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]

Table of handwritten notes

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              <div xml:id="echoid-div662" type="letter" level="4" n="7">
                <p>
                  <s xml:id="echoid-s4168" xml:space="preserve">
                    <pb o="344" rhead="IO. BAPT. BENED." n="356" file="0356" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0356"/>
                  refleyus ſecabit cathetum
                    <var>.b.o.</var>
                  in puncto
                    <var>.c.</var>
                  intra ſpeculum, nec dubitandum eſt quin
                    <lb/>
                  linea
                    <var>.e.b.</var>
                  ſectura ſit
                    <var>.b.o.</var>
                  eo quod cum angulus
                    <var>.o.e.c.</var>
                  ſit maior angulo
                    <var>.e.o.c.</var>
                  ex .19.
                    <lb/>
                  primi, & ſimiliter angulus
                    <var>.b.e.f.</var>
                  ſequitur ex .13. dicti, angulos
                    <var>.b.e.o.</var>
                  et
                    <var>.e.o.b.</var>
                  eſſe mi
                    <lb/>
                  nores duobus rectis, vnde ex penultima petitione primi, duæ lineæ
                    <var>.b.e.</var>
                  et
                    <var>.o.b.</var>
                    <reg norm="inuicem" type="context">inuicẽ</reg>
                    <lb/>
                  concurrent. </s>
                  <s xml:id="echoid-s4169" xml:space="preserve">Quare poſſumus ex hoc, quoddam corollarium extrahere, hoc eſt
                    <lb/>
                    <reg norm="neceſſarium" type="context">neceſſariũ</reg>
                    <reg norm="semper" type="context">sẽper</reg>
                  exiſtat, vt linea
                    <var>.c.e.</var>
                  minor eſſe linea
                    <var>.c.o</var>
                  . </s>
                  <s xml:id="echoid-s4170" xml:space="preserve">Sed vnde eueniat quod ip
                    <lb/>
                  ſa neceſſariò debeat ſemper maior eſſe ipſa
                    <var>.c.g.</var>
                  clarum eſt ex .7. tertij Eucli. </s>
                  <s xml:id="echoid-s4171" xml:space="preserve">Nunc
                    <lb/>
                  imaginemur ductas eſſe duas
                    <reg norm="tangentes" type="context">tãgentes</reg>
                    <var>.b.d.</var>
                  et
                    <var>.b.h.</var>
                  & ab
                    <var>.e.</var>
                    <reg norm="ipsam" type="context">ipsã</reg>
                    <var>.e.i.</var>
                  vnde certi erimus,
                    <lb/>
                  quod ab interuallo inter
                    <var>.h.</var>
                  et
                    <var>.d.</var>
                  punctum
                    <var>.b.</var>
                    <reg norm="ponſſibile" type="context">põſſibile</reg>
                  ſit vt reflectatur. </s>
                  <s xml:id="echoid-s4172" xml:space="preserve">Accipiamus
                    <lb/>
                  nunc
                    <var>.p.c.</var>
                  minorem medietate ipſius
                    <var>.b.c.</var>
                  & à puncto
                    <var>.p.</var>
                  imaginemur tangentem
                    <var>.p.q.</var>
                    <lb/>
                  in puncto
                    <var>.q.</var>
                  prorractaq́ue ſit
                    <var>.b.q.</var>
                  vt radius incidentiæ, </s>
                  <s xml:id="echoid-s4173" xml:space="preserve">tunc dico, radium reflexum
                    <lb/>
                  ipſius
                    <var>.b.q.</var>
                    <reg norm="non" type="context">nõ</reg>
                  concurrere in eodem puncto
                    <var>.c.</var>
                  ipſius catheti, ſi vero dixeris
                    <reg norm="quod" type="simple">ꝙ</reg>
                  ſic. </s>
                  <s xml:id="echoid-s4174" xml:space="preserve">Eſto
                    <lb/>
                    <reg norm="igitur" type="simple">igit̃</reg>
                  radius dictus
                    <var>.c.q.s</var>
                  . </s>
                  <s xml:id="echoid-s4175" xml:space="preserve">Imaginemur
                    <reg norm="tangentem" type="context context">tãgentẽ</reg>
                    <var>.e.i.</var>
                  in puncto
                    <var>.e.</var>
                  vnde ex .18. quinti Alha
                    <lb/>
                  zem, vel .12. ſexti Vitellionis proportio
                    <var>.b.i.</var>
                  ad
                    <var>.i.c.</var>
                  erit, vt
                    <var>.b.o.</var>
                  ad
                    <var>.o.c.</var>
                  & ſimiliter erit
                    <lb/>
                  ipſius
                    <var>.b.p.</var>
                  ad
                    <var>.p.c.</var>
                  vt
                    <var>.b.o.</var>
                  ad
                    <var>.o.c.</var>
                  ex eadem. </s>
                  <s xml:id="echoid-s4176" xml:space="preserve">Quare ex .11. quinti Eucli. proportio ip
                    <lb/>
                  ſius
                    <var>.b.p.</var>
                  ad
                    <var>.p.c.</var>
                  erit vt ipſius
                    <var>.b.i.</var>
                  ad
                    <var>.i.c.</var>
                  ſed quia
                    <var>.p.b.</var>
                  vt pars ipſius
                    <var>.b.i.</var>
                  minor eſt ip-
                    <lb/>
                  ſa, ergo ex .14. dicti
                    <var>.p.c.</var>
                  minor erit ipſa
                    <var>.c.i.</var>
                  hoc eſt totum minus ſua parte, quod eſt
                    <lb/>
                  impoſſibile, </s>
                  <s xml:id="echoid-s4177" xml:space="preserve">quare non in ipſo catheto videbitur imago ipſius obiecti.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4178" xml:space="preserve">Aliud notandum etiam cernere potes ex ipſis ſpeculis ſphæricis conuexis, hoc eſt
                    <lb/>
                  quod poſſibile ſit aliquoties, radium reflexum concurrere cum catheto incidentiæ
                    <lb/>
                  extra ſpeculum inter puncta
                    <var>.g.</var>
                  et
                    <var>.p.</var>
                  vt exempli gratia .ſi punctus
                    <var>.p.</var>
                  eſſet exactè
                    <lb/>
                  in medio inter
                    <var>.b.</var>
                  et g. </s>
                  <s xml:id="echoid-s4179" xml:space="preserve">tunc punctum
                    <var>.c.</var>
                  ipſius concurſus cum catheto incidentiæ eſſet
                    <lb/>
                  inter
                    <var>.g.</var>
                  et
                    <var>.p.</var>
                  eo quod
                    <reg norm="cum" type="context">cũ</reg>
                  linea
                    <var>.p.q.</var>
                  debeat @iui lere
                    <reg norm="angulum" type="context">angulũ</reg>
                    <var>.b.</var>
                  q, c.
                    <reg norm="per" type="simple">ꝑ</reg>
                  ęqualia, oportebit
                    <lb/>
                  c. poſitum eſſe inter
                    <var>.g.</var>
                  et
                    <var>.p.</var>
                  quia angulus
                    <var>.g.q.p.</var>
                  maior eſt angulo
                    <var>.p.q.b.</var>
                  vt per te faci
                    <lb/>
                  le potes ratiotinari, imaginando cir
                    <lb/>
                    <figure xlink:label="fig-0356-01" xlink:href="fig-0356-01a" number="388">
                      <image file="0356-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0356-01"/>
                    </figure>
                  culum circa
                    <reg norm="triangulum" type="context">triãgulum</reg>
                    <var>.g.q.b.</var>
                  & dia
                    <lb/>
                  merrum perpendicularem .ad
                    <var>.g.b.</var>
                    <lb/>
                  in puncto
                    <var>.p.</var>
                  producendo poſtea
                    <var>.q.
                      <lb/>
                    p.</var>
                    <reg norm="vſque" type="simple">vſq;</reg>
                  ad
                    <reg norm="alteram" type="context">alterã</reg>
                    <reg norm="partem" type="context">partẽ</reg>
                  circunferen-
                    <lb/>
                  tiæ ipſius circuli. </s>
                  <s xml:id="echoid-s4180" xml:space="preserve">
                    <reg norm="argumentando" type="context context">argumẽtãdo</reg>
                  dein-
                    <lb/>
                  de mediante vltima ſexti, illud
                    <reg norm="idem" type="context">idẽ</reg>
                    <lb/>
                  po@es etiam ſcire ex .22. quinti Alha
                    <lb/>
                  zeni. & ex .26. ſexti Vitellionis. </s>
                  <s xml:id="echoid-s4181" xml:space="preserve">vn-
                    <lb/>
                  de ſi ad ambas pupillas venerint ra
                    <lb/>
                  dij reflexi ipſius obiecti
                    <var>.b.a.</var>
                  duobus
                    <lb/>
                  punctis huiuſmodi ſpeculi, ita di-
                    <lb/>
                  ſtantibus à puncto
                    <var>.g.</var>
                  vt
                    <var>.q</var>
                  . </s>
                  <s xml:id="echoid-s4182" xml:space="preserve">tunc com
                    <lb/>
                  mune punctum concurſus axium vi
                    <lb/>
                  ſualium erit in catheto inter
                    <var>.g.p.</var>
                    <lb/>
                  vbi apparebit imago ex ſuperius di
                    <lb/>
                  ctis rationibus, ita vt
                    <reg norm="non" type="context">nõ</reg>
                  ſolum con
                    <lb/>
                  cauis, ſed etiam conuexis hoc accidere poſſit.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4183" xml:space="preserve">In planis autem
                    <reg norm="nunquam" type="context">nunquã</reg>
                  hoc poteſt euenire, vt tibi alias dixi, eo quod ſi
                    <reg norm="acceperi- mus" type="conjecture">accéperi-
                      <lb/>
                    mus</reg>
                    <reg norm="rectam" type="context">rectã</reg>
                    <var>.m.r.</var>
                  pro
                    <reg norm="coni" type="context">cõi</reg>
                  ſectione
                    <reg norm="ſuperficiei" type="simple">ſuꝑficiei</reg>
                    <var>.l.t.x.</var>
                  reflexionis &
                    <reg norm="ſuperficiei" type="simple">ſuꝑficiei</reg>
                  ſpeculi,
                    <reg norm="punctumque" type="context context simple">pũctũq́;</reg>
                    <lb/>
                  lucidum
                    <var>.l.</var>
                    <reg norm="protractoque" type="simple">protractoq́;</reg>
                  catheto
                    <var>.l.r.t.</var>
                    <reg norm="lineisque" type="simple">lineisq́;</reg>
                  incidentiæ
                    <var>.l.x.</var>
                  et
                    <var>.l.m.</var>
                  reflexionis etiam
                    <lb/>
                    <var>x.y.</var>
                  et
                    <var>.m.z.</var>
                  cum anguli
                    <var>.l.x.r.</var>
                  et
                    <var>.y.x.h.</var>
                  et
                    <var>.r.x.t.</var>
                  æquales inuicem ſint, & ſic anguli
                    <var>.l.m.
                      <lb/>
                    r.</var>
                  et
                    <var>.z.m.h.</var>
                  et
                    <var>.r.m.t.</var>
                  erit
                    <var>.r.t.</var>
                  tam pro triangulo
                    <var>.r.x.t.</var>
                  quam pro triangulo
                    <var>.r.m.t.</var>
                  æqua
                    <lb/>
                  lis
                    <var>.r.l.</var>
                  ex .26. primi, ita quod ſemper in puncto
                    <var>.t.</var>
                    <reg norm="conuenient" type="context">conueniẽt</reg>
                  omnes radij reflexi ipſius </s>
                </p>
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