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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EPISTOL AE.
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          <div xml:id="echoid-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div564" type="section" level="3" n="17">
              <div xml:id="echoid-div569" type="letter" level="4" n="3">
                <pb o="295" rhead="EPISTOL AE." n="307" file="0307" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0307"/>
                <p>
                  <s xml:id="echoid-s3655" xml:space="preserve">
                    <reg norm="Propoſitum" type="context">Propoſitũ</reg>
                  ſit nobis triangulum
                    <var>.a.b.g.</var>
                  cuius baſis data ſit cum area, ſeu perpendi-
                    <lb/>
                  culari
                    <var>.a.d.</var>
                  cum angulo etiam
                    <var>.a.</var>
                  ad cognoſcendum autem
                    <var>.a.b.</var>
                  et
                    <var>.b.g.</var>
                  cogitemus circu
                    <lb/>
                  lum
                    <var>.a.b.q.g.</var>
                  circunſcribere ipſum triangulum cuius diameter
                    <var>.p.q.</var>
                  ad rectos ſe-
                    <lb/>
                  cet baſim
                    <var>.b.g.</var>
                  in puncto
                    <var>.m.</var>
                  cogitemus etiam
                    <var>.b.p.</var>
                  et
                    <var>.p.g.</var>
                  vnde ex .20. ter-
                    <lb/>
                  tij Euclid. angulus
                    <var>.b.p.g.</var>
                  æqualis erit
                    <lb/>
                    <anchor type="figure" xlink:label="fig-0307-01a" xlink:href="fig-0307-01"/>
                  angulo
                    <var>.a.</var>
                  & angulus
                    <var>.m.p.b.</var>
                  erit eius di
                    <lb/>
                  midium, quod ex te ipſo cognoſces, &
                    <lb/>
                    <reg norm="angulus" type="simple">angulꝰ</reg>
                    <var>.p.b.m.</var>
                  ſimiliter cognoſcetur,
                    <lb/>
                  </s>
                  <s xml:id="echoid-s3656" xml:space="preserve">quare ex .29. primi eiuſdem Montere
                    <lb/>
                  gij cognoſcemus
                    <var>.p.m.</var>
                  et
                    <var>.p.b.</var>
                  (nam
                    <var>.b.
                      <lb/>
                    m.</var>
                  datum fuit, vt dimidium totius ba-
                    <lb/>
                  ſis
                    <var>.b.g.</var>
                  ) ducta poſtea
                    <var>.b.q.</var>
                  ex
                    <reg norm="eadem" type="context">eadẽ</reg>
                  .29.
                    <lb/>
                  cognoſcemus
                    <var>.p.q.</var>
                  cum
                    <var>.p.b.</var>
                  iam cogni
                    <lb/>
                  ta fuerit, à qua
                    <var>.p.q.</var>
                  (diametro)
                    <reg norm="dempta" type="context">dẽpta</reg>
                    <lb/>
                    <var>p.m.</var>
                  remanebit
                    <var>.q.m.</var>
                  cognita,
                    <reg norm="cum" type="context">cũ</reg>
                  qua
                    <lb/>
                  iuncta cum fuerit
                    <var>.m.t.</var>
                  æquali
                    <var>.a.d.</var>
                  per
                    <lb/>
                  pendiculari, dabitur
                    <var>.q.t.</var>
                  et
                    <var>.t.p.</var>
                  inter
                    <lb/>
                  quas
                    <var>.a.t.</var>
                  media proportionalis loca-
                    <lb/>
                  tur, </s>
                  <s xml:id="echoid-s3657" xml:space="preserve">quare cognoſcemus
                    <var>.a.t.</var>
                  quæ ſinus
                    <lb/>
                  eſt arcus
                    <var>.a.p.</var>
                  vnde cognitus erit arcus
                    <lb/>
                    <var>a.p.</var>
                  ſed arcus
                    <var>.p.g.</var>
                  cognitus eſt median
                    <lb/>
                  te angulo
                    <var>.p.b.g.</var>
                  cognito, qui quidem
                    <lb/>
                  arcus
                    <var>.p.g.</var>
                  ſi coniunctus fuerit cum arcu
                    <var>.p.a.</var>
                  cognoſcemus compoſitum
                    <var>.a.g.</var>
                  & eius
                    <lb/>
                  chorda ſimiliter (hoc eſt
                    <reg norm="ſecundum" type="context">ſecundũ</reg>
                  latus) qua cognita, illico cognoſcemus chordam
                    <lb/>
                    <var>a.b.</var>
                  hoc eſt tertium latus trianguli propoſiti.</s>
                </p>
                <div xml:id="echoid-div569" type="float" level="5" n="1">
                  <figure xlink:label="fig-0307-01" xlink:href="fig-0307-01a">
                    <image file="0307-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0307-01"/>
                  </figure>
                </div>
              </div>
              <div xml:id="echoid-div571" type="letter" level="4" n="4">
                <head xml:id="echoid-head436" style="it" xml:space="preserve">Quæ
                  <unsure/>
                dam not and a in Federicum Comandinum.</head>
                <head xml:id="echoid-head437" xml:space="preserve">AD EVNDEM.</head>
                <p>
                  <s xml:id="echoid-s3658" xml:space="preserve">PVtabas enim me ioco dixiſſe Federicum Comandinum non omnino irrepræ-
                    <lb/>
                  henſibilem eſſe, vide igitur, quod ſcribit in quinto lemmate in decimam
                    <lb/>
                  propoſitionem libr .2. de inſidentibus aquæ Archimedis, volens demonſtra-
                    <lb/>
                  re eandem eſſe proportionem
                    <var>.l.b.</var>
                  ad
                    <var>.b.m.</var>
                  quæ
                    <var>.c.e.</var>
                  ad
                    <var>.e.a.</var>
                  vbi eſt aliquo modo pro-
                    <lb/>
                  lixum, mediante linea
                    <var>.c.p.</var>
                  cum ſuis partibus, citans etiam antecedens lemma extra
                    <lb/>
                  propoſitum, eo quod nec in antecedente lemmate, nec in alio, ipſe vnquam proba
                    <lb/>
                  uerit proportionem
                    <var>.c.d.</var>
                  ad
                    <var>.d.q.</var>
                  eſſe, vt
                    <var>.l.b.</var>
                  @d.b.m. ſed ne putes me falli, tibi demon
                    <lb/>
                  ſtrabo non eſſe neceſſarium ducere lineam
                    <var>.c.m.p.</var>
                  vel
                    <var>.q.p.</var>
                  eo quod
                    <reg norm="cum" type="context">cũ</reg>
                  per quintam
                    <lb/>
                  lib. de quadratura parabolę Archimedis, ita ſit
                    <var>.c.d.</var>
                  ad
                    <var>.d.e.</var>
                  vt
                    <var>.l.b.</var>
                  ad
                    <var>.b.m.</var>
                  exiſtente
                    <lb/>
                    <var>a.c.</var>
                  dupla ipſi
                    <var>.d.c.</var>
                  et
                    <var>.e.c.</var>
                  dupla ipſi
                    <var>.g.c.</var>
                  et
                    <var>.l.d.</var>
                  dupla ipſi
                    <var>.l.b</var>
                  : erit, primo componen-
                    <lb/>
                  do
                    <var>.c.e.</var>
                  ad
                    <var>.e.d.</var>
                  vt
                    <var>.l.d.</var>
                  ad
                    <var>.d.m.</var>
                  & per æqualitatem proportionum, ita erit
                    <var>.e.g.</var>
                  ad
                    <var>.e.d.</var>
                    <lb/>
                  vt
                    <var>.b.d.</var>
                  2d.d.m. & per .19. quinti Eucli. ita erit
                    <var>.e.g.</var>
                  ideſt
                    <var>.g.c.</var>
                  ad
                    <var>.g.d.</var>
                  vt
                    <var>.b.d.</var>
                  ideſt
                    <var>.l.b.</var>
                    <lb/>
                  ad
                    <var>.b.m.</var>
                  ſed
                    <var>.c.g.</var>
                  ad
                    <var>.g.d.</var>
                  eft vt
                    <var>.c.e.</var>
                  ad
                    <var>.e.a.</var>
                  ratio eſt, quia componendo ita eſt
                    <var>.c.d.</var>
                  ad
                    <var>.d.
                      <lb/>
                    g.</var>
                  vt
                    <var>.c.a.</var>
                  ad
                    <var>.a.e.</var>
                  & hoc eſt, quia permutando, ita eſt
                    <var>.a.c.</var>
                  ad
                    <var>.d.c.</var>
                  vt
                    <var>.a.e.</var>
                  ad
                    <var>.d.g.</var>
                  & hoc
                    <lb/>
                  verum eſt ex .19. quinti eo quod totius
                    <var>.a.c.</var>
                  ad totum
                    <var>.d.c.</var>
                  eft vt abſciſſi
                    <var>.e.c.</var>
                  ad abſciſ
                    <lb/>
                  ſum
                    <var>.g.c.</var>
                  vt ſupradixi.</s>
                </p>
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