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-div495" type="section" level="3" n="5">
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                <p>
                  <s xml:id="echoid-s3195" xml:space="preserve">
                    <pb o="254" rhead="IO. BABPT. BENED." n="266" file="0266" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0266"/>
                  cundam verò ex .37. et .38. eiuſdem, </s>
                  <s xml:id="echoid-s3196" xml:space="preserve">propterea quod in .37. probat mediante maiori
                    <lb/>
                  diametro ipſius hyperbolis & defectionis, In .38. autem mediante minori diametro
                    <lb/>
                  ordinatè ad maiorem.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3197" xml:space="preserve">Tertia autem paſſio, non niſi circulo conuenit; </s>
                  <s xml:id="echoid-s3198" xml:space="preserve">pace ipſius Cardani dictum ſit.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3199" xml:space="preserve">Quapropter ſit circulus
                    <var>.q.o.b.</var>
                  cuius diameter ſit
                    <var>.q.b.</var>
                  contingentes vero ab extre
                    <lb/>
                  mitate diametri ſint
                    <var>.d.b.</var>
                  et
                    <var>.q.g.</var>
                  per punctum autem
                    <var>.o.</var>
                  quoduis, ipſius
                    <reg norm="circunferentiæ" type="context">circũferentiæ</reg>
                  ,
                    <lb/>
                  tranſeant
                    <var>.b.o.g.</var>
                  et
                    <var>.q.o.d</var>
                  . </s>
                  <s xml:id="echoid-s3200" xml:space="preserve">tunc dico productum
                    <var>.q.o.</var>
                  in
                    <var>.q.d.</var>
                  vel
                    <var>.b.o.</var>
                  in
                    <var>.b.g.</var>
                  ęquale eſ-
                    <lb/>
                  ſe quadrato
                    <var>.q.b.</var>
                  quod ita probo.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3201" xml:space="preserve">Nam angulus
                    <var>.q.b.d.</var>
                  ſeu
                    <var>.b.q.g.</var>
                  rectus eſt ex .17. tertij Eucli. et
                    <var>.b.o.q.</var>
                  ſimiliter re-
                    <lb/>
                  ctus ex .30. ipſius lib. angulus verò
                    <var>.b.q.d.</var>
                  ſeu
                    <var>.q.b.g.</var>
                  communis eſt. </s>
                  <s xml:id="echoid-s3202" xml:space="preserve">quare
                    <var>.b.q.</var>
                  media
                    <lb/>
                  proportionalis erit inter dictas lineas
                    <var>.q.d.</var>
                  et
                    <var>.q.o.</var>
                  & inter
                    <var>.b.g.</var>
                  et
                    <var>.b.o</var>
                  . </s>
                  <s xml:id="echoid-s3203" xml:space="preserve">Vnde ſequetur
                    <lb/>
                  propoſitum ex .16.6. Eucli.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3204" xml:space="preserve">Sed ſi circa diametrum
                    <var>.q.b.</var>
                  mente fingamus aliquam elipſim, quætangat ipſum
                    <lb/>
                    <figure xlink:label="fig-0266-01" xlink:href="fig-0266-01a" number="299">
                      <image file="0266-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0266-01"/>
                    </figure>
                  circulum duobus punctis me-
                    <lb/>
                  diantibus
                    <var>.q.</var>
                  et
                    <var>.b.</var>
                  (nam pluribus
                    <lb/>
                  eſſet impoſſibile, ex .27. quarti
                    <lb/>
                  Pergei) clarè patebit, quod
                    <reg norm="pum" type="context">pũ</reg>
                    <lb/>
                  ctus
                    <var>.o.</var>
                  erit extra
                    <reg norm="circunferentiam" type="context">circunferentiã</reg>
                    <lb/>
                  ipſius defectionis, </s>
                  <s xml:id="echoid-s3205" xml:space="preserve">quare ipſa cir
                    <lb/>
                  cunferentia ſecabit
                    <var>.b.g.</var>
                  vel
                    <var>.q.
                      <lb/>
                    d.</var>
                  in alio puncto, vnde ipſi non
                    <lb/>
                  occurret id quod probauimus
                    <lb/>
                  de circulo.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3206" xml:space="preserve">Admiratus etiam ſum, ipſum
                    <lb/>
                  Cardanum dicere hyperbolem
                    <lb/>
                  ita vocari, eo quod angulus con
                    <lb/>
                  tentus ab axe ipſius figuræ, & à
                    <lb/>
                  latere trigoni in hyperbole ma-
                    <lb/>
                  ior ſit quam in parabole, quod
                    <lb/>
                  eriam confirmat paulo inferius,
                    <lb/>
                  nam hoc verum non eſt, imo fal
                    <lb/>
                  ſiſſimum. </s>
                  <s xml:id="echoid-s3207" xml:space="preserve">Talis enim ſectio ita
                    <lb/>
                  nominata fuit, hoc eſt hyperbo
                    <lb/>
                  les, ſimili ratione, qua elipſis ſeu
                    <lb/>
                  defectio etiam vocata fuit, nam
                    <lb/>
                  ſicut in ipſa defectione quadra-
                    <lb/>
                  tum ordinatę
                    <var>.l.m.</var>
                  minor eſt pro
                    <lb/>
                  ducto lineæ
                    <var>.e.m.</var>
                  in
                    <var>.e.t.</var>
                  per figu
                    <lb/>
                  ram ſimilcm producto
                    <var>.d.e.</var>
                  in
                    <var>.e.
                      <lb/>
                    t.</var>
                  quæ eandem obtineat
                    <reg norm="altitu- dinem" type="context">altitu-
                      <lb/>
                    dinẽ</reg>
                  ipſius
                    <var>.e.m.</var>
                  vt ipſe Pergeus
                    <lb/>
                  monſtrat in .13. primi lib. ita in
                    <lb/>
                  hyperbole
                    <reg norm="dictum" type="context">dictũ</reg>
                  quadratum ex
                    <lb/>
                  cedit quantitatem illius figuræ,
                    <lb/>
                  per ſimilem dictæ vt in .12.
                    <reg norm="ipſius" type="simple">ipſiꝰ</reg>
                    <lb/>
                  Pergei facilè videre eſt. </s>
                  <s xml:id="echoid-s3208" xml:space="preserve">ſed
                    <reg norm="prae­ ter" type="simple">prę­
                      <lb/>
                    ter</reg>
                  illas paſſiones, quas notat </s>
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