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]

Table of Notes

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              <p>
                <pb o="102" rhead="IO. BAPT. BENED." n="114" file="0114" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0114"/>
                <s xml:id="echoid-s1325" xml:space="preserve">Quapropter non tacebo quod mihi in mentem venit circa hoc problema.</s>
              </p>
              <p>
                <s xml:id="echoid-s1326" xml:space="preserve">Sit ergo linea
                  <var>.a.b.</var>
                diuiſibilis in puncto
                  <var>.c.</var>
                ita vt cubum totius dictæ
                  <var>.a.b.</var>
                lineæ ad
                  <lb/>
                ſummam cuborum
                  <reg norm="ſuarum" type="context">ſuarũ</reg>
                partium
                  <var>.a.c.</var>
                et
                  <var>.c.b.</var>
                oporteat eam proportionem
                  <reg norm="habere" type="conjecture">habére</reg>
                ,
                  <lb/>
                exempli gratia, vt .125. ad .65. vt vitemus fracta pro nunc, notantes talem propor-
                  <lb/>
                tionem quadrupla nunquam maiorem eſſe poſſe, vt quilibet ex ſe contemplari po-
                  <lb/>
                teſt, conſtituendo punctum
                  <var>.c.</var>
                in medio loco inter
                  <var>.a.</var>
                et
                  <var>.b.</var>
                vnde proportio totalis
                  <lb/>
                cubi ad ſummam partialium eſſet omnium maxima quæ poſſint eſſe, collocando
                  <var>.c.</var>
                  <lb/>
                vbi volueris in dicta linea
                  <var>.a.b.</var>
                & hæc eſſet quadrupla.</s>
              </p>
              <p>
                <s xml:id="echoid-s1327" xml:space="preserve">Sed vt ad propoſitum reuertamur, conſiderabimus cubum totalem ipſius
                  <var>.a.b.</var>
                  <lb/>
                eſſe vt .125. & ſummam partialium vt .65. quam detrahemus ex cubo totali & nobis
                  <lb/>
                remanebit .60. pro ſumma trium ſolidorum inuicem æqualium, quorum longitu-
                  <lb/>
                do vniuſcuiuſque erit tota linea
                  <var>.a.b.</var>
                nobis cognita vt radix dati cubi totalis, quæ erit
                  <lb/>
                in hoc exemplo quinque partium, latitudo verò vniuſcuiuſque dictorum
                  <reg norm="ſolidorum" type="context">ſolidorũ</reg>
                  <lb/>
                erit
                  <var>.a.c.</var>
                pars maior ipſius
                  <var>.a.b.</var>
                quæ quidem
                  <var>.a.c.</var>
                adhuc nobis ignota eſt, profunditas
                  <lb/>
                ſeu altitudo vniuſcuiuſque illorum ſolidorum, erit
                  <var>.c.b.</var>
                pars reliqua ipſius
                  <var>.a.b.</var>
                &
                  <reg norm="etiam" type="context">etiã</reg>
                  <lb/>
                nobis incognita, ſed quia ſumma horum trium ſolidorum nobis manifeſta ſuperius
                  <lb/>
                fuit, quæ erat .60. propterà nobis cognita erit quantitas vniuſcuiuſque illorum ſoli-
                  <lb/>
                dorum, vt tertia pars totius ſummæ ipſorum quæ erit .20. in propoſito
                  <reg norm="exemplo" type="context">exẽplo</reg>
                , dein
                  <lb/>
                de cum vnumquodque illorum ſolidorum producatur à ſuperficie contenta ſeu pro
                  <lb/>
                ducta ab
                  <var>.c.a.</var>
                in
                  <var>.c.b.</var>
                in tota linea
                  <var>.a.b.</var>
                ſequitur quòd ſi diuiſerimus hoc ſolidum .20.
                  <lb/>
                per lineam
                  <var>.a.b.</var>
                quinque partium proueniet nobis cognita ſuperficies producta ab
                  <var>.
                    <lb/>
                  a.c.</var>
                in
                  <var>.c.b.</var>
                quatuor partium, ſed cum quadratum totius
                  <var>.a.b.</var>
                nobis cognitum ſit, eo
                  <lb/>
                quod
                  <var>.a.b.</var>
                vt eius latus etiam cognitum eſt. </s>
                <s xml:id="echoid-s1328" xml:space="preserve">Tunc dictum quadratum erit .25. quod
                  <lb/>
                quidem æquale eſt quadruplo illius quod fit ex
                  <var>.a.c.</var>
                in
                  <var>.c.b.</var>
                ſimul cum quadrato diffe
                  <lb/>
                rentiæ inter
                  <var>.a.c.</var>
                et
                  <var>.c.b.</var>
                per .8. ſecundi Eucli. </s>
                <s xml:id="echoid-s1329" xml:space="preserve">Vnde quia quadruplum illius quod fit
                  <lb/>
                ex
                  <var>.a.c.</var>
                in
                  <var>.c.b.</var>
                nobis cognitum eſt, vt
                  <lb/>
                16. eo quod ſimplum quod eſt .4.
                  <reg norm="iam" type="context">iã</reg>
                  <lb/>
                  <figure xlink:label="fig-0114-01" xlink:href="fig-0114-01a" number="157">
                    <image file="0114-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0114-01"/>
                  </figure>
                inuentum fuit, ideo ſi hoc quadru-
                  <lb/>
                plum .16. demptum fuerit ex totali
                  <lb/>
                quadrato .25. reliquum erit .9. qua
                  <lb/>
                  <reg norm="dratum" type="context">dratũ</reg>
                ſcilicet vnius partis
                  <var>.a.c.</var>
                ipſius
                  <lb/>
                hoc eſt illius partis, quæ differentia
                  <lb/>
                eſt inter
                  <var>a.c.</var>
                et
                  <var>.c.b.</var>
                quæ quidem erit
                  <num value="3">.
                    <lb/>
                  3.</num>
                partium quæ differentia cum ſub-
                  <lb/>
                tracta fuerit ex
                  <var>.a.b.</var>
                reliquum erit du
                  <lb/>
                plum ipſius
                  <var>.c.b.</var>
                duo ſcilicet. </s>
                <s xml:id="echoid-s1330" xml:space="preserve">Quare
                  <var>.
                    <lb/>
                  c.b.</var>
                erit vt
                  <var>.I.</var>
                et
                  <var>.a.c.</var>
                vt .4. & productum
                  <var>.a.c.</var>
                in
                  <var>.c.b.</var>
                erit .4. vnitatum ſuperficialium.
                  <lb/>
                </s>
                <s xml:id="echoid-s1331" xml:space="preserve">& c.</s>
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