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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IO. BAPT. BENED.
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          <div xml:id="echoid-div7" type="chapter" level="2" n="1">
            <div xml:id="echoid-div138" type="math:theorem" level="3" n="70">
              <p>
                <s xml:id="echoid-s605" xml:space="preserve">
                  <pb o="46" rhead="IO. BAPT. BENED." n="58" file="0058" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0058"/>
                  <var>g.m.</var>
                  <reg norm="cogiteturque" type="simple">cogiteturq́;</reg>
                rectangulum
                  <var>.y.x.</var>
                & rectangulum
                  <var>.k.x</var>
                . </s>
                <s xml:id="echoid-s606" xml:space="preserve">Itaque dabitur eadem pro
                  <lb/>
                portio
                  <var>.k.m.</var>
                ad
                  <var>.m.x.</var>
                nempe
                  <var>.k.x.</var>
                rectanguli ad
                  <var>.m.g.</var>
                quæ eſt
                  <var>.b.a.</var>
                ad
                  <var>.o.e.</var>
                et
                  <var>.y.x.</var>
                ad
                  <var>.m.
                    <lb/>
                  g.</var>
                quæ
                  <var>.b.a.</var>
                ad
                  <var>.a.o.</var>
                ſed ex prima ſexti aut .18. vel .19. ſeptimi, ſic ſe habet rectangu-
                  <lb/>
                lum
                  <var>.k.y.</var>
                ad
                  <var>.x.y.</var>
                ſicut
                  <var>.k.m.</var>
                ad
                  <var>.m.x.</var>
                </s>
                <s xml:id="echoid-s607" xml:space="preserve">quare ſicut
                  <var>.b.a.</var>
                ad
                  <var>.o.e.</var>
                ex .11. quinti, & eiuſdem
                  <lb/>
                rectanguli
                  <var>.k.y.</var>
                ad rectangulum
                  <var>.k.x.</var>
                ſicut
                  <var>.y.m.</var>
                ad
                  <var>.x.m.</var>
                nempe
                  <var>.b.a.</var>
                ad
                  <var>.a.o</var>
                . </s>
                <s xml:id="echoid-s608" xml:space="preserve">Quare
                  <lb/>
                ex communi ſcientia, ſic ſe habebit duplum rectanguli
                  <var>.k.y.</var>
                ad ſummam
                  <var>.y.x.</var>
                cum
                  <var>.
                    <lb/>
                  k.x.</var>
                rectangulorum, ſicut duplum
                  <var>.b.a.</var>
                ad ſummam
                  <var>.a.o.e.</var>
                et proportio ſummæ re-
                  <lb/>
                ctangulorum
                  <var>.y.x.</var>
                et
                  <var>.k.x.</var>
                duplo
                  <var>.g.m.</var>
                ſicut duplum
                  <var>.b.a.</var>
                ad
                  <var>.a.o.e</var>
                . </s>
                <s xml:id="echoid-s609" xml:space="preserve">Igitur ſumma duo-
                  <lb/>
                rum rectangulorum
                  <var>.y.x.</var>
                et
                  <var>.x.k.</var>
                media proportionalis erit inter duplum rectanguli
                  <var>.
                    <lb/>
                  k.y.</var>
                & duplum vnitatis ſuperſicialis
                  <var>.g.m</var>
                . </s>
                <s xml:id="echoid-s610" xml:space="preserve">Nunc terminetur rectangulum
                  <var>.a.r.</var>
                ex quo
                  <lb/>
                dabitur eadem proportio dupli
                  <var>.a.s.</var>
                ad
                  <var>.a.r.</var>
                ſicut dupli
                  <var>.b.a.</var>
                ad
                  <var>.a.e.</var>
                ex propoſitioni-
                  <lb/>
                bus notatis, ſexti aut ſeptimi. </s>
                <s xml:id="echoid-s611" xml:space="preserve">Quare etiam ſicut dupli rectanguli
                  <var>.k.y.</var>
                ad
                  <reg norm="ſummam" type="context">ſummã</reg>
                  <lb/>
                rectangulorum
                  <var>.y.x.</var>
                et
                  <var>.k.x</var>
                . </s>
                <s xml:id="echoid-s612" xml:space="preserve">Iam verò ſi conſtituatur
                  <var>.e.c.</var>
                pro vnitate lineari ipſius
                  <var>.
                    <lb/>
                  e.r.</var>
                certi erimus numerum
                  <var>.a.c.</var>
                æqualem eſſe
                  <var>.a.e.</var>
                & proportionem
                  <var>.r.e.</var>
                ad
                  <var>.e.c.</var>
                hoc
                  <lb/>
                eſt
                  <var>.a.r.</var>
                ad
                  <var>.a.c.</var>
                eandem quæ
                  <var>.y.x.</var>
                et
                  <var>.x.k.</var>
                rectangulorum ad
                  <var>.m.g.</var>
                ex prædictis rationi-
                  <lb/>
                bus, & ex hypotheſi, nempe quòd
                  <var>.
                    <lb/>
                  e.r.</var>
                æqualis ſit numero
                  <var>.k.m.y.</var>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0058-01a" xlink:href="fig-0058-01"/>
                hoc eſt rectangulorum
                  <var>.y.x.</var>
                et
                  <var>.x.
                    <lb/>
                  k</var>
                . </s>
                <s xml:id="echoid-s613" xml:space="preserve">Quamobrem
                  <var>.a.r.</var>
                ex communi
                  <lb/>
                ſcientia
                  <reg norm="medium" type="context">mediũ</reg>
                proportionale erit
                  <lb/>
                inter duplum
                  <var>.a.s.</var>
                & duplum
                  <var>.a.c.</var>
                  <reg norm="ea­ demque" type="context simple">ea­
                    <lb/>
                  dẽq́;</reg>
                  <reg norm="proportio" type="simple">ꝓportio</reg>
                dupli prędicti
                  <var>.a.s.</var>
                ad
                  <lb/>
                duplum
                  <var>.a.c.</var>
                ex æqualitate propor-
                  <lb/>
                tionum ſimul collectarum, eadem
                  <lb/>
                erit qùæ proportio dupli rectangu-
                  <lb/>
                li
                  <var>.k.y.</var>
                ad duplum
                  <var>.m.g.</var>
                hoc eſt
                  <var>.a.s.</var>
                  <lb/>
                ſimplicis ad ſimplicem
                  <var>.a.c.</var>
                quæ ſim
                  <lb/>
                plicis rectanguli
                  <var>.k.y.</var>
                ad ſimplicem
                  <lb/>
                vnitatem
                  <var>.g.m.</var>
                ſic enim ſe habet ſim
                  <lb/>
                plex ad ſimplex, ſicut duplum ad
                  <lb/>
                duplum. </s>
                <s xml:id="echoid-s614" xml:space="preserve">Sed pariter ita ſe habet
                  <var>.a.s.</var>
                ad
                  <var>.a.</var>
                c
                  <unsure/>
                . cogitato
                  <var>.a.c.</var>
                tamquam proueniente
                  <lb/>
                ex diuiſione
                  <var>.a.s.</var>
                per rectangulum
                  <var>.k.y.</var>
                vt conſtitutum eſt, ſicut
                  <var>.k.y.</var>
                ad
                  <var>.m.g.</var>
                ex defi-
                  <lb/>
                nitione diuiſionis vt iam dictum eſt, </s>
                <s xml:id="echoid-s615" xml:space="preserve">quare numerus
                  <var>.a.c.</var>
                æqualis erit numero
                  <var>.a.o.e</var>
                .</s>
              </p>
              <div xml:id="echoid-div138" type="float" level="4" n="1">
                <figure xlink:label="fig-0057-02" xlink:href="fig-0057-02a">
                  <image file="0057-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0057-02"/>
                </figure>
                <figure xlink:label="fig-0058-01" xlink:href="fig-0058-01a">
                  <image file="0058-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0058-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div140" type="math:theorem" level="3" n="71">
              <head xml:id="echoid-head87" xml:space="preserve">THEOREMA
                <num value="71">LXXI</num>
              .</head>
              <p>
                <s xml:id="echoid-s616" xml:space="preserve">CVR propoſitis .4. numeris, duobus nempe diuidentibus ac duobus diuiden-
                  <lb/>
                dis, ſi
                  <reg norm="adinuicem" type="context">adinuicẽ</reg>
                diuiſi fuerint,
                  <reg norm="duoque" type="simple">duoq́;</reg>
                  <reg norm="prouenientia" type="context">proueniẽtia</reg>
                  <reg norm="inuicem" type="context">inuicẽ</reg>
                multiplicata
                  <reg norm="quenuis" type="context">quẽuis</reg>
                nu
                  <lb/>
                merum producant, qui ſeruetur, ſi deinde ijdem numeri verſa vice mutuo diuiſi fue
                  <lb/>
                rint, & inter ſe multiplicata prouenientia,
                  <reg norm="productum" type="context">productũ</reg>
                hoc, primo ſeruato numero
                  <lb/>
                æquale erit.</s>
              </p>
              <p>
                <s xml:id="echoid-s617" xml:space="preserve">Exempli gratia propoſitis his .4. numeris .20. 30. 5. 10. duo autem .20. ſcilicet
                  <lb/>
                et .30. ſint numeri diuidendi, porrò .5. et .10. numeri diuidentes,
                  <reg norm="nempe" type="context">nẽpe</reg>
                vt primo .20
                  <lb/>
                per .5. diuidatur, tum .30. per .10. producetur .4. et .3. qui ſimul multiplicati
                  <reg norm="proferent" type="context">proferẽt</reg>
                  <num value="12">.
                    <lb/>
                  12.</num>
                tum .20. per .10. d iuiſo et .30. per .5. prouenientia erunt .2. 6. quæ inter ſe multi-
                  <lb/>
                plicata producent etiam .12.</s>
              </p>
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