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-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-s4599" xml:space="preserve">
                    <pb o="388" rhead="IO. BABPT. BENED." n="400" file="0400" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0400"/>
                  conuerſa proportionalitate in .19. quinti, cum
                    <var>.a.d.</var>
                  ad
                    <var>.d.o.</var>
                  iam probatum fuit (vbi
                    <lb/>
                  B.) ita eſſe ut
                    <var>.A.</var>
                  ad
                    <var>.B</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4600" xml:space="preserve">Sed in principio huius ſpeculationis probatum iam fuit ita eſſe ipſius
                    <var>.d.a.</var>
                  ad
                    <var>.d.e.</var>
                    <lb/>
                  vt ipſius
                    <var>.D.</var>
                  ad
                    <var>.B.</var>
                  vbi notatum eſt
                    <var>.M</var>
                  . </s>
                  <s xml:id="echoid-s4601" xml:space="preserve">quare ex .23. quinti, Archimedes verum dicit,
                    <lb/>
                  qu od
                    <var>.d.o.</var>
                  ad
                    <var>.d.e.</var>
                  erit vt
                    <var>.D.</var>
                  ad
                    <var>.A</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4602" xml:space="preserve">Sed cum
                    <var>.d.o.</var>
                  ad
                    <var>.d.e.</var>
                  ſe habeat ut
                    <var>.D.</var>
                  ad
                    <var>.A.</var>
                  erit ex conuerſa proportionalitate iam
                    <lb/>
                  dicta
                    <var>.d.e.</var>
                  ad
                    <var>.d.o.</var>
                  vt
                    <var>.A.</var>
                  ad
                    <var>.D.</var>
                  per euerſam vero erit
                    <var>.d.e.</var>
                  ad
                    <var>.a.o.</var>
                  vt
                    <var>.A.</var>
                  ad ſuum reſi-
                    <lb/>
                    <note xlink:label="note-0400-01" xlink:href="note-0400-01a" position="left" xml:space="preserve">λ</note>
                  duum. </s>
                  <s xml:id="echoid-s4603" xml:space="preserve">quod reſiduum componitur ex ſimplo
                    <var>.b.c.</var>
                  cum triplo
                    <var>.b.</var>
                  cum duplo
                    <var>.b.o.</var>
                  quod
                    <lb/>
                  à te ipſo videre poteris detrahendo numeros ipſarum quantitatum quæ in
                    <var>.D.</var>
                    <lb/>
                  reperiuntur, ex numeris earundem, quæ in
                    <var>.A.</var>
                  quod quidem reſiduum ſigniſicetur
                    <lb/>
                  à charactere
                    <var>.E</var>
                  . </s>
                  <s xml:id="echoid-s4604" xml:space="preserve">Vnde ex conuerſa proportionalitate verum dicit Archime. hoc eſt
                    <lb/>
                  quod ita ſe hab ebit
                    <var>.o.e.</var>
                  ad
                    <var>.d.e.</var>
                  vt
                    <var>.E.</var>
                  ad
                    <var>.A</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4605" xml:space="preserve">Cum autem ſit
                    <var>.a.b.</var>
                  ad
                    <var>.c.b.</var>
                  vt
                    <var>.c.b.</var>
                  ad
                    <var>.d.b.</var>
                  & ita
                    <var>.d.b.</var>
                  ad
                    <var>.e.b.</var>
                  ex ſuppoſito, ideo ex
                    <lb/>
                  17. quinti verum dicit Archim. </s>
                  <s xml:id="echoid-s4606" xml:space="preserve">hoc eſt quod ita erit ipſius
                    <var>.d.e.</var>
                  ad
                    <var>.e.b.</var>
                  vt
                    <var>.a.c.</var>
                  ad
                    <var>.c.b.</var>
                    <lb/>
                  & vt
                    <var>.c.d.</var>
                  ad
                    <var>.d.b.</var>
                  & ex .13. eiuſdem eadem proportio erit tripli ipſius
                    <var>.c.d.</var>
                  ad triplum
                    <lb/>
                  ipſius
                    <var>.d.b.</var>
                  quæ dupli ipſius
                    <var>.d.e.</var>
                  ad. duplum ipſius
                    <var>.e.b.</var>
                  vt inquit Archi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4607" xml:space="preserve">Ex qua .13. compoſitum ex
                    <var>.a.c.</var>
                  cum triplo ipſius
                    <var>.c.d.</var>
                  cum duplo ipſius
                    <var>.d.e.</var>
                  ean-
                    <lb/>
                  dem proportionem habebit ad
                    <reg norm="compoſitum" type="context">compoſitũ</reg>
                  ipſius
                    <var>.c.b.</var>
                  cum triplo
                    <reg norm="ipſius" type="simple">ipſiꝰ</reg>
                    <var>.d.b.</var>
                  cum duplo
                    <lb/>
                  ipſius
                    <var>.e.b.</var>
                  quam ipſius
                    <var>.d.e.</var>
                  ad
                    <var>.e.b</var>
                  . </s>
                  <s xml:id="echoid-s4608" xml:space="preserve">Sed horum compoſitorum primum ſignificetur
                    <lb/>
                  per
                    <var>.H.</var>
                  ſecundum verò ſignificatum fuit per
                    <var>.E.</var>
                  vnde
                    <var>.H.</var>
                  ad
                    <var>.E.</var>
                  ſe habebit vt
                    <var>.d.e.</var>
                  ad
                    <lb/>
                    <var>e.b.</var>
                  ſed
                    <var>.E.</var>
                  ad
                    <var>.A.</var>
                  iam dictum eſt eſſe vt
                    <var>.o.e.</var>
                  ad
                    <var>.d.e.</var>
                  vbi ſignatum eſt
                    <var>.λ</var>
                  . </s>
                  <s xml:id="echoid-s4609" xml:space="preserve">quare ex .23.
                    <lb/>
                  quinti eadem proportio erit ipſius
                    <var>.o.e.</var>
                  ad
                    <var>.e.b.</var>
                  quæ
                    <var>.H.</var>
                  ad
                    <var>.A.</var>
                  vt ipſe inquit.</s>
                </p>
                <note position="left" xml:space="preserve">X</note>
                <p>
                  <s xml:id="echoid-s4610" xml:space="preserve">Ex .18. poſtea eiuſdem ita erit
                    <var>.o.b.</var>
                  ad
                    <var>.e.b.</var>
                  vt
                    <var>.H.A.</var>
                  ad
                    <var>.A</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4611" xml:space="preserve">Notandum etiam eſt quod ſi collectæ fuerint omnes partes compoſiti
                    <var>.H.A.</var>
                  hoc
                    <lb/>
                  eſt duplum
                    <var>.a.b.</var>
                  cum duplo
                    <var>.b.e.</var>
                  cum quadruplo
                    <var>.b.c.</var>
                  cum quadruplo
                    <var>.b.d.</var>
                  cum ſimplo
                    <lb/>
                    <var>a.c.</var>
                  cum triplo
                    <var>.c.d.</var>
                  cum duplo
                    <var>.d.e.</var>
                  habebitur triplum
                    <var>.a.b.</var>
                  triplum
                    <var>.b.d.</var>
                  & ſexcuplum
                    <lb/>
                    <var>b.c.</var>
                  vt ipſe dixit. </s>
                  <s xml:id="echoid-s4612" xml:space="preserve">Quod autem hoc verum ſit, cum diſtinctæ fuerint omnes partes,
                    <lb/>
                  vt in ſubſcriptis his lineis videre eſt, videbis quod ſi ex
                    <var>.H.</var>
                  detracta fuerit ſimplex
                    <var>.a.
                      <lb/>
                    c.</var>
                  quæ quidem poſtea iuncta vni ex partibus quadrupli
                    <var>.b.c.</var>
                  ipſius
                    <var>.A.</var>
                  reſultabit nobis
                    <lb/>
                  vna inte gra
                    <var>.a.b</var>
                  . </s>
                  <s xml:id="echoid-s4613" xml:space="preserve">Vnde habebimus triplum ipſius
                    <var>.a.b.</var>
                  & in
                    <var>.A.</var>
                  remanebit triplum ip
                    <lb/>
                  ſius
                    <var>.c.b</var>
                  . </s>
                  <s xml:id="echoid-s4614" xml:space="preserve">Deinde ſi ex
                    <var>.H.</var>
                  auferatur triplum ipſius
                    <var>.c.d.</var>
                  & ipſum addatur tribus parti-
                    <lb/>
                  bus quadrupli
                    <var>.b.d.</var>
                  ipſius
                    <var>.A.</var>
                  habebimus tres vices
                    <var>.b.c.</var>
                  quæ ſi iungantur tribus, quæ
                    <lb/>
                  remanebant in
                    <var>.A.</var>
                  vt dixi, habebimus ſexcuplum ipſius
                    <var>.b.c.</var>
                  & in
                    <var>.A.</var>
                  remanebit ſim
                    <lb/>
                  plum
                    <var>.b.d.</var>
                  cum duplo ipſius
                    <var>.b.e</var>
                  . </s>
                  <s xml:id="echoid-s4615" xml:space="preserve">Vnde ſi ex
                    <var>.H.</var>
                  demptum fuerit duplum ipſius
                    <var>.d.
                      <lb/>
                    e.</var>
                  quod quidem iungatur cum duplo ipſius
                    <var>.b.e.</var>
                  habebimus duplum ipſius
                    <var>.b.d.</var>
                  quod
                    <lb/>
                  coniunctum cum ſimplo
                    <var>.b.d.</var>
                  quod in
                    <var>.A.</var>
                  relictum fuerat, habebimus triplum ipſius
                    <lb/>
                    <var>d.b</var>
                  . </s>
                  <s xml:id="echoid-s4616" xml:space="preserve">Verum igitur eſt quod inquit Archimedes, hoc eſt, quod
                    <var>.H.A.</var>
                  eſt triplum ip-
                    <lb/>
                  ſius
                    <var>.a.b.</var>
                  ſexcuplum ipſius
                    <var>.b.c.</var>
                  & triplum ipſius
                    <var>.b.d</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4617" xml:space="preserve">Verum etiam dicit ex eo (vt ſupra probatum eſt) quod
                    <var>.a.c</var>
                  :
                    <var>c.d</var>
                  : et
                    <var>.d.e.</var>
                  ſe
                    <reg norm="habebant" type="context">habebãt</reg>
                    <lb/>
                  in continua proportionalitate, </s>
                  <s xml:id="echoid-s4618" xml:space="preserve">quare ex conuerſa proportionalitate erunt ſibi inui-
                    <lb/>
                  cem continuæ proportionales.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4619" xml:space="preserve">Nunc autem cum
                    <var>.a.c</var>
                  :
                    <var>c.d.</var>
                  et
                    <var>.d.e.</var>
                  ſint continuæ proportionales in ea proportione
                    <lb/>
                  in qua ſunt
                    <var>.a.b</var>
                  :
                    <var>c.b</var>
                  :
                    <var>d.b</var>
                  : et
                    <var>.e.b.</var>
                  vt in principio diximus, erit ex .22. quinti
                    <var>.a.c.</var>
                  ad
                    <var>.d.
                      <lb/>
                    e.</var>
                  vt
                    <var>.a.b.</var>
                  ad
                    <var>.d.b.</var>
                  & ſic etiam
                    <var>.c.b.</var>
                  ad
                    <var>.e.b</var>
                  . </s>
                  <s xml:id="echoid-s4620" xml:space="preserve">Vnde ex .24. eiuſdem
                    <var>.a.d.</var>
                  ad
                    <var>.d.e.</var>
                  erit vt
                    <var>.a.
                      <lb/>
                    b.</var>
                  cum
                    <var>.b.e.</var>
                  ad
                    <var>.d.b.</var>
                  & vt
                    <var>.c.b.</var>
                  cum
                    <var>.b.d.</var>
                  ad
                    <var>.e.b.</var>
                  & ex .13. dicti vt
                    <var>.a.b.</var>
                  cum
                    <var>.b.e.</var>
                  bis
                    <lb/>
                  ſumpto, & cum
                    <var>.b.d.</var>
                  ad
                    <var>.e.b</var>
                  . </s>
                  <s xml:id="echoid-s4621" xml:space="preserve">Quare ex conuerſa proportionalitate, vt ſe habet
                    <var>.e.d.</var>
                    <lb/>
                  ad
                    <var>.d.a.</var>
                  ita ſe habebit
                    <var>.e.b.</var>
                    <reg norm="cum" type="context">cũ</reg>
                    <var>.d.b.</var>
                  ad
                    <var>d.b.</var>
                    <reg norm="cum" type="context">cũ</reg>
                    <var>.b.c.</var>
                  duplicato &
                    <reg norm="cum" type="context">cũ</reg>
                    <var>.b.a.</var>
                  vt inquit Archi
                    <lb/>
                  medes. </s>
                  <s xml:id="echoid-s4622" xml:space="preserve">Nunc antecedens vocetur
                    <var>.M.</var>
                  hoc eſt
                    <var>.b.e.</var>
                  cum
                    <var>.d.b.</var>
                  conſequens verò, hoc </s>
                </p>
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