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="body" level="1" n="1">
          <div xml:id="echoid-div7" type="chapter" level="2" n="1">
            <div xml:id="echoid-div24" type="math:theorem" level="3" n="10">
              <p>
                <s xml:id="echoid-s147" xml:space="preserve">
                  <pb o="10" rhead="IO. BAPT. BENED." n="22" file="0022" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0022"/>
                dabunt duodecim? </s>
                <s xml:id="echoid-s148" xml:space="preserve">nempe dabunt decemocto, numerum quæſitum ſcilicet,
                  <lb/>
                </s>
                <s xml:id="echoid-s149" xml:space="preserve">Tunc autem nil aliud pręſtamus quam quòd quærimus numerum ad quem ita ſe
                  <lb/>
                habeant duodecim, ſicut duo ad tria. </s>
                <s xml:id="echoid-s150" xml:space="preserve">Ita etiam ſi quis quærat, cuius numeri duo
                  <lb/>
                tertia ſint tres quintę, dicet, ſi tria dant
                  <reg norm="quinque" type="simple">quinq;</reg>
                , quid dabunt duo tertia? </s>
                <s xml:id="echoid-s151" xml:space="preserve">nempe da-
                  <lb/>
                bunt integrum cum fracto nono. </s>
                <s xml:id="echoid-s152" xml:space="preserve">Hoc erit
                  <reg norm="itaque" type="simple">itaq;</reg>
                quęrere numerum ad quem ſic ſe
                  <lb/>
                habeant duo tertia ſicut tria ad
                  <reg norm="quinque" type="simple">quinq;</reg>
                , quod manifeſtum eſt per ſe.</s>
              </p>
              <p>
                <s xml:id="echoid-s153" xml:space="preserve">Eadem ratione qui ſcire vellet, cuius numeri duæ ſeptimæ, eſſent octo integra-
                  <lb/>
                rum cum duabus quintis, diceret, ſi duo dant ſeptem quid dabunt octo integra cum
                  <lb/>
                duabus quintis? </s>
                <s xml:id="echoid-s154" xml:space="preserve">nempe dabunt .29. integra cum duabus quintis numerum quæſi-
                  <lb/>
                tum. </s>
                <s xml:id="echoid-s155" xml:space="preserve">Sic etiam qui transferre uellet fractum numerum in fractum, id perficeret
                  <lb/>
                ex regula de tribus.</s>
              </p>
              <p>
                <s xml:id="echoid-s156" xml:space="preserve">Exempli gratia ſi proponerentur vnde cim tertiædecimæ vnius totius, toto diui-
                  <lb/>
                ſo in .13. partes,
                  <reg norm="deſideraremusque" type="simple">deſideraremusq́;</reg>
                ſcire, quot partes totius
                  <reg norm="eſsent" type="context">eſsẽt</reg>
                vndecim
                  <reg norm="tertiaedeci- mæ" type="simple">tertiędeci-
                    <lb/>
                  mæ</reg>
                , toto in .4. partes diuiſo, diceremus ſi .13. dant .11. quid dabunt quatuor? </s>
                <s xml:id="echoid-s157" xml:space="preserve">nem
                  <lb/>
                pe
                  <reg norm="dabunt" type="context">dabũt</reg>
                tres quartas
                  <reg norm="cum" type="context">cũ</reg>
                  <reg norm="quinque" type="simple">quinq;</reg>
                tertijsdecimis unius quartæ, hoc verò nihil aliud eſt
                  <lb/>
                quam querere numerum, ad quem ſic ſe habeat totum in 4. partes diuiſum, ſicut
                  <lb/>
                idem totum diuiſum in tredecim ſe habet ad undecim tertiasdecimas, Porrò ad
                  <lb/>
                alia etiam multa hæc regula accommodata eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s158" xml:space="preserve">Hæc enim
                  <reg norm="non" type="context">nõ</reg>
                ſine propoſito dicta ſunt, ſed ut
                  <reg norm="quiſque" type="simple">quiſq;</reg>
                videat cauſam ſimilium ope-
                  <lb/>
                rationum, quæ à practicis circa fractos numeros ſcriptæ ſunt, omnem à diuina illa
                  <lb/>
                regula de tribus originem trahere ut etiam in ſequentibus videbimus.</s>
              </p>
            </div>
            <div xml:id="echoid-div28" type="math:theorem" level="3" n="11">
              <head xml:id="echoid-head27" xml:space="preserve">THEOREMA
                <num value="11">XI</num>
              .</head>
              <p>
                <s xml:id="echoid-s159" xml:space="preserve">
                  <emph style="sc">CVr</emph>
                productum ex eo quod oritur in diuidente, ſemper æquale eſt numero
                  <lb/>
                diuiſibili ſi queras ita accipe.</s>
              </p>
              <p>
                <s xml:id="echoid-s160" xml:space="preserve">Sit numerus diuiſibilis
                  <var>.b.</var>
                quod oritur ſit
                  <var>.c.</var>
                diuidens
                  <var>.d.</var>
                & vnitas diuidentis
                  <var>.t.</var>
                cum
                  <lb/>
                igitur, vt in præcedenti theoremate dictum
                  <lb/>
                fuit, eadem ſit proportio
                  <var>.b.</var>
                ad
                  <var>.c.</var>
                quæ eſt
                  <var>.d.</var>
                  <lb/>
                  <figure xlink:label="fig-0022-01" xlink:href="fig-0022-01a" number="18">
                    <image file="0022-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0022-01"/>
                  </figure>
                ad
                  <var>.t.</var>
                manifeſte deprehenditur ex .20. ſepti
                  <lb/>
                mi, productum ex
                  <var>.b.</var>
                in
                  <var>.t.</var>
                æquale eſſe pro-
                  <lb/>
                ducto
                  <var>.c.</var>
                in
                  <var>d</var>
                .</s>
              </p>
            </div>
            <div xml:id="echoid-div30" type="math:theorem" level="3" n="12">
              <head xml:id="echoid-head28" xml:space="preserve">THEOREMA
                <num value="12">XII</num>
              .</head>
              <p>
                <s xml:id="echoid-s161" xml:space="preserve">ID ipſum alia ratione contemplari licet.</s>
              </p>
              <p>
                <s xml:id="echoid-s162" xml:space="preserve">Numerus diuiſibilis ſignificetur per lineam
                  <var>.n.e.</var>
                diuidens verò per lineam
                  <var>.a.e.</var>
                  <lb/>
                quod oritur linea
                  <var>.u.e.</var>
                vnitas diuidentis
                  <var>.o.e.</var>
                  <reg norm="quam" type="context">quã</reg>
                cogitamus eſſe vnitatem linearem;
                  <lb/>
                </s>
                <s xml:id="echoid-s163" xml:space="preserve">ad hæc productum ex
                  <var>.u.e.</var>
                in
                  <var>.a.e.</var>
                ſit ſuperficies
                  <var>.u.a</var>
                . </s>
                <s xml:id="echoid-s164" xml:space="preserve">Dico ſuperficiem
                  <var>.u.a.</var>
                componi
                  <lb/>
                ex tot vnitatibus ſuperficialibus quot linearibus conſtat linea
                  <var>.n.e.</var>
                nam ex ijs quæ
                  <lb/>
                diuidendi ratione notauimus,
                  <reg norm="conſtituitur" type="context">cõſtituitur</reg>
                  <lb/>
                eandem proportionem eſſe
                  <var>.n.e.</var>
                ad
                  <var>.u.e.</var>
                  <lb/>
                  <figure xlink:label="fig-0022-02" xlink:href="fig-0022-02a" number="19">
                    <image file="0022-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0022-02"/>
                  </figure>
                quę eſt
                  <var>.a.e.</var>
                ad
                  <var>.o.e</var>
                . </s>
                <s xml:id="echoid-s165" xml:space="preserve">At ex prima ſexti aut
                  <lb/>
                18. ſeptimi ſic ſe habet totale
                  <reg norm="productum" type="context">productũ</reg>
                  <var>.
                    <lb/>
                  u.a.</var>
                ad partiale
                  <var>.u.o.</var>
                ſicut
                  <var>.a.e.</var>
                ad
                  <var>.o.e</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s166" xml:space="preserve">quare ſic ſe habebit
                  <var>.u.a.</var>
                ad
                  <var>.u.o.</var>
                ſicut
                  <var>.n.
                    <lb/>
                  e.</var>
                ad
                  <var>.u.e.</var>
                ſed
                  <var>.u.e.</var>
                et
                  <var>.u.o.</var>
                numero non differunt, cum ſint vnius & eiuſdem ſpeciei, (ta-
                  <lb/>
                met ſi numerus
                  <var>.u.o.</var>
                ſit ſuperficialis et
                  <var>.u.e.</var>
                linearis). </s>
                <s xml:id="echoid-s167" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                ex nona quinti numerus
                  <var>.
                    <lb/>
                  u.a.</var>
                æqualis erit numero
                  <var>.n.e</var>
                .</s>
              </p>
            </div>
          </div>
        </div>
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