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]
page |< < (15) of 445 > >|
THEOREM. ARIT.
    <echo version="1.0">
      <text type="book" xml:lang="la">
        <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-div48" type="math:theorem" level="3" n="21">
              <p>
                <s xml:id="echoid-s216" xml:space="preserve">
                  <pb o="15" rhead="THEOREM. ARIT." n="27" file="0027" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0027"/>
                eſſe
                  <reg norm="producto" type="simple">ꝓducto</reg>
                  <var>.q.p.</var>
                in
                  <var>.g.k.</var>
                  <reg norm="quod" type="wordlist">qđ</reg>
                  <reg norm="autem" type="context">autẽ</reg>
                ſit
                  <var>.o</var>
                . </s>
                <s xml:id="echoid-s217" xml:space="preserve">Patet enim
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                  <var>.o.</var>
                ad
                  <var>.q.p.</var>
                  <reg norm="eandem" type="context">eandẽ</reg>
                eſſe
                  <lb/>
                cum proportione
                  <var>.g.k.</var>
                ad ſuam vnitatem linearem, ex decimaoctaua, aut decima-
                  <lb/>
                nona ſeptimi, hæc vero vnitas linearis ſit
                  <var>.t.</var>
                cuius ſuperficialis ſit
                  <var>.u.</var>
                vnitas ſcilicet to-
                  <lb/>
                ties in ſeipſam multiplicata quoties propoſita dignitas patitur, tametſi in præſen
                  <lb/>
                ti exemplo quadrata dignitas ſumatur. </s>
                <s xml:id="echoid-s218" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                ex eiſdem propoſitionibus decimaocta
                  <lb/>
                ua aut decimanona, ſic ſe habet
                  <var>.m.</var>
                ad
                  <var>.n.</var>
                ſicut
                  <var>.i.</var>
                ad
                  <var>.u</var>
                . </s>
                <s xml:id="echoid-s219" xml:space="preserve">Scimus pręterea
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                  <var>.
                    <lb/>
                  m.</var>
                ad
                  <var>.n.</var>
                (eo quod in propoſito exemplo ſint quadrata) duplam eſſe proportioni
                  <var>.b.
                    <lb/>
                  d.</var>
                ad
                  <var>.q.p.</var>
                et ipſius
                  <var>.i.</var>
                ad
                  <var>.u.</var>
                pariter duplam proportioni
                  <var>.g.k.</var>
                ad
                  <var>.t.</var>
                iam autem dictum
                  <lb/>
                fuit ſic ſe habere
                  <var>.m.</var>
                ad
                  <var>.n.</var>
                ſicut
                  <var>.i.</var>
                ad
                  <var>.u</var>
                . </s>
                <s xml:id="echoid-s220" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                  <var>.
                    <lb/>
                  b.d.</var>
                ſic ſe habebit ad
                  <var>.q.p.</var>
                ſicut
                  <var>.g.k.</var>
                ad
                  <var>.t.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0027-01a" xlink:href="fig-0027-01"/>
                quandoquidem ſic ſe habeattotum ad
                  <reg norm="to- tum" type="context">to-
                    <lb/>
                  tũ</reg>
                , ſicut pars ad
                  <reg norm="partem" type="context">partẽ</reg>
                ,
                  <reg norm="dum" type="context">dũ</reg>
                ſimiles ſint, proba
                  <lb/>
                  <reg norm="tum" type="context">tũ</reg>
                  <reg norm="autem" type="context">autẽ</reg>
                eſt ſuperius ita ſe habere
                  <var>.o.</var>
                ad
                  <var>.q.p.</var>
                  <lb/>
                ſicut
                  <var>.g.k.</var>
                ad
                  <var>.t.</var>
                  <reg norm="itaque" type="simple">itaq;</reg>
                  <var>.o.</var>
                ſic ſe habebit ad
                  <var>.q.p.</var>
                  <lb/>
                ſicut
                  <var>.b.d.</var>
                ad
                  <var>.q.p.</var>
                vnde
                  <var>.o.</var>
                æqualis erit
                  <var>.b.d.</var>
                  <lb/>
                Hocipſum cęteris dignitatibus conueniet,
                  <lb/>
                mutatis tantummodo proportionibus
                  <var>.m.
                    <lb/>
                  n.</var>
                ad proportionem
                  <var>.b.d</var>
                :
                  <var>q.p.</var>
                ſic propor-
                  <lb/>
                tionibus duarum dignitatum
                  <var>.i.u.</var>
                ad pro-
                  <lb/>
                portionem ſuarum radicum
                  <var>.g.k.t</var>
                .</s>
              </p>
              <div xml:id="echoid-div48" type="float" level="4" n="1">
                <figure xlink:label="fig-0027-01" xlink:href="fig-0027-01a">
                  <image file="0027-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0027-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div50" type="math:theorem" level="3" n="22">
              <head xml:id="echoid-head38" xml:space="preserve">THEOREMA
                <num value="22">XXII</num>
              .</head>
              <p>
                <s xml:id="echoid-s221" xml:space="preserve">
                  <emph style="sc">DOcent</emph>
                veteres, quòd ſi quilibet numerus in duas partes inæquales diuiſus
                  <lb/>
                fuerit,
                  <reg norm="totumque" type="simple">totumq́</reg>
                diuiſum per
                  <reg norm="vnam" type="context">vnã</reg>
                partium, & per eandem pars altera diuiſa fue-
                  <lb/>
                rit: </s>
                <s xml:id="echoid-s222" xml:space="preserve">differentia prouenientium ſemper vnitas erit. </s>
                <s xml:id="echoid-s223" xml:space="preserve">quodquidem veriſſimum eſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s224" xml:space="preserve">Detur enim
                  <var>.b.d.</var>
                propoſitus numerus in duas partes inæquales diuiſus
                  <var>.b.c.</var>
                et
                  <var>.c.d.</var>
                  <lb/>
                & in primis
                  <reg norm="totum" type="context">totũ</reg>
                  <var>.b.d.</var>
                per
                  <var>.c.d.</var>
                diuidatur, ex quo oriatur
                  <var>e.o.</var>
                vnitas autem
                  <reg norm="per" type="punctuation simple">.ꝑ</reg>
                  <var>.i.o.</var>
                ſigni-
                  <lb/>
                ficetur, tum pars ipſa
                  <var>.b.c.</var>
                  <reg norm="per" type="simple punctuation">ꝑ.</reg>
                  <reg norm="eandem" type="context">eãdem</reg>
                  <var>.c.d.</var>
                diuidatur,
                  <reg norm="ſitque" type="simple">ſitq́;</reg>
                  <reg norm="proueniens" type="context">proueniẽs</reg>
                  <var>.a</var>
                . </s>
                <s xml:id="echoid-s225" xml:space="preserve">Sanè ex defini-
                  <lb/>
                tione diuiſionis, eadem erit proportio
                  <var>.b.d.</var>
                ad
                  <var>.e.o.</var>
                quæ eſt
                  <var>.c.d.</var>
                ad
                  <var>.i.o.</var>
                et ita
                  <var>.b.c.</var>
                ad
                  <var>.a.</var>
                  <lb/>
                ſicut
                  <var>.c.d.</var>
                ad
                  <var>.i.o</var>
                . </s>
                <s xml:id="echoid-s226" xml:space="preserve">Ex
                  <ref id="ref-0009">.19. autem quinti</ref>
                , ita ſe habet
                  <var>.b.c.</var>
                ad
                  <var>.e.i.</var>
                ſicut
                  <var>.b.d.</var>
                ad
                  <var>.e.o.</var>
                at
                  <var>.b.d.</var>
                  <lb/>
                ad
                  <var>.e.o.</var>
                ſic ſe habet ſicut
                  <var>.c.d.</var>
                ad
                  <var>.i.o.</var>
                hoc eſt ſicut
                  <var>.b.c.</var>
                ad
                  <var>.a</var>
                . </s>
                <s xml:id="echoid-s227" xml:space="preserve">Quare ex .11. quinti ſic ſe
                  <lb/>
                habebit
                  <var>.b.c.</var>
                ad
                  <var>.e.i.</var>
                ſicut .ad
                  <var>.a.</var>
                ex quo ex .9.
                  <reg norm="praedi­ cti" type="simple">prędi­
                    <lb/>
                  cti</reg>
                  <var>.a.</var>
                æqualis erit
                  <var>.e.i.</var>
                ſed
                  <var>.e.i.</var>
                minor eſt
                  <var>.e.o.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0027-02a" xlink:href="fig-0027-02"/>
                per
                  <var>.i.o</var>
                . </s>
                <s xml:id="echoid-s228" xml:space="preserve">Quare ſequitur propoſitum verum eſ­
                  <lb/>
                ſe. </s>
                <s xml:id="echoid-s229" xml:space="preserve">Quod ipſum pauciſſimis verbis ſic definiri
                  <lb/>
                poteſt, ſi dixerimus, eiuſmodi diuidens .in par-
                  <lb/>
                te diuiſibili,
                  <reg norm="quam" type="context">quã</reg>
                in toto, ſemel minus ingredi,
                  <lb/>
                quandoquidem altera pars eſt, ex qua totum integrum perficitur.</s>
              </p>
              <div xml:id="echoid-div50" type="float" level="4" n="1">
                <figure xlink:label="fig-0027-02" xlink:href="fig-0027-02a">
                  <image file="0027-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0027-02"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div52" type="math:theorem" level="3" n="23">
              <head xml:id="echoid-head39" xml:space="preserve">THEOREMA
                <num value="23">XXIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s230" xml:space="preserve">HOcipſum alia ratione contemplari po­
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0027-03a" xlink:href="fig-0027-03"/>
                terimus.</s>
              </p>
              <div xml:id="echoid-div52" type="float" level="4" n="1">
                <figure xlink:label="fig-0027-03" xlink:href="fig-0027-03a">
                  <image file="0027-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0027-03"/>
                </figure>
              </div>
              <p>
                <s xml:id="echoid-s231" xml:space="preserve">Significetur enim totalis numerus per
                  <var>.a.e.</var>
                  <lb/>
                in duas partes diuiſus
                  <var>.a.u.</var>
                et
                  <var>.u.e.</var>
                totius autem diuidens ſit
                  <var>.u.e.</var>
                & partis alterius
                  <var>.a.u.</var>
                  <lb/>
                totius verò
                  <reg norm="proueniens" type="context">proueniẽs</reg>
                ſit
                  <var>.a.c.</var>
                partis
                  <reg norm="autem" type="context">autẽ</reg>
                , ſit
                  <reg norm="proueniens" type="context">proueniẽs</reg>
                  <var>.a.n.</var>
                tum differentia ſit
                  <var>.n.c.</var>
                vni </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>