ex prima ſexti vel 18. ſeptimi, ſed vt
                  <var>.f.d.</var>
                ad id
                  <reg norm="quod" type="simple">ꝙ</reg>
                  <lb/>
                fit ex
                  <var>.f.c.</var>
                in
                  <var>.c.t.</var>
                eſt vt
                  <var>.c.d.</var>
                ad
                  <var>.c.t.</var>
                & vt eius
                  <reg norm="quod" type="simple">ꝙ</reg>
                fit ex
                  <lb/>
                  <var>f.c.</var>
                in
                  <var>.c.t.</var>
                ad
                  <var>.n.t.</var>
                eſt vt
                  <var>.f.c.</var>
                ad
                  <var>.c.n.</var>
                ex dictis pro-
                  <lb/>
                poſitionibus </s>
                <s xml:id="echoid-s92" xml:space="preserve">quare ex æqua proportionalitate, eodem
                  <lb/>
                modo diſcurrendo in figura
                  <var>.o.a.e.</var>
                ita ſe habebit
                  <var>.f.d.</var>
                  <lb/>
                ad
                  <var>.n.t.</var>
                vt
                  <var>.o.e.</var>
                ad
                  <var>.u.i</var>
                . </s>
                <s xml:id="echoid-s93" xml:space="preserve">Porrò ex ijs, quæ hactenus de
                  <lb/>
                fractorum multiplicatione conſiderata fuerunt, apertè
                  <lb/>
                ratio deprehenditur, cur productum, ſingulis producen
                  <lb/>
                tibus ſemper minus ſit, cum producta ſint ſuperficialia
                  <lb/>
                producentia verò ſemper linearia, omiſſis productis
                  <lb/>
                corporeis, quæ omnia ad ſuperficialia reducuntur.</s>
              </p>
            </div>
            <div xml:id="echoid-div22" type="math:theorem" level="3" n="9">
              <head xml:id="echoid-head25" xml:space="preserve">THEOREMA
                <num value="9">IX</num>
              .</head>
              <p>
                <s xml:id="echoid-s94" xml:space="preserve">
                  <emph style="sc">IN Ipsa</emph>
                fractorum diuiſione, animaduertendum eſt, denominantes numeros
                  <lb/>
                ſemper æquales inuicem eſſe debere, vnius ſcilicet ſpeciei, quòd ſi æquales non
                  <lb/>
                fuerint, neceſſe eſt via multiplicationis ipſorum denominantium adinuicem effice-
                  <lb/>
                re æquales vt ſint, ex quo productum oritur eiuſmodi, vt aptum ſit habere partes
                  <lb/>
                fractorum, quæ deſiderabantur.</s>
              </p>
              <p>
                <s xml:id="echoid-s95" xml:space="preserve">Exempli gratia, ſi proponerentur diuidenda ſeptem octaua per tria quarta præ-
                  <lb/>
                cipit antiquorum regula, vt ad vnam tantum denominationem reducantur. </s>
                <s xml:id="echoid-s96" xml:space="preserve">quare
                  <lb/>
                multiplicant denominantes inuicem. </s>
                <s xml:id="echoid-s97" xml:space="preserve">ex quo productum in materia propoſita ori-
                  <lb/>
                tur triginta duarum partium commune denominans, cuius duo numerantes ſunt vi-
                  <lb/>
                gintiquatuor & vigintiocto, producti ex multiplicatione vnius numerantis in deno
                  <lb/>
                minantem alterius, ex quo dantur vigintiquatuor tamquam tria quarta trigintaduo
                  <lb/>
                rum, & vigintiocto tanquam ſeptem octaua particularum vniformium, prout ope
                  <lb/>
                primæ ſexti aut decimæoctauæ ſeptimi in ſubſcripta figura cognoſci poteſt.</s>
              </p>
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