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-div495" type="section" level="3" n="5">
              <div xml:id="echoid-div505" type="letter" level="4" n="5">
                <p>
                  <s xml:id="echoid-s3184" xml:space="preserve">
                    <pb o="253" rhead="EPISTOLAE." n="265" file="0265" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/pageimg/0265"/>
                  angulo
                    <var>.b.u.e.</var>
                  vnde ex .4. ſexti eadem proportio erit ipſius
                    <var>.b.n.</var>
                  ad
                    <var>.b.e.</var>
                  quæ
                    <var>.b.e.</var>
                  ad
                    <lb/>
                    <var>b.u</var>
                  . </s>
                  <s xml:id="echoid-s3185" xml:space="preserve">quare ex .16. eiuſdem patebit propoſitum.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3186" xml:space="preserve">Secundus autem modus ita ſe habet, ducta
                    <var>.q.n.</var>
                  habebimus duo triangula ortho-
                    <lb/>
                  gonia ſimilia inuicem
                    <var>.b.q.n.</var>
                  et
                    <var>.b.u.o.</var>
                  eo quod angulus
                    <var>.b.</var>
                  communis ambobus exi-
                    <lb/>
                  ſtit, </s>
                  <s xml:id="echoid-s3187" xml:space="preserve">quare ex .4. ſexti ita ſe habebit
                    <var>.u.b.</var>
                  ad
                    <var>.b.o.</var>
                  vt
                    <var>.q.b.</var>
                  ad
                    <var>.b.n.</var>
                  vnde ex .15. eiuſdem
                    <lb/>
                  quod fit ex
                    <var>.u.b.</var>
                  in
                    <var>.b.n.</var>
                  æquale erit ei, quod fit ex
                    <var>.q.b.</var>
                  in
                    <var>.b.o</var>
                  . </s>
                  <s xml:id="echoid-s3188" xml:space="preserve">Sed ex .16. eiuſdem,
                    <reg norm="quod" type="simple">ꝙ</reg>
                    <lb/>
                  fit ex
                    <var>.q.b.</var>
                  in
                    <var>.b.o.</var>
                  ęquatur quadrato
                    <var>.b.e.</var>
                  quia
                    <var>.b.e.</var>
                  media proportionalis eſt inter dia
                    <lb/>
                  metrum & ſemidiametrum eiuſdem circuli. ex .4. eiuſdem, </s>
                  <s xml:id="echoid-s3189" xml:space="preserve">quare quod fit ex
                    <var>.u.b.</var>
                  in
                    <lb/>
                    <var>b.n.</var>
                  æquale erit quadrato ipſius
                    <var>.b.e</var>
                  .</s>
                </p>
                <p>
                  <s xml:id="echoid-s3190" xml:space="preserve">Tertius modus adiungitur, & eſt quod cum quadratum
                    <var>.u.b.</var>
                  exiſtente
                    <var>.u.</var>
                  extra cir-
                    <lb/>
                  culum æquale ſit ei, quod ſit ex
                    <var>.u.b.</var>
                  in
                    <var>.b.n.</var>
                  ſimul ſumpto cum eo,
                    <reg norm="quod" type="simple">ꝙ</reg>
                  fit ex
                    <var>.u.b.</var>
                  in
                    <var>.u.n.</var>
                    <lb/>
                  ex ſecunda ſecundi, & idem quadratum
                    <var>.u.b.</var>
                  æquale duobus quadratis
                    <var>.u.o.</var>
                  et
                    <var>.o.b.</var>
                  ex
                    <lb/>
                  penultima primi, ideo duo dicta producta æqualia erunt dictis duobus quadratis
                    <var>.o.</var>
                    <lb/>
                    <figure xlink:label="fig-0265-01" xlink:href="fig-0265-01a" number="298">
                      <image file="0265-01" xlink:href="http://echo.mpiwg-berlin.mpg.de/zogilib?fn=/permanent/library/163127KK/figures/0265-01"/>
                    </figure>
                  u. ſcilicet et
                    <var>.o.b.</var>
                  ſed quadratum
                    <lb/>
                  o u. æquatur ei, quod fit ex
                    <var>.a.u.</var>
                    <lb/>
                  in
                    <var>.e.u.</var>
                  & ei quod fit. ex
                    <var>.o.e.</var>
                  in ſe
                    <lb/>
                  ipſam ex .6. ſecundi, </s>
                  <s xml:id="echoid-s3191" xml:space="preserve">quare duo
                    <lb/>
                    <reg norm="iam" type="context">iã</reg>
                  dicta producta æqualia erunt
                    <lb/>
                  duobus dictis quadratis,
                    <var>o.b.</var>
                  ſci
                    <lb/>
                  licet. et
                    <var>.o.e.</var>
                  & ei quod fit ex
                    <var>.a.
                      <lb/>
                    u.</var>
                  in
                    <var>.u.e.</var>
                  ſed quod fit ex
                    <var>b.u.</var>
                  in
                    <var>.u
                      <lb/>
                    n.</var>
                  æquale eſt ei quod fit ex
                    <var>.a.u.</var>
                    <lb/>
                  in
                    <var>.u.e.</var>
                  ex .35. 3.
                    <reg norm="relinquitur" type="simple">relinquit̃</reg>
                  ergo
                    <lb/>
                  vt id
                    <reg norm="quod" type="wordlist">qđ</reg>
                  fit ex
                    <var>.u.b.</var>
                  in
                    <var>.b.n.</var>
                  æqua-
                    <lb/>
                  le ſit
                    <reg norm="duobus" type="simple">duobꝰ</reg>
                  quadratis
                    <var>.o.b.</var>
                  et
                    <var>.o.
                      <lb/>
                    e</var>
                  . </s>
                  <s xml:id="echoid-s3192" xml:space="preserve">quare & quadrato ipſius
                    <var>.b.e.</var>
                    <lb/>
                  ex Pitagorica.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3193" xml:space="preserve">Siautem
                    <reg norm="punctum" type="context context">pũctũ</reg>
                    <var>.u.</var>
                  fuiſſet intra
                    <lb/>
                  circulum idem eueniret. </s>
                  <s xml:id="echoid-s3194" xml:space="preserve">Nam
                    <lb/>
                  quadrato
                    <var>.b.e.</var>
                    <reg norm="æquantur" type="context">æquãtur</reg>
                  duo qua
                    <lb/>
                  drata
                    <var>.o.b.</var>
                  et
                    <var>.o.e.</var>
                  ſed vice qua-
                    <lb/>
                  drati
                    <var>.o.e.</var>
                  dicemus
                    <reg norm="quadratum" type="context">quadratũ</reg>
                    <var>.o.
                      <lb/>
                    u.</var>
                  cum eo quod fit ex
                    <var>.a.u.</var>
                  in
                    <var>.u.e.</var>
                    <lb/>
                  ex .5. ſecundi, id eſt quadratum
                    <var>.
                      <lb/>
                    o.u.</var>
                    <reg norm="cum" type="context">cũ</reg>
                  eo quod fit ex
                    <var>.b.u.</var>
                  in
                    <var>.u.
                      <lb/>
                    n.</var>
                  ex .34. tertij, vnde quadratum
                    <lb/>
                    <var>b.e.</var>
                  æquale erit quadrato
                    <var>.o.b.</var>
                    <lb/>
                  & quadrato
                    <var>.o.u.</var>
                  ideſt quadrato
                    <lb/>
                    <var>b.u.</var>
                  ex Pitagorica ſimul
                    <reg norm="cum" type="context">cũ</reg>
                  pro-
                    <lb/>
                  ducto
                    <var>.b.u.</var>
                  in
                    <var>.u.n.</var>
                  ideſt producto
                    <lb/>
                    <var>n.b.</var>
                  in
                    <var>.b.u.</var>
                  quod æquale eſt qua
                    <lb/>
                  drat
                    <var>o.b.u.</var>
                  cum producto
                    <var>.b.u.</var>
                  in
                    <lb/>
                    <var>u.n.</var>
                  ex .3. ſecundi.</s>
                </p>
                <p>
                  <s xml:id="echoid-s3195" xml:space="preserve">Circa tres paſſiones commu-
                    <lb/>
                  nes poſtea circulo hyperboli, &
                    <lb/>
                  defectioni notandum eſt
                    <reg norm="primam" type="context">primã</reg>
                    <lb/>
                  patere ex .36: primi Pergei, ſe- </s>
                </p>
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