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">
                <pb o="392" rhead="IO. BAPT. BENED." n="404" file="0404" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0404"/>
                <p>
                  <s xml:id="echoid-s4640" xml:space="preserve">In vltima verò propoſitione ſecundi lib. de ponderibus Archi. hoc modo intelli­
                    <lb/>
                  gendus eſt, vt ſi diceret,
                    <lb/>
                  Sit paraboles
                    <var>.a.</var>
                  cuius baſis ſit
                    <var>.a.c.</var>
                    <reg norm="ſitque" type="simple">ſitq́;</reg>
                    <var>.d.e.</var>
                  recta parallela dictæ baſi
                    <var>.a.c.</var>
                    <reg norm="diameterque" type="simple">diameterq́;</reg>
                    <lb/>
                    <var>b.f</var>
                  .
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4641" xml:space="preserve">Inquit deinde quod linea contingens in
                    <var>.b.</var>
                  parallela erit ipſi
                    <var>.a.c.</var>
                  et
                    <var>.e.d.</var>
                  quod proba
                    <lb/>
                  bimus hoc modo.
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4642" xml:space="preserve">Cum
                    <var>.b.f.</var>
                  diameter ſit et
                    <var>.a.c.</var>
                  baſis, clarum erit ex definitione quod
                    <var>.b.f.</var>
                  diuidet
                    <var>.a.c.</var>
                    <lb/>
                  per æqualia in
                    <var>.g</var>
                  . </s>
                  <s xml:id="echoid-s4643" xml:space="preserve">Vnde ex .7. vel etiam ex .46. primi Pergei
                    <var>.d.e.</var>
                  diuiſa erit per æqua
                    <lb/>
                  lia à diametro
                    <var>.b.f</var>
                  . </s>
                  <s xml:id="echoid-s4644" xml:space="preserve">Quare verum dicit ex quinta ſecundi ipſius Pergei hoc eſt quod
                    <lb/>
                  dicta contingens in puncto. b parallela erit ambobus
                    <var>.a.c.</var>
                  et
                    <var>.e.d</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4645" xml:space="preserve">Inquit poſtea quod diuiſa cum fuerit pars diametri quę inter
                    <var>.d.e.</var>
                  et
                    <var>.a.c.</var>
                  poſita eſt
                    <lb/>
                  (hoc eſt
                    <var>.g.f.</var>
                  ) per quinque partes æquales,
                    <reg norm="quarum" type="context">quarũ</reg>
                  partium media ſit
                    <var>.h.k.</var>
                  diuiſa etiam
                    <lb/>
                  imaginatione ſit in puncto
                    <var>.i.</var>
                  ita quod proportio ipſius
                    <var>.h.i.</var>
                  ad
                    <var>.i.K.</var>
                  eadem ſit quæ in-
                    <lb/>
                  ter duo ſolida quorum vnum (illud ſcilicet à quo relatio incipit, hoc eſt antecedens)
                    <lb/>
                  pro ſua baſi teneat quadratum ipſius
                    <var>.a.f.</var>
                  cuius etiam ſolidi altitudo compoſita ſit ex
                    <lb/>
                    <anchor type="note" xlink:label="note-0404-01a" xlink:href="note-0404-01"/>
                  duplo ipſius
                    <var>.d.g.</var>
                  cum ſimplo
                    <var>.a.f</var>
                  . </s>
                  <s xml:id="echoid-s4646" xml:space="preserve">Aliud verò ſolidum habeat pro ſua baſi quadra-
                    <lb/>
                  tum ipſius
                    <var>.d.g.</var>
                  eius verò altitudo compoſita ſit ex duplo ipſius
                    <var>.a.f.</var>
                  cum ſimplo
                    <var>.d.g</var>
                  .</s>
                </p>
                <div xml:id="echoid-div748" type="float" level="5" n="12">
                  <note xlink:label="note-0404-01" xlink:href="note-0404-01a" position="left" xml:space="preserve">R</note>
                </div>
                <p>
                  <s xml:id="echoid-s4647" xml:space="preserve">Inquit nunc Archi. quod cum ita factum fuerit, oſtendet punctum
                    <var>.i.</var>
                  centrum eſſe
                    <lb/>
                  portionis abſciſſę à tota ſectione, quod
                    <reg norm="fruſtum" type="context">fruſtũ</reg>
                    <reg norm="nominatur" type="simple">nominat̃</reg>
                    <reg norm="ſignatum" type="context">ſignatũ</reg>
                  characteribus
                    <var>.a.d.e.c</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4648" xml:space="preserve">Sit igitur num@.
                    <var>m.n.</var>
                  inquit, æqualis diametro
                    <var>.b.f.</var>
                  et
                    <var>.n.o.</var>
                  æqualis
                    <var>.b.g.</var>
                    <reg norm="ſitque" type="simple">ſitq́;</reg>
                    <var>.x.n.</var>
                  me
                    <lb/>
                  dia proportionalis inter
                    <var>.n.m.</var>
                  et
                    <var>.n.o.</var>
                  et
                    <var>.t.n.</var>
                  in continua proportionalitate poſt
                    <var>.o.n.</var>
                    <lb/>
                  hoc eſt quod ea proportio quæ eſt ipſius
                    <var>.o.n.</var>
                  ad
                    <var>.n.t.</var>
                  eadem ſit ipſius
                    <var>.x.n.</var>
                  ad
                    <var>.n.o</var>
                  . </s>
                  <s xml:id="echoid-s4649" xml:space="preserve">Hinc
                    <lb/>
                  habebimus .4. lineas in continua proportionalitate ſibi inuicem coniunctas
                    <var>.m.n</var>
                  :
                    <var>x.
                      <lb/>
                    n</var>
                  :
                    <var>o.n.</var>
                  et
                    <var>.t.n</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4650" xml:space="preserve">Vult etiam quod à linea
                    <var>.i.b.</var>
                  incipiens ab
                    <var>.i.</var>
                  verſus
                    <var>.g.</var>
                  alia linea abſciſſa ſit, cui li-
                    <lb/>
                    <anchor type="note" xlink:label="note-0404-02a" xlink:href="note-0404-02"/>
                  neæ, ita proportionata ſit
                    <var>.f.h.</var>
                  vt
                    <var>.t.m.</var>
                  eſt ad
                    <var>.t.n.</var>
                  quæ quidem linea ſignata ſit
                    <var>.i.r</var>
                  .</s>
                </p>
                <div xml:id="echoid-div749" type="float" level="5" n="13">
                  <note xlink:label="note-0404-02" xlink:href="note-0404-02a" position="left" xml:space="preserve">A</note>
                </div>
                <p>
                  <s xml:id="echoid-s4651" xml:space="preserve">Dicit poſtea quod diameter
                    <var>.b.f.</var>
                  erit fortaſſe a xis vel aliqua reliquarum diame-
                    <lb/>
                  trorum, quod quidem in .46. primi Pergei videre eſt, cum omnes diametri ſint in-
                    <lb/>
                  uicem paralleli ipſi axi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4652" xml:space="preserve">Cum poſtea dicit, quod
                    <var>.a.f.</var>
                  et
                    <var>.d.g.</var>
                  ſunt intentæ ductæq́ue, ibi vult id em infer-
                    <lb/>
                  re, quod Pergeus vocat ordinatè, vt ex .11. et .49. primi ipſius Pergei videre li-
                    <lb/>
                  cet, vnde ex .20. eiuſdem proportio
                    <var>.b.f.</var>
                  ad
                    <var>.b.g.</var>
                  erit vt quadrati
                    <var>.a.f.</var>
                  ad quadratum
                    <lb/>
                  ipſius
                    <var>.d.g.</var>
                  vt ipſe dicit.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4653" xml:space="preserve">Sed ita erit quadrati
                    <var>.m.n.</var>
                  ad qua
                    <reg norm="dratum" type="context">dratũ</reg>
                    <var>.x.n.</var>
                  ex .18. ſexti Eucli. </s>
                  <s xml:id="echoid-s4654" xml:space="preserve">Quare ex .11. quin-
                    <lb/>
                    <anchor type="note" xlink:label="note-0404-03a" xlink:href="note-0404-03"/>
                  ti quadratum ipſius
                    <var>.m.n.</var>
                  ad quadratum ipſius
                    <var>.n.x.</var>
                  eandem habebit proportionem,
                    <lb/>
                  quam quadratum ipſius
                    <var>.a.f.</var>
                  ad quadratum ipſius
                    <var>.d.g</var>
                  . </s>
                  <s xml:id="echoid-s4655" xml:space="preserve">Vnde ex .18. & ex communi
                    <lb/>
                    <reg norm="ſcientia" type="context">ſciẽtia</reg>
                  , eadem proportio erit ipſius
                    <var>.m.n.</var>
                  ad
                    <var>.n.x.</var>
                  quę ipſius
                    <var>.a.f.</var>
                  ad
                    <var>.d.g.</var>
                  vt inquit Arch.</s>
                </p>
                <div xml:id="echoid-div750" type="float" level="5" n="14">
                  <note xlink:label="note-0404-03" xlink:href="note-0404-03a" position="left" xml:space="preserve">α</note>
                </div>
                <p>
                  <s xml:id="echoid-s4656" xml:space="preserve">Quaptopter proportio cubi ipſius
                    <var>.m.n.</var>
                  ad cubum ipſius
                    <var>.n.x.</var>
                  erit vt cubi ipſius
                    <var>.a.
                      <lb/>
                    f.</var>
                  ad cubum ipſius
                    <var>.d.g.</var>
                  vt etiam dicit ex communi ſcientia, nec non ex .36. vndecimi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4657" xml:space="preserve">Inquit poſtea quod proportio totius ſectionis
                    <var>.a.b.c.</var>
                  ad portionem
                    <var>.d.b.e.</var>
                  eadem
                    <lb/>
                  eſt quæ cubi ipſius
                    <var>.a.f.</var>
                  ad cubum ipſius
                    <var>.d.g.</var>
                  quod verum eſt, vt aliàs tibi monſtraui in
                    <lb/>
                  diuiſione parabolæ ſecundum aliquam propoſitam proportionem.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4658" xml:space="preserve">Quando autem dicit quod proportio cubi ipſius
                    <var>.m.n.</var>
                  ad cubum ipſius
                    <var>.n.x.</var>
                  eadem
                    <lb/>
                    <anchor type="note" xlink:label="note-0404-04a" xlink:href="note-0404-04"/>
                  eſt quæ ipſius
                    <var>.m.n.</var>
                  ad
                    <var>.n.t.</var>
                  verum dicit ex .36. vndecimi. </s>
                  <s xml:id="echoid-s4659" xml:space="preserve">Vnde ex .11. quinti ita ſe
                    <lb/>
                  habebit totalis ſectio
                    <var>.a.b.c.</var>
                  ad portionem
                    <var>.d.b.c.</var>
                  vt
                    <var>.m.n.</var>
                  ad
                    <var>.n.t.</var>
                  & ex .17. eiuſdem ita
                    <lb/>
                  erit ipſius
                    <var>.m.t.</var>
                  ad
                    <var>.t.n.</var>
                  vt fruſti
                    <var>.a.d.e.c.</var>
                  ad ſectionem
                    <var>.d.b.e.</var>
                  quemadmodum ipſe di-
                    <lb/>
                  cit. </s>
                  <s xml:id="echoid-s4660" xml:space="preserve">Sed quia ſuperius, vbi
                    <var>.A.</var>
                  ipſa
                    <var>.f.h.</var>
                  (quæ eſt tres quintæ ipſius
                    <var>.f.g.</var>
                  ) ad
                    <var>.i.r.</var>
                  ita rela- </s>
                </p>
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