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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THEOR. ARITH.
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            <div xml:id="echoid-div16" type="math:theorem" level="3" n="6">
              <pb o="5" rhead="THEOR. ARITH." n="17" file="0017" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0017"/>
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            <div xml:id="echoid-div18" type="math:theorem" level="3" n="7">
              <head xml:id="echoid-head23" xml:space="preserve">THEOREMA
                <num value="7">VII</num>
              .</head>
              <p>
                <s xml:id="echoid-s76" xml:space="preserve">
                  <emph style="sc">CVr</emph>
                multiplicaturi integros numeros & fractos, cum integris & fractis, de-
                  <lb/>
                beant integros reducere ad ſpecies fractorum, eos colligendo cum fractis:
                  <lb/>
                </s>
                <s xml:id="echoid-s77" xml:space="preserve">deinde multiplicare hos vltimos numerantes adinuicem & productum partiri
                  <lb/>
                per productum denominantium.</s>
              </p>
              <p>
                <s xml:id="echoid-s78" xml:space="preserve">Vt (exempli cauſa) ſi volumus multiplicare vnum & duo tertia, per duo & tria
                  <lb/>
                quarta, reducentur omnia in fractos, ex quo vna ex parte eſſent quinque ter-
                  <lb/>
                tia, multiplicanda cum vndecim quartis ex altera, quo facto oriretur productum
                  <lb/>
                quinquagintaquinque fractorum, quod diuiſum per
                  <lb/>
                productum ternarijin quaternarium, videlicet per duode
                  <lb/>
                cim, quatuor integri proferentur cum ſeptem duodeci-
                  <lb/>
                mis fractis vnius integri.</s>
              </p>
              <figure position="here">
                <image file="0017-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0017-01"/>
              </figure>
              <p>
                <s xml:id="echoid-s79" xml:space="preserve">Detur ſubſequens figura in qua linea
                  <var>a.i.</var>
                æqualis ſit li-
                  <lb/>
                neæ
                  <var>.u.a.</var>
                quarum
                  <reg norm="vnaquæque" type="simple">vnaquæq;</reg>
                  <reg norm="conſideretur" type="context">cõſideretur</reg>
                pro integro nume
                  <lb/>
                ro: </s>
                <s xml:id="echoid-s80" xml:space="preserve">
                  <reg norm="cogiteturque" type="simple">cogiteturq́;</reg>
                  <var>.a.i.</var>
                valere quatuor in pręſenti
                  <reg norm="exemplo" type="context">exẽplo</reg>
                , &
                  <var>.a.
                    <lb/>
                  u.</var>
                tria: </s>
                <s xml:id="echoid-s81" xml:space="preserve">detur deinde linea
                  <var>.a.o.</var>
                æquipollens vni integro
                  <reg norm="cum" type="context">cũ</reg>
                  <lb/>
                duobus tertijs, &
                  <var>a.e.</var>
                æquipollens duobus integris & tri-
                  <lb/>
                bus quartis. </s>
                <s xml:id="echoid-s82" xml:space="preserve">Iam ſi hæ duæ lineæ in ſuos fractos redu-
                  <lb/>
                cantur, multiplicata (vt in ſequenti figura apparet.)
                  <var>a.o.</var>
                  <reg norm="cum" type="context">cũ</reg>
                  <lb/>
                  <var>a.e.</var>
                orietur productum
                  <var>o.e.</var>
                fractorum ſuperficialium
                  <lb/>
                  <reg norm="quinquagintaquinque" type="simple">quinquagintaquinq;</reg>
                , quorum integer ſuperficialis va-
                  <lb/>
                let duodecim, ſcilicet
                  <var>.u.i.</var>
                vt cuique manifeſtum eſt, ex
                  <lb/>
                quo, quærenti media partitione, quoties duodecim in-
                  <lb/>
                grediatur quinquagintaquinque, citra errorem, quæſitum
                  <lb/>
                occurret.</s>
              </p>
            </div>
            <div xml:id="echoid-div19" type="math:theorem" level="3" n="8">
              <head xml:id="echoid-head24" xml:space="preserve">THEOREMA
                <num value="8">VIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s83" xml:space="preserve">
                  <emph style="sc">ID ipsvm</emph>
                accideret ſi fractiad vnam eandemq́ue denominationem reduceren-
                  <lb/>
                tur, qui poſtmodum ſimul multiplicarentur, productumq́ue partiremur per qua-
                  <lb/>
                dratum denominantis communis.</s>
              </p>
              <p>
                <s xml:id="echoid-s84" xml:space="preserve">Exempli cauſa, ſint eadem quinque tertia, & vndecim quarta adinuicem multi-
                  <lb/>
                plicanda, quæ ſi reducantur ad vnam & eandem denominationem quinarius
                  <lb/>
                numerans vnius, multiplicabitur cum quaternario deno-
                  <lb/>
                minante alterius, & vndenarius ſecundi cum ternario de-
                  <lb/>
                nominante primi. ex quo vna ex parte eſſent viginti, ex
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0017-02a" xlink:href="fig-0017-02"/>
                altera 33. numerantia vnius
                  <reg norm="communis" type="context">cõmunis</reg>
                denominantis, quod
                  <lb/>
                eſſet productum ternarij in quaternarium, videlicet duo-
                  <lb/>
                decim, vt ex veteri regula patet. </s>
                <s xml:id="echoid-s85" xml:space="preserve">Iam ſi multiplicentur vi
                  <lb/>
                ginti cum trigintatribus, dabuntur 660. fracti, quorum in-
                  <lb/>
                teger erit quadratum duodenarij, nempe 144. quibus qui-
                  <lb/>
                dem 660. diuiſis per 144. proferentur quatuor integri &
                  <lb/>
                ſeptem duodecimi.</s>
              </p>
              <div xml:id="echoid-div19" type="float" level="4" n="1">
                <figure xlink:label="fig-0017-02" xlink:href="fig-0017-02a">
                  <image file="0017-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0017-02"/>
                </figure>
              </div>
              <p>
                <s xml:id="echoid-s86" xml:space="preserve">Cuius rei gratia ſit in ſubſcripta figura linea
                  <var>.a.i.</var>
                & ei
                  <lb/>
                æqualis
                  <var>.a.u.</var>
                pro integro lineari, quæ
                  <var>.a.i.</var>
                diuidatur in qua-
                  <lb/>
                tuor partes, &
                  <var>.a.u.</var>
                in tres, & linea
                  <var>.a.e.</var>
                ſit vndecim
                  <reg norm="partium" type="context">partiũ</reg>
                  <lb/>
                talium qualium
                  <var>.a.i.</var>
                eſt quatuor, &
                  <var>.a.o.</var>
                ſit quinque pro-
                  <lb/>
                ut
                  <var>.a.u.</var>
                eſt trium. </s>
                <s xml:id="echoid-s87" xml:space="preserve">nunc multiplicato
                  <var>.a.o.</var>
                &
                  <var>.a.i.</var>
                orietur pro-
                  <lb/>
                ductum
                  <var>.o.i.</var>
                viginti partium ſuperficialium. </s>
                <s xml:id="echoid-s88" xml:space="preserve">tum multipli- </s>
              </p>
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