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-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div642" type="section" level="3" n="28">
              <div xml:id="echoid-div657" type="letter" level="4" n="5">
                <pb o="342" rhead="IO. BAPT. BENED." n="354" file="0354" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0354"/>
                <p>
                  <s xml:id="echoid-s4152" xml:space="preserve">Alia etiam via poſſumus idem concludere. </s>
                  <s xml:id="echoid-s4153" xml:space="preserve">Imaginemur maiorem axem alicu-
                    <lb/>
                  ius ellipſis tranſire per duo puncta
                    <var>.r.</var>
                  et
                    <var>.b.</var>
                  ſupponendo ipſa puncta, ea eſle, quæ ita
                    <lb/>
                  axem diuidunt, vt ſingula produ-
                    <lb/>
                    <figure xlink:label="fig-0354-01" xlink:href="fig-0354-01a" number="385">
                      <image file="0354-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0354-01"/>
                    </figure>
                  cta fectionum ſint, vt inquit Per-
                    <lb/>
                  geus. </s>
                  <s xml:id="echoid-s4154" xml:space="preserve">imaginemur, etiam
                    <var>.p.h.</var>
                  con
                    <lb/>
                  tiguam eſſe ipſi ellipſi in
                    <reg norm="puncto" type="context">pũcto</reg>
                    <var>.a.</var>
                    <lb/>
                  vnde ſi protractæ fuerint duæ
                    <var>.r.a.</var>
                    <lb/>
                  et
                    <var>.b.a.</var>
                  habebimus ex .48. tertijip-
                    <lb/>
                  ſius Pergei angulos
                    <var>.b.a.h.</var>
                  et
                    <var>.r.a.
                      <lb/>
                    p.</var>
                  inuicem æquales. </s>
                  <s xml:id="echoid-s4155" xml:space="preserve">Ducendo
                    <lb/>
                  poſtea ad quoduis punctum ipſius
                    <lb/>
                    <var>p.h.</var>
                  duas
                    <var>.b.o.</var>
                  et
                    <var>.r.o.</var>
                  certi erimus,
                    <lb/>
                  quod ſecabuntur à gyro oxygo-
                    <lb/>
                  nio, quarum vna ſecta ſit in pun-
                    <lb/>
                  cto
                    <var>.i.</var>
                  ducta poſtea
                    <var>.i.r.</var>
                  clarum erit ex .52. dicti, quod longitudo
                    <var>.b.i.r.</var>
                  æqualis erit lon
                    <lb/>
                  gitudini
                    <var>.b.a.r.</var>
                  & minor ipſa
                    <var>.b.o.r.</var>
                  ex .21. primi Euclid.</s>
                </p>
              </div>
              <div xml:id="echoid-div660" type="letter" level="4" n="6">
                <head xml:id="echoid-head505" style="it" xml:space="preserve">Deerrore Euclidis circa ſpeculum vstorium.</head>
                <head xml:id="echoid-head506" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s4156" xml:space="preserve">VErum ſpeculum vſtorium, illud non eſt, quod ab Euclide traditum fuit, &
                    <reg norm="quod" type="simple">ꝙ</reg>
                    <lb/>
                  tu etiam putas, Nam Euclides errat, cum credat radios reflexos à ſuperficie
                    <lb/>
                  ſphærica concaua ſeinuicem in centro ſpeculi interſecare. </s>
                  <s xml:id="echoid-s4157" xml:space="preserve">Nam cum omnes lineę
                    <lb/>
                  recte à centro, & cir cunferentia alicuius ſphæræ terminatæ, ſint eidem circunferen-
                    <lb/>
                  tiæ perpendiculares, ſequeretur ex neceſſitate radios incidentiæ etiam perpendicu
                    <lb/>
                  lares eidem ſuperficiei eſſe, cum anguli incidentiæ ſemper æquales ſint angulis re-
                    <lb/>
                  flexionis, vnde etiam ex neceſſitate ſequeretur punctum corporis lucidi, à quo radij
                    <lb/>
                  luminoſi excunt, in centro ſpeculi reperiri. </s>
                  <s xml:id="echoid-s4158" xml:space="preserve">quod quidem falſiſſimum eſt.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4159" xml:space="preserve">Alia etiam via poſſum hanc oſtendere impoſſibilitatem, & tibi probabo, quod
                    <lb/>
                  in nullo aliquo puncto poſſunt inuicem conuenire ipſi radijrefle xi omnes.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4160" xml:space="preserve">Sit igitur
                    <var>.l.a.c.</var>
                    <reg norm="conis" type="context">cõis</reg>
                  ſectio ſuperficiei reflexionis cum ſpeculo, cuius centrum ſit
                    <var>.o.</var>
                    <lb/>
                  punctum verò lucidum ſit
                    <var>.g.</var>
                    <reg norm="protrahaturque" type="simple">protrahaturq́</reg>
                    <var>.g.o.a</var>
                  . </s>
                  <s xml:id="echoid-s4161" xml:space="preserve">Nunc autem primum dico, quod
                    <lb/>
                  radij reflexi à punctis diuerſarum
                    <reg norm="diſtantiarum" type="context">diſtantiarũ</reg>
                  ab
                    <var>.a.</var>
                  non
                    <reg norm="coincident" type="context">coincidẽt</reg>
                  inuicem in aliquo
                    <lb/>
                  puncto lineę
                    <var>.g.o.a</var>
                  : ſint ergo duo puncta
                    <var>.u.</var>
                  et
                    <var>.r.</var>
                  diuerſarum
                    <reg norm="diſtantiarum" type="context">diſtantiarũ</reg>
                  ab
                    <var>.a.</var>
                  à quibus
                    <lb/>
                  veniant duo radij incidentiæ
                    <var>.g.r.</var>
                  et
                    <var>.g.u.</var>
                  radius verò reflexus ab
                    <var>.r.</var>
                  ſit
                    <var>.r.e.</var>
                  protrahatur
                    <lb/>
                    <var>u.e.</var>
                  quam dico effe non poſſe radium reflexum ab
                    <var>.u.</var>
                  quotieſcunque eius incidens
                    <lb/>
                  deſcendat ab
                    <var>.g</var>
                  . </s>
                  <s xml:id="echoid-s4162" xml:space="preserve">Protrahantur ergo duæ lineæ
                    <var>.o.r.</var>
                  et
                    <var>.o.u.</var>
                  vnde cum dixerit aliquis
                    <lb/>
                    <var>u.e.</var>
                    <reg norm="reflexum" type="context">reflexũ</reg>
                  eſſe ipſius
                    <var>.g.u.</var>
                  igitur anguli
                    <var>.g.u.o.</var>
                  et
                    <var>.o.u.e.</var>
                  erunt inuicem æquales, & ſic
                    <lb/>
                  etiam erunt duo
                    <var>.g.r.o.</var>
                  et
                    <var>.o.r.e.</var>
                  vnde ex tertia ſexti & .11. quinti Eucli. proportio
                    <var>.g.
                      <lb/>
                    u.</var>
                  ad
                    <var>.u.e.</var>
                  æqualis eſſet ei, quæ
                    <var>.g.r.</var>
                  ad
                    <var>.r.e.</var>
                  quod quidem impoſſibile eſſe demonſtra-
                    <lb/>
                  bo, eo quod cum
                    <var>.g.u.</var>
                  maior ſit
                    <var>.g.r.</var>
                  ex .8. tertij, erit ex .8. quinti proportio ipſius
                    <var>.g.u.</var>
                    <lb/>
                  ad
                    <var>.r.e.</var>
                  maior proportione ipſius
                    <var>.g.r.</var>
                  ad
                    <var>.r.e.</var>
                  ſed ex .7. tertij
                    <var>.u.e.</var>
                  minor eſt
                    <var>.r.e.</var>
                  erit igi-
                    <lb/>
                  tur ex dicta .8. quinti maior proportio
                    <reg norm="ipſius" type="simple">ipſiꝰ</reg>
                    <var>.g.u.</var>
                  ad
                    <var>.u.e.</var>
                  quam
                    <var>.g.u.</var>
                  ad
                    <var>.r.e.</var>
                  vnde eo ma­ </s>
                </p>
              </div>
            </div>
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