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-div106" type="math:theorem" level="3" n="50">
              <p>
                <s xml:id="echoid-s444" xml:space="preserve">
                  <pb o="33" rhead="THEOR. ARITH." n="45" file="0045" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0045"/>
                numero, verbi gratia .92. præcepit regula detrahi primum numerum ex ſecundo,
                  <lb/>
                nempe .20. ex .92. cuius reſiduum, ſcilicet .72. conſeruetur, tum detrahi iubet bi
                  <lb/>
                narium ex primo, ſic in propoſito exemplo remanebunt .18. huius autem .18. dimi
                  <lb/>
                dium in ſeipſum multiplicari iubet, quod cum ſit .9. datur numerus .81. ex quo .81.
                  <lb/>
                primum numerum conſeruatum, nempe .72. vult regula detrahi, ſic remanebit .9.
                  <lb/>
                tum huius .9. quadrata radix detrahenda eſt ex dimidio ipſius .18. quod fuit ante qua
                  <lb/>
                dratum, ſic ſupererit .6. hoc eſt .9. excepta radice quadrata, qui .6. erit minor pars
                  <lb/>
                quæſita, maior verò .14. quarum productum .84. coniunctum cum partium differen
                  <lb/>
                tia præbet exactè .92.</s>
              </p>
              <p>
                <s xml:id="echoid-s445" xml:space="preserve">Cuius rei hæc eſt ſpeculatio. </s>
                <s xml:id="echoid-s446" xml:space="preserve">Primus numerus minor, qui proponitur diuiſibilis
                  <lb/>
                ſignificetur linea
                  <var>.q.g.</var>
                maior vero linea
                  <var>.x.</var>
                tum cogitemus
                  <var>.q.g.</var>
                diuiſam, cuius maior
                  <lb/>
                pars ſit
                  <var>.q.o.</var>
                minor
                  <var>.o.g.</var>
                differentia
                  <var>.q.p.</var>
                ex quo
                  <var>.p.o.</var>
                æqualis erit
                  <var>.o.g.</var>
                ſit autem produ-
                  <lb/>
                ctum
                  <var>.b.o</var>
                . </s>
                <s xml:id="echoid-s447" xml:space="preserve">Oportet igitur, ut
                  <var>.b.o.</var>
                ſimul cum differentia
                  <var>.q.p.</var>
                æquale ſit numero
                  <var>.x.</var>
                ſe-
                  <lb/>
                cundò propoſito, qui notus eſt, </s>
                <s xml:id="echoid-s448" xml:space="preserve">quare etiam ſumma producti
                  <var>.b.o.</var>
                cum differentia
                  <lb/>
                  <var>q.p.</var>
                cognita erit, ex qua detracto primo numero
                  <var>.q.g.</var>
                reſiduum cognitum erit, nunc
                  <lb/>
                igitur quodnam erit hoc reſiduum? </s>
                <s xml:id="echoid-s449" xml:space="preserve">attendamus qua ratione ex ſumma
                  <var>.b.o.</var>
                et
                  <var>.q.p.</var>
                  <lb/>
                detrahenda ſit
                  <var>.q.g</var>
                . </s>
                <s xml:id="echoid-s450" xml:space="preserve">In primis ſi ſubtraxerimus ex dicta ſumma
                  <var>.q.p.</var>
                quę pars eſt
                  <var>.q.g.</var>
                  <lb/>
                ſupererit detrahenda
                  <var>.p.g.</var>
                ex
                  <var>.b.o.</var>
                pars inquam ipſius
                  <var>.q.g.</var>
                quod fiet quotieſcunque
                  <lb/>
                cogitauerimus
                  <var>.q.o.</var>
                duabus vnitatibus diminutam, et per
                  <var>.o.g.</var>
                multiplicatam, ſit au-
                  <lb/>
                tem productum
                  <var>.b.e.</var>
                nam cum
                  <var>.o.g.</var>
                toties
                  <var>.b.o.</var>
                ingrediatur, quot ſunt in
                  <var>.q.o.</var>
                vnitates
                  <lb/>
                ex prima ſexti aut .18. vel .19. ſeptimi,
                  <reg norm="detrahendaque" type="simple">detrahendaq́;</reg>
                ſit
                  <var>.p.g.</var>
                ex
                  <var>.b.o.</var>
                quæ
                  <var>.p.g.</var>
                dupla
                  <lb/>
                eſt
                  <var>.o.g.</var>
                patebit
                  <var>.o.c.</var>
                æqualem eſſe
                  <var>.p.g.</var>
                fu-
                  <lb/>
                pererit ita que
                  <var>.b.e.</var>
                productum
                  <var>.q.e.</var>
                in
                  <var>.e.</var>
                  <lb/>
                  <figure xlink:label="fig-0045-01" xlink:href="fig-0045-01a" number="62">
                    <image file="0045-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0045-01"/>
                  </figure>
                i. cognitum, erutis autem ex
                  <var>.q.g.</var>
                ijſdem
                  <lb/>
                duabus vnitatibus, remanebit
                  <var>.q.i.</var>
                nobis
                  <lb/>
                nota, ex quo
                  <var>.e.i.</var>
                æqualis erit
                  <var>.e.c</var>
                . </s>
                <s xml:id="echoid-s451" xml:space="preserve">Cum
                  <lb/>
                igitur productum
                  <var>.q.e.</var>
                in
                  <var>.e.i.</var>
                cognoſcamus
                  <lb/>
                ſimul cum
                  <var>.q.i</var>
                : Sivoluerimus partes
                  <var>.q.e.</var>
                  <lb/>
                et
                  <var>.e.i.</var>
                cognoſcere, vtemur .45. theorema-
                  <lb/>
                te huius libri, & propoſitum obtinebimus, nam cognoſcemus
                  <var>.e.i.</var>
                & ex conſequen-
                  <lb/>
                ti
                  <var>.o.g.</var>
                eius æqualem.</s>
              </p>
            </div>
            <div xml:id="echoid-div108" type="math:theorem" level="3" n="51">
              <head xml:id="echoid-head67" xml:space="preserve">THEOREMA
                <num value="51">LI</num>
              .</head>
              <p>
                <s xml:id="echoid-s452" xml:space="preserve">
                  <emph style="sc">DIvidere</emph>
                numerum in duas eiuſmodi partes, quæ pro medio proportionali
                  <lb/>
                alterum numerum propoſitum recipiant, primi dimidio minorem, aliud ni
                  <lb/>
                hil eſt, quàm binas primi numeri partes inuenire, quæ inter ſe multiplicatæ quadra
                  <lb/>
                to ſecundi numeri numerum æqualem proferant, ex .16. ſexti aut .20. ſeptimi, quod
                  <lb/>
                tamen .45. theoremate fuit à nobis ſpeculatum.</s>
              </p>
            </div>
            <div xml:id="echoid-div109" type="math:theorem" level="3" n="52">
              <head xml:id="echoid-head68" xml:space="preserve">THEOREMA
                <num value="52">LII</num>
              .</head>
              <p>
                <s xml:id="echoid-s453" xml:space="preserve">CVR pro poſitis tribus numeris quibuſcunque, ſi productum primi in ſecun-
                  <lb/>
                dum per tertium multiplicetur, atque ſecundum hoc productum
                  <reg norm="corporeum" type="context">corporeũ</reg>
                ,
                  <lb/>
                per primum numerum diuidatur, proueniens erit numerus æqualis producto ſe-
                  <lb/>
                cundi in tertium.</s>
              </p>
              <p>
                <s xml:id="echoid-s454" xml:space="preserve">Exempli cauſa, proponantur hi tres numeri .10. 11. 12.
                  <reg norm="multiplicenturque" type="simple">multiplicenturq́;</reg>
                .10.
                  <reg norm="cum" type="context">cũ</reg>
                .</s>
              </p>
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