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">
              <div xml:id="echoid-div737" type="letter" level="4" n="1">
                <p>
                  <s xml:id="echoid-s4580" xml:space="preserve">
                    <pb o="386" rhead="IO. BAPT. BENED." n="398" file="0398" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0398"/>
                  ipſius
                    <var>.b.e.</var>
                  ad compoſitum quintupli ipſius
                    <var>.a.b.</var>
                  cum decuplo ipſius
                    <var>.c.b.</var>
                  cum decuplo
                    <lb/>
                  ipſius
                    <var>.b.d.</var>
                  cum quintuplo ipſius
                    <var>.b.e.</var>
                  eadem ſit quæ ipſius
                    <var>.g.h.</var>
                  ad
                    <var>.a.d.</var>
                  & vult proba-
                    <lb/>
                  re
                    <var>.f.h.</var>
                  eſſe duas quintas ipſius
                    <var>.a.b</var>
                  .</s>
                </p>
                <div xml:id="echoid-div740" type="float" level="5" n="4">
                  <figure xlink:label="fig-0397-01" xlink:href="fig-0397-01a">
                    <image file="0397-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0397-01"/>
                  </figure>
                </div>
                <p>
                  <s xml:id="echoid-s4581" xml:space="preserve">Cum autem dicit proportionem ipſius
                    <var>.a.c.</var>
                  ad
                    <var>.c.d.</var>
                  & ipſius
                    <var>.c.d.</var>
                  ad
                    <var>.d.e.</var>
                  eſſe vt ipſius
                    <lb/>
                    <var>a.b.</var>
                  a
                    <unsure/>
                  d.b.c. & cętera, verum dicit ex .19. quinti Eucli. eo quod cum ex hypotheſi ſit
                    <lb/>
                  ipſius,
                    <var>a.b.</var>
                  totalis ad
                    <var>.c.b.</var>
                  totalom vt ipſius
                    <var>.c.b.</var>
                  partialis (ſumptæ vt pars abſciſa ab
                    <var>.a.
                      <lb/>
                    b.</var>
                  pro nunc) ad
                    <var>.d.b.</var>
                  partialem (abſciſam ab
                    <var>.c.b.</var>
                  ) erit ex .19. dicta ipſius
                    <var>.a.c.</var>
                  (reſidui
                    <lb/>
                  ex
                    <var>.a.b.</var>
                  ) ad
                    <var>.c.d.</var>
                  (reſiduum ex
                    <var>.c.b.</var>
                  ) vt ipſius
                    <var>.a.b.</var>
                  ad
                    <var>.c.b.</var>
                  & ita probabitur de pro-
                    <lb/>
                  portione ipſiuas
                    <var>.c.d.</var>
                  ad
                    <var>.d.e.</var>
                  eadem ratione.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4582" xml:space="preserve">Cum verò ex .18. quinti ſit ipſius
                    <var>.a.b.</var>
                  cum
                    <var>.c.b.</var>
                  ad
                    <var>.c.b.</var>
                  vt ipſius
                    <var>.a.d.</var>
                  ad
                    <var>.d.e.</var>
                  ergo ex
                    <lb/>
                  22. eiuſdem, ita erit ipſius
                    <var>.a.b.</var>
                  cum
                    <var>.c.b.</var>
                  ad
                    <var>.d.b.</var>
                  vt
                    <var>.a.d.</var>
                  ad
                    <var>.d.e.</var>
                  & exijſdem rationibus
                    <lb/>
                  eadem proportio erit ipſius
                    <var>.c.b.</var>
                  cum
                    <var>.d.b.</var>
                  ad
                    <var>.b.e.</var>
                  vt
                    <var>.a.d.</var>
                  ad
                    <var>.d.e.</var>
                  quod inquit Archi.
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4583" xml:space="preserve">Verum etiam erit (ex .13. quinti) cum dicit eandem proportionem eſſe ipſius
                    <var>.a.d.</var>
                    <lb/>
                  ad
                    <var>.d.e.</var>
                  quę dupli primi antecedentis cum ſimplo ſecundi antecedentis ad duplum
                    <lb/>
                  primi conſequentis cum ſimplo ſecundi conſequentis, hoc eſt dupli ipſius
                    <var>.a.b.c.</var>
                    <reg norm="cum" type="context">cũ</reg>
                    <lb/>
                  ſimplo
                    <var>.c.b.d.</var>
                  ad duplum ipſius
                    <var>.d.b.</var>
                  cum ſimplo
                    <var>.e.b.</var>
                  hoc eft dupli
                    <var>.a.b.</var>
                  cum triplo ip-
                    <lb/>
                  ſius
                    <var>.b.c.</var>
                  cum ſimplo
                    <var>.d.b.</var>
                  ad duplum ipſius
                    <var>.d.b.</var>
                  cum ſimplo
                    <var>.e.b</var>
                  . </s>
                  <s xml:id="echoid-s4584" xml:space="preserve">Nunc duplum
                    <var>.a.b.</var>
                    <lb/>
                  cum triplo
                    <var>.b.c.</var>
                  cum ſimplo
                    <var>.b.d.</var>
                  ſignatum ſit charactere
                    <var>.D.</var>
                  ſuum verò conſequens,
                    <lb/>
                  hoc eſt duplum
                    <var>.d.b.</var>
                    <reg norm="cum" type="context">cũ</reg>
                  ſimplo
                    <var>.e.b.</var>
                  ſignificetur à charactere
                    <var>.B.</var>
                  hinc proportio ipſius
                    <lb/>
                    <anchor type="note" xlink:label="note-0398-01a" xlink:href="note-0398-01"/>
                    <var>a.d.</var>
                  ad
                    <var>.d.e.</var>
                  erit vt
                    <var>.D.</var>
                  ad
                    <var>.B</var>
                  .</s>
                </p>
                <div xml:id="echoid-div741" type="float" level="5" n="5">
                  <note xlink:label="note-0398-01" xlink:href="note-0398-01a" position="left" xml:space="preserve">M</note>
                </div>
                <p>
                  <s xml:id="echoid-s4585" xml:space="preserve">Inquit nunc Archimedes, ſi quis ſumeret aliquod maius antecedens æquale ſci-
                    <lb/>
                  licet duplo ipſius
                    <var>.a.b.</var>
                  cum quadruplo ipſius
                    <var>.b.e.</var>
                  cum quadruplo ipſius
                    <var>.b.d.</var>
                  cum du-
                    <lb/>
                  plo ipſius
                    <var>.b.e.</var>
                    <reg norm="compararetque" type="simple">compararetq́;</reg>
                  illud cum
                    <reg norm="conſequente" type="context">cõſequente</reg>
                    <var>.B.</var>
                  clarum eſſet ex .8. quinti quod
                    <lb/>
                  tale antecedens maiorem proportionem haberet ad
                    <var>.B.</var>
                  quam ad
                    <var>.D.</var>
                  hoc eſt maiorem
                    <lb/>
                  quàm ipſius
                    <var>.a.d.</var>
                  ad
                    <var>.d.e.</var>
                  ex .12. quinti.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4586" xml:space="preserve">Nunc ſi ſumpta fuerit aliqua linea, puta
                    <var>.d.o.</var>
                  cui
                    <var>.a.d.</var>
                    <reg norm="dictam" type="context">dictã</reg>
                  habeat proportionem
                    <lb/>
                  maiorem, larum erit ex ſecunda parte decimę quinti quod
                    <var>.d.o.</var>
                  minor erit ipſa
                    <var>.d.e.</var>
                    <lb/>
                  Corrige igitur impreſſionem Baſileę locando characterem
                    <var>.o.</var>
                  inter
                    <var>.d.</var>
                  et
                    <var>.e.</var>
                  eo quod
                    <lb/>
                  ibi poſitum non fuit.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4587" xml:space="preserve">Volo nunc quod dictum maius antecedens æquale ſcilicet duplo ipſius
                    <var>.a.b.</var>
                  cum
                    <lb/>
                  quadruplo ipſius
                    <var>.b.c.</var>
                  cum quadruplo ipſius
                    <var>.b.d.</var>
                  cum duplo ipſius
                    <var>.b.c.</var>
                  ſignificetur à
                    <lb/>
                  charactere
                    <var>.A</var>
                  . </s>
                  <s xml:id="echoid-s4588" xml:space="preserve">Hinc habebimus proportionem ipſius
                    <var>.a.d.</var>
                  ad
                    <var>.d.o.</var>
                  ut
                    <var>.A.</var>
                  ad
                    <var>.B</var>
                  .</s>
                </p>
                <note position="left" xml:space="preserve">β</note>
                <p>
                  <s xml:id="echoid-s4589" xml:space="preserve">Ex .18. quinti poſtea habebimus
                    <var>.A.B.</var>
                  ad
                    <var>.B.</var>
                  vt
                    <var>.a.o.</var>
                  ad
                    <var>.d.o.</var>
                  & proportionalitate
                    <lb/>
                  euerſa in .19. dicti ita erit
                    <var>.A.B.</var>
                  ad
                    <var>.A.</var>
                  vt
                    <var>.a.o.</var>
                  ad
                    <var>.a.d</var>
                  . </s>
                  <s xml:id="echoid-s4590" xml:space="preserve">Sed hoc vltimum antecedens in
                    <lb/>
                    <anchor type="note" xlink:label="note-0398-03a" xlink:href="note-0398-03"/>
                  ſe continet id quod Archimedes ſcribit, hoc eſt duplum ipſius
                    <var>.a.b.</var>
                    <reg norm="quadruplum" type="context">quadruplũ</reg>
                  ipſius
                    <lb/>
                    <var>b.c.</var>
                  ſexcuplum ipſius
                    <var>.b.d.</var>
                  & triplum ipſius
                    <var>.b.e</var>
                  . </s>
                  <s xml:id="echoid-s4591" xml:space="preserve">Conſequens verò
                    <var>.A.</var>
                  continet du
                    <lb/>
                  plum ipſius
                    <var>.a.b.</var>
                  quadruplum ipſius
                    <var>.b.c.</var>
                  quadruplum ipſius
                    <var>.b.d.</var>
                  & duplum ipſius
                    <var>.b.e</var>
                  .</s>
                </p>
                <div xml:id="echoid-div742" type="float" level="5" n="6">
                  <note xlink:label="note-0398-03" xlink:href="note-0398-03a" position="left" xml:space="preserve">T</note>
                </div>
                <p>
                  <s xml:id="echoid-s4592" xml:space="preserve">Ex ſuppoſito deinde ipſius Archimedis & ex conuerſa proportionalitate in .19.
                    <lb/>
                  dicta, verum eſt id quod dicit Archimedes, videlicet quod eadem proportio eſt
                    <lb/>
                  ipſius
                    <var>.a.d.</var>
                  ad
                    <var>.g.h.</var>
                  quod quintupli ipſius
                    <var>.a.b.</var>
                  cum quintuplo ipſius
                    <var>.b.e.</var>
                  cum decuplo
                    <lb/>
                  ipſius
                    <var>.b.c.</var>
                  cum decuplo ipſius
                    <var>.b.d.</var>
                  (quod quidem antecedens ſignificetur per
                    <var>.V.</var>
                  )
                    <lb/>
                  ad duplum ipſius
                    <var>.a.b.</var>
                  cum quadruplo ipſius
                    <var>.b.c.</var>
                  cum ſexcuplo ipſius
                    <var>.b.d.</var>
                  cum triplo
                    <lb/>
                  ipſius
                    <var>.b.e.</var>
                  hoc eſt ad
                    <var>.A.B</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4593" xml:space="preserve">Erit igitur
                    <var>.V.</var>
                  ad
                    <var>.A.B.</var>
                  vt ipſius
                    <var>.a.d.</var>
                  ad
                    <var>.g.h.</var>
                  ſed ſuperius vbi ſignatum eſt
                    <var>.T.</var>
                  iam
                    <lb/>
                  probatum fuit ita eſſe
                    <var>.A.B.</var>
                  ad
                    <var>.A.</var>
                  vt ipſius
                    <var>.a.o.</var>
                  ad
                    <var>.a.d</var>
                  . </s>
                  <s xml:id="echoid-s4594" xml:space="preserve">Ergò ex .23. quinti Archime
                    <lb/>
                  des verum ſcribit, hoc eſt quod ita erit ipſius
                    <var>.V.</var>
                  ad
                    <var>.A.</var>
                  vt ipſius
                    <var>.a.o.</var>
                  ad
                    <var>.g.h</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4595" xml:space="preserve">Clarum per ſe etiam eſt, id quod Archimed. dicit hoc eſt quod
                    <var>.V.</var>
                  ad
                    <var>.A.</var>
                  eſt vt
                    <lb/>
                    <anchor type="note" xlink:label="note-0398-04a" xlink:href="note-0398-04"/>
                  </s>
                </p>
              </div>
            </div>
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