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-div46" type="math:theorem" level="3" n="20">
              <p>
                <s xml:id="echoid-s206" xml:space="preserve">
                  <var>
                    <pb o="14" rhead="IO. BAPT. BENED." n="26" file="0026" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0026"/>
                  q.</var>
                radicem eſſe quadratam producti
                  <var>.l.e.</var>
                in
                  <var>.e.p.</var>
                quod
                  <reg norm="productum" type="context">productũ</reg>
                ſit quadratuni
                  <unsure/>
                  <lb/>
                corporeum
                  <var>.c.g.</var>
                cogitemus pariter duo quadrata
                  <var>.l.e.</var>
                et
                  <var>.e.p.</var>
                eſſe pariter corpo-
                  <lb/>
                rea, tantę profunditatis, quantam, vnitas linearis radicum
                  <var>.m.e.</var>
                et
                  <var>.e.q.</var>
                requirit.
                  <lb/>
                </s>
                <s xml:id="echoid-s207" xml:space="preserve">Hæc duo corpora producentur à ſuperficie in vnitatem,
                  <reg norm="vocenturque" type="simple">vocenturq́;</reg>
                  <var>.l.x.</var>
                et
                  <var>.x.p.</var>
                quo
                  <lb/>
                facto, cogitemus corpus
                  <var>.a.g.</var>
                tamquam productum cubi
                  <var>.l.b.</var>
                in quadratum
                  <var>.e.p</var>
                . </s>
                <s xml:id="echoid-s208" xml:space="preserve">Vn-
                  <lb/>
                de ex decimaoctaua, aut decimanona ſeptimi, eadem erit proportio
                  <var>.a.g.</var>
                ad
                  <var>.c.g.</var>
                  <lb/>
                quæ eſt
                  <var>.l.b.</var>
                ad
                  <var>.l.x.</var>
                corporeum, ſed ex .25. vndecimi & prima ſexti, ita ſe habet
                  <var>.a.K.</var>
                  <lb/>
                ad
                  <var>.K.c.</var>
                vnitatem linearé ſicut
                  <var>.a.g.</var>
                ad
                  <var>.c.g.</var>
                & ex
                  <reg norm="eiſdem" type="context">eiſdẽ</reg>
                ita ſe habebit
                  <var>.b.e.</var>
                ad
                  <var>.e.x.</var>
                vnita-
                  <lb/>
                tem linearem, ſicut
                  <var>.l.b.</var>
                ad quadratum
                  <var>.l.x.</var>
                corporeum. </s>
                <s xml:id="echoid-s209" xml:space="preserve">Itaque ſic ſe habebit
                  <var>.b.e.</var>
                ad
                  <lb/>
                vnitatem linearem
                  <var>.e.x.</var>
                videlicet
                  <var>.K.c.</var>
                ſicut
                  <var>.a.K.</var>
                ad ipſam
                  <var>.K.c</var>
                . </s>
                <s xml:id="echoid-s210" xml:space="preserve">Vnde ex nona quinti
                  <var>.
                    <lb/>
                  a.K.</var>
                æqualis erit
                  <var>.e.b.</var>
                & conſequenter æqualis
                  <var>.m.e</var>
                . </s>
                <s xml:id="echoid-s211" xml:space="preserve">Iam verò ſit
                  <var>.u.g.</var>
                productum
                  <var>.l.b.</var>
                  <lb/>
                cubi, in cubum
                  <var>.o.p.</var>
                vt ſupra dictum eſt, Hinc patebit ex quauis duarum propoſitio-
                  <lb/>
                num, decimaoctaua, aut decimanona ſeptimi, eandem futuram proportionem
                  <var>.u.g.</var>
                  <lb/>
                ad
                  <var>.a.g.</var>
                quæ eſt
                  <var>.o.p.</var>
                ad
                  <var>.x.p.</var>
                quadratum corporeum. </s>
                <s xml:id="echoid-s212" xml:space="preserve">Quare ex poſtremis, dictis ratio-
                  <lb/>
                nibus, eadem erit proportio
                  <var>.u.K.</var>
                ad
                  <var>.a.K.</var>
                quæ eſt
                  <var>.o.e.</var>
                ad vnitatem linearem
                  <var>.e.x.</var>
                at
                  <lb/>
                ex dictis decimaoctaua & decimanona ſeptimi, ita ſe habet
                  <reg norm="numerus" type="simple">numerꝰ</reg>
                  <var>.m.q.</var>
                ad
                  <reg norm="numerum" type="context">numerũ</reg>
                  <lb/>
                  <reg norm="ſuperficialem" type="context">ſuperficialẽ</reg>
                  <var>.m.e.</var>
                qui
                  <reg norm="producitur" type="simple">ꝓducitur</reg>
                à lineari
                  <var>.m.e.</var>
                in vnitaté
                  <reg norm="linearem" type="context">linearẽ</reg>
                ipſius
                  <var>.e.q.</var>
                ſicut nume
                  <lb/>
                rus
                  <var>.q.e.</var>
                ad ſuam vnitaté, ſed
                  <reg norm="cum" type="context">cũ</reg>
                numerus
                  <var>.a.K.</var>
                æqualis ſit numero
                  <var>.m.e.</var>
                vt
                  <reg norm="probatum" type="context">probatũ</reg>
                eſt
                  <lb/>
                erit ergo ex vndecima & nona quinti, numerus
                  <var>.u.K.</var>
                æqualis numero
                  <var>.m.q</var>
                . </s>
                <s xml:id="echoid-s213" xml:space="preserve">At
                  <var>.f.g.</var>
                  <lb/>
                pariter æqualis eſt numero
                  <var>.m.q.</var>
                ex præcedenti theoremate, vnde
                  <var>.K.u.</var>
                pariter æqua
                  <lb/>
                lis erit
                  <var>.f.g</var>
                . </s>
                <s xml:id="echoid-s214" xml:space="preserve">Itaque ſequitur
                  <var>.u.g.</var>
                cubum eſſe, &
                  <var>f.g.</var>
                radicem ipſius, æqualem numero
                  <var>.
                    <lb/>
                  m.q.</var>
                quod quærebatur.</s>
              </p>
              <div xml:id="echoid-div46" type="float" level="4" n="1">
                <figure xlink:label="fig-0025-03" xlink:href="fig-0025-03a">
                  <image file="0025-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-03"/>
                </figure>
                <figure xlink:label="fig-0025-04" xlink:href="fig-0025-04a">
                  <image file="0025-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0025-04"/>
                </figure>
              </div>
              <figure position="here">
                <image file="0026-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0026-01"/>
              </figure>
              <figure position="here">
                <image file="0026-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0026-02"/>
              </figure>
            </div>
            <div xml:id="echoid-div48" type="math:theorem" level="3" n="21">
              <head xml:id="echoid-head37" xml:space="preserve">THEOREMA
                <num value="21">XXI</num>
              .</head>
              <p>
                <s xml:id="echoid-s215" xml:space="preserve">VT autem in uniuerſum ſciri poſſit totum
                  <reg norm="infinitum" type="context">infinitũ</reg>
                dignitatum, hoc eſt radicem
                  <lb/>
                producti duarum dignitatum ſimilium, productum eſſe duarum radicum ea-
                  <lb/>
                rundem dignitatum.</s>
              </p>
              <p>
                <s xml:id="echoid-s216" xml:space="preserve">Ponamus, exempli gratia, duas radices quadratas
                  <var>.q.p.</var>
                et
                  <var>.g.K.</var>
                incognitas, quas
                  <lb/>
                qui velit adinuicem multiplicare, cogatur earum quadrata cognita
                  <var>.n.</var>
                cum
                  <var>.i.</var>
                multi-
                  <lb/>
                plicare, quorum productum ſit quadratum
                  <var>.m.</var>
                radix cuius ſit
                  <var>.b.d.</var>
                quam dico æqualé </s>
              </p>
            </div>
          </div>
        </div>
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