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-div650" type="letter" level="4" n="3">
                <p>
                  <s xml:id="echoid-s4103" xml:space="preserve">
                    <pb o="338" rhead="IO. BAPT. BENED." n="350" file="0350" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0350"/>
                  clarum erit ex rationibus ſupradictis nos ipſam videre in
                    <reg norm="communi" type="context">cõmuni</reg>
                  concurſu ipſorum
                    <lb/>
                  axium viſualium, qui axes cum reperiantur vnà cum ipſis radijs reflexis
                    <var>.n.a.</var>
                  et
                    <var>.t.u.</var>
                    <lb/>
                  ex neceſſitate ſeinuicem
                    <reg norm="ſecabunt" type="context">ſecabũt</reg>
                  in catheto
                    <var>.b.c.</var>
                  cum extendantur in ipſis ſuperficie-
                    <lb/>
                  bus reflexionum, quæ ſuperficies nihil aliud commune inuicem habent, quam cathe
                    <lb/>
                  tum dictum
                    <var>.b.c.</var>
                  ſit igitur in puncto
                    <var>.d</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4104" xml:space="preserve">Ex his dictis alia oritur neceſſitas, hoc eſt, quod quotieſcunque vnam tantummo
                    <lb/>
                  do imaginem obiecti
                    <var>.b.</var>
                  videmus, dato quod duæ ſuperficies reflexionis ſint, & non
                    <lb/>
                  vna tantum, tunc angulos
                    <var>.n.</var>
                  et
                    <var>.t.</var>
                  ſemper inuicem æquales eſſe oportebit. </s>
                  <s xml:id="echoid-s4105" xml:space="preserve">Vnde ar-
                    <lb/>
                  cus
                    <var>.n.c.</var>
                  et
                    <var>.t.c.</var>
                  ex neceſſitate inuicem æquales erunt.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4106" xml:space="preserve">Scimas enim ex .3. ſexti Euclid. quod eadem proportio erit ipſius
                    <var>.b.n.</var>
                  ad
                    <var>.n.
                      <lb/>
                    d.</var>
                  quę ipſius
                    <var>.b.x.</var>
                  ad
                    <var>.x.d.</var>
                  & ipſius
                    <var>.b.t.</var>
                  ad
                    <var>.t.d.</var>
                  ſimiliter, </s>
                  <s xml:id="echoid-s4107" xml:space="preserve">quare ipſiusb
                    <var>.n.</var>
                  ad
                    <var>.n.
                      <lb/>
                    d.</var>
                  erit vt ipſius
                    <var>.b.t.</var>
                  ad
                    <var>.t.d</var>
                  . </s>
                  <s xml:id="echoid-s4108" xml:space="preserve">Vnde ſequitur
                    <var>.b.n.</var>
                  æqualem eſſe ipſi
                    <var>.b.t.</var>
                  et
                    <var>.n.d.</var>
                    <lb/>
                  ipſi
                    <var>.t.d.</var>
                  vt à medio circulo
                    <var>.E.</var>
                  potes videre, quamuis etiam
                    <var>.b.</var>
                  non eſſet extremum
                    <lb/>
                  diametri, ſed vbicunque volueris in ipſo diametro, vel
                    <reg norm="etiam" type="context">etiã</reg>
                  protracta, eo quod pun-
                    <lb/>
                  ctum
                    <var>.n.</var>
                  & punctum
                    <var>.t.</var>
                  in eodem ſemicirculo, vel in æqualibus ſemicirculis, non
                    <reg norm="poſsent" type="context">poſsẽt</reg>
                    <lb/>
                  aliter in ipſa circunferentia locari,
                    <reg norm="eandem" type="context">eãdem</reg>
                  ſeruando proportionem
                    <var>.b.n.</var>
                  ad
                    <var>.n.d.</var>
                  vt
                    <var>.b.
                      <lb/>
                    t.</var>
                  ad
                    <var>.t.d.</var>
                  </s>
                  <s xml:id="echoid-s4109" xml:space="preserve">propterea quod in omni alio ſitu exiſtente puncto
                    <var>.t.</var>
                  ipſa
                    <var>.b.t.</var>
                  eſſet aut maior
                    <lb/>
                  aut minor ipſa
                    <var>.b.n.</var>
                  et
                    <var>.t.d.</var>
                  aut minor, aut maior ipſa
                    <var>.t.d.</var>
                  ex .7. & 14. tertij Eucli. </s>
                  <s xml:id="echoid-s4110" xml:space="preserve">vnde
                    <lb/>
                  aut maior, aut minor proportio eſſet ipſius
                    <var>.b.t.</var>
                  ad
                    <var>.t.d.</var>
                  quam ipſius
                    <var>.b.n.</var>
                  ad
                    <var>.n.d.</var>
                  & non
                    <lb/>
                  eadem.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4111" xml:space="preserve">Nunc è conuerſo ſi
                    <var>.b.n.</var>
                  et
                    <var>.b.t.</var>
                  ſunt ſibi inuicem æquales, & ſic
                    <var>.n.d.</var>
                  cum
                    <var>.t.d.</var>
                  ſequi-
                    <lb/>
                  tur ex .8. primi Eucli. angulos
                    <var>.n.</var>
                  et
                    <var>.t.</var>
                  inuicem æquales eſſe.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4112" xml:space="preserve">Ab ijſdem ſpeculationibus potes etiam videre vnde accidat quod partes ſuperio
                    <lb/>
                  res alicuius obiecti reflexæ à tali ſpeculo concauo videntur nobis inferiores eſſe, &
                    <lb/>
                  inferiores appareant ſuperiores, & dextræ ſiniſtræ, & ſiniſtræ dextræ. </s>
                  <s xml:id="echoid-s4113" xml:space="preserve">quod autem
                    <lb/>
                  hucuſque demonſtraui de ſpeculis planis, & ſphæricis concauis, ratiocinare tu ijſdem
                    <lb/>
                  medijs circa ſphærica conuexa, vbi clarè videbis puncta huiuſmodi ſpeculi conuexi,
                    <lb/>
                  à quibus reflectitur imago obiecti ad ambos oculos, ſemper oportere æquidiſtantia
                    <lb/>
                  eſſe à
                    <reg norm="puncto" type="context">pũcto</reg>
                  communi ipſius ſuperficiei ſpeculi, & catheto incidentiæ, dum unam tan
                    <lb/>
                  tummodo imaginem ipſius obiecti videmus, & à diuerſis ſuperficiebus reflexionum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4114" xml:space="preserve">Nolo etiam prætermittere, quod nunc mihi ſuccurrit, hoc eſt quod poſſet ali-
                    <lb/>
                  quis duos ſitus inuenire, vnum pro oculo, alterum verò pro obiecto, reſpectu alicu-
                    <lb/>
                  ius ſpeculi concaui, ſphęroidis prolatæ, vt reflexio ipſius obiecti videretur, vt linea
                    <lb/>
                  diuidens per æqualia ipſum ſpeculum. </s>
                  <s xml:id="echoid-s4115" xml:space="preserve">Reſpectu verò alicuius ſpeculi concaui ſphæ-
                    <lb/>
                  roidis oblongæ, vt reflexio obiecti ad oculum veniret à tota ſuperficie ipſius ſpecu-
                    <lb/>
                  li, vnde tota ſuperficies ipſius ſpeculi videretur colorata illo colore cuius eſſet
                    <lb/>
                  obiectum, quæ quidem paſſiones
                    <reg norm="pendent" type="context">pendẽt</reg>
                  à .48. tertij lib. ipſius Pergei, vt ex te ipſo fa
                    <lb/>
                  cile videre potes, propter æqualitatem angulorum reflexionis, & incidentiæ.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4116" xml:space="preserve">Opinio autem mea, quam ſcire cupis de imagine obiecti reflexa, quam putas eſ-
                    <lb/>
                  ſe in ſuperficie ſpeculi, hæc eſt, quod nec in ſuperficie, nec ultra, nec citra eam eſt ip
                    <lb/>
                  ſa imago, quod autem vltra non ſit, hoc puto nulli dubium eſſe. </s>
                  <s xml:id="echoid-s4117" xml:space="preserve">eadem etiam ra-
                    <lb/>
                  tione non erit citra ſuperficiem ſpeculi concaui, quamuis ipſam nos compræhenda-
                    <lb/>
                  mus in concurſu radiorum viſualium, tam ab vno ſpeculo quam ab alio reflexione
                    <lb/>
                  facta. </s>
                  <s xml:id="echoid-s4118" xml:space="preserve">Sed quòd ipſa neque ſit in ipſa ſpeculi ſuperficie, manifeſtum erit ex hoc,
                    <reg norm="quod" type="simple">ꝙ</reg>
                    <lb/>
                  duo ſpectantes in eodem ſpeculo, duas diuerſas imagines vident, tres,
                    <reg norm="autem" type="wordlist">aũt</reg>
                  tres, qua-
                    <lb/>
                  tuor, quatuor, & ſic deinceps, vnde tot eſſent imagines ſupra ſuperficiem ſpeculi,
                    <lb/>
                  quot obiecta,
                    <reg norm="quod" type="simple">ꝙ</reg>
                  tamen ita non eſt, nec plus eſt in vno loco ipſa imago, quam in alio, </s>
                </p>
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