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-div24" type="math:theorem" level="3" n="10">
              <p>
                <s xml:id="echoid-s107" xml:space="preserve">
                  <var>
                    <pb o="8" rhead="IO. BAPT. BENED." n="20" file="0020" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0020"/>
                  n.c.</var>
                ad
                  <var>.a.e.</var>
                ſint æquales inuicem quandoqui-
                  <lb/>
                  <figure xlink:label="fig-0020-01" xlink:href="fig-0020-01a" number="14">
                    <image file="0020-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0020-01"/>
                  </figure>
                dem vnaquæque earum ex triangulorum ſimi
                  <lb/>
                litudine æqualis eſt proportioni
                  <var>.o.n.</var>
                ad
                  <var>.o.a</var>
                .
                  <lb/>
                </s>
                <s xml:id="echoid-s108" xml:space="preserve">itaque
                  <var>.n.t.</var>
                hoc eſt
                  <var>.a.i.</var>
                tanto maior erit
                  <var>.a.x.</var>
                  <lb/>
                quanto
                  <var>.n.c.</var>
                maior eſt
                  <var>.a.e.</var>
                vnde ficut
                  <var>.a.e.</var>
                con-
                  <lb/>
                ſtat octo nonis ipſius
                  <var>.n.c.</var>
                ita pars
                  <var>.a.x.</var>
                ipſius
                  <var>.
                    <lb/>
                  a.e.</var>
                octo nonis conſtabit ipſius
                  <var>.a.i</var>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s109" xml:space="preserve">Hinc patet ratio cur partituri numerum mino
                  <lb/>
                rem per maiorem collocent minorem fupra
                  <lb/>
                virgulam & maiorem infra & zerum ad
                  <reg norm="læuam" type="context">læuã</reg>
                .</s>
              </p>
              <p>
                <s xml:id="echoid-s110" xml:space="preserve">Sciendum eſt præterea diuidere numerum
                  <lb/>
                per numerum: </s>
                <s xml:id="echoid-s111" xml:space="preserve">eſſe inuenire
                  <reg norm="alterum" type="context">alterũ</reg>
                latus à quo
                  <lb/>
                producitur, ſuppoſito ſemper quòd numerus
                  <lb/>
                diuifibilis ſuperſicialis ſit, & rectangulus.</s>
              </p>
              <p>
                <s xml:id="echoid-s112" xml:space="preserve">Exempli gratia, ſi proponantur triginta diuidenda per quinarium, nihil aliud erit
                  <lb/>
                hæc diuiſio, quam inuentio alterius numeri, qui multiplicatus per quinarium produ-
                  <lb/>
                cat triginta ſuperficies rectangulas, huiuſmodi verò eſt ſenarius, cuius ſingulæ vnita-
                  <lb/>
                tes ſuperficiales erunt.</s>
              </p>
              <p>
                <s xml:id="echoid-s113" xml:space="preserve">Cuius rei gratia ſit ſubſcriptum rectangulum
                  <var>.a.e.</var>
                triginta vnitatum
                  <reg norm="ſuperſicialium" type="context">ſuperſicialiũ</reg>
                ,
                  <lb/>
                cuius latus
                  <var>.e.n.</var>
                ſit quinque vnitatum. </s>
                <s xml:id="echoid-s114" xml:space="preserve">hinc latus
                  <var>.a.n.</var>
                erit ſex vnitatum; </s>
                <s xml:id="echoid-s115" xml:space="preserve">ita diuiden-
                  <lb/>
                tes rectangulum
                  <var>.e.a.</var>
                nihil a iud faciemus, quam vt inue-
                  <lb/>
                  <figure xlink:label="fig-0020-02" xlink:href="fig-0020-02a" number="15">
                    <image file="0020-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0020-02"/>
                  </figure>
                nia mus quantum valeat latus
                  <var>.a.n.</var>
                quod erit ſex vnitatum.
                  <lb/>
                </s>
                <s xml:id="echoid-s116" xml:space="preserve">Sin verò diuiſerimus per latus
                  <var>.a.n.</var>
                quæremus latus
                  <var>.e.n.</var>
                  <lb/>
                quinque vnitatum. </s>
                <s xml:id="echoid-s117" xml:space="preserve">ex quo, proportio totius numeri diuifi-
                  <lb/>
                bilis ad numerum qui oritur, erit ſicut diuidentis ad vnita-
                  <lb/>
                tem, ex prima ſexti, aut .18. vel .19. ſeptimi, & permutatim
                  <lb/>
                ita ſe habebit diuiſibile ad diuidentem, ſicut numerus qui
                  <lb/>
                oritur ad vnitatem.</s>
              </p>
              <p>
                <s xml:id="echoid-s118" xml:space="preserve">Partiri igitur nihil aliud eſt, quam inuenire latus rectanguli, quod productum in
                  <lb/>
                diuidente, numerum diuiſibilem compl at, ex quo numerus diuiſibilis ſuperficialis
                  <lb/>
                eſt, diuidens autem, & qui oritur, numeri lineares & latera producentia huiuſcemodi
                  <lb/>
                numerum diuiſibilem. </s>
                <s xml:id="echoid-s119" xml:space="preserve">nam multiplicare & diuidere opponuntur inuicem, cum au-
                  <lb/>
                tem ex multiplicatione laterum ſiue linearum generatur ſuperficies, ex diuiſione po-
                  <lb/>
                ſtea ipſius ſuperficiei inuenitur alterum latus. </s>
                <s xml:id="echoid-s120" xml:space="preserve">quare mirum non eſt, ſi proueniens ex
                  <lb/>
                vna diuiſione (via fractorum) ſit ſemper maius numero diuiſibili.</s>
              </p>
              <p>
                <s xml:id="echoid-s121" xml:space="preserve">Exempli gratia, diuidendo dimidium per tertiam partem, reſultat vnus integer nu
                  <lb/>
                merus cum dimidio pro numero qui oritur. </s>
                <s xml:id="echoid-s122" xml:space="preserve">Sit itaque dimidium ſuperſiciale diuiſi-
                  <lb/>
                bile
                  <var>.b.c.</var>
                cuius totum ſit
                  <var>.b.p.</var>
                quadratum. </s>
                <s xml:id="echoid-s123" xml:space="preserve">tertium verò lineare diuidens,
                  <var>b.n.</var>
                cuius to-
                  <lb/>
                tum lineare ſit
                  <var>.b.d.</var>
                quærendum nobis eſt latus
                  <var>.b.s.</var>
                quod cum latere
                  <var>.b.n.</var>
                producat re
                  <lb/>
                ctangulum
                  <var>.n.s.</var>
                æquale dimidio ſuperſiciali propoſito
                  <var>.b.c.</var>
                quod ſi ſiat, ex .15. ſexti,
                  <lb/>
                aut .20. ſeptimi. erit eadem proportio
                  <var>.b.n.</var>
                ad
                  <var>.b.q.</var>
                quæ eſt
                  <var>.q.c.</var>
                ad
                  <var>.b.s.</var>
                dicemus itaque
                  <lb/>
                ſi
                  <var>.n.b.</var>
                dat
                  <var>.b.q.</var>
                quid dabit
                  <var>.q.c</var>
                ? </s>
                <s xml:id="echoid-s124" xml:space="preserve">certè
                  <var>.b.s.</var>
                ſed
                  <var>.n.b.</var>
                eſt tertium lineare et
                  <var>.b.q.</var>
                lineare
                  <reg norm="in- tegrum" type="context">in-
                    <lb/>
                  tegrũ</reg>
                , &
                  <var>b.s.</var>
                proueniens lineare. </s>
                <s xml:id="echoid-s125" xml:space="preserve">& quia
                  <var>.b.c.</var>
                dimidium ſuperficiale, producitur à
                  <var>.q.c.</var>
                  <lb/>
                dimidio lineari in
                  <var>.q.b.</var>
                integro lineari. </s>
                <s xml:id="echoid-s126" xml:space="preserve">quare cum
                  <var>.n.s.</var>
                ſit ęqualis
                  <var>.b.c.</var>
                & productum ex
                  <var>.
                    <lb/>
                  b.n.</var>
                minori
                  <var>.q.c.</var>
                neceſſe eſt, vt producatur in
                  <var>.b.s.</var>
                maiore
                  <var>.q.b.</var>
                quod
                  <var>.q.b.</var>
                maius eſt
                  <var>.q.c.</var>
                  <lb/>
                quod quidem
                  <var>.q.c.</var>
                ita appellatur ſicut
                  <var>.b.c</var>
                . </s>
                <s xml:id="echoid-s127" xml:space="preserve">quare mirum non eſt ſi proueniens per fra-
                  <lb/>
                ctos numeros ex diuiſione, maior ſit numero diuiſibili.</s>
              </p>
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