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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IO. BAPT. BENED.
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          <div xml:id="echoid-div477" type="chapter" level="2" n="6">
            <div xml:id="echoid-div480" type="section" level="3" n="2">
              <div xml:id="echoid-div480" type="letter" level="4" n="1">
                <p>
                  <s xml:id="echoid-s2658" xml:space="preserve">
                    <pb o="212" rhead="IO. BAPT. BENED." n="224" file="0224" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0224"/>
                  diametri ſint
                    <var>.q.d.</var>
                  et
                    <var>.a.b.</var>
                  quæ ſe inuicem interſecent in puncto
                    <var>.o.</var>
                  vnde cum anguli
                    <lb/>
                  contra ſe poſiti circa
                    <var>.o.</var>
                  æquales inuicem ſint ex .15. primi Eucli. </s>
                  <s xml:id="echoid-s2659" xml:space="preserve">& angulus
                    <var>.a.q.d.</var>
                  æ-
                    <lb/>
                  qualis angulo
                    <var>.a.b.d.</var>
                  & angulus
                    <var>.q.b.a.</var>
                  æqualis angulo
                    <var>.q.d.a.</var>
                  et
                    <var>.b.q.d.</var>
                  angulo
                    <var>.b.a.d.</var>
                    <lb/>
                  ex .20. tertij tunc triangulus
                    <var>.a.o.q.</var>
                  ſimilis erit triangulo
                    <var>.d.o.b.</var>
                  et
                    <var>.q.o.b.</var>
                  ſimilis trian-
                    <lb/>
                  gulo
                    <var>.a.o.d.</var>
                  ex definitione. </s>
                  <s xml:id="echoid-s2660" xml:space="preserve">Vnde eadem proportio erit ipſius
                    <var>.q.o.</var>
                  ad
                    <var>.b.o.</var>
                  quæ ipſius
                    <lb/>
                    <var>q.a.</var>
                  ad
                    <var>.b.d.</var>
                  & ipſius
                    <var>.b.o.</var>
                  ad
                    <var>.o.d.</var>
                  eadem quæ
                    <var>.q.b.</var>
                  ad
                    <var>.a.d.</var>
                  & ipſius
                    <var>.q.o.</var>
                  ad
                    <var>.o.a.</var>
                  eadem
                    <lb/>
                  quæ
                    <var>.q.b.</var>
                  ad
                    <var>.a.d.</var>
                  proportio igitur
                    <var>.q.o.</var>
                  ad
                    <var>.o.d.</var>
                  cognita nobis erit, vt compoſita ex
                    <lb/>
                  ea quæ eſt
                    <var>.q.o.</var>
                  ad
                    <var>.o.b.</var>
                  ex
                    <var>.o.b.</var>
                  ad
                    <var>.o.d.</var>
                  quæ nobis cognitę ſunt, mediante
                    <lb/>
                  proportione ipſius
                    <var>.q.a.</var>
                  ad
                    <var>.b.d.</var>
                  & ipſius
                    <var>.q.b.</var>
                  ad
                    <var>.a.d.</var>
                  proportio ſimiliter ipſius
                    <var>.b.o.</var>
                    <lb/>
                  ad
                    <var>.o.a.</var>
                  nobis cognita erit, vt compoſita ex proportione ipſius
                    <var>.b.o.</var>
                  ad
                    <var>.o.q.</var>
                  &
                    <lb/>
                  ipſius
                    <var>.o.q.</var>
                  ad
                    <var>.o.a.</var>
                  cognitis, mediante proportione ipſius
                    <var>.b.d.</var>
                  ad
                    <var>.q.a.</var>
                  & ipſius
                    <var>.q.b.</var>
                  ad
                    <lb/>
                    <var>a.d.</var>
                  cum
                    <reg norm="autem" type="context">autẽ</reg>
                  proportio ipſius
                    <var>.q.o.</var>
                  ad
                    <var>.o.b.</var>
                  nobis cognita ſit, </s>
                  <s xml:id="echoid-s2661" xml:space="preserve">tunc nobis cognita erit
                    <lb/>
                  proportio ipſius
                    <var>.q.d.</var>
                  ad
                    <var>.a.b</var>
                  . </s>
                  <s xml:id="echoid-s2662" xml:space="preserve">Nam ut
                    <var>.q.o.</var>
                  ad
                    <var>.o.b.</var>
                  eſt vt
                    <var>.a.o.</var>
                  ad
                    <var>.o.d.</var>
                  ex ſimilitudine,
                    <lb/>
                  </s>
                  <s xml:id="echoid-s2663" xml:space="preserve">quare proportio compoſiti ex primo, & quarto terminorum ad compoſitum ex .2. &
                    <lb/>
                  tertio, cognita erit. </s>
                  <s xml:id="echoid-s2664" xml:space="preserve">ſed quod fit ex
                    <var>.q.d.</var>
                  in
                    <var>.a.b.</var>
                  cognitum nobis eſt, vt æquale duobus
                    <lb/>
                  productis, hoc eſt ex
                    <var>.q.a.</var>
                  in
                    <var>.d.b.</var>
                  & ex
                    <var>.q.b.</var>
                  in
                    <var>.d.a.</var>
                  ex ſecunda primi Almageſti. </s>
                  <s xml:id="echoid-s2665" xml:space="preserve">quæ
                    <lb/>
                  producta nobis cognita ſunt, cum nobis data ſint eorum latera. </s>
                  <s xml:id="echoid-s2666" xml:space="preserve">Quapropter facta
                    <lb/>
                  cum fuerit figura quadrilatera rectangula ſimilis alicui alterirectangulæ figuræ pro
                    <lb/>
                  ductæ à duobus lateribus inuicem ita proportionatis, vt ſe habet
                    <var>.q.d.</var>
                  ad
                    <var>.a.b.</var>
                  æqua-
                    <lb/>
                  lis tamen duobus productis, hoc eſt producto ex
                    <var>.q.a.</var>
                  in
                    <var>.d.b.</var>
                  & ex
                    <var>.q.b.</var>
                  in
                    <var>.d.a.</var>
                  ex
                    <lb/>
                  doctrina, 25. ſexti Eucli quæ quidem figura, exempli gratia, ſit
                    <var>.u.t.</var>
                  eius verò latera
                    <lb/>
                  ſint
                    <var>.u.n.</var>
                  et
                    <var>.n.t.</var>
                  Hæc enim dico æqualia eſſe
                    <var>.q.d.</var>
                  et
                    <var>.b.a.</var>
                  hoc eſt
                    <var>.n.t.</var>
                  maius maio-
                    <lb/>
                  ri
                    <var>.b.a.</var>
                  et
                    <var>.u.n.</var>
                  minus minori
                    <var>.q.d</var>
                  . </s>
                  <s xml:id="echoid-s2667" xml:space="preserve">Quod ita probabo. </s>
                  <s xml:id="echoid-s2668" xml:space="preserve">cogitemus rectangulum
                    <var>.s.r.</var>
                    <lb/>
                  productum eſſe ex duobus lateribus
                    <var>.q.d.</var>
                  et
                    <var>.a.b.</var>
                  ſed,
                    <var>s.n.</var>
                  æqualis ſit
                    <var>.q.d.</var>
                  et
                    <var>.n.r.</var>
                  æqua-
                    <lb/>
                  lis
                    <var>.a.b.</var>
                    <reg norm="ſintque" type="simple">ſintq́;</reg>
                  duæ lineæ
                    <var>.s.n.</var>
                  et
                    <var>.n.t.</var>
                  inuicem directè coniunctæ, vnde
                    <var>.u.n.</var>
                  directè
                    <lb/>
                  coniuncta etiam erit cum
                    <var>.n.r.</var>
                  ex quo rectangulum
                    <var>.u.t.</var>
                  æquale erit rectangulo
                    <var>.s.r.</var>
                  ex
                    <lb/>
                  communi conceptu,
                    <reg norm="eademque" type="simple">eademq́</reg>
                  proportio erit
                    <var>.u.n.</var>
                  ad
                    <var>.n.t.</var>
                  quę
                    <var>.s.n.</var>
                  ad
                    <var>.n.r.</var>
                  eo
                    <reg norm="quod" type="simple">ꝙ</reg>
                  ita fa-
                    <lb/>
                  ctum fuit, cum autem ita ſit
                    <var>.u.n.</var>
                  ad
                    <var>.n.t.</var>
                  vt
                    <var>.s.n.</var>
                  ad
                    <var>.n.r.</var>
                  </s>
                  <s xml:id="echoid-s2669" xml:space="preserve">tunc permutando ita erit
                    <var>.n.t.</var>
                  ad
                    <lb/>
                    <var>n.r.</var>
                  vt
                    <var>.u.n.</var>
                  ad
                    <var>.n.s.</var>
                  ſed quia ita eſt
                    <var>.u.n.</var>
                  ad
                    <var>.n.r.</var>
                  vt
                    <var>.s.n.</var>
                  ad
                    <var>.n.t.</var>
                  ex 15. ſexti, </s>
                  <s xml:id="echoid-s2670" xml:space="preserve">tunc permutan
                    <lb/>
                  do ita erit
                    <var>.n.r.</var>
                  ad
                    <var>.n.t.</var>
                  vt
                    <var>.n.u.</var>
                  ad
                    <var>.n.s.</var>
                  </s>
                  <s xml:id="echoid-s2671" xml:space="preserve">quare ex 11. quinti ita erit
                    <var>.n.t.</var>
                  ad
                    <var>.n.r.</var>
                  vt
                    <var>.n.r.</var>
                  ad
                    <var>.n.
                      <lb/>
                    t.</var>
                  quapropter ex neceſſitate ſequitur
                    <var>.n.t.</var>
                  et
                    <var>.n.r.</var>
                  inuicem æquales eſſe, et
                    <var>.u.n.</var>
                  ſimiliter
                    <lb/>
                  cum
                    <var>.n.s</var>
                  .</s>
                </p>
                <div xml:id="echoid-div480" type="float" level="5" n="1">
                  <figure xlink:label="fig-0223-01" xlink:href="fig-0223-01a">
                    <image file="0223-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-01"/>
                  </figure>
                  <figure xlink:label="fig-0223-02" xlink:href="fig-0223-02a">
                    <image file="0223-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-02"/>
                  </figure>
                  <figure xlink:label="fig-0223-03" xlink:href="fig-0223-03a">
                    <image file="0223-03" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-03"/>
                  </figure>
                  <figure xlink:label="fig-0223-04" xlink:href="fig-0223-04a">
                    <image file="0223-04" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-04"/>
                  </figure>
                  <figure xlink:label="fig-0223-05" xlink:href="fig-0223-05a">
                    <image file="0223-05" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0223-05"/>
                  </figure>
                </div>
                <p>
                  <s xml:id="echoid-s2672" xml:space="preserve">Inuentæ nunc cum fuerint duæ diametri
                    <var>.q.d.</var>
                  et
                    <var>.a.b.</var>
                  ipſius quadrilateri, difficile
                    <lb/>
                  non erit eius angulos inuenire, eo
                    <reg norm="quod" type="simple">ꝙ</reg>
                  mediante
                    <var>.a.b.</var>
                  cognita, ſimul cum
                    <var>.b.d.</var>
                  et
                    <var>.a.d.</var>
                  da
                    <lb/>
                  tis, faciemus triangulum
                    <var>.a.b.d.</var>
                  vel
                    <reg norm="mediante" type="context">mediãte</reg>
                    <var>.q.d.</var>
                  et
                    <var>.q.a.</var>
                  et
                    <var>.a.d.</var>
                  cognitis faciemus
                    <reg norm="triam" type="context">triã</reg>
                    <lb/>
                  gulum
                    <var>.a.q.d.</var>
                  ex .22. primi. </s>
                  <s xml:id="echoid-s2673" xml:space="preserve">Vnde cum centrum circuli circunſcriptibilis cuiuſuis di-
                    <lb/>
                  ctorum triangulorum ex quinta quarti inuentum fuerit, triangulum reliquum, ab eo
                    <lb/>
                  dem circulo circunſcriptum erit, ex communi ſcientia.</s>
                </p>
                <p>
                  <s xml:id="echoid-s2674" xml:space="preserve">SEd vt ipſa operatio facilior fiat, Sint eędem lineæ
                    <var>.b.d</var>
                  :
                    <var>b.q</var>
                  :
                    <var>a.q.</var>
                  et
                    <var>.a.d.</var>
                  ex quibus
                    <lb/>
                  poſſit
                    <reg norm="quadrilaterum" type="context">quadrilaterũ</reg>
                  effici. </s>
                  <s xml:id="echoid-s2675" xml:space="preserve">Videatur deinde primò quas volumus oppoſitas ſibi
                    <lb/>
                  inuicem eſſe, ponatur ergò ut
                    <var>.q.a.</var>
                  et
                    <var>.b.d.</var>
                  velimus oppoſitas inuicem facere, et
                    <var>.q.b.</var>
                    <lb/>
                  cum
                    <var>.a.d.</var>
                  ſimiliter, accipiemus nunc
                    <var>.K.</var>
                  cuiuſuis magnitudinis, cui comparetur
                    <var>.e.</var>
                    <lb/>
                  ita proportionata, vt
                    <var>.q.b.</var>
                  eſt ipſi
                    <var>.a.d.</var>
                  ex doctrina .10. ſexti Eucli. vel accipiatur
                    <var>.a.d.</var>
                    <lb/>
                  vice
                    <var>.K.</var>
                  et
                    <var>.q.b.</var>
                  vice
                    <var>.e.</var>
                  quod idem erit, & expeditius, inuenietur ſimiliter
                    <var>.h.</var>
                  ita pro-
                    <lb/>
                  portionata ad
                    <var>.e.</var>
                  et
                    <var>.g.</var>
                  ad
                    <var>.k.</var>
                  vt
                    <var>.b.d.</var>
                  eſt ad
                    <var>.q.a.</var>
                  vel
                    <var>.g.</var>
                  ad
                    <var>.h.</var>
                  vt
                    <var>.a.d.</var>
                  ipſi
                    <var>.q.b.</var>
                  quod
                    <reg norm="idem" type="context">idẽ</reg>
                  erit.</s>
                </p>
                <p>
                  <s xml:id="echoid-s2676" xml:space="preserve">Hoc facto coniungantur inuicem directè
                    <var>.g.</var>
                  et
                    <var>.e.</var>
                  quarum compoſitum ſit
                    <var>.g.e.</var>
                  &
                    <lb/>
                  ita duæ
                    <var>.K.</var>
                  et
                    <var>.h.</var>
                  ex quibus ſit
                    <var>.K.h</var>
                  . </s>
                  <s xml:id="echoid-s2677" xml:space="preserve">Nunc ex iſtis duabus lineis
                    <var>.e.g.</var>
                  et
                    <var>K.h.</var>
                  fiat paral- </s>
                </p>
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