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-div76" type="math:theorem" level="3" n="34">
              <pb o="22" rhead="IO. BAPT. BENED." n="34" file="0034" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0034"/>
            </div>
            <div xml:id="echoid-div77" type="math:theorem" level="3" n="35">
              <head xml:id="echoid-head51" xml:space="preserve">THEOREMA
                <num value="35">XXXV</num>
              .</head>
              <p>
                <s xml:id="echoid-s309" xml:space="preserve">
                  <emph style="sc">QVivis</emph>
                numerus per alterum multiplicatus, & diuiſus, medius eſt propor-
                  <lb/>
                tionalis inter productum multiplicationis, & proueniens diaiſionis.</s>
              </p>
              <p>
                <s xml:id="echoid-s310" xml:space="preserve">Exempli gratia, ſi .20.
                  <reg norm="multiplicentur" type="context">multiplicẽtur</reg>
                per quinque & inde per quinque diuidantur
                  <lb/>
                productum erit .100. proueniens .4. inter quos numeros .20. medius eſt propor-
                  <lb/>
                tionalis.</s>
              </p>
              <p>
                <s xml:id="echoid-s311" xml:space="preserve">Hoc vt ſpeculemur, proponatur numerus multiplicandus & diuidendus, qui ſi-
                  <lb/>
                gnificetur linea
                  <var>.u.e.</var>
                multiplicans autem & diuidens linea
                  <var>.a.u.</var>
                multiplicationis
                  <lb/>
                productum ſit
                  <var>.e.a.</var>
                proueniens ex diuiſione ſit
                  <var>.o.e</var>
                . </s>
                <s xml:id="echoid-s312" xml:space="preserve">Nunc proueniens
                  <var>.e.o.</var>
                per
                  <reg norm="nu- merum" type="context">nu-
                    <lb/>
                  merũ</reg>
                  <var>.a.u.</var>
                diuidentem multiplicetur, cuius multiplicationis productum ſit
                  <var>.e.i.</var>
                  <lb/>
                quare, eadem erit proportio numeri
                  <var>.a.e.</var>
                  <lb/>
                ad numerum
                  <var>.e.i.</var>
                quæ eſt numeri
                  <var>.u.e.</var>
                ad
                  <lb/>
                  <figure xlink:label="fig-0034-01" xlink:href="fig-0034-01a" number="47">
                    <image file="0034-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0034-01"/>
                  </figure>
                numerum
                  <var>.e.o.</var>
                ex prima ſextiaut .18. vel
                  <lb/>
                19. ſeptimi. </s>
                <s xml:id="echoid-s313" xml:space="preserve">Sed cum numerus
                  <var>.u.e.</var>
                ex
                  <ref id="ref-0010">.11. theoremate præſentis libri</ref>
                , numero
                  <var>.e.
                    <lb/>
                  i.</var>
                æqualis ſit. </s>
                <s xml:id="echoid-s314" xml:space="preserve">verum eſſe, quod propoſi-
                  <lb/>
                tum fuit conſequetur.</s>
              </p>
            </div>
            <div xml:id="echoid-div79" type="math:theorem" level="3" n="36">
              <head xml:id="echoid-head52" xml:space="preserve">THEOREMA
                <num value="36">XXXVI</num>
              .</head>
              <p>
                <s xml:id="echoid-s315" xml:space="preserve">CVR ij, qui propoſitum numerum ita multiplicare & diuidere cupiunt, vt pro
                  <lb/>
                ductum multiplicationis, tam ſit multiplex prouenienti ex diuiſione, quam
                  <lb/>
                quæritur, rectè ſumant aliquem numerum pro multiplicante & diuidente, qui ſit ra
                  <lb/>
                dix quadrata denominantis quęſitę multiplicitatis.</s>
              </p>
              <p>
                <s xml:id="echoid-s316" xml:space="preserve">Exempli gratia, proponuntur .20. multiplicanda atque diuidenda, ita vt pro-
                  <lb/>
                ductum multiplicationis nonuplum ſit prouenienti ex diuiſione, nempè, vt pro-
                  <lb/>
                ueniens, nona pars ſit eiuſmodi producti, </s>
                <s xml:id="echoid-s317" xml:space="preserve">quare quadratam radicem ipſorum no-
                  <lb/>
                uem, ideſt denominantis ſumunt, tria ſcilicet, multiplicant igitur & diuidunt
                  <lb/>
                data .20. ex quo productum erit .60. proueniens autem .6. cum duabus tertijs. </s>
                <s xml:id="echoid-s318" xml:space="preserve">&
                  <lb/>
                propoſitum ſequitur.</s>
              </p>
              <p>
                <s xml:id="echoid-s319" xml:space="preserve">Cuius ſpeculationis cauſa, ſignificetur numerus propoſitus linea
                  <var>.u.e.</var>
                multipli-
                  <lb/>
                cans autem & diuidens linea
                  <var>.u.a.</var>
                productum ſit
                  <var>.e.a.</var>
                proueniens
                  <var>.e.o.</var>
                quadratum
                  <lb/>
                verò
                  <var>.a.u.</var>
                ſit
                  <var>.x.a.</var>
                erit igitur proportio
                  <var>.a.e.</var>
                ad
                  <var>.e.o.</var>
                dupla proportioni
                  <var>.a.e.</var>
                ad nume
                  <lb/>
                rum
                  <var>.u.e.</var>
                ex præcedenti theoremate: </s>
                <s xml:id="echoid-s320" xml:space="preserve">Adhæc, cogitemus in linea
                  <var>.u.a.</var>
                vnitatem
                  <var>.
                    <lb/>
                  u.i.</var>
                  <reg norm="terminenturque" type="simple">terminenturq́;</reg>
                duo producta
                  <var>.e.i.</var>
                et
                  <var>.x.i.</var>
                </s>
                <s xml:id="echoid-s321" xml:space="preserve">quare eadem erit proportio
                  <var>.a.e.</var>
                ad
                  <var>.e.i.</var>
                  <lb/>
                quæ eſt
                  <var>.a.e.</var>
                ad
                  <var>.u.e.</var>
                numerus enim
                  <var>.e.i.</var>
                (quamuis ſuperficialis) idem eſt cum nume-
                  <lb/>
                ro lineari
                  <var>.u.e.</var>
                ſed
                  <var>.a.e.</var>
                ad
                  <var>.e.i.</var>
                ſic ſe habet ſicut
                  <var>.a.u.</var>
                ad
                  <var>.u.i.</var>
                ex prima ſexti aut .18.
                  <lb/>
                vel .19. ſeptimi, (quod ipſum dico de
                  <var>.a.x.</var>
                ad
                  <var>.x.i.</var>
                ) </s>
                <s xml:id="echoid-s322" xml:space="preserve">quare proportio
                  <var>.a.x.</var>
                ad
                  <var>.x.i.</var>
                hoc
                  <lb/>
                eſt
                  <var>.x.u.</var>
                ęqualis erit
                  <reg norm="proportioni" type="simple">ꝓportioni</reg>
                  <var>.a.e.</var>
                ad
                  <var>.u.e.</var>
                at trigeſimotertio & trigeſimoquarto theo
                  <lb/>
                remate probatum eſt proportionem numeri
                  <var>.a.x.</var>
                ad vnitatem, duplam eſſe propor-
                  <lb/>
                tioni eiuſdem numeri
                  <var>.a.x.</var>
                ad
                  <var>.u.x.</var>
                ſequitur
                  <lb/>
                igitur cum dimidia ſint æqualia, tota etiam
                  <lb/>
                æqualia eſſe: </s>
                <s xml:id="echoid-s323" xml:space="preserve">hoc eſt proportionem numeri
                  <var>.
                    <lb/>
                    <figure xlink:label="fig-0034-02" xlink:href="fig-0034-02a" number="48">
                      <image file="0034-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0034-02"/>
                    </figure>
                  a.e.</var>
                ad numerum
                  <var>.e.o.</var>
                æqualem eſſe propor
                  <lb/>
                tioni numeri
                  <var>.a.x.</var>
                ad vnitatem. </s>
                <s xml:id="echoid-s324" xml:space="preserve">Itaque rectè
                  <lb/>
                ſumitur numerus
                  <var>.a.u.</var>
                eiuſmodi vt
                  <reg norm="quadratum" type="context">quadratũ</reg>
                </s>
              </p>
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