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-div134" type="math:theorem" level="3" n="67">
              <p>
                <s xml:id="echoid-s568" xml:space="preserve">
                  <pb o="43" rhead="THEOREM. ARIT." n="55" file="0055" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0055"/>
                ita vt ſimul prouenientibus in ſummam collectis huius fummæ ad primum nume-
                  <lb/>
                rum propoſitum proportio futura ſit ea quæ eſt tertij ad ſecundum. </s>
                <s xml:id="echoid-s569" xml:space="preserve">Rectè dimidium
                  <lb/>
                primi numeri in ſeipſum multiplicant, ex quo quadrato ſecundum numerum detra
                  <lb/>
                hunt, tum reſidui radicem ſumunt, quam iungentes, & detrahentes ex dimidio
                  <lb/>
                primi, partes quæſitas habent, cætera ex neceſsitate ſubſequuntur, prout nunc a
                  <lb/>
                me docebitur.</s>
              </p>
              <p>
                <s xml:id="echoid-s570" xml:space="preserve">Exempli gratia, proponitur numerus .20. in duas partes diuidendus, quibus po
                  <lb/>
                ſtea mutuò diuiſis, & per ſummam prouenientium diuiſa ſumma quadratorum,
                  <lb/>
                dent
                  <reg norm="ſecundum" type="context">ſecundũ</reg>
                numerum propoſitum .36. nam reliqua conſequuntur. </s>
                <s xml:id="echoid-s571" xml:space="preserve">Itaque .10.
                  <lb/>
                dimidium primi in ſeipſum multiplicatur, & ex quadrato .100. eruitur numerus .36.
                  <lb/>
                nempe ſecundus propoſitus reſidui porrò .64. quadrata radix .8. fumitur, quam con
                  <lb/>
                iungimus & detrahimus ex dimidio primi ſcilicet .10. ex quo partes quæſitæ dabun
                  <lb/>
                tur .18. et .2. quæ mutuo diuiſæ dabunt ſuorum prouenientium ſummam .9. cum no-
                  <lb/>
                na parte, per quam diuidentes .328. ſummam quadratorum ipſarum partium,
                  <lb/>
                exactè dabitur numerus .36. qui fuit ſecundò propoſitus. </s>
                <s xml:id="echoid-s572" xml:space="preserve">Tum ſi per ſingu-
                  <lb/>
                las iam inuentas partes quilibet numerus diuiſus fuerit, verbi gratia .72. ſumma pro
                  <lb/>
                uenientium erit .40. qui num@rus eandem proportionem cum primo nempe .20. ſer
                  <lb/>
                uabit, quam tertius propoſitus .72. cum ſecundo .36.</s>
              </p>
              <p>
                <s xml:id="echoid-s573" xml:space="preserve">Quod vt ſpeculemur, primus numerus ſignificetur linea
                  <var>.n.e.</var>
                ita diuidendus à
                  <lb/>
                puncto
                  <var>.o.</var>
                vt diuiſa parte
                  <var>.n.o.</var>
                per
                  <var>.o.e.</var>
                et
                  <var>.o.e.</var>
                per
                  <var>.n.o.</var>
                & per ſummam prouenien-
                  <lb/>
                tium diuiſa ſumma quadratorum
                  <var>.n.o.</var>
                et
                  <var>.o.e.</var>
                detur ſecundus numerus notatus linea
                  <var type="line">.
                    <lb/>
                  q.K</var>
                . </s>
                <s xml:id="echoid-s574" xml:space="preserve">Porrò meminiſſe oportet quòd .26. theoremate probatum fuit vltimum hoc
                  <lb/>
                proueniens æquale producto partium inter ſe futurum, nempe producto
                  <var>.n.o.</var>
                in
                  <var>.o.
                    <lb/>
                  e.</var>
                quod ſignificetur rectangulo
                  <var>.n.e</var>
                . </s>
                <s xml:id="echoid-s575" xml:space="preserve">Itaque datis
                  <var>.n.e.</var>
                et
                  <var>.q.K.</var>
                ſi .45. theorema conſu-
                  <lb/>
                luerimus, partes
                  <var>.n.o.</var>
                et
                  <var>.o.e.</var>
                cognoſcemus.</s>
              </p>
              <p>
                <s xml:id="echoid-s576" xml:space="preserve">Proponitur deinde tertius quilibetnumerus, verbi gratia
                  <var>.x.</var>
                diuidendus per
                  <var>.o.e.</var>
                  <lb/>
                et
                  <var>.o.n.</var>
                qui ſi diuidatur per
                  <var>.o.e.</var>
                dabit pro
                  <lb/>
                ueniens
                  <var>.b.o</var>
                . </s>
                <s xml:id="echoid-s577" xml:space="preserve">Si verò per
                  <var>.n.o.</var>
                proueniens
                  <lb/>
                  <figure xlink:label="fig-0055-01" xlink:href="fig-0055-01a" number="75">
                    <image file="0055-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0055-01"/>
                  </figure>
                erit
                  <var>.d.n.</var>
                nunc aſſerimus
                  <reg norm="ſummam" type="context">ſummã</reg>
                duorum
                  <lb/>
                horum prouenientium, ſic primo nume-
                  <lb/>
                ro
                  <var>.n.e.</var>
                dato proportionatam eſſe, ſicut
                  <lb/>
                tertius
                  <var>.x.</var>
                  <reg norm="ſecundo" type="context">ſecũdo</reg>
                  <var>.q.K</var>
                . </s>
                <s xml:id="echoid-s578" xml:space="preserve">Producatur enim li-
                  <lb/>
                nea
                  <var>.d.n.</var>
                donec
                  <var>.n.q.</var>
                æqualis ſit
                  <var>.o.b.</var>
                ex
                  <lb/>
                quo
                  <var>.q.d.</var>
                erit ſumma vltimò prouenien-
                  <lb/>
                tium: </s>
                <s xml:id="echoid-s579" xml:space="preserve">item producatur
                  <var>.e.n.</var>
                donec
                  <var>.n.u.</var>
                æ-
                  <lb/>
                qualis ſit
                  <var>.o.e.</var>
                  <reg norm="termineturque" type="simple">termineturq́</reg>
                rectangulum
                  <var>.
                    <lb/>
                  q.u.</var>
                quod tertio numero propoſito
                  <var>.x.</var>
                vt
                  <lb/>
                patet, æquale erit, </s>
                <s xml:id="echoid-s580" xml:space="preserve">quare ex .15. ſexti aut .
                  <lb/>
                20. ſeptimi eadem erit proportio
                  <var>.d.n.</var>
                ad
                  <lb/>
                  <var>n.q.</var>
                quæ
                  <var>.u.n.</var>
                nempe
                  <var>.o.e.</var>
                ad
                  <var>.o.n.</var>
                & com-
                  <lb/>
                ponendo
                  <var>.d.q.</var>
                ad
                  <var>.q.n.</var>
                ſicut
                  <var>.e.n.</var>
                ad
                  <var>.n.o.</var>
                &
                  <lb/>
                permutando
                  <var>.d.q.</var>
                ad
                  <var>.e.n.</var>
                quæ
                  <var>.q.n.</var>
                hoc eſt
                  <var>.
                    <lb/>
                  b.o.</var>
                ad
                  <var>.o.n.</var>
                nempe ſicut
                  <var>.b.e.</var>
                ad
                  <var>.e.n.</var>
                ſuperficialem, ex prima ſexti aut .18. vel .19.
                  <lb/>
                ſeptimi, ſed rectangulum
                  <var>.e.n.</var>
                conſtitutum fuit æquale numero
                  <var>.q.K</var>
                . </s>
                <s xml:id="echoid-s581" xml:space="preserve">itaque verum
                  <lb/>
                eſt propoſitum.</s>
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