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-div61" type="math:theorem" level="3" n="27">
              <pb o="18" rhead="IO. BAPT. BENED." n="30" file="0030" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0030"/>
              <p>
                <s xml:id="echoid-s266" xml:space="preserve">Proponunt hi numerum in binas eiuſmodi partes diuidendum, vt ſumma qua-
                  <lb/>
                dratorum dictarum partium, alteri numero poſsibili propoſito æqualis ſit, poſſi-
                  <lb/>
                bili inquam, etenim ſi eiuſmodi numerus propoſitus, minor eſſet producto totius
                  <lb/>
                primi in ſuum dimidium, eſſet huiuſmodi factum impoſſibile. </s>
                <s xml:id="echoid-s267" xml:space="preserve">Quod nos exequi
                  <lb/>
                cupientes, ſumamus primum
                  <reg norm="numerum" type="context">numerũ</reg>
                propoſitum, quem in ſe ipſum multiplice-
                  <lb/>
                mus. </s>
                <s xml:id="echoid-s268" xml:space="preserve">ab hoc quadrato deducamus ſecundum numerum propoſitum, tum quod re-
                  <lb/>
                manſerit duplicemus, quod duplum denuo iubeo ex eodem primo quadrato detra-
                  <lb/>
                hi, accepta poſtea radice quadrata reſidui & dempta ex priori numero propoſito,
                  <lb/>
                </s>
                <s xml:id="echoid-s269" xml:space="preserve">tunc dimidium reſidui vna pars erit ex duabus primi numeri quæſita.</s>
              </p>
              <p>
                <s xml:id="echoid-s270" xml:space="preserve">Exempli gratia proponantur .20. diuidenda in duas eiuſmodi partes, vt ſumma
                  <lb/>
                quadratorum ipſarum partium æqualis ſit .272. qui numerus maior eſt .200. maior
                  <lb/>
                inquam dimidio quadrati .400. ipſorum .20. hic autem numerus .272. è quadra-
                  <lb/>
                to .400. deducatur,
                  <reg norm="remanebunt" type="context">remanebũt</reg>
                enim .128. quod duplicari iubeo,
                  <reg norm="producentur" type="context">producẽtur</reg>
                  <reg norm="ſiquidem" type="context">ſiquidẽ</reg>
                  <num value="256">.
                    <lb/>
                  256.</num>
                quæ pariter deducta è quadrato totali, remanebunt .144. cuius radicem ſumi
                  <lb/>
                volo, quæ erit .12. & dempta ex .20. priori numero dato remanebit .8. cuius di-
                  <lb/>
                midium erit .4: pars vna ex quæſitis, quæ ex primo numero propoſito .20. detra-
                  <lb/>
                hetur,
                  <reg norm="remanebitque" type="simple">remanebitq́</reg>
                .16. pro altera parte.</s>
              </p>
              <p>
                <s xml:id="echoid-s271" xml:space="preserve">Cuius demonſtrationis cauſa, in primis cogitemus quadratum
                  <var>.a.c.</var>
                cognitum nu-
                  <lb/>
                meri
                  <var>.a.b.</var>
                primò propoſiti, qui cogitetur diuiſus in duo quadrata
                  <var>.d.e.</var>
                et
                  <var>.e.b.</var>
                  <reg norm="duo- que" type="simple">duo-
                    <lb/>
                  q́</reg>
                ſupplementa
                  <var>.a.e.</var>
                et
                  <var>.e.c.</var>
                numerus autem ſummæ duorum quadratorum
                  <var>.d.e.
                    <lb/>
                  b.</var>
                pro ſecundo propoſito datur; </s>
                <s xml:id="echoid-s272" xml:space="preserve">ex quo, ſumma duorum ſupplementorum
                  <var>.a.e.c.</var>
                  <lb/>
                conſequenter erit cognita, quę cum duplicata fuerit, & quatuor hæc ſupplementa
                  <unsure/>
                  <lb/>
                cogitatione accommodata, prout in
                  <lb/>
                quadrato
                  <var>.f.g.</var>
                apparet (
                  <reg norm="quanuis" type="context">quãuis</reg>
                idipſum
                  <lb/>
                  <figure xlink:label="fig-0030-01" xlink:href="fig-0030-01a" number="39">
                    <image file="0030-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0030-01"/>
                  </figure>
                proueniret ſi modo Eucl. octaua
                  <reg norm="ſecundi" type="context">ſecũdi</reg>
                  <lb/>
                aptaretur) æquali quadrato
                  <var>.a.c.</var>
                ita vt
                  <lb/>
                cogitatis quatuor ſupplementis numeri
                  <lb/>
                cogniti in quadrato
                  <var>.f.g.</var>
                ex conſequen-
                  <lb/>
                ti cognoſcetur numerus quadrati partia
                  <lb/>
                lis
                  <var>.h.i.</var>
                & vna etiam eius radix qua de-
                  <lb/>
                tracta ex numero
                  <var>.a.b.</var>
                aut
                  <var>.f.n.</var>
                (quod
                  <lb/>
                idem eſt) primo propoſiti, relinquetur numerus cognitus duplum
                  <var>.x.k.n.</var>
                aut
                  <var>.t.b.</var>
                  <lb/>
                pars vna totius
                  <var>.a.b.</var>
                ex quo uerum erit hoc meum problema.</s>
              </p>
            </div>
            <div xml:id="echoid-div63" type="math:theorem" level="3" n="28">
              <head xml:id="echoid-head44" xml:space="preserve">THEOREMA
                <num value="28">XXVIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s273" xml:space="preserve">SI quis & aliam rationem perficiendæ
                  <lb/>
                  <figure xlink:label="fig-0030-02" xlink:href="fig-0030-02a" number="40">
                    <image file="0030-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0030-02"/>
                  </figure>
                huius rei quærat, hoc præſtet inuen-
                  <lb/>
                to numero huius ſupplementi, cum in
                  <lb/>
                præcedenti theoremate dictum fuerit,
                  <lb/>
                qua ratione manifeſtetur duplum ſupple-
                  <lb/>
                menti ipſius.</s>
              </p>
              <p>
                <s xml:id="echoid-s274" xml:space="preserve">Cogitemus in ſubſcripta figura lineam
                  <var>.
                    <lb/>
                  a.b.</var>
                tanquam primum numerum propoſi-
                  <lb/>
                tum, & productum
                  <var>.a.e.</var>
                ſupplemento
                  <var>.a.e.</var>
                primæ præcedentis figuræ æquale ſit,
                  <lb/>
                ac deinde ordine ab antiquis tradito procedatur, ad quadratum reducto dimidio
                  <var>.
                    <lb/>
                  a.b.</var>
                videlicet
                  <var>.b.c.</var>
                quod erit
                  <var>.b.d.</var>
                ex quo detrahatur deinde
                  <var>.a.e</var>
                . </s>
                <s xml:id="echoid-s275" xml:space="preserve">quare remane- </s>
              </p>
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