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="math:theorem" level="3" n="1">
              <p>
                <s xml:id="echoid-s44" xml:space="preserve">
                  <pb o="2" rhead="IO. BAPT. BENED." n="14" file="0014" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0014"/>
                quis, qua ratione fractus numerus
                  <var>.c.i.</var>
                minor ſit in ſuo integro
                  <var>.d.b.</var>
                fracto
                  <var>.a.i.</var>
                in
                  <lb/>
                ſuo integro
                  <var>.a.b.</var>
                aut fracto
                  <var>.a.c.</var>
                in ſuo integro
                  <var>.a.d.</var>
                conſideret is quo pacto pro-
                  <lb/>
                portio
                  <var>.c.i.</var>
                ad
                  <var>.d.b.</var>
                minor ſit proportione
                  <var>.a.i.</var>
                ad
                  <var>.a.b.</var>
                et
                  <var>.a.c.</var>
                ad
                  <var>.a.d.</var>
                hac ratione. </s>
                <s xml:id="echoid-s45" xml:space="preserve">Ma-
                  <lb/>
                nifeſtum eſt ex
                  <ref id="ref-0003">prima ſexti de quantitate
                    <lb/>
                  continua</ref>
                , aut
                  <ref id="ref-0004">.18. ſeptimi Euclidis</ref>
                de diſcre
                  <lb/>
                  <figure xlink:label="fig-0014-01" xlink:href="fig-0014-01a" number="2">
                    <image file="0014-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0014-01"/>
                  </figure>
                ta, proportionem ipſius
                  <var>.d.i.</var>
                ad
                  <var>.d.b.</var>
                eſſe ſi-
                  <lb/>
                cut
                  <var>.a.i.</var>
                ad
                  <var>.a.b.</var>
                & cum
                  <var>.c.i.</var>
                minor ſit
                  <var>.d.i.</var>
                  <lb/>
                velut pars ſuo toto, proportio,
                  <var>c.i.</var>
                ad
                  <var>.d.b.</var>
                  <lb/>
                minor erit proportione
                  <var>.d.i.</var>
                ad
                  <var>.d.b.</var>
                ex .8.
                  <lb/>
                quinti, </s>
                <s xml:id="echoid-s46" xml:space="preserve">quare minor erit pariter proportio-
                  <lb/>
                ne
                  <var>.a.i.</var>
                ad
                  <var>.a.b.</var>
                ex
                  <ref id="ref-0005">.12.
                    <reg norm="eiuſdem" type="context">eiuſdẽ</reg>
                  </ref>
                vnà etiam pro-
                  <lb/>
                portio
                  <var>.c.i.</var>
                ad
                  <var>.d.b.</var>
                minor erit
                  <var>.a.c.</var>
                ad
                  <var>.a.d.</var>
                  <lb/>
                ex eiſdem cauſis, medio
                  <var>.c.b</var>
                . </s>
                <s xml:id="echoid-s47" xml:space="preserve">Ex quibus pa-
                  <lb/>
                tet ratio, cur fracti diuerſarum denomina-
                  <lb/>
                tionum ad vnicam reducantur. </s>
                <s xml:id="echoid-s48" xml:space="preserve">Cur etiam
                  <lb/>
                numeros integros in partes fractis ſimiles
                  <lb/>
                frangere liceat, quæ omnia ex ſubſequenti
                  <lb/>
                figura facilè cognoſci poſſunt.</s>
              </p>
            </div>
            <div xml:id="echoid-div9" type="math:theorem" level="3" n="2">
              <head xml:id="echoid-head18" xml:space="preserve">THEOREMA
                <num value="2">II</num>
              .</head>
              <p>
                <s xml:id="echoid-s49" xml:space="preserve">
                  <emph style="sc">QVae</emph>
                ſit ratio, cur hi, qui numeros, fractos diuerſarum denominationum col-
                  <lb/>
                ligere volunt, & in ſummam redigere, multiplicent vnum ex numerantibus
                  <lb/>
                per denominatorem alterius, & poſtmodum denominatores adinuicem, quorum
                  <lb/>
                vltimum productum, commune eſt denominans duorum priorum productorum,
                  <lb/>
                quæ collecta in ſummam efficiunt quod quærebatur.</s>
              </p>
              <p>
                <s xml:id="echoid-s50" xml:space="preserve">Qua in re ſciendum eſt, denominantes conſiderari tanquam partes vnius
                  <reg norm="eiuſdem- q́ue" type="context">eiuſdẽ-
                    <lb/>
                  q́ue</reg>
                magnitudinis quantitatis continuæ, linearum (verbigratia)
                  <var>a.b.</var>
                et
                  <var>.a.d.</var>
                  <reg norm="æqualium" type="context">æqualiũ</reg>
                  <lb/>
                in longitudine,
                  <reg norm="quarum" type="context">quarũ</reg>
                  <var>.a.b.</var>
                in quatuor partes diuidatur, et
                  <var>.a.d.</var>
                in tres. </s>
                <s xml:id="echoid-s51" xml:space="preserve">Quare ſi colli-
                  <lb/>
                gere voluerimus duo tertia cum tribus quartis, multiplicabimus
                  <var>.a.c.</var>
                duo tertia,
                  <lb/>
                cum
                  <var>.a.b.</var>
                diuiſa in 4. partes, produceturq́ue
                  <var>.c.b.</var>
                octo partium ſuperficialium, de-
                  <lb/>
                hinc multiplicando
                  <var>.a.i.</var>
                tres quartas cum
                  <var>.a.d.</var>
                diuiſa in .3. partes producetur
                  <var>.i.d.</var>
                pri
                  <lb/>
                mis ſingulis æqualis, nouem partium ſuper
                  <lb/>
                ficialium, multiplicata deinde
                  <var>a.b.</var>
                diui-
                  <lb/>
                  <figure xlink:label="fig-0014-02" xlink:href="fig-0014-02a" number="3">
                    <image file="0014-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0014-02"/>
                  </figure>
                ſa in .4. partes per
                  <var>.a.d.</var>
                in .3. diuiſa, produ-
                  <lb/>
                cetur quadratum
                  <var>.d.b.</var>
                in continuo, in 12.
                  <lb/>
                partes diuiſum, quod erit totum commune
                  <lb/>
                ſingulis productis, quorum primum erat
                  <var>.c.
                    <lb/>
                  b</var>
                . </s>
                <s xml:id="echoid-s52" xml:space="preserve">Quare
                  <var>.c.b.</var>
                ita ſe habet ad totum
                  <var>.d.b.</var>
                ſi-
                  <lb/>
                cut
                  <var>.a.c.</var>
                ad
                  <var>.a.d.</var>
                ex prima ſexti in continuis,
                  <lb/>
                aut .18. ſeptimi in diſcretis quantitatibus,
                  <lb/>
                et
                  <var>.d.i.</var>
                ad
                  <var>.d.b.</var>
                ſicut
                  <var>.a.i.</var>
                ad
                  <var>.a.b.</var>
                ex eiſdem
                  <lb/>
                propoſitionibus. </s>
                <s xml:id="echoid-s53" xml:space="preserve">Collectis deinde parti-
                  <lb/>
                bus producti
                  <var>.c.b.</var>
                cum partibus producti
                  <var>.
                    <lb/>
                  d.i.</var>
                manifeſtè depræhendetur eiuſmodi
                  <lb/>
                ſummam componi ex partibus vnius totius
                  <lb/>
                communis ſingulis earum.</s>
              </p>
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