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-div564" type="section" level="3" n="17">
              <div xml:id="echoid-div564" type="letter" level="4" n="1">
                <p>
                  <s xml:id="echoid-s3629" xml:space="preserve">
                    <pb o="293" rhead="EPISTOL AE." n="305" file="0305" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0305"/>
                  ideo vnaquæque eius pars
                    <var>.a.d.</var>
                  et
                    <var>.d.g.</var>
                  ſimiliter nobis cognita erit ex quinta ſecundi
                    <lb/>
                  Eucl. </s>
                  <s xml:id="echoid-s3630" xml:space="preserve">vnde ex penultima primi habebimus propoſitum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3631" xml:space="preserve">Poſſumus item circulum mente concipere cuius
                    <var>.a.g.</var>
                  ſit diameter, & ab eius cen-
                    <lb/>
                  tro
                    <var>.e.</var>
                  protracta cum fuerit
                    <var>.e.b.</var>
                  quæ nobis cognita erit, vt medietas ipſius
                    <var>.a.g.</var>
                  de cu
                    <lb/>
                  ius potentia, dempta
                    <reg norm="cum" type="context">cũ</reg>
                  fuerit potentia
                    <reg norm="ipſius" type="simple">ipſiꝰ</reg>
                    <var>b.o.</var>
                  remanebit nobis potentia ipſius
                    <var>.d.
                      <lb/>
                    e.</var>
                  & ita eius longitudo, quæ addita medietati
                    <var>.e.g.</var>
                  & detracta à dimidio
                    <var>.e.d.</var>
                  erunt
                    <lb/>
                  nobis cognitæ
                    <var>.a.d.</var>
                  et
                    <var>.d.g.</var>
                  vnde
                    <var>.b.g.</var>
                  et
                    <var>.b.d.</var>
                  remanebunt nobis cognitæ ex dicta pe-
                    <lb/>
                  nultima primi Eucli. </s>
                  <s xml:id="echoid-s3632" xml:space="preserve">huiuſmodi figuram videbis in dicto .25. problemate .2. li. Mon-
                    <lb/>
                  tisregij.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3633" xml:space="preserve">Aliter etiam poſſumus hoc idem efficere.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3634" xml:space="preserve">Sit rectangulus hic ſubſcriptus
                    <var>.a.b.c.u.</var>
                  ſuperficiei cognitę ſimul cum diametro
                    <var>.a.
                      <lb/>
                    c.</var>
                  extendatur imaginatione
                    <var>.b.c.</var>
                  vſque ad, f. ita quod
                    <var>.c.f.</var>
                  æqualis ſit
                    <var>.c.u.</var>
                  intelligan-
                    <lb/>
                    <reg norm="turque" type="simple">turq́;</reg>
                  quadrata
                    <var>.g.f</var>
                  :
                    <var>g.u.</var>
                  ct
                    <var>.u.f.</var>
                  vnde
                    <reg norm="summa" type="context">sũma</reg>
                    <reg norm="quadratorum" type="context">quadratorũ</reg>
                    <var>.g.u</var>
                  :u.f. cognita nobis erit ex
                    <lb/>
                  penultima primi. </s>
                  <s xml:id="echoid-s3635" xml:space="preserve">nam
                    <var>.a.c.</var>
                  data nobis fuit, quare
                    <reg norm="ſummam" type="context">ſummã</reg>
                    <var>.g.u</var>
                  :u.b: et
                    <var>.u.f.</var>
                  cognoſce-
                    <lb/>
                  mus, cui
                    <reg norm="summæ" type="context">sũmæ</reg>
                  addito ſuplemento
                    <var>.d.e.</var>
                  æ quali
                    <var>.u.
                      <lb/>
                    b.</var>
                  dabit nobis
                    <reg norm="cognitum" type="context">cognitũ</reg>
                  quadrarum
                    <var>.g.f.</var>
                  totale, qua
                    <lb/>
                    <figure xlink:label="fig-0305-01" xlink:href="fig-0305-01a" number="327">
                      <image file="0305-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0305-01"/>
                    </figure>
                  re cognoſcetur eius radix
                    <var>.b.f.</var>
                  cognita igitur
                    <var>.b.f.</var>
                    <lb/>
                  cum pro ducto
                    <var>.b.u.</var>
                  illico ex .5. ſecundi cognoſce-
                    <lb/>
                  tur
                    <var>.b.c.</var>
                  et
                    <var>.c.f.</var>
                  forte cognita
                    <var>.b.f.</var>
                  diuiſa
                    <reg norm="per" type="simple">ꝑ</reg>
                  æqualia
                    <lb/>
                  in puncto
                    <var>.t.</var>
                  & per inæqualiz in
                    <reg norm="puncto" type="context">pũcto</reg>
                    <var>.c</var>
                  . </s>
                  <s xml:id="echoid-s3636" xml:space="preserve">Nam qua
                    <lb/>
                    <reg norm="dratum" type="context">dratũ</reg>
                  ipſius
                    <var>.t.f.</var>
                  cognitum, ęquatur
                    <reg norm="rectangulo" type="context">rectãgulo</reg>
                    <var>.b.u.</var>
                    <lb/>
                    <reg norm="cum" type="context">cũ</reg>
                  quadrato ipſius
                    <var>.t.c.</var>
                    <reg norm="dempto" type="context">dẽpto</reg>
                  igitur rectangulo,
                    <var>b.
                      <lb/>
                    u.</var>
                  ex quadrato ipſius
                    <var>.t.f.</var>
                  relinquetur quadratum
                    <lb/>
                    <reg norm="ipſius" type="simple">ipſiꝰ</reg>
                    <var>.t.c.</var>
                  cognitum & eius radix
                    <var>.t.c.</var>
                  qua addita ipſi
                    <lb/>
                  medietati
                    <var>.b.t.</var>
                  &
                    <reg norm="dempta" type="context">dẽpta</reg>
                  ex medietate
                    <var>.f.t.</var>
                  relinque-
                    <lb/>
                  tur propoſitum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3637" xml:space="preserve">Similiter de tertio exemplo eiuſdem Stifelij
                    <lb/>
                  infero.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3638" xml:space="preserve">Sit rectangulus
                    <var>.a.b.c.u.</var>
                  cuius diametri
                    <var>.a.c.</var>
                  quantitas, ſimul cum proportione late
                    <lb/>
                  rum
                    <var>.b.c.</var>
                  et
                    <var>.b.a.</var>
                  nobis data ſit. </s>
                  <s xml:id="echoid-s3639" xml:space="preserve">cum autem ſcire voluerimus eius ſuperficiem
                    <var>.b.u.</var>
                  cla-
                    <lb/>
                  rum eſt, quod cum nobis data ſit proportio
                    <var>.b.c.</var>
                  ad
                    <var>.b.a.</var>
                  illico cognoſcemus
                    <reg norm="etiam" type="context">etiã</reg>
                  pro-
                    <lb/>
                  portionem quadrati ipſius
                    <var>.b.c.</var>
                  ad quadratum ip-
                    <lb/>
                    <figure xlink:label="fig-0305-02" xlink:href="fig-0305-02a" number="328">
                      <image file="0305-02" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0305-02"/>
                    </figure>
                  ſius
                    <var>.b.a.</var>
                  cum dupla ſit ei quæ
                    <var>.b.c.</var>
                  ad
                    <var>.b.a.</var>
                  ita etiam
                    <lb/>
                  & aggregati dictorum quadratorum ad quadra-
                    <lb/>
                  tum ipſius
                    <var>.b.a.</var>
                  hoc eſt nota erit nobis proportio
                    <lb/>
                  quadrati ipſius
                    <var>.a.c.</var>
                  diagonalis ad quadratum ip-
                    <lb/>
                  ſius
                    <var>.a.b.</var>
                  idem dico de quadrato
                    <var>.b.c.</var>
                  ideſt quod
                    <lb/>
                  proportio quadrati ipſius
                    <var>.a.c.</var>
                  ad quadratum
                    <var>.b.c.</var>
                    <lb/>
                  cognita nobis erit, ſed
                    <var>.a.c.</var>
                  data nobis fuit, qua-
                    <lb/>
                  re cognoſcemus etiam omnia dicta quadrata eo-
                    <lb/>
                    <reg norm="rumque" type="simple">rumq́;</reg>
                  radices
                    <var>.a.b.</var>
                  et
                    <var>.b.c.</var>
                  </s>
                  <s xml:id="echoid-s3640" xml:space="preserve">quare & ſuperficiem re-
                    <lb/>
                  ctanguli quæſitam.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3641" xml:space="preserve">Quartum exemplum etiam faciliori via poteſt
                    <lb/>
                  ſolui, propterea, quod cum nobis cognita ſit ba-
                    <lb/>
                  ſis trianguli cum ſumma reliquorum laterum, &
                    <lb/>
                    <reg norm="cum" type="context">cũ</reg>
                  angulo oppoſito baſi ipſius reliqua cognita no
                    <lb/>
                  bis emergunt ex .15. problemate ſecundi lib. de Triangulis ipſius Monteregii.</s>
                </p>
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