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-div137" type="math:theorem" level="3" n="69">
              <p>
                <s xml:id="echoid-s599" xml:space="preserve">
                  <pb o="45" rhead="THEOREM. ARIT." n="57" file="0057" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0057"/>
                componendo ſic ſe habebit
                  <var>.k.y.</var>
                ad
                  <var>.m.y.</var>
                ſicut
                  <var>.e.a.</var>
                ad
                  <var>.o.a.</var>
                & permutando
                  <var>.k.y.</var>
                ad
                  <var>.e.
                    <lb/>
                  a.</var>
                ſicut
                  <var>.m.y.</var>
                ad
                  <var>.o.a.</var>
                & ex .19. quinti ita
                  <var>.k.m.</var>
                ad
                  <var>.e.o.</var>
                ſicut
                  <var>.k.y.</var>
                ad
                  <var>.e.a.</var>
                & permutando
                  <var>.
                    <lb/>
                  k.m.</var>
                ad
                  <var>.k.y.</var>
                ſicut
                  <var>.e.o.</var>
                ad
                  <var>.e.a</var>
                . </s>
                <s xml:id="echoid-s600" xml:space="preserve">Nunc producatur
                  <var>.f.t.</var>
                donec
                  <var>.t.i.</var>
                æqualis ſit
                  <var>.k.y.</var>
                  <reg norm="produ- ctaque" type="simple">produ-
                    <lb/>
                  ctaq́;</reg>
                  <var>.m.t.</var>
                done
                  <var>c.t.s.</var>
                æqualis ſit vnitati
                  <var>.x.</var>
                  <reg norm="termineturque" type="simple">termineturq́;</reg>
                rectangulum
                  <var>.s.i.</var>
                ex quo da-
                  <lb/>
                bitur proportio numeri
                  <var>.f.m.</var>
                ad numerum
                  <var>.s.i.</var>
                compoſita ex
                  <var>.m.t.</var>
                ad
                  <var>.t.s.</var>
                et
                  <var>.f.t.</var>
                ad
                  <var>.t.i.</var>
                  <lb/>
                ex .24. ſexti, aut quinta octaui, ſed ita etiam proportio
                  <var>.q.b.</var>
                ad
                  <var>.a.e.</var>
                componitur ex
                  <lb/>
                eiſdem proportionibus, nempe ex
                  <var>.q.b.</var>
                ad
                  <var>.o.e.</var>
                æquali
                  <var>.m.t.</var>
                ad
                  <var>.t.s.</var>
                & ex proportione
                  <var>.
                    <lb/>
                  o.e.</var>
                ad
                  <var>.a.e.</var>
                æquali
                  <var>.f.t.</var>
                ad
                  <var>.t.i.</var>
                ita que proportio numeri
                  <var>.f.m.</var>
                ad
                  <var>.s.i.</var>
                hoc eſt ad
                  <reg norm="numerum" type="context">numerũ</reg>
                  <lb/>
                ipſius
                  <var>.k.y.</var>
                ęqualis eſt proportioni numeri
                  <var>.q.b.</var>
                ad
                  <var>.a.e.</var>
                  <reg norm="nempe" type="context">nẽpe</reg>
                  <var>.k.g.</var>
                ad
                  <var>.k.u.</var>
                hoc eſt
                  <var>.k.p.</var>
                ad
                  <lb/>
                  <var>x.y.</var>
                ex quo ſequitur
                  <var>.k.p.</var>
                conſtare numero ęquali
                  <var>.f.m.</var>
                proueniens igitur ex diuiſione
                  <lb/>
                numeri
                  <var>.k.z.</var>
                per
                  <var>.f.m.</var>
                æquale eſt numero ipſius
                  <var>.a.e</var>
                .</s>
              </p>
              <figure position="here" number="77">
                <image file="0057-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0057-01"/>
              </figure>
            </div>
            <div xml:id="echoid-div138" type="math:theorem" level="3" n="70">
              <head xml:id="echoid-head86" xml:space="preserve">THEOREMA
                <num value="70">LXX</num>
              .</head>
              <p>
                <s xml:id="echoid-s601" xml:space="preserve">HAEC porrò concluſio alia etiam via demonſtrari poteſt.</s>
              </p>
              <p>
                <s xml:id="echoid-s602" xml:space="preserve">Significetur numerus diuidendus atque multiplicandus linea
                  <var>.b.a</var>
                . </s>
                <s xml:id="echoid-s603" xml:space="preserve">Deinde
                  <lb/>
                diuidentes &
                  <reg norm="multiplicantes" type="context">multiplicãtes</reg>
                ſint
                  <var>.k.m.</var>
                et
                  <var>.m.y.</var>
                prouenientia ex diuiſione ſint
                  <var>.a.o.</var>
                et
                  <var>.o.
                    <lb/>
                  e.</var>
                atque
                  <var>.a.o.</var>
                ex
                  <var>.m.y</var>
                :
                  <var>o.e.</var>
                verò ex
                  <var>.k.m.</var>
                proueniat, quorum ſumma ſit
                  <var>.a.e</var>
                : productum
                  <lb/>
                autem
                  <var>.b.a.</var>
                in
                  <var>.k.m.</var>
                ſit
                  <var>.b.p.</var>
                et
                  <var>.p.s.</var>
                productum
                  <var>.b.a.</var>
                in
                  <var>.m.y.</var>
                ad hæc rectangulum
                  <var>.k.y.</var>
                ſit
                  <lb/>
                productum
                  <var>.k.m.</var>
                in
                  <var>.m.y</var>
                : quo to-
                  <lb/>
                tum productum
                  <var>.a.s.</var>
                diuidatur, pro
                  <lb/>
                  <figure xlink:label="fig-0057-02" xlink:href="fig-0057-02a" number="78">
                    <image file="0057-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0057-02"/>
                  </figure>
                  <reg norm="ueniensque" type="simple">ueniensq́;</reg>
                ſit
                  <var>.a.c.</var>
                cui,
                  <var>a.c</var>
                :
                  <reg norm="productum" type="context">productũ</reg>
                  <var>.
                    <lb/>
                  a.s.</var>
                  <reg norm="eandem" type="context context">eãdẽ</reg>
                  <reg norm="proportionem" type="context">proportionẽ</reg>
                ſeruabit,
                  <reg norm="quam" type="context">quã</reg>
                  <lb/>
                  <var>k.y.</var>
                rectangulum ad vnitatem ex
                  <lb/>
                definitione diuiſionis, hoc autem
                  <lb/>
                proueniens
                  <var>.a.c.</var>
                  <reg norm="conſtare" type="context">cõſtare</reg>
                numero æ-
                  <lb/>
                quali aſſero ſummæ
                  <var>.a.e</var>
                . </s>
                <s xml:id="echoid-s604" xml:space="preserve">Primum
                  <lb/>
                enim ex dicta definitione diuiſio-
                  <lb/>
                nis habemus eandem eſſe propor-
                  <lb/>
                tionem
                  <var>.b.a.</var>
                ad
                  <var>.a.o.</var>
                quæ
                  <var>.m.y.</var>
                ad
                  <lb/>
                vnitatem, & quod ſic ſe habet
                  <var>.b.a.</var>
                  <lb/>
                ad
                  <var>.o.e.</var>
                ſicut
                  <var>.k.m.</var>
                ad eandem vnita
                  <lb/>
                tem. </s>
                <s xml:id="echoid-s605" xml:space="preserve">Itaque vnitas hæc linearis ſi-
                  <lb/>
                gnificetur per
                  <var>.m.x.</var>
                in ſingulis late-
                  <lb/>
                ribus
                  <var>.k.m.</var>
                et
                  <var>.m.y.</var>
                producentibus rectangulum
                  <var>.k.y</var>
                : ſuperficialis autem vnitas ſit. </s>
              </p>
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