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-div737" type="section" level="3" n="42">
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                <pb o="396" rhead="IO, BAPT. BENED." n="408" file="0408" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0408"/>
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                  <s xml:id="echoid-s4691" xml:space="preserve">Sed
                    <var>.q.b.</var>
                  ſimiliter tres quintæ eſt ipſius
                    <var>.d.b.</var>
                  ex .8. prædicta. </s>
                  <s xml:id="echoid-s4692" xml:space="preserve">Quare
                    <var>.q.r.</var>
                  tres quintæ
                    <lb/>
                  erit ipſius
                    <var>.f.g.</var>
                  ex .19. quinti.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4693" xml:space="preserve">Dicamus igitur hoc modo cum
                    <var>.f.b.</var>
                  totum ad totum
                    <var>.b.r.</var>
                  ita ſe habeat vt abſciſ-
                    <lb/>
                  ſum
                    <var>.b.g.</var>
                  ad abſciſſum
                    <var>.q.b.</var>
                  ex .7. et .8. dicti primi libri eiuſdem ideo reſiduum
                    <var>.f.g.</var>
                  ex
                    <lb/>
                    <var>f.b.</var>
                  ad reſiduum
                    <var>.r.q.</var>
                  ex
                    <var>.r.b.</var>
                  erit vt totum
                    <var>.f.b.</var>
                  ad. totum
                    <var>.r.b.</var>
                  ex .19. quinti Eucli.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4694" xml:space="preserve">Sed iam ſub. β. probauimus ita ſe habere fruſtum
                    <var>.a.d.e.c.</var>
                  ad parabolam
                    <var>.d.b.e.</var>
                  vt
                    <lb/>
                    <var>m.t.</var>
                  ad
                    <var>.t.n.</var>
                  ſed vt
                    <var>.m.t.</var>
                  ad
                    <var>.t.n.</var>
                  ita aſſ umpta fuit (vbi
                    <var>.A.</var>
                  ).
                    <var>i.r.</var>
                  ad quam ſic ſe haberet
                    <var>.f.
                      <lb/>
                    h.</var>
                  hoc eſt tres quintæ ipſius
                    <var>.f.g.</var>
                  hoc eſt
                    <var>.q.r</var>
                  . </s>
                  <s xml:id="echoid-s4695" xml:space="preserve">quare ex .11. quinti prop ortio fruſti
                    <var>.a.
                      <lb/>
                    d.e.c.</var>
                  ad parabolam partialem erit vt
                    <var>.q.r.</var>
                  ad
                    <var>.r.i</var>
                  . </s>
                  <s xml:id="echoid-s4696" xml:space="preserve">Exiſtente igitur
                    <var>.r.</var>
                  centro totius pa
                    <lb/>
                  rabolæ et
                    <var>.q.</var>
                  centro partialis, ergo
                    <var>.i.</var>
                  centrum erit fruſti propoſiti.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4697" xml:space="preserve">Sed ſi nullo ſolido intercedente, voluerimus centrum
                    <var>.i.</var>
                  fruſti
                    <var>.a.e.</var>
                  citius inuenire,
                    <lb/>
                  inueniemus primò centrum
                    <var>.r.</var>
                  totius figuræ ex .8. ſecundi eiuſdem conſtituendo
                    <var>.b.r.</var>
                    <lb/>
                  tres quintas totius axis
                    <var>.b.f.</var>
                  & centrum
                    <var>.q.</var>
                  parabolæ
                    <var>.d.b.e.</var>
                  partialis ſimiliter.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4698" xml:space="preserve">Nunc igitur manifeſtum eſt nobis, eandem proportionem fore ipſius
                    <var>.q.r.</var>
                    <lb/>
                  ad
                    <var>.r.i.</var>
                  quæ fruſti
                    <var>.a.e.</var>
                  ad portionem
                    <var>.d.b.e.</var>
                  ex .8. dicta. </s>
                  <s xml:id="echoid-s4699" xml:space="preserve">Vnde ex coniuncta pro-
                    <lb/>
                  portionalitate ita ſe habebit
                    <var>.q.i.</var>
                  ad
                    <var>.i.r.</var>
                  vt
                    <var>.a.b.c.</var>
                  ad
                    <var>.d.b.e.</var>
                  ſed vt
                    <var>.a.b.c.</var>
                  ad
                    <var>.d.b.e.</var>
                  ita ſe
                    <lb/>
                  habet
                    <var>.m.n.</var>
                  ad
                    <var>.n.t.</var>
                  eo quod vnaquæque harum duarum proportionum ſeſquialtera
                    <lb/>
                  eſt proportioni
                    <var>.f.b.</var>
                  ad
                    <var>.b.g.</var>
                  eo. quod
                    <var>.f.b.</var>
                  ad
                    <var>.b.g.</var>
                  ita ſe habet. vt
                    <var>.m.n.</var>
                  ad
                    <var>.o.n.</var>
                  </s>
                  <s xml:id="echoid-s4700" xml:space="preserve">quare
                    <lb/>
                    <var>m.n.</var>
                  ad
                    <var>.t.n.</var>
                  ita ſe habebit vt
                    <var>.g.i.</var>
                  ad
                    <var>.r.i.</var>
                  vnde diſiunctim
                    <var>.m.t.</var>
                  ad
                    <var>.t.n.</var>
                  ita ſe habebit vt
                    <lb/>
                    <var>q.r.</var>
                  ad
                    <var>.r.i</var>
                  . </s>
                  <s xml:id="echoid-s4701" xml:space="preserve">Iungatur igitur
                    <var>.r.i.</var>
                  quæ quidem
                    <var>.r.i.</var>
                  ita ſe habeat ad
                    <var>.r.q.</var>
                  vt
                    <var>.t.n.</var>
                  ad
                    <var>.t.m.</var>
                  vt
                    <lb/>
                  habeatur centrum fruſti.</s>
                </p>
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