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-div7" type="chapter" level="2" n="1">
            <div xml:id="echoid-div92" type="math:theorem" level="3" n="43">
              <p>
                <s xml:id="echoid-s375" xml:space="preserve">
                  <pb o="28" rhead="IO. BAPT. BENED." n="40" file="0040" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0040"/>
                quadrato dimidij, prout ex ſpeculatione huiuſmodi operis cognoſcetur,
                  <reg norm="cuiæquanda" type="context">cuiæquãda</reg>
                  <lb/>
                eſt
                  <reg norm="differentia" type="context">differẽtia</reg>
                inter
                  <reg norm="ſummam" type="context">ſummã</reg>
                  <reg norm="quadratorum" type="context">quadratorũ</reg>
                  <reg norm="duorum" type="context">duorũ</reg>
                qui
                  <reg norm="quæruntur" type="context">quærũtur</reg>
                  <reg norm="numerorum" type="context">numerorũ</reg>
                , ſimul
                  <reg norm="cum" type="context">cũ</reg>
                pro
                  <lb/>
                ducto
                  <reg norm="eorum" type="context">eorũ</reg>
                radicum. </s>
                <s xml:id="echoid-s376" xml:space="preserve">Dimidium numeri .20. in ſeipſum multiplicandum eſſet, qua-
                  <lb/>
                  <reg norm="dratumque" type="simple">dratumq́;</reg>
                detrahendum ex .208. vtremanerent .108. quorum .108. tertiæ partis qua
                  <lb/>
                drata radix eſſet .6. quæ ſi iuncta fuerit dimidio .20. nempe .10. daretur maior nu-
                  <lb/>
                merus quæſitus .16. quo detracto è .20. darentur .4.</s>
              </p>
              <p>
                <s xml:id="echoid-s377" xml:space="preserve">Cuius ſpeculationis cauſa, datus primus numerus ſignificetur linea
                  <var>.g.h.</var>
                in qua
                  <lb/>
                maior numerus incognitus ſit
                  <var>.g.h.</var>
                minor verò
                  <var>.b.h.</var>
                quorum quadrata ſint
                  <var>.y.t.</var>
                et
                  <var>.
                    <lb/>
                  b.l.</var>
                in quadrato maximo
                  <var>.g.p.</var>
                tum productum
                  <var>.g.b.</var>
                in
                  <var>.b.h.</var>
                ſit
                  <var>.g.c.</var>
                  <reg norm="cogitenturque" type="simple">cogitenturq́;</reg>
                duo
                  <lb/>
                diametri
                  <var>.q.h.</var>
                et
                  <var>.g.p.</var>
                diuiſi per medium in puncto
                  <var>.o.</var>
                per quod duę lineæ ducan-
                  <lb/>
                tur
                  <var>.f.d.</var>
                et
                  <var>.k.m.</var>
                parallelæ lateribus maximi quadrati. </s>
                <s xml:id="echoid-s378" xml:space="preserve">Hæ dictum quadratum in
                  <lb/>
                quatuor quadrata æqualia diuident, quorum
                  <reg norm="vnumquodque" type="simple">vnumquodq́;</reg>
                , æquale erit quadrato
                  <var>.
                    <lb/>
                  g.f.</var>
                dimidij ipſius
                  <var>.g.h.</var>
                datę, </s>
                <s xml:id="echoid-s379" xml:space="preserve">quare eorum
                  <reg norm="vnumquodque" type="simple">vnumquodq́;</reg>
                cognitum erit. </s>
                <s xml:id="echoid-s380" xml:space="preserve">Iterum co
                  <lb/>
                gitemus
                  <var>.s.x.</var>
                per
                  <var>.e.</var>
                  <reg norm="parallelam" type="context">parallelã</reg>
                  <var>.g.k.</var>
                tantum diſtan-
                  <lb/>
                tem à
                  <var>.g.k.</var>
                quantum
                  <var>.y.l.</var>
                ab
                  <var>.g.h.</var>
                diſtare inueni-
                  <lb/>
                  <figure xlink:label="fig-0040-01" xlink:href="fig-0040-01a" number="55">
                    <image file="0040-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0040-01"/>
                  </figure>
                tur. </s>
                <s xml:id="echoid-s381" xml:space="preserve">Cogitetur pariter
                  <var>.z.i.a.</var>
                per punctum
                  <var>.i.</var>
                  <lb/>
                parallela
                  <var>.d.p.</var>
                </s>
                <s xml:id="echoid-s382" xml:space="preserve">quare
                  <var>.a.t.</var>
                æqualis erit
                  <var>.f.c.</var>
                et
                  <var>.y.x.</var>
                  <lb/>
                æqualis
                  <var>.f.e.</var>
                et
                  <var>.y.s</var>
                :
                  <var>b.l.</var>
                æqualis. </s>
                <s xml:id="echoid-s383" xml:space="preserve">Ita ſubtractis è
                  <lb/>
                duobus quadratis ſuperius dictis
                  <var>.a.t.y.x.</var>
                et
                  <var>.b.l.</var>
                  <lb/>
                producto
                  <var>.y.b.</var>
                æqualibus, ſupererunt
                  <var>.k.d.</var>
                et
                  <var>.a.c.
                    <lb/>
                  x.</var>
                cognita, tanquam æqualia dato ſecundo nu-
                  <lb/>
                mero, ſed
                  <var>.k.d.</var>
                quadratum eſt medietatis
                  <var>.g.f.</var>
                  <lb/>
                cognitæ, cognoſcetur igitur reſiduum
                  <var>.a.c.x.</var>
                vnà
                  <lb/>
                etiam ſingulæ tertiæ partes nempe quadrata
                  <var>.o.
                    <lb/>
                  i.o.c.</var>
                et
                  <var>.o.e.</var>
                & radix
                  <var>.b.f.</var>
                vel
                  <var>.f.s.</var>
                ſingularum,
                  <lb/>
                qua coniuncta dimidio
                  <var>.g.f.</var>
                  <reg norm="rurfusque" type="simple">rurfusq́;</reg>
                ab
                  <reg norm="eodem" type="context">eodẽ</reg>
                de-
                  <lb/>
                tracta, propoſitum conſequemur.</s>
              </p>
            </div>
            <div xml:id="echoid-div94" type="math:theorem" level="3" n="44">
              <head xml:id="echoid-head60" xml:space="preserve">THEOREMA
                <num value="44">XLIIII</num>
              .</head>
              <p>
                <s xml:id="echoid-s384" xml:space="preserve">CVR ſi quis cupiat numerum propoſitum in duas eiuſmodi partes diuidere, vt
                  <lb/>
                quadratum maioris, quadratum minoris ſuperet quantitate alterius numeri
                  <lb/>
                propoſiti, rectè primum numerum in ſeipſum multiplicabit, & ab eodem ſecun-
                  <lb/>
                dum numerum detrahet, reſiduum verò per duplum primi diuidet, ex quo proue-
                  <lb/>
                niens primi pars minor erit, quæ ex illo primo detracta, partem maiorem
                  <lb/>
                proferet.</s>
              </p>
              <p>
                <s xml:id="echoid-s385" xml:space="preserve">Exempli gratia, ſi proponantur .20. diuiſa in duas eiuſmodi partes, vt
                  <reg norm="quadratum" type="context">quadratũ</reg>
                  <lb/>
                maioris ſuperet quadratum minoris numero æquali ipſi .240. oportebit primum
                  <lb/>
                numerum, qui quadratus cum fuerit, erit .400. in ſeipſum multiplicare, & ex hoc
                  <lb/>
                quadrato ſecundum numerum nempe .240. detrahere, </s>
                <s xml:id="echoid-s386" xml:space="preserve">tunc remanebunt .160. quę
                  <lb/>
                diuiſa per .40.
                  <reg norm="numerum" type="context">numerũ</reg>
                  <reg norm="duplum" type="context">duplũ</reg>
                primo, dabuntur quatuor pro minori numero, à reſi-
                  <lb/>
                duo verò .20. detractis quatuor, erunt .16. pro maiorinumero.</s>
              </p>
              <p>
                <s xml:id="echoid-s387" xml:space="preserve">Quod vt exactè conſideremus, primus numerus propoſitus ſignificetur linea
                  <var>.q.
                    <lb/>
                  h.</var>
                diuidendus in duas partes
                  <var>.q.p.</var>
                et
                  <var>.p.h.</var>
                tales quales quærimus. </s>
                <s xml:id="echoid-s388" xml:space="preserve">Poſtmodum eriga
                  <lb/>
                  <gap extent="2"/>
                r quadratum
                  <var>.q.e.</var>
                diuiſum diametro
                  <var>.f.h.</var>
                  <reg norm="ductisque" type="simple">ductisq́;</reg>
                  <var>.p.o.t.</var>
                et
                  <var>.a.o.c.</var>
                parallelis lateri-
                  <lb/>
                bus quadrati, dabuntur imaginaria quadrata
                  <var>.c.t.</var>
                et
                  <var>.p.a.</var>
                duarum partium
                  <var>.q.p.</var>
                et
                  <var>.p.
                    <lb/>
                  h.</var>
                incognitarum. </s>
                <s xml:id="echoid-s389" xml:space="preserve">Ad hæc cogitemus quadratum
                  <var>.u.n.</var>
                æquale quadrato
                  <var>.p.a.</var>
                è quadra­ </s>
              </p>
            </div>
          </div>
        </div>
      </text>
    </echo>
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