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-div134" type="math:theorem" level="3" n="67">
              <pb o="44" rhead="IO. BAPT. BENED." n="56" file="0056" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0056"/>
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            <div xml:id="echoid-div136" type="math:theorem" level="3" n="68">
              <head xml:id="echoid-head84" xml:space="preserve">THEOREMA
                <num value="68">LXVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s582" xml:space="preserve">CVR numero per numerum diuiſo,
                  <reg norm="productoque" type="simple">productoq́;</reg>
                duorum numerorum per pro-
                  <lb/>
                ueniens multiplicato, quod vltimò productum eſt, diuiſi numeri ſemper qua
                  <lb/>
                dratum exiſtat.</s>
              </p>
              <p>
                <s xml:id="echoid-s583" xml:space="preserve">Exempli gratia, ſi diuidamus .10. per .2. proueniens erit .5. quo producto ex duo
                  <lb/>
                bus numeris multiplicato, nempe .20. habe
                  <lb/>
                bimus .100. quadratum numeri diuiſi.</s>
              </p>
              <figure position="here">
                <image file="0056-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0056-01"/>
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              <p>
                <s xml:id="echoid-s584" xml:space="preserve">Cuius gratia duo numeri ſint
                  <var>.a.</var>
                et
                  <var>.e.</var>
                por
                  <lb/>
                  <var>.a.</var>
                per
                  <var>.e.</var>
                diuiſo detur
                  <var>.u.</var>
                tum
                  <var>.o.</var>
                produ-
                  <lb/>
                ctum
                  <var>.a.</var>
                in
                  <var>.e.</var>
                eſſe conſtituatur, quo per
                  <var>.u.</var>
                  <lb/>
                multiplicato dabitur
                  <var>.x.</var>
                quadratum
                  <var>.a.</var>
                pro-
                  <lb/>
                ptereà quòd
                  <var>.a.</var>
                medium eſt proportionale
                  <lb/>
                inter
                  <var>.o.</var>
                et
                  <var>.u.</var>
                ex .35. theoremate. </s>
                <s xml:id="echoid-s585" xml:space="preserve">itaque
                  <lb/>
                ex .16. ſexti aut .20. ſeptimi, propoſiti veri-
                  <lb/>
                tas eluceſcet.</s>
              </p>
            </div>
            <div xml:id="echoid-div137" type="math:theorem" level="3" n="69">
              <head xml:id="echoid-head85" xml:space="preserve">THEOREMA
                <num value="69">LXIX</num>
              .</head>
              <p>
                <s xml:id="echoid-s586" xml:space="preserve">CVR numero aliquo per duos alios multiplicato & diuiſo, ſi per horum duo-
                  <lb/>
                rum productum, ſumma duorum primorum productorum diuiſa fuerit, vl-
                  <lb/>
                timum proueniens, ſummæ duorum primorum prouenientium æquale ſit.</s>
              </p>
              <p>
                <s xml:id="echoid-s587" xml:space="preserve">Exempli gratia, proponitur numerus .24. per .8. et .6. multiplicandus & diuiden
                  <lb/>
                dus ſumma productorum crit .336. prouenientium autem .7. ſi igitur ſummam .336.
                  <lb/>
                productorum per productum duorum ſecundorum numerorum nempe .48. diuiſe-
                  <lb/>
                rimus, proueniens pariter erit .7.</s>
              </p>
              <p>
                <s xml:id="echoid-s588" xml:space="preserve">In cuius
                  <reg norm="gratiam" type="context">gratiã</reg>
                primus numerus ſignificetur linea
                  <var>.q.b.</var>
                multiplicandus & diuiden-
                  <lb/>
                dus numeris deſignatis per
                  <var>.k.m.</var>
                et
                  <var>.y.m.</var>
                productorum ſumma ſit
                  <var>.k.z.</var>
                prouenien-
                  <lb/>
                tium autem
                  <var>.a.e</var>
                : et
                  <var>.a.o.</var>
                ex
                  <var>.k.m.</var>
                et
                  <var>.o.e.</var>
                ex
                  <var>.y.m</var>
                : tum productum
                  <var>.k.m.</var>
                in
                  <var>.m.y.</var>
                ſit
                  <var>.f.
                    <lb/>
                  m</var>
                . </s>
                <s xml:id="echoid-s589" xml:space="preserve">Dico quòd ſi
                  <var>.k.z.</var>
                per
                  <var>.f.m.</var>
                diuiſerimus proueni et
                  <var>.a.e</var>
                . </s>
                <s xml:id="echoid-s590" xml:space="preserve">Quod cum ſic fuerit, erit
                  <lb/>
                quoque verum quòd diuiſa
                  <var>.k.z.</var>
                per
                  <var>.a.e.</var>
                proueniet
                  <var>.f.m.</var>
                numerus ſcilicet æqualis
                  <lb/>
                numero
                  <var>.f.m.</var>
                ex .13. theoremate huius. </s>
                <s xml:id="echoid-s591" xml:space="preserve">Itaque quotieſcunque probauero quòd di-
                  <lb/>
                uiſa
                  <var>.k.z.</var>
                per
                  <var>.a.e.</var>
                proueniat numerus æqualis ipſi
                  <var>.f.m.</var>
                propoſitum verum eſſe con
                  <lb/>
                ſequetur. ex .13. theoremate. </s>
                <s xml:id="echoid-s592" xml:space="preserve">Quòd ſi proueniens ex diuiſione
                  <var>.k.z.</var>
                per
                  <var>.a.e.</var>
                æqua
                  <lb/>
                le fuerit
                  <var>.f.m.</var>
                patet ex .7. quinti quòd
                  <reg norm="eadem" type="context">eadẽ</reg>
                erit proportio numeri
                  <var>.k.m.y.</var>
                ad ipſum
                  <lb/>
                proueniens, quæ ad numerum
                  <var>.f.m</var>
                . </s>
                <s xml:id="echoid-s593" xml:space="preserve">Cogitemus
                  <reg norm="itaque" type="simple">itaq;</reg>
                  <var>.k.u.</var>
                æqualem
                  <var>.a.e.</var>
                ſuper quam
                  <lb/>
                mente concipiamus rectangulum
                  <var>.u.p.</var>
                æqualem
                  <var>.k.z.</var>
                ex quo eadem erit proportio
                  <var>.
                    <lb/>
                  k.p.</var>
                ad
                  <var>.k.y.</var>
                quæ
                  <var>.g.k.</var>
                ad
                  <var>.k.u.</var>
                ex .15. ſexti, aut, 20. ſeptimi, numerus autem
                  <var>.k.p.</var>
                erit
                  <lb/>
                proueniens, quod probandum eſt æquale eſſe
                  <var>.f.m</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s594" xml:space="preserve">Probabitur autem ſic, ex .9. quinti, nempe demonſtrato quòd numerus
                  <var>.k.p.</var>
                ean
                  <lb/>
                dem proportionem habeat ad numerum
                  <var>.k.y.</var>
                quam habet numerus
                  <var>.f.m.</var>
                ad eundem
                  <lb/>
                  <var>k.y</var>
                . </s>
                <s xml:id="echoid-s595" xml:space="preserve">Sed probatum eſt ſic ſe habere
                  <var>.k.g.</var>
                ad
                  <var>.k.u.</var>
                ſicut
                  <var>.k.p.</var>
                ad
                  <var>.k.y.</var>
                ſufficiet igitur pro-
                  <lb/>
                bare ſic ſe habere
                  <var>.k.g.</var>
                ad
                  <var>.k.u.</var>
                ſicut
                  <var>.f.m.</var>
                ad
                  <var>.k.y</var>
                . </s>
                <s xml:id="echoid-s596" xml:space="preserve">Sed
                  <var>.k.g.</var>
                dicitur æqualis eſſe
                  <var>.q.b</var>
                : et
                  <var>.k.</var>
                  <lb/>
                u; </s>
                <s xml:id="echoid-s597" xml:space="preserve">a.e. ſatis erit igitur probare ita ſe habere
                  <var>.q.b.</var>
                ad
                  <var>.a.e.</var>
                ſicut
                  <var>.f.m.</var>
                ad
                  <var>.k.y</var>
                . </s>
                <s xml:id="echoid-s598" xml:space="preserve">Scimus au-
                  <lb/>
                tem quòd eadem eſt proportio
                  <var>.q.b.</var>
                ad
                  <var>.a.o.</var>
                quæ
                  <var>.m.k.</var>
                ad vnitatem, quæ ſit
                  <var>.x.</var>
                & quod
                  <lb/>
                proportio
                  <var>.o.e.</var>
                ad
                  <var>.q.b.</var>
                eadem eſt, quæ
                  <var>.x.</var>
                ad
                  <var>.m.y.</var>
                ex definitione diuiſionis. </s>
                <s xml:id="echoid-s599" xml:space="preserve">Quare
                  <lb/>
                ex æqualitate proportionum eadem erit proportio
                  <var>.k.m.</var>
                ad
                  <var>.m.y.</var>
                quæ
                  <var>.e.o.</var>
                ad
                  <var>.o.a.</var>
                & </s>
              </p>
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