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">
                <p>
                  <s xml:id="echoid-s4554" xml:space="preserve">
                    <pb o="384" rhead="IO. BAPT. BENED." n="396" file="0396" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0396"/>
                  eiuſdem erit vt
                    <var>.a.d.</var>
                  ad
                    <var>.d.b</var>
                  . </s>
                  <s xml:id="echoid-s4555" xml:space="preserve">Idem etiam dico in ſecunda parabola, ſed ipſius
                    <var>.x.o.</var>
                  ad
                    <lb/>
                    <var>o.r.</var>
                  eſt vt
                    <var>.a.b.</var>
                  ad
                    <var>.b.d.</var>
                  ex .6. ſexti Eucli. </s>
                  <s xml:id="echoid-s4556" xml:space="preserve">vnde ex .11. quinti
                    <var>.n.f.</var>
                  ad
                    <var>.f.x.</var>
                  erit vt
                    <var>.ω.y.</var>
                    <lb/>
                  ad
                    <var>.y.g</var>
                  . </s>
                  <s xml:id="echoid-s4557" xml:space="preserve">Sed in precedenti iam tibi dixi
                    <var>.a.b.</var>
                  mediam proportionalem eſſe inter
                    <var>.h.</var>
                    <lb/>
                  et
                    <var>.b.d</var>
                  . </s>
                  <s xml:id="echoid-s4558" xml:space="preserve">Sit nunc
                    <var>.z.</var>
                  pro ſecunda parabola, ita ut
                    <var>.h.</var>
                  eſt pro prima, vnde
                    <var>.o.x.</var>
                  crit media
                    <lb/>
                  proportionalis inter
                    <var>.z.</var>
                  et
                    <var>.o.r.</var>
                  & ex .11. quinti ita erit
                    <var>.h.</var>
                  ad
                    <var>.a.b.</var>
                  vt
                    <var>.z.</var>
                  ad
                    <var>.x.o.</var>
                  & ex .22.
                    <lb/>
                  h. ad
                    <var>.a.x.</var>
                  ut z. ad
                    <var>.x.g.</var>
                  & quia ex .16. ſexti
                    <var>.a.x.</var>
                  media proportionalis eſt inter
                    <var>.h.</var>
                  et
                    <var>.f.
                      <lb/>
                    x.</var>
                  cum ſupponatur productum
                    <var>.h.</var>
                  in
                    <var>.f.x.</var>
                  æquale eſſe quadrato
                    <var>.a.x</var>
                  . </s>
                  <s xml:id="echoid-s4559" xml:space="preserve">Idem dico
                    <var>.x.g.</var>
                    <lb/>
                  mediam eſſe proportionalem inter
                    <var>.z.</var>
                  et
                    <var>.g.y.</var>
                  </s>
                  <s xml:id="echoid-s4560" xml:space="preserve">quare ex .11. iam dicta, ita erit
                    <var>.a.x.</var>
                  ad
                    <var>.f.
                      <lb/>
                    x.</var>
                  vt
                    <var>.y.g.</var>
                  ad
                    <var>.x.o.</var>
                  & ex eadem, ita erit ipſius
                    <var>.f.n.</var>
                  ad
                    <var>.a.b.</var>
                  ut
                    <var>.y.ω.</var>
                  ad
                    <var>.x.o.</var>
                  & ſic
                    <var>.f.n.</var>
                  ad
                    <var>.d.a.</var>
                    <lb/>
                  vt
                    <var>.y.ω.</var>
                  ad
                    <var>.x.r.</var>
                  ſed
                    <var>.f.m.</var>
                  ad
                    <var>f.n.</var>
                  eſt vt
                    <var>.y.t.</var>
                  ad
                    <var>.y.ω.</var>
                  ex .18. quinti vnde
                    <var>.f.m.</var>
                  ad
                    <var>.a.d.</var>
                  erit vt
                    <lb/>
                    <var>y.t.</var>
                  ad
                    <var>.x.r</var>
                  . </s>
                  <s xml:id="echoid-s4561" xml:space="preserve">Idem dico de eorum duplis.</s>
                </p>
                <div xml:id="echoid-div739" type="float" level="5" n="3">
                  <figure xlink:label="fig-0395-02" xlink:href="fig-0395-02a">
                    <image file="0395-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0395-02"/>
                  </figure>
                </div>
                <p>
                  <s xml:id="echoid-s4562" xml:space="preserve">Ex ijſdem rationibus dico ita eſſe
                    <var>.b.d.</var>
                  ad
                    <var>.b.m.</var>
                  vt
                    <var>.o.r.</var>
                  ad
                    <var>.o.t.</var>
                  & ex .17. quinti
                    <var>.d.m.</var>
                    <lb/>
                  ad
                    <var>.b.m.</var>
                  vt
                    <var>.r.t.</var>
                  ad
                    <var>.t.o</var>
                  . </s>
                  <s xml:id="echoid-s4563" xml:space="preserve">Reliqua tibi conſideranda relinquo.</s>
                </p>
                <figure position="here">
                  <image file="0396-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0396-01"/>
                </figure>
                <p>
                  <s xml:id="echoid-s4564" xml:space="preserve">In reliquis verò propoſitionibus illius lib. nullo pacto poteris dubitare: </s>
                  <s xml:id="echoid-s4565" xml:space="preserve">Verum ne
                    <lb/>
                  in .4. aliquid tibi noui exurgat, te ſcire volo
                    <ref id="ref-0025">corollarium .20. in libr. de quadratu­
                      <lb/>
                    ra parabolę</ref>
                  docere poſſibile eſſe inſcriptionem rectilineæ, ea tamen conditione
                    <reg norm="quam" type="context">quã</reg>
                    <lb/>
                  dicit Archimedes.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4566" xml:space="preserve">In quinta poſtea animaduertendum eſt, quod prima pars, probat tantummodo de
                    <lb/>
                  centro trianguli, et .2. pars probat de centro pentagoni, à te ipſo deinde potes pro-
                    <lb/>
                  bare de centro nonanguli: </s>
                  <s xml:id="echoid-s4567" xml:space="preserve">& ſic de cæteris: </s>
                  <s xml:id="echoid-s4568" xml:space="preserve">eo quod cum probatum fuerit de centro
                    <lb/>
                  figuræ in medio locatæ ſi conſtitutæ poſtea fuerint ſimiles figuræ in portionibus la-
                    <lb/>
                  teralibus habebitur propoſitum in infinitum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4569" xml:space="preserve">Idem intelligendum eſt in .3. propoſitione quamuis exemplum vlterius non ex-
                    <lb/>
                  tendatur quam ad pentagonos.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4570" xml:space="preserve">Sexta verò
                    <reg norm="propoſitio" type="simple">ꝓpoſitio</reg>
                  tibi ſacilis erit, quæ nihilominus
                    <reg norm="pont" type="context">põt</reg>
                    <reg norm="demonſtrari" type="context">demõſtrari</reg>
                  hoc
                    <reg norm="mon" type="context">mõ</reg>
                  ſcili­
                    <lb/>
                  cet. </s>
                  <s xml:id="echoid-s4571" xml:space="preserve">Sint .4.
                    <reg norm="quantitates" type="context">quãtitates</reg>
                    <var>.a.b.c.d.</var>
                  ipſius Archimedis
                    <reg norm="ſupponendo" type="context">ſupponẽdo</reg>
                    <var>.a.</var>
                  pro figura rectilinea
                    <lb/>
                  inſcripta in parabola, et
                    <var>.b.</var>
                  pro reſiduo ipſius parabolę et
                    <var>.c.</var>
                  pro triangulo
                    <var>.a.b.c.</var>
                  in me
                    <lb/>
                  dio ipſius parabolę et
                    <var>.d.</var>
                  pro triangulo
                    <var>.r</var>
                  . </s>
                  <s xml:id="echoid-s4572" xml:space="preserve">Nunc cum
                    <var>.a.</var>
                  maior ſit
                    <var>.c.</var>
                  prout totum ma-
                    <lb/>
                  ius eſt ſua parte, ideo ex .8. quinti maior proportio habebit
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  quam
                    <var>.c.</var>
                  ad
                    <var>.b.</var>
                    <lb/>
                  Cum autem
                    <var>.b.</var>
                  minor ſit
                    <var>.d.</var>
                  ex ſuppoſito, ideo ex eadem dicta, maior proportio habe
                    <lb/>
                  bit
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  quam
                    <var>.c.</var>
                  ad
                    <var>.d.</var>
                  cum verò centrum cuiuſuis figuræ plenæ neceſſariò ſit intra
                    <lb/>
                  ipſam figuram, idcirco centrum reſidui ipſius parabolę intra ipſam reperietur. </s>
                  <s xml:id="echoid-s4573" xml:space="preserve">quod
                    <lb/>
                  ita
                    <reg norm="clarum" type="context">clarũ</reg>
                    <reg norm="per" type="simple">ꝑ</reg>
                  ſe eſt,
                    <reg norm="quemadmodum" type="wordlist">quẽadmodũ</reg>
                  quoduis aliud axioma, & quia
                    <reg norm="dictum" type="context">dictũ</reg>
                    <reg norm="centrum" type="context">centrũ</reg>
                  ex .8. primi
                    <lb/>
                  de centris, neceſſariò eſt in linea
                    <var>.b.h.</var>
                  inter
                    <var>.b.</var>
                  et
                    <var>.h</var>
                  . </s>
                  <s xml:id="echoid-s4574" xml:space="preserve">Sit igitur
                    <var>.g.</var>
                  vnde ex eadem .8. ita
                    <lb/>
                  erit
                    <var>.g.h.</var>
                  ad
                    <var>.h.e.</var>
                  vt
                    <var>.a.</var>
                  ad
                    <var>.b.</var>
                  ergo
                    <var>.g.h.</var>
                  ad
                    <var>.h.e.</var>
                  maior proportio erit
                    <reg norm="quam" type="context">quã</reg>
                    <var>.c.</var>
                  ad
                    <var>.d.</var>
                  hoc eſt
                    <lb/>
                  quam
                    <var>.b.h.</var>
                  ad
                    <var>.f.</var>
                  ex .12. quinti. </s>
                  <s xml:id="echoid-s4575" xml:space="preserve">Sed
                    <reg norm="cum" type="context">cũ</reg>
                    <var>.h.b.</var>
                  maior ſit ipſa
                    <var>.h.g.</var>
                  prout omne totum ma-
                    <lb/>
                  ius eſt ſua parte, ideo maior proportio habebit
                    <var>.h.b.</var>
                  ad
                    <var>.h.e.</var>
                  quam
                    <var>.h.g.</var>
                  ad
                    <var>.h.e.</var>
                  vnde
                    <lb/>
                  multo
                    <reg norm="maiorem" type="context">maiorẽ</reg>
                    <reg norm="quam" type="context">quã</reg>
                    <var>.h.b.</var>
                  ad
                    <var>.f.</var>
                  ex
                    <reg norm="coni" type="context">cõi</reg>
                    <reg norm="conceptu" type="context">cõceptu</reg>
                  , </s>
                  <s xml:id="echoid-s4576" xml:space="preserve">quare
                    <var>.h.e.</var>
                  erit minor ipſa
                    <var>.f.</var>
                  ex .10.
                    <reg norm="quinti" type="context">quĩti</reg>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4577" xml:space="preserve">Septima verò et .8. propoſitio nullius tibi erit difficultatis.</s>
                </p>
              </div>
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