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.
    <echo version="1.0">
      <text type="book" xml:lang="la">
        <div xml:id="echoid-div7" type="body" level="1" n="1">
          <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>
                  <s xml:id="echoid-s4665" xml:space="preserve">
                    <pb o="394" rhead="IO. BAPT. BENED." n="406" file="0406" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0406"/>
                  iam dicti ad cubum inſius
                    <var>.a.f.</var>
                  ex .11. quinti erit vt dupli
                    <var>.x.n.</var>
                    <reg norm="cum" type="context">cũ</reg>
                  ſimplo
                    <var>.m.n.</var>
                  ad
                    <var>.m.n</var>
                  .</s>
                </p>
                <div xml:id="echoid-div752" type="float" level="5" n="16">
                  <figure xlink:label="fig-0405-01" xlink:href="fig-0405-01a">
                    <image file="0405-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0405-01"/>
                  </figure>
                </div>
                <note position="left" xml:space="preserve">δ</note>
                <p>
                  <s xml:id="echoid-s4666" xml:space="preserve">Superius autem vbi. β. demonſtratum fuit ita eſſe ipſius
                    <var>.m.n.</var>
                  ad
                    <var>.n.t.</var>
                  vt cubi
                    <var>.m.n.</var>
                    <lb/>
                  ad cubum
                    <var>.x.n.</var>
                  & inter. α et. β probatum fuit ita eſſe cubi
                    <var>.a.f.</var>
                  ad cubum
                    <var>.d.g.</var>
                  vt
                    <lb/>
                  cubi
                    <var>.m.n.</var>
                  ad cubum
                    <var>.x.n</var>
                  . </s>
                  <s xml:id="echoid-s4667" xml:space="preserve">Vnde ex .11. quinti
                    <var>.m.n.</var>
                  ad
                    <var>.n.t.</var>
                  erit vt cubi
                    <var>.a.f.</var>
                  ad cubum
                    <lb/>
                    <var>d.g</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4668" xml:space="preserve">Dicit poſtea quod eadem proportio erit inter cubum
                    <var>.d.g.</var>
                  & corpus illud quod
                    <lb/>
                  pro baſi habeat quadratum inſius
                    <var>.d.g.</var>
                  altitudinem verò vt dictum eſt, quæ eſt inter
                    <lb/>
                    <var>d.g.</var>
                  & compoſitum ex duplo
                    <var>.a.f.</var>
                  cum ſimplo
                    <var>.d.g.</var>
                  quod compoſitum eſt altitudo di
                    <lb/>
                  cta, &
                    <reg norm="verum" type="context">verũ</reg>
                  dicit ex ratione ſuperius allegata pro reliquo corpore & cubo ipſius
                    <var>.a.f</var>
                  .
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4669" xml:space="preserve">Quare etiam quemadmodum
                    <var>.t.n.</var>
                  ſe habet ad duplum ipſius
                    <var>.o.n.</var>
                  cum ſimplo
                    <var>.t.n.</var>
                    <lb/>
                  ex ijſdem rationibus ſupradictis, vbiloquuti ſumus de
                    <var>.x.n.</var>
                  cum
                    <var>.m.n</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4670" xml:space="preserve">Diſponantur
                    <reg norm="nunc" type="context">nũc</reg>
                  omnia tali ordine, ita vt
                    <var>.u.</var>
                  primum ſit corpus quod pro ſua ba
                    <lb/>
                  ſi habeat quadratum ipſius
                    <var>.a.f.</var>
                  & c.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4671" xml:space="preserve">Et
                    <var>.y.</var>
                  ſit cubus ipſius
                    <var>.a.f.</var>
                  et
                    <var>.s.</var>
                  ſit cubus ipſius
                    <var>.d.g.</var>
                  et
                    <var>.z.</var>
                  ſit corpus quod baſim ha-
                    <lb/>
                  bet quadratum ipſius
                    <var>.d.g.</var>
                  altitudinem verò vt ſupradictum eſt, et
                    <var>.p.</var>
                  ſit compoſitum
                    <lb/>
                  dupli
                    <var>.n.x.</var>
                  cum ſimplo
                    <var>.m.n.</var>
                  et
                    <var>.l.</var>
                  ſit compoſitum dupli ipſius
                    <var>.n.o.</var>
                  cum ſimplo
                    <var>.t.n.</var>
                    <lb/>
                  Sed
                    <var>.u.</var>
                  locata ſit è regione
                    <var>.p.</var>
                  et
                    <var>.y.</var>
                  è regione
                    <var>.m.n.</var>
                  et
                    <var>.s.</var>
                  è regione
                    <var>.n.t.</var>
                  et
                    <var>.z.</var>
                  è regione
                    <var>.l.</var>
                    <lb/>
                  & habebimus proportionem ipſius
                    <var>.u.</var>
                  ad
                    <var>.y.</var>
                  vt
                    <var>.y.</var>
                  ad
                    <var>.m.n.</var>
                  & ipſius
                    <var>.y.</var>
                  ad
                    <var>.s.</var>
                  vt
                    <var>.m.n.</var>
                  ad
                    <var>.
                      <lb/>
                    n.t.</var>
                  quod ſuperius iam demonſtratum fuit, vbi, δ. et
                    <var>.s.</var>
                  ad
                    <var>.z.</var>
                  ita ſe habebit vt
                    <var>.n.t.</var>
                  ad
                    <var>.
                      <lb/>
                    l.</var>
                  vt vltimò probatum fuit. </s>
                  <s xml:id="echoid-s4672" xml:space="preserve">Quare ex .22. quinti ita ſe habebit
                    <var>.u.</var>
                  ad
                    <var>.z.</var>
                  vt
                    <var>.p.</var>
                  ad
                    <var>.l.</var>
                    <lb/>
                  quemadmodum dicit Archi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4673" xml:space="preserve">Et quia vt ſe habet
                    <var>.u.</var>
                  ad
                    <var>.z.</var>
                  ita facta fuit
                    <var>.h.i.</var>
                  ad
                    <var>.i.K.</var>
                  vbi
                    <var>.R.</var>
                  ideo ex .11. quinti vt ſe
                    <lb/>
                  habet
                    <var>.h.i.</var>
                  ad
                    <var>.i.K.</var>
                  ita ſe habebit
                    <var>.p.</var>
                  ad
                    <var>.l.</var>
                  vt ipſe dicit: </s>
                  <s xml:id="echoid-s4674" xml:space="preserve">Et ex .18. quinti ita erit
                    <var>.h.K.</var>
                    <lb/>
                  ad
                    <var>.K.i.</var>
                  vt
                    <var>.p.l.</var>
                  ad
                    <var>.l.</var>
                  & ex communi conceptu
                    <var>.g.f.</var>
                  ſe habebit ad
                    <var>.h.K.</var>
                  vt quintuplum
                    <lb/>
                  ipſius
                    <var>.p.l.</var>
                  ad
                    <var>.p.l.</var>
                  & ex .22. eiuſdem ita ſe habebit
                    <var>.f.g.</var>
                  ad
                    <var>.i.k.</var>
                  vt quintuplum ipſius
                    <var>.p.
                      <lb/>
                    l.</var>
                  ad
                    <var>.l.</var>
                  quintuplum autem ipſius
                    <var>.p.l.</var>
                  compoſitum eſt ex quintuplo ipſius
                    <var>.n.m.</var>
                  cum
                    <lb/>
                  decuplo ipſius
                    <var>.n.x.</var>
                  cum quintuplo ipſius
                    <var>.n.t.</var>
                  cum decuplo ipſius
                    <var>.n.o.</var>
                  vt à te facilè
                    <lb/>
                  computare potes.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4675" xml:space="preserve">Verum etiam erit ex communi ſcientia quod
                    <var>.g.f.</var>
                  ad
                    <var>.f.k.</var>
                  eſt ut quintuplum ipſius
                    <lb/>
                    <var>p.l.</var>
                  ad duplum ipſius
                    <var>.p.l.</var>
                  eo quod ſuperius ſuppoſitum fuit
                    <var>.h.K.</var>
                  eſſe
                    <reg norm="quintam" type="context">quintã</reg>
                  mediam,
                    <lb/>
                  vnde
                    <var>.k.f.</var>
                  relinquebatur pro duabus quintis inferioribus, duplum autem
                    <var>.p.l.</var>
                  com-
                    <lb/>
                  poſitum eſt ex duplo ipſius
                    <var>.m.n.</var>
                  cum duplo ipſius
                    <var>.n.t.</var>
                  cum quadruplo ipſius
                    <var>.n.x.</var>
                  &
                    <lb/>
                  cum quadruplo ipſius
                    <var>.x.o</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4676" xml:space="preserve">Ex conuerſa proportionalitate deinde ita ſe habet,
                    <var>i.K.</var>
                  ad
                    <var>.i.k.</var>
                  ad
                    <var>.f.g.</var>
                  vt
                    <var>.l.</var>
                  ad quin-
                    <lb/>
                  tuplum ipſius
                    <var>.p.l.</var>
                  et
                    <var>.k.f.</var>
                  ad
                    <var>.f.g.</var>
                  vt duplum ipſius
                    <var>.p.l.</var>
                  ad quintuplum ipſius
                    <var>.p.l</var>
                  . </s>
                  <s xml:id="echoid-s4677" xml:space="preserve">Vnde
                    <lb/>
                  ex .24. quinti
                    <var>.i.f.</var>
                  ſe habebit ad
                    <var>.f.g.</var>
                  vt
                    <reg norm="duplum" type="context">duplũ</reg>
                  ipſius
                    <var>.p.l.</var>
                  cum ſimplo
                    <var>.l.</var>
                  ad quintuplum
                    <lb/>
                  ipſius
                    <var>.p.l</var>
                  . </s>
                  <s xml:id="echoid-s4678" xml:space="preserve">Deinde ex conuerſa proportionalitate quintuplum ipſius
                    <var>.p.l.</var>
                  ſe habebit
                    <lb/>
                    <anchor type="note" xlink:label="note-0406-02a" xlink:href="note-0406-02"/>
                  ad duplum ipſius
                    <var>.p.l.</var>
                  cum ſimplo
                    <var>.l.</var>
                  vt
                    <var>.f.g.</var>
                  ad
                    <var>.f.i</var>
                  . </s>
                  <s xml:id="echoid-s4679" xml:space="preserve">Sed compoſitum dupli ipſius
                    <var>.p.l.</var>
                    <lb/>
                  cum ſimplo
                    <var>.l.</var>
                  æquale eſt duplo ipſius
                    <var>.m.n.</var>
                  cum quadruplo ipſius
                    <var>.x.n.</var>
                  cum ſexcuplo
                    <lb/>
                  ipſius
                    <var>.o.n.</var>
                  cum triplo ipſius
                    <var>.n.t.</var>
                  vt per te computare potes.</s>
                </p>
                <div xml:id="echoid-div753" type="float" level="5" n="17">
                  <note xlink:label="note-0406-02" xlink:href="note-0406-02a" position="left" xml:space="preserve">θ</note>
                </div>
                <p>
                  <s xml:id="echoid-s4680" xml:space="preserve">Superius enim ſumpta fuit
                    <var>.i.r.</var>
                  ad quam ita ſe haberet
                    <var>.f.h.</var>
                  hoc eſt tres quintæ ip-
                    <lb/>
                  ſius
                    <var>.f.g.</var>
                  vt
                    <var>.m.t.</var>
                  ad
                    <var>.t.n</var>
                  . </s>
                  <s xml:id="echoid-s4681" xml:space="preserve">Quare ex conuerſa proportionalitate ita ſe habebit
                    <var>.i.r.</var>
                  ad tres
                    <lb/>
                  quintas ipſius
                    <var>.f.g.</var>
                  vt
                    <var>.t.n.</var>
                  ad
                    <var>.t.m</var>
                  . </s>
                  <s xml:id="echoid-s4682" xml:space="preserve">Et quia
                    <var>.o.n.</var>
                  ſumpta fuit æqualis ipſi
                    <var>.b.g.</var>
                  et
                    <var>.m.n.</var>
                  ipſi
                    <lb/>
                    <var>b.f.</var>
                  ideo
                    <var>.m.o.</var>
                  ex communi ſcientia æ qualis erit ipſi
                    <var>.g.f</var>
                  . </s>
                  <s xml:id="echoid-s4683" xml:space="preserve">Vnde proportio
                    <var>.r.i.</var>
                  ad tres
                    <lb/>
                  quintas ipſius
                    <var>.m.o.</var>
                  erit vt
                    <var>.n.t.</var>
                  ad
                    <var>.t.m.</var>
                  vt inquit Archi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4684" xml:space="preserve">Sed vbi. θ. iam probauimus ita ſe habere
                    <var>.i.f.</var>
                  ad
                    <var>.f.g.</var>
                  vt duplum
                    <reg norm="ipſius" type="simple">ipſiꝰ</reg>
                    <var>.p.l.</var>
                  cum ſim-
                    <lb/>
                  plo
                    <var>.l.</var>
                  ſe habet ad quintuplum ipſius
                    <var>.p.l.</var>
                  hoc eſt
                    <var>.i.f.</var>
                  ad
                    <var>.m.o.</var>
                  vt duplum ipſius
                    <var>.p.l.</var>
                  cum
                    <lb/>
                  ſimplo
                    <var>.l.</var>
                  ad quintuplum ipſius
                    <var>.p.l</var>
                  .</s>
                </p>
              </div>
            </div>
          </div>
        </div>
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