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-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div737" type="section" level="3" n="42">
              <div xml:id="echoid-div737" type="letter" level="4" n="1">
                <p>
                  <pb o="390" rhead="IO. BAPT. BENED." n="402" file="0402" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0402"/>
                  <s xml:id="echoid-s4626" xml:space="preserve">Vbiautem ſcriptum eſt
                    <lb/>
                    <q open="" close="">ad vtrunque ſimul
                      <var>.b.d</var>
                    :
                      <var>d.a.</var>
                    cum dupla
                      <var>.b.c.</var>
                    </q>
                    <lb/>
                  dicendum eſt ita,
                    <lb/>
                  ad vtranque ſimul
                    <var>.b.d.b.a.</var>
                  cum dupla
                    <var>.b.c</var>
                  .</s>
                </p>
                <div xml:id="echoid-div746" type="float" level="5" n="10">
                  <figure xlink:label="fig-0401-01" xlink:href="fig-0401-01a">
                    <image file="0401-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0401-01"/>
                  </figure>
                </div>
                <p>
                  <s xml:id="echoid-s4627" xml:space="preserve">Inquit deinde Archi. quod ſicut ſe haber
                    <var>.e.a.</var>
                  ad
                    <var>.d.a.</var>
                  ita ſe habebit duplum
                    <var>.M.N.</var>
                    <lb/>
                  ad duplum
                    <var>.N</var>
                  . </s>
                  <s xml:id="echoid-s4628" xml:space="preserve">Quod quidem verum eſt ex .13. quinti, huiuſmodi verò antecedons
                    <lb/>
                  & conſequens, Archi. manifeſtat ex ſuis partibus, ſumendo duplum
                    <var>.e.b.</var>
                  cum duplo
                    <lb/>
                    <var>b.d.</var>
                  pro duplo
                    <var>.M.</var>
                  & duplum
                    <var>.b.d.</var>
                  cum duplo
                    <var>.a.b.</var>
                  cum quadruplo
                    <var>.b.c.</var>
                  pro duplo
                    <var>.N.</var>
                    <lb/>
                  quę ſimul iuncta æquantur duplo
                    <var>.e.b.</var>
                  cum duplo
                    <var>.a.b.</var>
                  cum quadruplo
                    <var>.b.d.</var>
                  cum qua-
                    <lb/>
                  druplo
                    <var>.b.c.</var>
                  ex quo æquabuntur
                    <var>.A.</var>
                  vocentur igitur hæc omnia
                    <var>.A.</var>
                  potius quàm du-
                    <lb/>
                  plum ipſius
                    <var>.M.N</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4629" xml:space="preserve">Verum etiam ſcribit, vbi dicit, quod proportio
                    <var>.e.a.</var>
                  ad tres quintas ipſius
                    <var>.a.d.</var>
                  erit
                    <lb/>
                  vt
                    <var>.A.</var>
                  ad tres quintas dupli
                    <var>.N.</var>
                  ex .22. quinti. </s>
                  <s xml:id="echoid-s4630" xml:space="preserve">Sed cum ex ſuppoſito ita ſe habeat
                    <var>.f.
                      <lb/>
                    g.</var>
                  ad tres quintas ipſius
                    <var>.a.d.</var>
                  quemadmodum
                    <var>.b.e.</var>
                  ad
                    <var>.e.a.</var>
                  erit ex .16. quinti verum
                    <reg norm="quod" type="simple">ꝙ</reg>
                    <lb/>
                  dicit Archimed. </s>
                  <s xml:id="echoid-s4631" xml:space="preserve">hoc eſt, ita ſe habere
                    <var>.b.e.</var>
                  ad
                    <var>.f.g.</var>
                  vt
                    <var>.e.a.</var>
                  ad tres quintas ipſius
                    <var>.a.d</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4632" xml:space="preserve">Et per .11. eiuſdem verum etiam erit quod ſicut ſe habet
                    <var>.e.b.</var>
                  ad
                    <var>.f.g.</var>
                  ita ſe habe-
                    <lb/>
                  bit
                    <var>.A.</var>
                  ad tres quintas dupli
                    <var>.N.</var>
                  quod quidem duplum
                    <var>.N.</var>
                  ſignificetur per
                    <var>.Q</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4633" xml:space="preserve">Sed ſuperius iam demonſtratum fuit (vbi
                    <var>.X.</var>
                  ) quod
                    <var>.o.b.</var>
                  ad
                    <var>.b.e.</var>
                  ita ſe habebat vt
                    <lb/>
                    <var>H.A.</var>
                  ad
                    <var>.A.</var>
                  &
                    <reg norm="nunc" type="context">nũc</reg>
                  demum probatum fuit ita eſſe
                    <var>.A.</var>
                  ad tres quintas ipſius
                    <var>.Q.</var>
                  vt
                    <var>.e.b.</var>
                    <lb/>
                  ad
                    <var>.f.g</var>
                  . </s>
                  <s xml:id="echoid-s4634" xml:space="preserve">Quare ex .22. quinti ita erit
                    <var>.H.A.</var>
                  ad tres quintas ipſius
                    <var>.Q.</var>
                  vt
                    <var>.o.b.</var>
                  ad
                    <var>.f.g.</var>
                  vt
                    <lb/>
                    <anchor type="note" xlink:label="note-0402-01a" xlink:href="note-0402-01"/>
                  idem inquit.</s>
                </p>
                <div xml:id="echoid-div747" type="float" level="5" n="11">
                  <note xlink:label="note-0402-01" xlink:href="note-0402-01a" position="left" xml:space="preserve">Y</note>
                </div>
                <p>
                  <s xml:id="echoid-s4635" xml:space="preserve">Sed
                    <var>.H.A.</var>
                  ad
                    <var>.Q.</var>
                  (vt ex ſuis partibus videre eſt) ita ſe habet vt tres ad duo ex .13.
                    <lb/>
                  quinti, vt inquit Archimedes.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4636" xml:space="preserve">Ipſe etiam dicit proportionem
                    <var>.H.A.</var>
                  ad tres quintas ipſius
                    <var>.Q.</var>
                  eſſe vt quinque
                    <lb/>
                  ad duo. </s>
                  <s xml:id="echoid-s4637" xml:space="preserve">Pro cuius rei euidentia imaginemur tam
                    <var>.H.A.</var>
                  quam
                    <var>.Q.</var>
                  diuiſa per
                    <reg norm="quinque" type="simple">quinq;</reg>
                    <lb/>
                  partes æquales, vnde ex .16. quinti habebimus quamlibet quintam
                    <reg norm="partem" type="context">partẽ</reg>
                  ipſius
                    <var>.Q.</var>
                    <lb/>
                    <reg norm="æqualem" type="context">æqualẽ</reg>
                  eſſe duabus tertijs vniuſcuiuſque quintæ partis
                    <var>.H.A.</var>
                  vnde tres quintæ ipſius
                    <lb/>
                  Q. erunt, ex communi conceptu, ſex tertiæ vnius quintæ ipſius
                    <var>.H.A.</var>
                  hoc eſt duæ
                    <lb/>
                  quintæ. ipſius
                    <var>.H.A</var>
                  . </s>
                  <s xml:id="echoid-s4638" xml:space="preserve">Quare
                    <var>.o.b.</var>
                  ita ſe habebit ad
                    <var>.f.g.</var>
                  vt quinque ad duo ex commu
                    <lb/>
                  ni
                    <reg norm="conceptu" type="context">cõceptu</reg>
                  , cum
                    <var>.o.b.</var>
                  ad
                    <var>.f.g.</var>
                  probatum fuerit ſe habere vt
                    <var>.H.A.</var>
                  ad tres quintas ipſius
                    <lb/>
                  Q. (vbi
                    <var>.Y.</var>
                  ) ſed iam probatum fuit (vbi. ω) quod
                    <var>.o.a.</var>
                  ad
                    <var>.h.g.</var>
                  erat etiam vt
                    <lb/>
                  quinque ad duo, hoc eſt quod
                    <var>.f.h.</var>
                  erit duæ quintę ipſius
                    <var>.a.b</var>
                  . </s>
                  <s xml:id="echoid-s4639" xml:space="preserve">Quod eſt propoſitum.</s>
                </p>
              </div>
            </div>
          </div>
        </div>
      </text>
    </echo>