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-div9" type="math:theorem" level="3" n="2">
              <pb o="3" rhead="THEOR. ARITH." n="15" file="0015" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0015"/>
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            <div xml:id="echoid-div11" type="math:theorem" level="3" n="3">
              <head xml:id="echoid-head19" xml:space="preserve">THEOREMA
                <num value="3">III</num>
              .</head>
              <p>
                <s xml:id="echoid-s54" xml:space="preserve">
                  <emph style="sc">CVr</emph>
                reperturi qualis ſit fractus aliquis numerus reſpectu alterius; </s>
                <s xml:id="echoid-s55" xml:space="preserve">multiplicare
                  <lb/>
                debeant numeratores adinuicem & ita etiam denominatores, ex quo produ-
                  <lb/>
                ctum ex numeratoribus nomen capiat à producto denominatorum.</s>
              </p>
              <p>
                <s xml:id="echoid-s56" xml:space="preserve">Huius ſi cauſam noſce vis, ſume
                  <var>.o.i.</var>
                &
                  <var>.o.u.</var>
                pro totis denominatoribus, tum
                  <var>.o.e.</var>
                  <lb/>
                &
                  <var>.o.a.</var>
                pro numeratoribus (exempli cauſa) ſit
                  <var>.o.i.</var>
                ſenarius
                  <var>.o.u.</var>
                quaternarius
                  <var>.o.e.</var>
                  <lb/>
                quinarius
                  <var>.o.a.</var>
                ternarius. </s>
                <s xml:id="echoid-s57" xml:space="preserve">Si noſce vis quæ ſint tres quartę partes quinque ſextarum,
                  <lb/>
                patet ex regulis practicis oriri quindecim vigeſimaſquartas. </s>
                <s xml:id="echoid-s58" xml:space="preserve">Id quomodo fiat, ex
                  <lb/>
                ſubſcripta ſigura depræhendetur, memores tamen eſſe oportet, quodlibet
                  <reg norm="productum" type="context">productũ</reg>
                  <lb/>
                conſiderari
                  <reg norm="tanquam" type="context">tanquã</reg>
                ſuperficiem, producentia
                  <reg norm="autem" type="context">autẽ</reg>
                tan-
                  <lb/>
                quam lineas. </s>
                <s xml:id="echoid-s59" xml:space="preserve">In hac igitur ſigura productum ex totis
                  <lb/>
                  <figure xlink:label="fig-0015-01" xlink:href="fig-0015-01a" number="4">
                    <image file="0015-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0015-01"/>
                  </figure>
                linearibus eſt
                  <var>.u.i.</var>
                aggregatum ex .24. partibus, &
                  <var>.u.e.</var>
                  <lb/>
                productum aggregatum ex .20. </s>
                <s xml:id="echoid-s60" xml:space="preserve">Quodita ſe habebit
                  <lb/>
                ad productum totale
                  <var>.u.i.</var>
                ſicut
                  <var>.o.e.</var>
                ad
                  <var>o.i.</var>
                ex prima
                  <lb/>
                ſexti aut .18. ſeptimi, ita
                  <var>.u.e.</var>
                erunt quinque ſextæ par
                  <lb/>
                tes
                  <var>.u.i.</var>
                quarum in propoſito exemplo, tres quartæ
                  <lb/>
                  <reg norm="quæruntur" type="context">quærũtur</reg>
                . </s>
                <s xml:id="echoid-s61" xml:space="preserve">Si
                  <reg norm="itaque" type="simple">itaq;</reg>
                multiplicabitur
                  <var>.o.e.</var>
                  <reg norm="cum" type="context">cũ</reg>
                  <var>.o.a.</var>
                orietur
                  <lb/>
                productum
                  <var>.a.e.</var>
                ita
                  <reg norm="proportionatum" type="context">proportionatũ</reg>
                ad
                  <var>.u.e.</var>
                ſicut
                  <var>.o.a.</var>
                ad
                  <lb/>
                  <var>o.u.</var>
                reperitur, ex prædictis rationibus. </s>
                <s xml:id="echoid-s62" xml:space="preserve">Quòd ſi
                  <reg norm="ſtatutum" type="context">ſtatutũ</reg>
                  <lb/>
                eſt
                  <var>.o.a.</var>
                tres quartas partes eſſe ipſius
                  <var>.u.o.</var>
                  <reg norm="etiam" type="context">etiã</reg>
                  <var>.a.e.</var>
                tres
                  <lb/>
                quartæ partes
                  <reg norm="erunt" type="context">erũt</reg>
                  <var>.u.e.</var>
                ſed
                  <var>.u.e.</var>
                quinque ſextæ ſunt ip-
                  <lb/>
                ſius
                  <var>.u.i.</var>
                ex quo ſequitur bonum eſſe huiuſmodi opus.</s>
              </p>
            </div>
            <div xml:id="echoid-div13" type="math:theorem" level="3" n="4">
              <head xml:id="echoid-head20" xml:space="preserve">THEOREMA
                <num value="4">IIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s63" xml:space="preserve">
                  <emph style="sc">CVr</emph>
                multiplicaturi fractos cum integris, rectè multiplicent numerantem fra-
                  <lb/>
                cti per numerum integrorum, partianturq́ue productum per
                  <reg norm="denominantem" type="context">denominantẽ</reg>
                  <lb/>
                fracti, ex quo numerus quæſitus colligitur.</s>
              </p>
              <p>
                <s xml:id="echoid-s64" xml:space="preserve">Propter quod mente concipiamus in ſubſequenti figura, numerum integrorum
                  <lb/>
                tanquam lineam
                  <var>.a.e.</var>
                qui, verbigratia, ſit denarius, quorum vnuſquiſque ſit æqualis
                  <lb/>
                  <var>a.i.</var>
                cogiteturq́ue productum ipſius
                  <var>.a.e.</var>
                in
                  <var>.a.i.</var>
                ſitq́ue
                  <var>.u.e.</var>
                quod quidem erit dena-
                  <lb/>
                rius ſuperficialis, conſtituta prius
                  <var>.a.u.</var>
                æqualis
                  <var>.a.i.</var>
                &
                  <var>.a.o.</var>
                ſint duæ tertiæ
                  <var>.a.u.</var>
                  <reg norm="quarum" type="context">quarũ</reg>
                  <lb/>
                duarum tertiarum productum in numerum
                  <var>.a.e.</var>
                ſit
                  <var>.o.e.</var>
                pariter
                  <var>.u.i.</var>
                vnitas ſit ſuper-
                  <lb/>
                ficialis prout
                  <var>.a.i.</var>
                vnitas eſt linearis, quam
                  <var>.u.i.</var>
                reſpicere debet productum
                  <var>.o.e.</var>
                ex
                  <lb/>
                quo integer ſuperficialis
                  <var>.u.i.</var>
                erit tanquam ternarius, & productum
                  <var>.o.i.</var>
                tanquam bi
                  <lb/>
                narius, & quia quælibet pars è viginti ipſius
                  <var>.o.e.</var>
                æqualis eſt tertiæ parti
                  <var>.u.i.</var>
                vnita-
                  <lb/>
                tis ſuperficialis; </s>
                <s xml:id="echoid-s65" xml:space="preserve">ſi cupiamus ſcire quot integræ vnitates ſint in partibus
                  <var>.o.e.</var>
                conſul-
                  <lb/>
                tum eſt eaſdem diuidere per denominantem
                  <var>.u.i.</var>
                compoſitum ex tribus partibus ſu
                  <lb/>
                perficialibus, & cum tam linea
                  <var>u.a.</var>
                quam ſuperficies
                  <var>.u.i.</var>
                diuidatur in 3. partes
                  <reg norm="aequa­ les" type="simple">ęqua­
                    <lb/>
                  les</reg>
                noſce peroportunum eſt eiuſmodi partitionem numeri
                  <var>.o.e.</var>
                fieri per numerum
                  <lb/>
                ipſius
                  <var>.u.i.</var>
                non
                  <var>.u.a.</var>
                ex prædictis cauſis.</s>
              </p>
              <figure position="here" number="5">
                <image file="0015-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0015-02"/>
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