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">
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                <p>
                  <s xml:id="echoid-s4018" xml:space="preserve">
                    <pb o="332" rhead="IO. BAPT. BENED." n="344" file="0344" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0344"/>
                  lineas
                    <var>.b.q.</var>
                  et
                    <var>.b.n.</var>
                  ſimul ſumptas longiores eſſe omnibus alijs lineis exeuntibus ab ip
                    <lb/>
                  ſis punctis
                    <var>.q.n.</var>
                  quæ in aliquo puncto dictæ circunferentiæ ſimul concurrant.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4019" xml:space="preserve">Sint igitur aliæ duæ
                    <var>.q.o.</var>
                  et
                    <var>.n.o.</var>
                  quas probare volo ſimul ſumptas, eſſe minores dua
                    <lb/>
                  bus ſimul ſumptis
                    <var>.q.b.</var>
                  et
                    <var>.n.b</var>
                  . </s>
                  <s xml:id="echoid-s4020" xml:space="preserve">Nam ex .20. tertij Eucli. cognoſcimus angulos
                    <var>.q.b.n.</var>
                    <lb/>
                  et
                    <var>.q.o.n.</var>
                  inuicem æquales eſſe, & ſimiliter angulos
                    <var>.b.n.o.</var>
                  et
                    <var>.b.q.o</var>
                  . </s>
                  <s xml:id="echoid-s4021" xml:space="preserve">deinde ex .15. pri
                    <lb/>
                  mi eiuſdem habemus angulos contra ſe poſitos,
                    <lb/>
                  circa
                    <var>.a.</var>
                  eſſe etiam inuicem ęquales. </s>
                  <s xml:id="echoid-s4022" xml:space="preserve">Vnde ex .4
                    <lb/>
                    <figure xlink:label="fig-0344-01" xlink:href="fig-0344-01a" number="368">
                      <image file="0344-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-01"/>
                    </figure>
                  ſexti, habebimus proportionem
                    <var>.a.b.</var>
                  ad .a
                    <lb/>
                  o. eandem eſſe, quæ
                    <var>.a.n.</var>
                  ad
                    <var>.a.q.</var>
                  & ſic .b
                    <lb/>
                  n. ad
                    <var>.o.q</var>
                  . </s>
                  <s xml:id="echoid-s4023" xml:space="preserve">Quare ita erit
                    <var>.a.b.n.</var>
                  ad
                    <var>.a.o.q.</var>
                  vt
                    <var>.a.n</var>
                    <lb/>
                  ad
                    <var>.a.q.</var>
                  ſed cum
                    <var>.a.n.</var>
                  maior ſit
                    <var>.q.a.</var>
                  ex .18. primi,
                    <lb/>
                  eo quod angulus
                    <var>.b.q.n.</var>
                  (qui æqualis eſt angulo
                    <var>.
                      <lb/>
                    b.n.q.</var>
                  ex .5. eiuſdem) maior eſt angulo
                    <var>.a.n.q.</var>
                    <lb/>
                  qui pars eſt ipſius
                    <var>.b.n.q.</var>
                  ergo latera ſimul ſum-
                    <lb/>
                  pta
                    <var>.a.b.n.</var>
                  maiora erunt lateribus
                    <var>.a.o.q.</var>
                  ſed ex
                    <num value="20">.
                      <lb/>
                    20.</num>
                  primi
                    <var>.a.b.n.</var>
                    <reg norm="etiam" type="context">etiã</reg>
                  maior erit
                    <var>.a.n.</var>
                  vnde ex .25.
                    <lb/>
                  quinti
                    <var>.q.a.b.n.</var>
                  maior erit
                    <var>.n.a.o.q</var>
                  . </s>
                  <s xml:id="echoid-s4024" xml:space="preserve">quare ſequi-
                    <lb/>
                  tur verum eſſe propofitum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s4025" xml:space="preserve">Sed ſi oculus eſſet in
                    <var>.u.</var>
                  quemadmodum in ſubſcripta hic
                    <reg norm="ſecunda" type="context">ſecũda</reg>
                  figura videre eſt,
                    <lb/>
                  res autem viſibilis in
                    <var>.n.</var>
                  ambo extra dictum circulum, eſto etiam primum
                    <var>.b.u.</var>
                  æqua-
                    <lb/>
                  lis
                    <var>.b.n.</var>
                  probabo ſimiliter
                    <var>.u.b.n.</var>
                  maiores eſſe
                    <var>.u.o.n</var>
                  . </s>
                  <s xml:id="echoid-s4026" xml:space="preserve">Nam angulus
                    <var>.o.</var>
                  maior eſt angu-
                    <lb/>
                  lo
                    <var>.b.</var>
                  eo quod ſi circulum
                    <var>.u.b.n.</var>
                  cogitemus circunſcribere triangulum
                    <var>.u.b.n.</var>
                  ducen-
                    <lb/>
                  do vſque ad ſuam circunferentiam
                    <var>.o.n.</var>
                  in puncto
                    <var>.s.</var>
                  deinde ducendo
                    <var>.u.s.</var>
                  habebimus
                    <lb/>
                  ex .20. tertij angulum
                    <var>.u.s.n.</var>
                    <reg norm="æqualem" type="context">æqualẽ</reg>
                  angulo
                    <var>.u.b.n.</var>
                  ſed
                    <reg norm="cum" type="context">cũ</reg>
                  angulus
                    <var>.u.o.n.</var>
                  exterior trian
                    <lb/>
                  guli
                    <var>.u.o.s.</var>
                  exiſtat, ipſe maior erit angulo
                    <var>.s.</var>
                  ex .16. primi. </s>
                  <s xml:id="echoid-s4027" xml:space="preserve">duco poſtea
                    <var>.o.q.</var>
                  parallelam
                    <lb/>
                  ad
                    <var>.u.s.</var>
                  quæ ſecabit
                    <var>.a.u.</var>
                  in puncto
                    <var>.q.</var>
                  & habebimus angulum
                    <var>.a.o.q.</var>
                  ęqualem angulo
                    <var>.
                      <lb/>
                    n.s.u.</var>
                  ex .29. eiuſdem, hoc eſt angulo
                    <var>.n.b.u.</var>
                  fed ex ſu-
                    <lb/>
                    <figure xlink:label="fig-0344-02" xlink:href="fig-0344-02a" number="369">
                      <image file="0344-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-02"/>
                    </figure>
                  pradictis rationibus, lineæ
                    <var>.q.b.n.</var>
                  ſimul ſumptæ maio-
                    <lb/>
                  rem efficient longitudinem, quam
                    <var>.q.o.n</var>
                  . </s>
                  <s xml:id="echoid-s4028" xml:space="preserve">Nunc cum
                    <lb/>
                  ipſi
                    <var>.q.b.</var>
                  addita fuerit
                    <var>.u.q.</var>
                  & vice
                    <var>.q.o.</var>
                  ſumpta fuerit ali-
                    <lb/>
                  qua linea minor ipſa
                    <var>.u.q.o.</var>
                  eo amplius
                    <var>.u.q.b.n.</var>
                  maior
                    <lb/>
                  erit, quod quidem hoc modo faciendum. </s>
                  <s xml:id="echoid-s4029" xml:space="preserve">Acci-
                    <lb/>
                  piatur
                    <var>.o.u.</var>
                  vt comes
                    <var>.o.n.</var>
                  quæ minor eſt ambabus
                    <var>.o.
                      <lb/>
                    q.</var>
                  et
                    <var>.q.u.</var>
                  ex .20. primi, ita enim habebimus
                    <reg norm="propoſitum" type="context">propoſitũ</reg>
                  .
                    <lb/>
                  </s>
                  <s xml:id="echoid-s4030" xml:space="preserve">ſed breuiori modo hoc ipſum videbis ex pręcedenti,
                    <lb/>
                  & ex .21. primi Euclid. </s>
                  <s xml:id="echoid-s4031" xml:space="preserve">Nam ex præcedenti
                    <var>.u.b.n.</var>
                  lon-
                    <lb/>
                  gior eſt ipſa
                    <var>.u.s.n.</var>
                  ex .21. autem primi
                    <var>.u.s.n.</var>
                  longior eſt
                    <lb/>
                  ipſa
                    <var>.u.o.n.</var>
                  ergo verum eſt propoſitum.</s>
                </p>
                <figure position="here" number="370">
                  <image file="0344-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0344-03"/>
                </figure>
                <p>
                  <s xml:id="echoid-s4032" xml:space="preserve">Si verò radius incidentiæ
                    <reg norm="non" type="context">nõ</reg>
                  fuerit æqualis radio
                    <lb/>
                  reflexionis, ſit vt in hac ſubſcripta tertia figura vide
                    <lb/>
                  re eſt
                    <var>.u.b.p</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s4033" xml:space="preserve">Cum autem probauerim longitudinem
                    <var>.u.b.n.</var>
                  ma
                    <lb/>
                  iorem eſſe longitudine
                    <var>.u.o.n.</var>
                  coniungatur
                    <var>.n.p.</var>
                  cum
                    <lb/>
                    <var>u.b.n</var>
                  . </s>
                  <s xml:id="echoid-s4034" xml:space="preserve">deinde. ab
                    <var>.o.</var>
                  ad
                    <var>.p.</var>
                  ducatur
                    <var>.o.p.</var>
                  quæ minor
                    <lb/>
                  erit longitudine
                    <var>.o.n.p.</var>
                  ex .20. primi, & illicò
                    <lb/>
                  manifeſtabitur verum eſſe propoſitum, etiam hoc
                    <lb/>
                  tertio modo.</s>
                </p>
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