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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EPISTOL AE.
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          <div xml:id="echoid-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div676" type="section" level="3" n="30">
              <div xml:id="echoid-div676" type="letter" level="4" n="1">
                <pb o="353" rhead="EPISTOL AE." n="365" file="0365" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0365"/>
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            <div xml:id="echoid-div680" type="section" level="3" n="31">
              <div xml:id="echoid-div680" type="letter" level="4" n="1">
                <head xml:id="echoid-head517" xml:space="preserve">DE MODO DIVIDENDI PARABOLAM
                  <lb/>
                propoſitam ſecundum datam proportionem.</head>
                <head xml:id="echoid-head518" style="it" xml:space="preserve">Pamphilo Gothfrid.</head>
                <p>
                  <s xml:id="echoid-s4244" xml:space="preserve">
                    <emph style="sc">QVod</emph>
                  à me quæris, eſt quidem poſſibile, non tamen adhuc inuentum, quo
                    <lb/>
                  niam nemo ad
                    <reg norm="hunc" type="context">hũc</reg>
                  vſque diem diuiſit vnam datam proportionem in tres
                    <lb/>
                  æquales partes, ſed ſi hoc pro facto conceſſeris, nunc tibi morem geram.
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4245" xml:space="preserve">Nam proponis n. ihi parabolem
                    <var>.x.b.e.</var>
                  cum proportione
                    <var>.p.</var>
                  ad
                    <var>.q.</var>
                    <reg norm="cupiſque" type="simple">cupiſq́;</reg>
                    <lb/>
                  ſcire modum diuidendi ipſam parabolem vna mediante linea parallela ipſi baſi, ita
                    <lb/>
                  vt eandem habeat proportionem tota parabola ad partem abſciſſam, quæ eſt inter
                    <var>.
                      <lb/>
                    p.</var>
                  et
                    <var>.q</var>
                  . </s>
                  <s xml:id="echoid-s4246" xml:space="preserve">Ad quod faciendum, ſupponendum primò datam proportionem inter
                    <var>.
                      <lb/>
                    p.</var>
                  et
                    <var>.q.</var>
                  diuiſam eſſe in tres partes æquales, duabus lineis mediantibus
                    <var>.n.</var>
                  et
                    <var>.u.</var>
                  quæ me
                    <lb/>
                  diæ proportionales vocabuntur inter
                    <var>.p.</var>
                  et
                    <var>.q</var>
                  . </s>
                  <s xml:id="echoid-s4247" xml:space="preserve">deinde à quouis puncto circunferentię
                    <lb/>
                  ipſius figuræ ducatur parallela baſi
                    <var>.x.e.</var>
                  poſtea verò per puncta media harum dua-
                    <lb/>
                  rum
                    <reg norm="æquidiſtantium" type="context">æquidiſtantiũ</reg>
                  protrahatur
                    <var>.g.b.</var>
                  quæ diameter erit ſectionis, ex 28. ſecundi Per-
                    <lb/>
                  gei, </s>
                  <s xml:id="echoid-s4248" xml:space="preserve">diuidatur deinde hæc diameter in puncto
                    <var>.a.</var>
                  ita quod eadem proportio ſit ipſius
                    <lb/>
                    <var>b.g.</var>
                  ad
                    <var>.b.a.</var>
                  quæ ipſius
                    <var>.p.</var>
                  ad
                    <var>.u.</var>
                  quod tibi facile erit, ſecando à linea
                    <var>.p.</var>
                  partem
                    <var>.i.</var>
                  æqua
                    <lb/>
                  lem ipſi
                    <var>.u.</var>
                  tali modo poſtea diuidendo
                    <var>.b.g.</var>
                  ex .12. ſexti, ducatur a puncto
                    <var>.a.</var>
                  ipſa
                    <var>.d.
                      <lb/>
                    h.</var>
                  parallclam ipſi
                    <var>.x.e.</var>
                  & habebitur propoſitum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4249" xml:space="preserve">Pro cuius reiratione, ſcies primum quod
                    <var>.h.d.</var>
                  diuiſa erit à diametro
                    <var>.b.g.</var>
                  per æqua
                    <lb/>
                  lia ex .7. primi Pergei, vel ſi cogitabimus aliquam lineam tangentem ipſam parabo
                    <lb/>
                  lam in puncto
                    <var>.b</var>
                  . </s>
                  <s xml:id="echoid-s4250" xml:space="preserve">tunc ex quinta ſecundi ipſius Pergei habebimus ipſam eſſe paralle-
                    <lb/>
                  lam
                    <var>.e.x.</var>
                  & ex .30. primi Eucli. erit ſimiliter æquidiſtans
                    <var>.d.h.</var>
                  vnde ex .46. primi eiuſ-
                    <lb/>
                  dem Pergei
                    <var>.h.a.</var>
                  æqualis erit
                    <var>.d.a</var>
                  . </s>
                  <s xml:id="echoid-s4251" xml:space="preserve">Protrahatur deinde
                    <var>.e.b</var>
                  : d b:
                    <var>x.b.</var>
                  et
                    <var>.h.b.</var>
                  vnde ex .17
                    <lb/>
                  lib. de quadratura parabolæ Archimedis, habebimus eandem proportionem ſuper
                    <lb/>
                  ficiei totalis parabolæ
                    <var>.x.b.e.</var>
                  ad trigonum
                    <var>.x.b.e.</var>
                  quæ portionis
                    <var>.h.b.d.</var>
                  ad ſuum
                    <reg norm="tri- gonum" type="context">tri-
                      <lb/>
                    gonũ</reg>
                  , eo quod
                    <reg norm="tam" type="context">tã</reg>
                  vna quàm alia erit ſeſquitertia,
                    <reg norm="eius" type="simple">eiꝰ</reg>
                    <reg norm="etiam" type="context">etiã</reg>
                  medietates ſic ſe
                    <reg norm="habebunt" type="context">habebũt</reg>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4252" xml:space="preserve">Vnde permutando, proportio medietatis totalis parabolę ad medietatem partia
                    <lb/>
                  lem ipſius, æqualis erit proportioni trianguli
                    <lb/>
                    <var>g.b.e.</var>
                  ad triangulum
                    <var>.a.b.d.</var>
                  ſed ex .20. primi
                    <lb/>
                  Pergei, eadem eſt proportio quadrati ipſius
                    <var>.
                      <lb/>
                      <anchor type="figure" xlink:label="fig-0365-01a" xlink:href="fig-0365-01"/>
                    g.e.</var>
                  ad quadratum ipſius
                    <var>.a.d.</var>
                  quæ
                    <var>.b.g.</var>
                  ad
                    <var>.b.a.</var>
                    <lb/>
                  hoc eſt, vt
                    <var>.g.e.</var>
                  ad
                    <var>.a.o.</var>
                  ex ſimilitudine triangu-
                    <lb/>
                  lorum, & quia
                    <var>.b.g.</var>
                  ad
                    <var>.b.a.</var>
                  eſt ſicut
                    <var>.p.</var>
                  ad
                    <var>.u.</var>
                  ita
                    <lb/>
                  igitur erit quadrati ipſius
                    <var>.g.e.</var>
                  ad quadratum
                    <lb/>
                  ipſeus
                    <var>.a.d.</var>
                  </s>
                  <s xml:id="echoid-s4253" xml:space="preserve">quare
                    <var>.g.e.</var>
                  ad
                    <var>.a.d.</var>
                  erit ut p. ad
                    <var>.n.</var>
                    <lb/>
                  ex .18. ſexti Euclid. </s>
                  <s xml:id="echoid-s4254" xml:space="preserve">ſed cum ex .24. eiuſdem
                    <lb/>
                  proportio trianguli
                    <var>.b.g.e.</var>
                  ad triangulum
                    <var>.b.
                      <lb/>
                    a.d.</var>
                  compoſita ſit ex proportione
                    <var>.g.e.</var>
                  ad
                    <var>.a.
                      <lb/>
                    d.</var>
                  er. ex
                    <var>.g.b.</var>
                  ad
                    <var>.b.a.</var>
                  hoc eſt
                    <var>.g.e.</var>
                  ad
                    <var>.a.o.</var>
                  &
                    <lb/>
                  quia
                    <reg norm="proportio" type="simple">ꝓportio</reg>
                    <var>.g.e.</var>
                  ad
                    <var>.a.o.</var>
                  æqualis eſt ei quæ
                    <var>.p.</var>
                    <lb/>
                  ad. u ex .11. quinti Euclid. </s>
                  <s xml:id="echoid-s4255" xml:space="preserve">& proportio
                    <var>.g.e.</var>
                    <lb/>
                  ad
                    <var>.a.d.</var>
                  æqualis eſt ei quæ
                    <var>.p.</var>
                  ad
                    <var>.n.</var>
                  hoc eſt vt
                    <var>.u.</var>
                    <lb/>
                  ad
                    <var>.q.</var>
                  ergo proportio trianguli
                    <var>.b.g.e.</var>
                  ad trian-
                    <lb/>
                  gulum
                    <var>.b.a.d.</var>
                  compoſita erit ex ca quę
                    <var>.p.</var>
                  ad
                    <var>.u.</var>
                    <lb/>
                  & ex ea quæ
                    <var>.u.</var>
                  ad
                    <var>.q.</var>
                  æqualis ergo erit ei, quæ
                    <lb/>
                  p. ad
                    <var>.q.</var>
                  & ita medietates parabolarum, & eorum dupla.</s>
                </p>
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                  <figure xlink:label="fig-0365-01" xlink:href="fig-0365-01a">
                    <image file="0365-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0365-01"/>
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