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-div126" type="math:theorem" level="3" n="62">
              <p>
                <s xml:id="echoid-s537" xml:space="preserve">
                  <pb o="40" rhead="IO. BAPT. BENED." n="52" file="0052" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0052"/>
                g. æqualis erit
                  <var>.g.d.</var>
                tum productum
                  <var>.a.g.</var>
                in
                  <var>.g.d.</var>
                ſit
                  <var>.a.i.</var>
                et
                  <var>.t.i.</var>
                æqualis
                  <var>.a.i.</var>
                et
                  <var>.l.i.</var>
                pariter
                  <lb/>
                ſecetur æqualis
                  <var>.t.i.</var>
                quæ omnia ex diametro
                  <var>.q.d.</var>
                cogitari poſſunt: </s>
                <s xml:id="echoid-s538" xml:space="preserve">erit igitur
                  <var>.u.i.</var>
                æ-
                  <lb/>
                qualis
                  <var>.i.d.</var>
                  <reg norm="ſupereritque" type="simple">ſupereritq́;</reg>
                quadratum
                  <var>.q.u.</var>
                differentiæ
                  <var>.a.h.</var>
                cognitum, hoc verò cogi-
                  <lb/>
                temus diuiſum eſſe in .4. partes æquales medijs diametris
                  <var>.p.r.</var>
                et
                  <var>.n.e.</var>
                </s>
                <s xml:id="echoid-s539" xml:space="preserve">quare
                  <reg norm="vnaquæque" type="simple">vnaquæq;</reg>
                  <lb/>
                partium cognoſcetur, &
                  <reg norm="quadratum" type="context">quadratũ</reg>
                erit ipſius
                  <var>.a.K.</var>
                aut ipſius
                  <var>.K.h.</var>
                dimidij
                  <var>.a.h</var>
                . </s>
                <s xml:id="echoid-s540" xml:space="preserve">Quòd
                  <lb/>
                ſi aliquod iſtorum quadratorum detrahere voluerimus, nempe
                  <var>.n.r.</var>
                ex dimidio ſum
                  <lb/>
                  <var>.b.</var>
                duorum quadratorum
                  <var>.q.i.</var>
                et
                  <var>.i.d.</var>
                cognitæ, hac via procedemus, primum con
                  <lb/>
                ſiderabimus
                  <var>.t.r.</var>
                coniunctam
                  <var>.t.i.</var>
                quæ quantitates erunt ſumma dimidij
                  <reg norm="duorum" type="context">duorũ</reg>
                qua-
                  <lb/>
                dratorum
                  <var>.q.i.</var>
                et
                  <var>.i.d.</var>
                quando quidem
                  <var>.t.r.</var>
                  <lb/>
                  <reg norm="dimidium" type="context">dimidiũ</reg>
                eſt quadrati
                  <var>.t.l.</var>
                et
                  <var>.t.i.</var>
                  <reg norm="dimidium" type="context">dimidiũ</reg>
                  <lb/>
                  <anchor type="figure" xlink:label="fig-0052-01a" xlink:href="fig-0052-01"/>
                gnomonis
                  <var>.t.i.l.</var>
                coniunctum dimidio
                  <lb/>
                quadrati
                  <var>.i.d.</var>
                ex quo
                  <var>.i.t.r.</var>
                dimidium erit
                  <var>.
                    <lb/>
                  b.</var>
                ex qua quantitate
                  <var>.i.t.r.</var>
                cogitare debe
                  <lb/>
                mus detrahi quadratum ipſius
                  <var>.K.h.</var>
                nem
                  <lb/>
                pe
                  <var>.n.r</var>
                : </s>
                <s xml:id="echoid-s541" xml:space="preserve">quare quod ſupereſt cognitum
                  <lb/>
                erit nempe
                  <var>.y.s.</var>
                cum
                  <var>.n.i.</var>
                ſed
                  <var>.y.m.</var>
                æqualis
                  <lb/>
                eſt
                  <var>.n.i.</var>
                et
                  <var>.y.m.</var>
                cum
                  <var>.y.s.</var>
                conſtituunt qua-
                  <lb/>
                dratum
                  <var>.p.m</var>
                . </s>
                <s xml:id="echoid-s542" xml:space="preserve">
                  <reg norm="Itaque" type="simple">Itaq;</reg>
                  <var>.p.m.</var>
                quadratum &
                  <lb/>
                conſequenter
                  <var>.p.s.</var>
                eius radix cognoſce-
                  <lb/>
                tur, ita etiam & productum huius
                  <var>.p.s.</var>
                in
                  <var>.
                    <lb/>
                  s.x.</var>
                æqualis
                  <var>.c.</var>
                nempe
                  <var>.p.x</var>
                :
                  <reg norm="eſtque" type="simple">eſtq́;</reg>
                produ-
                  <lb/>
                ctum huiuſmodi ſemper minus quantita
                  <lb/>
                te
                  <var>.r.t.i</var>
                : per
                  <var>.u.i.</var>
                æquale quadrato minori
                  <var>.
                    <lb/>
                  i.d</var>
                . </s>
                <s xml:id="echoid-s543" xml:space="preserve">quare
                  <var>.i.d.</var>
                cognoſcetur, conſequen-
                  <lb/>
                ter
                  <var>.i.</var>
                @q. tanquam reſiduum ex
                  <var>.b.</var>
                & eo-
                  <lb/>
                rum radices quadratæ cognoſcentur
                  <var>.a.
                    <lb/>
                  g.</var>
                et
                  <var>.g.d</var>
                .</s>
              </p>
              <div xml:id="echoid-div127" type="float" level="4" n="2">
                <figure xlink:label="fig-0052-01" xlink:href="fig-0052-01a">
                  <image file="0052-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0052-01"/>
                </figure>
              </div>
            </div>
            <div xml:id="echoid-div129" type="math:theorem" level="3" n="63">
              <head xml:id="echoid-head79" xml:space="preserve">THEOREMA
                <num value="63">LXIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s544" xml:space="preserve">IDEM præſtari hac alia via, meo iudicio poteſt. </s>
                <s xml:id="echoid-s545" xml:space="preserve">Secundus numerus in
                  <reg norm="ſuum" type="context">ſuũ</reg>
                dimi
                  <lb/>
                  <reg norm="dium" type="context">diũ</reg>
                multiplicetur,
                  <reg norm="productum" type="context">productũ</reg>
                autem ex dimidio primi detrahatur, ex quo re-
                  <lb/>
                manens erit productum vnius quadratæ radicis in alteram partium primi numeri
                  <lb/>
                quæſitarum, deinde productum hoc duplicetur, & primo numero dato coniunga-
                  <lb/>
                tur,
                  <reg norm="ſicque" type="simple">ſicq́;</reg>
                huius ſummæ quadrata radix erit ſumma radicum quadratarum dictarum
                  <lb/>
                partium, cui iuncto producto ex quadrageſimoquinto theoremate ſingulæ radices
                  <lb/>
                proferentur.</s>
              </p>
              <p>
                <s xml:id="echoid-s546" xml:space="preserve">Exempli gratia, primus numerus diuiſibilis erat .50. alter verò .6. </s>
                <s xml:id="echoid-s547" xml:space="preserve">Iam ſi multi-
                  <lb/>
                plicemus .6. per .3. nempe dimidium proferetur numerus .18. quo ex dimidio pri-
                  <lb/>
                mi, nempe .25. detracto, ſupererit .7. productum vnius radicis in alteram, quod du
                  <lb/>
                plicatum dabit .14. quo coniuncto cum primo numero .50. dabitur numerus .64.
                  <lb/>
                cuius quadrata radix ſcilicet .8. erit ſumma radicum duarum partium quæſitarum,
                  <lb/>
                qua & producto .7. ex quadrag eſimoquinto theoremate dictæ radices diſtinguen,
                  <lb/>
                tur, quarum vna erit .7. & altera
                  <var>.I</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s548" xml:space="preserve">Vtautem hocſpeculemur, præcedenti figura vti poterimus, in qua patet
                  <var>.t.r.</var>
                pro
                  <lb/>
                ductum eſſe ſecundi numeri
                  <var>.c.</var>
                nempe
                  <var>.a.h.</var>
                hoc eſt
                  <var>.t.u.</var>
                in dimidio
                  <var>.a.e.</var>
                ſcilicet
                  <var>.p.t.</var>
                re-
                  <lb/>
                ſiduum autem dimidij primi
                  <var>.b.</var>
                eſſe
                  <var>.t.i.</var>
                nempe
                  <var>.a.i.</var>
                productum radicum, quod ſupple­ </s>
              </p>
            </div>
          </div>
        </div>
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