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-div285" type="math:theorem" level="3" n="149">
              <p>
                <pb o="100" rhead="IO. BAPT. BENED." n="112" file="0112" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0112"/>
                <s xml:id="echoid-s1302" xml:space="preserve">a.b:
                  <var>c.d</var>
                :
                  <var>e.f.</var>
                et
                  <var>.g.h.</var>
                quorum
                  <var>.a.b.</var>
                et
                  <var>.g.h.</var>
                nobis tantummodo cogniti ſint,
                  <reg norm="ſitque" type="simple">ſitq́</reg>
                imagina
                  <lb/>
                tione deſcriptus cubus
                  <var>.a.q.</var>
                primi termini,
                  <reg norm="cubusque" type="simple">cubusq́</reg>
                  <var>.d.k.</var>
                ſecundi rermini, conſidere-
                  <lb/>
                mus etiam baſim
                  <var>.a.i.</var>
                quadratam ipſius cubi
                  <var>.a.q.</var>
                hoc eſt præcedentem dignitatem ip
                  <lb/>
                ſius cubi eiuſdem radicis, quæ quidem baſis
                  <var>.a.i.</var>
                multiplicetur per quartum
                  <reg norm="terminum" type="context">terminũ</reg>
                  <lb/>
                  <var>g.h.</var>
                productum autem ſit
                  <var>.g.a.</var>
                vnde eadem proportio erit
                  <var>.a.q.</var>
                ad
                  <var>.a.g.</var>
                quæ
                  <var>.b.q.</var>
                ad
                  <var>.b.
                    <lb/>
                  g.</var>
                per .25. vndecimi, ſed per primam ſexti, vel .18. aut .19. ſeptimi ita eſt
                  <var>.q.i.</var>
                ad
                  <var>.i.g.</var>
                  <lb/>
                vt
                  <var>.b.q.</var>
                ad
                  <var>.b.g.</var>
                </s>
                <s xml:id="echoid-s1303" xml:space="preserve">quare per .11. quinti
                  <lb/>
                ita erit
                  <var>.a.q.</var>
                ad
                  <var>.a.g.</var>
                vt
                  <var>.q.i.</var>
                ad
                  <var>.i.g.</var>
                ideſt
                  <lb/>
                  <figure xlink:label="fig-0112-01" xlink:href="fig-0112-01a" number="155">
                    <image file="0112-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0112-01"/>
                  </figure>
                vt
                  <var>.a.b.</var>
                ad
                  <var>.g.h.</var>
                ſed vt eſt
                  <var>.a.b.</var>
                ad
                  <var>.g.h.</var>
                  <lb/>
                ſic eſt
                  <var>.a.q.</var>
                ad
                  <var>.k.d.</var>
                per .36. vndecimi,
                  <lb/>
                ſeu per .11. octaui, vnde per .11. quin
                  <lb/>
                ti ſic erit
                  <var>.a.q.</var>
                ad
                  <var>.a.g.</var>
                vt ad
                  <var>.k.d</var>
                . </s>
                <s xml:id="echoid-s1304" xml:space="preserve">Qua-
                  <lb/>
                re per .9. eiuſdem
                  <var>.a.g.</var>
                ęqualis erit
                  <var>.k.
                    <lb/>
                  d</var>
                . </s>
                <s xml:id="echoid-s1305" xml:space="preserve">Vnde rectè erit accipere radicem
                  <lb/>
                cubam
                  <var>.a.g.</var>
                pro
                  <reg norm="ſecundo" type="context">ſecũdo</reg>
                termino
                  <var>.c.d.</var>
                  <lb/>
                id, quod nobis inſeruit ad inueniendam tertiam partem vnius propoſitæ propor-
                  <lb/>
                tionis.</s>
              </p>
            </div>
            <div xml:id="echoid-div287" type="math:theorem" level="3" n="150">
              <head xml:id="echoid-head169" xml:space="preserve">THEOREMA
                <num value="150">CL</num>
              .</head>
              <p>
                <s xml:id="echoid-s1306" xml:space="preserve">
                  <emph style="sc">Sed</emph>
                vt ſpeculatio iſta ita vniuerſalis fiat vt ad
                  <reg norm="oens" type="context">oẽs</reg>
                dignitates applicari poſſit;
                  <lb/>
                </s>
                <s xml:id="echoid-s1307" xml:space="preserve">Supponamus
                  <var>.a.q.</var>
                et
                  <var>.k.d.</var>
                eſſe duas dignitates quas volueris vnius, ſed eiuſdem
                  <lb/>
                ſpeciei, et
                  <var>.a.i.</var>
                dignitas præcedens dignitatem
                  <var>.a.q.a.</var>
                cuius multiplicatione in
                  <var>.a.b.</var>
                  <lb/>
                eius radix producitur dignitas
                  <var>.a.q.</var>
                & ab ipſius
                  <var>.a.i.</var>
                multiplicatione in
                  <var>.g.h.</var>
                reſultet
                  <var>.a.
                    <lb/>
                  g.</var>
                vnde ex .18. vel .19. ſeptimi eadem proportio erit
                  <var>.a.q.</var>
                ad
                  <var>.a.g.</var>
                quæ
                  <var>.a.b.</var>
                ad
                  <var>.g.h.</var>
                ſed
                  <lb/>
                eadem etiam eſt
                  <var>.a.q.</var>
                ad
                  <var>.k.d.</var>
                ex ijs, quæ in .17. theoremare dixi, vnde ex .11. quinti,
                  <lb/>
                ita erit
                  <var>.a.q.</var>
                ad
                  <var>.a.g.</var>
                vt ad
                  <var>.k.d</var>
                . </s>
                <s xml:id="echoid-s1308" xml:space="preserve">Quapropter
                  <var>.a.g.</var>
                æqualis erit
                  <var>.k.d.</var>
                & ideo cum inuenta
                  <lb/>
                fuerit radix huiuſmodi dignitatis ex quantitate
                  <var>.a.g.</var>
                habebimus
                  <var>.c.d.</var>
                ſecundum ter-
                  <lb/>
                minum quæſitum.</s>
              </p>
            </div>
            <div xml:id="echoid-div288" type="math:theorem" level="3" n="151">
              <head xml:id="echoid-head170" xml:space="preserve">THEOREMA
                <num value="151">CLI</num>
              .</head>
              <p>
                <s xml:id="echoid-s1309" xml:space="preserve">
                  <emph style="sc">Vnde</emph>
                verò fiat, quòd cum quis voluerit dimidium alicuius datæ proportio-
                  <lb/>
                nis inuenire, rectè faciat, ſi accipiat radices quadratas illorum datorum rer-
                  <lb/>
                minorum, etſi voluerit tertiam partem, accipiat radices cubas: </s>
                <s xml:id="echoid-s1310" xml:space="preserve">ſi autem quartam,
                  <lb/>
                accipereradices cenſicas cenſicas ipſorum, & ſic de ſingulis in .17. </s>
                <s xml:id="echoid-s1311" xml:space="preserve">Theoremate om-
                  <lb/>
                nia patent.</s>
              </p>
            </div>
            <div xml:id="echoid-div289" type="math:theorem" level="3" n="152">
              <head xml:id="echoid-head171" xml:space="preserve">THEOREMA
                <num value="152">CLII</num>
              .</head>
              <p>
                <s xml:id="echoid-s1312" xml:space="preserve">
                  <emph style="sc">Vnde</emph>
                autem fiat, vt cum quis voluerit multiplicare aliquam proportionem
                  <lb/>
                per fractos, rectè faciat prius multiplicando eam per numeratorem, dein-
                  <lb/>
                de productum diuiſerit per denominationem ipſorum fractorum.</s>
              </p>
              <p>
                <s xml:id="echoid-s1313" xml:space="preserve">Vt exempli gratia, cum aliquis voluerit multiplicare proportionem ſeſquiquar-
                  <lb/>
                tam per duo tertia, multiplicabit prius ipſam proportionem per numeratorem .2.
                  <lb/>
                & productum, erit proportio .25. ad .16. qua poſtea diuiſa per .3. denominatorem,
                  <lb/>
                prouentus erit proportio radicis cubæ .25. ad radicem cubam .16. vel vt proportio.</s>
              </p>
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