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-div120" type="math:theorem" level="3" n="59">
              <pb o="38" rhead="IO. BAPT. BENED." n="50" file="0050" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0050"/>
              <p>
                <s xml:id="echoid-s521" xml:space="preserve">Hoc vt demonſtremus, primus nu-
                  <lb/>
                  <figure xlink:label="fig-0050-01" xlink:href="fig-0050-01a" number="67">
                    <image file="0050-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0050-01"/>
                  </figure>
                merus linea
                  <var>.a.b.</var>
                ſignificetur, quam di-
                  <lb/>
                uiſam cogitemus in puncto
                  <var>.c.</var>
                in partes
                  <lb/>
                quæſitas, ex quo præſupponitur duas li-
                  <lb/>
                neas
                  <var>.a.c.</var>
                et
                  <var>.c.b.</var>
                duo quadrata eſſe, quæ
                  <lb/>
                in altera figura ſignificetur per
                  <var>.d.</var>
                et
                  <var>.e.</var>
                  <lb/>
                productum autem radicum cognitum
                  <var>.
                    <lb/>
                  f.</var>
                quandoquidem datum eſt, cuius qua-
                  <lb/>
                dratum æquale erit producto quadra-
                  <lb/>
                torum
                  <var>.d.e.</var>
                adinuicem, nempe
                  <var>.b.c.</var>
                in
                  <var>.a.c.</var>
                ex .19. theoremate huius. </s>
                <s xml:id="echoid-s522" xml:space="preserve">Quod verbi
                  <lb/>
                gratia ſit
                  <var>.x.</var>
                  <reg norm="itaque" type="simple">itaq;</reg>
                cognitum, quo facto, doctrinam .45. theorematis libri huius ſecuti,
                  <lb/>
                propoſitum conſequemur.</s>
              </p>
            </div>
            <div xml:id="echoid-div122" type="math:theorem" level="3" n="60">
              <head xml:id="echoid-head76" xml:space="preserve">THEOREMA
                <num value="60">LX</num>
              .</head>
              <p>
                <s xml:id="echoid-s523" xml:space="preserve">CVR productum differentiæ duarum radicum in ſummam ipſarum, ſemper
                  <lb/>
                differentia ſit quadratorum ipſarum radicum.</s>
              </p>
              <p>
                <s xml:id="echoid-s524" xml:space="preserve">
                  <reg norm="Exempli" type="context">Exẽpli</reg>
                gratia, quoslibet duos numeros pro radicibus ſumpſerimus, vt potè .3. et
                  <num value="5">.
                    <lb/>
                  5.</num>
                quorum differentia eſt .2. certè ſi differentiam hanc per ſummam radicum ſcili-
                  <lb/>
                cet .8. multiplicauerimus, dabitur numerus .16. quod productum differentia eſt
                  <lb/>
                ſuorum quadratorum, nempeinter .9. et .25.</s>
              </p>
              <p>
                <s xml:id="echoid-s525" xml:space="preserve">Hoc vt ſpeculemur, duæ radices in linea
                  <var>.n.i.</var>
                ſignificentur, quarum vna ſit
                  <var>.n.c.</var>
                &
                  <lb/>
                altera
                  <var>.c.i.</var>
                ipſarum autem differentia
                  <var>.n.t.</var>
                ex quo
                  <var>.t.
                    <lb/>
                  c.</var>
                æqualis erit
                  <var>.c.i</var>
                . </s>
                <s xml:id="echoid-s526" xml:space="preserve">Tum cogitato toto quadrato
                  <var>.d.i.</var>
                  <lb/>
                  <figure xlink:label="fig-0050-02" xlink:href="fig-0050-02a" number="68">
                    <image file="0050-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0050-02"/>
                  </figure>
                cum diametro
                  <var>.d.i.</var>
                  <reg norm="ductaque" type="simple">ductaq́</reg>
                parallela lateri
                  <var>.n.d.</var>
                à
                  <lb/>
                puncto
                  <var>.c.</var>
                & altera à puncto
                  <var>.t.</var>
                & à puncto
                  <var>.o.</var>
                tertia
                  <lb/>
                ipſi
                  <var>.n.i.</var>
                & à puncto
                  <var>.a.</var>
                quarta
                  <var>.x.a.e.</var>
                parallela ipſi
                  <var>.
                    <lb/>
                  o.</var>
                inueniemus
                  <var>.b.n.</var>
                productum eſſe differentiæ
                  <var>.n.
                    <lb/>
                  t.</var>
                in ſumma radicum
                  <var>.n.i.</var>
                & cum
                  <var>.d.o.</var>
                et
                  <var>.a.o.</var>
                ſint
                  <lb/>
                quadrata radicum prædictarum: </s>
                <s xml:id="echoid-s527" xml:space="preserve">b.e. æquale erit
                  <var>.
                    <lb/>
                  n.u.</var>
                cum vtrunque horum productorum æquale ſit
                  <var>.
                    <lb/>
                  x.u.</var>
                ex quo gnomon
                  <var>.e.d.u.</var>
                æqualis erit producto
                  <var>.
                    <lb/>
                  b.n.</var>
                quod ſcire cupiebamus.</s>
              </p>
            </div>
            <div xml:id="echoid-div124" type="math:theorem" level="3" n="61">
              <head xml:id="echoid-head77" xml:space="preserve">THEOREMA
                <num value="61">LXI</num>
              .</head>
              <p>
                <s xml:id="echoid-s528" xml:space="preserve">CVR propoſitum aliquem numerum diuiſuri in duas eiuſmodi partes, vt diffe-
                  <lb/>
                rentia radicum quadratarum æqualis ſit alteri numero propoſito, cuius ta-
                  <lb/>
                men quadratum dimidij primi quadratum non excedat. </s>
                <s xml:id="echoid-s529" xml:space="preserve">Rectè ſecundum numerum
                  <lb/>
                in ſeipſum multiplicant, productum verò ex primo numero detrahunt,
                  <reg norm="rurſusque" type="simple">rurſusq́;</reg>
                di
                  <lb/>
                midium reſidui quadrant, & quadratum hoc ex quadrato dimidij primi ſubtrahunt,
                  <lb/>
                atque ita radice quadrata reſidui, dimidio primi coniuncta, pars maior datur, qua
                  <lb/>
                ex ipſo dimidio detracta, pars minor relinquitur.</s>
              </p>
              <p>
                <s xml:id="echoid-s530" xml:space="preserve">Exempli gratia, propoſito numero .20. ita ut propoſitum eſt, diuidendo, nem-
                  <lb/>
                pe vt differentia radicum quadratarum dictarum partium æqualis ſit binario, bina-
                  <lb/>
                rium hocin ſeipſum multiplicabimus, cuius quadratum .4. è primo numero .20. de­ </s>
              </p>
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