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-div22" type="math:theorem" level="3" n="9">
              <pb o="7" rhead="THEOR. ARITH." n="19" file="0019" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0019"/>
              <p>
                <s xml:id="echoid-s98" xml:space="preserve">Sit itaque linea
                  <var>.a.i.</var>
                diuifa in partes octo, & ei æqualis in longitudine
                  <var>.a.u.</var>
                in qua-
                  <lb/>
                tuor, productum verò vnius in alteram
                  <lb/>
                  <figure xlink:label="fig-0019-01" xlink:href="fig-0019-01a" number="12">
                    <image file="0019-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0019-01"/>
                  </figure>
                ſit
                  <var>.u.i.</var>
                trigintaduarum particularum
                  <lb/>
                fuperficialium fimilium &
                  <reg norm="æqualium" type="context">æqualiũ</reg>
                ad-
                  <lb/>
                inuicem. </s>
                <s xml:id="echoid-s99" xml:space="preserve">fit deinde
                  <var>.a.e.</var>
                ſeptem
                  <reg norm="partium" type="context">partiũ</reg>
                  <lb/>
                lineæ
                  <var>.a.i.</var>
                &
                  <var>.a.o.</var>
                trium partium
                  <var>.a.u</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s100" xml:space="preserve">tunc productum
                  <var>.a.e.</var>
                in
                  <var>.a.u.</var>
                erit
                  <var>.u.e.</var>
                  <lb/>
                particularum ſuperficialium vigintiocto
                  <lb/>
                & productum
                  <var>.a.o.</var>
                in
                  <var>.a.i.</var>
                erit
                  <var>.o.i.</var>
                par
                  <lb/>
                ticularum
                  <reg norm="ſuperficialium" type="context">ſuperficialiũ</reg>
                vigintiquatuor
                  <lb/>
                eiuſdem naturæ cum partibus triginta-
                  <lb/>
                duabus totius denominantis communis.
                  <lb/>
                </s>
                <s xml:id="echoid-s101" xml:space="preserve">vnde diuifo numerante vigintiocto per-
                  <lb/>
                numerantem vigintiquatuor, dabitur
                  <lb/>
                vnum cum fexta parte illius vnius.</s>
              </p>
            </div>
            <div xml:id="echoid-div24" type="math:theorem" level="3" n="10">
              <head xml:id="echoid-head26" xml:space="preserve">THEOREMA
                <num value="10">X</num>
              .</head>
              <p>
                <s xml:id="echoid-s102" xml:space="preserve">
                  <emph style="sc">PArtiri</emph>
                ſeu diuidere vno numero alium numerum, eſt etiam quodammodo
                  <lb/>
                eiuſmodi partem numeri diuifibilis inuenire refpectu totius numeri diuifibilis,
                  <lb/>
                cuiuſmodi eſt vnitas in diuidente refpectu totius diuidentis, partem inquam numeri
                  <lb/>
                diuiſibilis ſic ſe habentem ad totum numerum diuiſibilem ſicut vnitas ad totum di-
                  <lb/>
                uidentem, quod ſimiliter ex regula de tribus præſtamus dicentes, ſi tantus numerus
                  <lb/>
                diuidens dat
                  <reg norm="vnitatem" type="context">vnitatẽ</reg>
                , quid dabit numerus diuifibilis, quemadmodum ex
                  <ref id="ref-0006">.15. ſexti</ref>
                  <lb/>
                ſeu
                  <ref id="ref-0007">.20. ſeptimi</ref>
                licet ſpeculari, Idcircò quotieſcunque minorem numerum per
                  <lb/>
                maiorem diuidimus, ſemper qui prouenit fractus eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s103" xml:space="preserve">Exempli gratia, ſi cogitaremus lineam
                  <var>.a.e.</var>
                diuiſam in octo partes æquales, qua
                  <lb/>
                rum vna ſcilicet vnitas effet
                  <var>.a.i.</var>
                & cupere-
                  <lb/>
                mus eam diuidere in nouem partes, ac ſcire
                  <lb/>
                  <figure xlink:label="fig-0019-02" xlink:href="fig-0019-02a" number="13">
                    <image file="0019-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0019-02"/>
                  </figure>
                quan a ſit nona illius pars; </s>
                <s xml:id="echoid-s104" xml:space="preserve">manifeſtum eſſet,
                  <lb/>
                nonam partem ipſius
                  <var>.a.e.</var>
                minorem futuram
                  <lb/>
                ipſa
                  <var>.a.i.</var>
                cum
                  <var>.a.i.</var>
                diminui debeat à ſua inte-
                  <lb/>
                gritate eadem proportione, qua
                  <var>.a.e.</var>
                minor
                  <lb/>
                reperitur vna linea nouem partium æqualium
                  <lb/>
                fingularum
                  <var>.a.i</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s105" xml:space="preserve">Quod vt dilucidè cuiuis innoteſcat, hoc
                  <lb/>
                etiam modo licebit videre ſitlinea
                  <var>.n.c.</var>
                no-
                  <lb/>
                nupla ad
                  <var>.a.i.</var>
                & parallela ad
                  <var>.a.e.</var>
                dubium non
                  <lb/>
                eſt quin
                  <var>.n.c.</var>
                maior futura ſit ipſa
                  <var>.a.e.</var>
                iam ſi
                  <lb/>
                earum extrema congiungantur medijs duabus
                  <lb/>
                lineis
                  <var>.n.a.</var>
                et
                  <var>.c.e.</var>
                quæ ſimul concurrant in
                  <lb/>
                puncto
                  <var>.o.</var>
                (quod eſt probatu facillimum) da-
                  <lb/>
                buntur certe duo trianguli fimiles
                  <var>.a.o.e.</var>
                et
                  <var>.n.o.c</var>
                . </s>
                <s xml:id="echoid-s106" xml:space="preserve">Sit deinde
                  <var>.n.t.</var>
                vna è partibus
                  <lb/>
                ipſius
                  <var>.n.c.</var>
                quæ
                  <var>.n.t.</var>
                æqualis erit
                  <var>.a.i.</var>
                ex præſuppoſito. </s>
                <s xml:id="echoid-s107" xml:space="preserve">ducatur deinde
                  <var>.o.t.</var>
                quę
                  <lb/>
                interſecet
                  <var>.a.e.</var>
                in puncto
                  <var>.x.</var>
                dico
                  <var>.a.x.</var>
                tanto minorem futuram
                  <var>.a.i.</var>
                quanto
                  <var>.a.e.</var>
                  <lb/>
                minor eſt
                  <var>.n.c.</var>
                neque enim dubium eſſe poteſt quin proportiones
                  <var>.n.t.</var>
                ad
                  <var>.a.x.</var>
                et
                  <var>. </var>
                </s>
              </p>
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